y
13> 6/Y
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
CONDUCTED BY
SIR OLIVER JOSEPH LODGE, D.Bc, LL.D., F.R.S.
SIR JOSEPH JOHN THOMSON, M.A., Sc.D., LL.D., F.R.S
JOHN JOLY, M.A, D.Sc, F.R.S., F.G.S.
GEORGE CAREY FOSTER, B.A, LL.D, F.R.S.
AND
WILLIAM FRANCIS, F.L.S.
"Nee aranearum sane textus ideo melior quia ex se fila gignunt, uec noster
vilior quia ex alienis libamus ut apes." Just. Lips. Polit. lib. i. cap. 1 . Not.
VOL. XX VIII.— SIXTH SERIES.
JULY— DECEMBER 1914.
>r. ,«• > /
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JUMMAM
CONTENTS OF VOL. XXVIII,
(SIXTH SEEIES).
NUMBER CLXIIL— JULY 1914.
Page
Prof. A. Gray : Notes on Hydrodynamics.— 1 1
Prof. A. Gray : Notes on Hydrodynamics. — II 13
Prof. W. M. Thornton on the Lost Pressure in Gaseous
Explosions 18
Mr. J. Proudman on the Motion of Viscous Liquids in
Channels 30
Dr. J. P. Dalton on a New ContinuousBalance Method of
Comparing an Inductance with a Capacity 37
Dr. Herbert Edmeston Watson and Mr. Gostabehari Pal on
the Radioactivity of the Rocks of the Kolar GoldFields . . 44
Mr. Gilbert Cook on the Collapse of Short Tubes by External
Pressure. (Plate I.) 51
Dr. I. J. Schwatt : Note on a Definite Integral 57
Dr. Max Planck on New Paths of Physical Knowledge .... 60
Dr. Eva von Bahr on the QuantumTheory and the Rotation
Energy of Molecules 71
Prof. J. S. Townsend on the Potentials required to Maintain
Currents between Coaxial Cylinders 83
Prof. J. W. Nicholson on Atomic Structure and the Spec
trum of Helium 90
Prof. W. C. McC. Lewis on the Relation of the Internal
Pressure of a Liquid to its Dielectric Capacity and Permea
bilility 104
Dr. W. Marshall Watts on the Spectra given by Carbon and
some of its Compounds ; and, in particular, the "Swan'*'
Spectrum. (Plate II.) 117
Mr. Allan Ferguson on the Shape of the Capillary Surface
inside a Tube of Small Radius, with other Allied Problems. 128
Prof. W. M. Hicks on Highfrequencv Spectra and the
Periodic Table * 139
Mr. J. A. Tomkins on the SlipCurves of an Amsler Plani
meter 142
Prof. J. R. Rydberg on the Ordinals of the Elements and
the Highfrequency Spectra 144
Mr. Allan Ferguson on the Forces acting on a Solid Sphere
in contact with a Liquid Surface (II.) 149
Notices respecting New Books : —
Mr. Frederick Soddy's The Chemistry of the Radio
elements, Part II 154
IV CONTENTS OF VOL. XXVIII. SIXTH SERIES.
Page
Proceedings of the Geological Society : —
Lady McBobert on Acid and Intermediate Intrusions in
the Neighbourhood of Melrose 155
Prof. J. W. Judd on the geology of Eockall 156
Dr. H. S. Washington on the Composition of Bockallite. 157
Prof. J. W. Gregory on the Evolution of the Essex
RiverSystem 158
Mr. J. B. Scrivenor on the Topazbearing Eocks of
Gunong Bakau (Federated Malay States) 160
NUMBEE OLXIV.— AUGUST.
Lord Eayleigh on the Equilibrium of Eevolving Liquid under
Capillary Force 161
Dr. Charles Sheard on the Positive Ionization from Heated
Platinum 170
Messrs. C. E. Magnusson and H. C. Stevens ort Visual Sen
sations caused by a Magnetic Field. (Plates III. & IV.) . 188
Dr. Albert C. Crehore on the, Theory of the String Galvano
meter of Einthoven 207
Prof. D. N. Mallik on the Dynamical Theory of Diffraction . 224
Mr. P. .T. Edmunds on the Discharge of Electricity from
Points 234
Dr. Frank Horton on the Action of a Wehnelt Cathode .... 244
Dr. C. V. Burton on the Possible Dependence of Gravitational
Attraction on Chemical Composition, and the Fluctuations
of the Moon's Longitude which might result therefrom . . 252
Mr. S. S. Eichardson on Polarizing Prisms for the Ultra
violet 256
Sir Ernest Eutherford and Dr. E. N. da C. Andrade on the
Spectrum of the Penetrating y Eays from Eadium B and
Eadium C. (Plate V.) 263
Mr. W. F. Eawlinson on the Xray Spectrum of Nickel .... 274
Messrs. H. Eobinson and W. F. Eawlinson on the Magnetic
Spectrum of the /3 Bays excited in Metals by Soft X Bays 277
Sir Ernest Eutherford and Messrs. H. Eobinson and W. F.
Eawlinson on the Spectrum of the /3 Eays excited by;iy Eays. 281
Dr. Norman Campbell on the Ionization of Platinum by
Cathode Eays 286
Dr. G. Bruhat on the Theories of the Eotational Optical
Activitv 302
NUMBEE CLXV.— SEPTEMBEE/
Sir Ernest Eutherford on the Connexion between the /3 and
y Eay Spectra 305
Sir Ernest Eutherford : Eadium Constants on the Inter
national Standard 320
Messrs. H. G. J. Moseley and H. Eobinson on the Number
of Ions produced by the (3 and y Eadiations from Eadium. 327
CONTENTS OF VOL. XXVIII. SIXTH SERIES. V
Page
Prof. A. LI. Hughes on the Contact Difference of Potential of
Distilled Metals 337
Prof. L. L. Campbell on Disintegration of the Aluminium
Cathode 347
Mr. W. Lawrence Bragg on the Crystalline Structure of
Copper 355
Prof. J. C. McLennan on the Absorption Spectrum of Zinc
Vapour. (Plate VI.) 360
Mr. D. C. H. Florance on Secondary y Radiation 363
Mr. H. A. McTaggart on Electrification at LiquidGas Surfaces 367
Mr. L. Isserlis : The Application of Solid Hypergeometrical
Series to Frequency Distributions in Space. (Plate VII.) . 379
Mr. Allan Ferguson on the Surfacetensions of Liquids in
contact with different Gases 403
Dr. T. Martin Lowry on an Oxidizable Variety of Nitrogen.
(Plates VILI.XI.) 412
Notices respecting New Books :—
Dr. A. LI. Hughes's Photoelectricity 416
NUMBER CLXVL— OCTOBER.
Prof. E. M. Wellisch : Experiments on the Active Deposit
of Radium 417
Mr. Gervaise Le Bas on the Theory of Molecular Volumes. —
Part IV 439
Dr. "W. F. G. Swann on the Electrical Resistance of Thin
Metallic Films, and a Theory of the Mechanism of Con
duction in such Films 467
Profs. F. C. Brown and L. P. Sieg on the Seat of Light
Action in Certain Crystals of Metallic Selenium, and some
New Properties in Matter 497
Mr. Edwin Edser on the Reflexion of Electromagnetic Waves
at the Surface of a Moving Mirror 508
Miss Jadwiga Szmidt on the Distribution of Energy in the
Different Types of y Rays emitted from Certain Radio
active Substances 597
Lieut. H. P. Walmsley on the Distribution of the Active
Deposit of Radium in Electric Fields 539
Sir Ernest Rutherford and Mr. H. Robinson on the Mass and
Velocities of the a Particles from Radioactive Substances
(Plate XII.) 550
Prof. Richard C. Tolman on the Relativity Theory : General
Dynamical Principles , 570
Prof. Richard C. Tolman on the Relativity Theory : The
Equipartition Law in a System of Particle's 583
Mr. Ernest Vanstone on the UnitStere Theory of Molecular
A^olume : A Criticism ' . 600
Mr. Gervaise Le Bas : A Reply to a criticism of the Unit
Stere Theory 6O7
vi CONTENTS OF VOL. XXVIII. SIXTH SERIES.
Page
Lord Eayleigh : Further Eeinarks on the Stability of Viscous
Fluid Motion 609
Sir J. J. Thomson on the Production of Soft Eontgen Eadia
tion by the Impact of Positive and Slow Cathode Eays . . 620
Prof. W. H. Bragg and Mr. S. E. Peirce on the Absorption
Coefficients of X Eays 626
Dr. A. van den Broek : Ordinals or Atomic Numbers ? . . . . 630
Proceedings of the Geological Society : —
Prof. J. W. Gregory on the Structure of the Carlisle
Solway Basin 632
NUMBEE CLXYII.— NOVEMBEE,
Prof. O. W. Eichardson on the Distribution of the Molecules
of a Gas in a Field of Force, with Applications to the
Theory of Electrons 633
Mr. G. H. Livens on the Statistical Relations of Eadiant
Energy 648
Mr. J. Eice : Note on the Form assumed by the Eed Cor
puscles of the Blood, or by the Suspended Particles in a
Lecithin Emulsion 5 . . 664
Dr. J. E. Wilton on Figures of Equilibrium of Eotating
Fluid under the restriction that the Figure is to be a
Surface of Eevolution e 671
Mr. W. H. Jenkinson on Concentration Cells in Ionized
Gases 685
Prof. E. P. Adams and Mr. Albert K. Chapman on the
Corbino Effect 692
Sir Joseph Larmor on the Eeflexion of Electromagnetic
Waves by a moving perfect Eeflector, and their Mechanical
Eeaction 702
Messrs. Herbert E. Ives and Edwin F. Kingsbury on the
Theory of the Flicker Photometer 708
Dr. G. A. Shakespear : Experiments on the Resistance of
the Air to Falling Spheres 728
Prof. W. M. Thornton on the Least Energy required to start
a Gaseous Explosion. (Plate XIII.) 734
Dr. Margaret B. Moir on Permanent Magnetism of Certain
Chrome and Tungsten Steels 738
Mr. J. G. Stewart on the Inapplicability of Boltzmann's
Equipartition Hypothesis to Gases in a State of Change
of Internal Energy ; and its bearing on the experimental
determination of the Specific Heat of Gases 748
Mr. F. H. Newman on the Ionization Potential of Mercury
Vapour 753
Mr. G. H. Livens on the Theories of Eotational Optical
Activity 756
CONTENTS OF VOL. XXVIII. SIXTH SERIES. Vll
Proceedings of the Geological Society : —
Mr. C. T. Trecbmann on the Scandinavian Drift of the
Durham Coast 758
Mr. F. W. Penny on the Eelationship of the Vredefort
Granite to the Witwatersrand System 759
Mr. E. B. Bailey on the Ballachulish Fold near the Head
of Loch Creran 760
Intelligence and Miscellaneous Articles : —
Correction in " Notes on the Motion of Viscous Liquids
in Channels " by J. Proudman 760
NUMBER CLXV1IL— DECEMBEK.
Dr. A. N. Lucian on the Distribution of the Active Deposit
of Actinium in an Electric Field 761
Mr. J. C. Buckley on the Bitilar Property of Twisted Strips . 778
Mr. Ivar Maimer on the HighFrequency Spectra of the
Elements. (Plate XIV.) 787
Mr. S. Lees on the Analysis of Energy Distribution for
Natural Radiation 794
Mr. T. Carlton Sutton on the Mechanism of Molecular
Action. (A Contribution to the Kinetic Theory of Gases.) 798
Mr. A. B. Wood on the Volatility of Thorium D. With
a Note on the Relative /3 Activities of Thorium C and D. . 808
Mr. R. W. Varder and Dr. E. Marsden on the Transforma
tions of Actinium C 818
Mr. T. Harris on the Reflexion of Electromagnetic Waves
from the Surface of a Moving Mirror 822
Messrs. Arthur Holmes and Robert W. Lawson on Lead and
the End Product of Thorium.— Part 1 823
Proceedings of the Geological Society : —
Mr. L. Richardson on the Inferior Oolite and Contiguous
Deposits of the DoultingMilbornePort District
(Somerset) 841
Intelligence and Miscellaneous Articles : —
Method of comparing SelfInductance and Capacity,
by A. Campbell 842
Index 843
P L A T E S.
I. Illustrative of Mr. Gilbert Cook's Paper on the Collapse of
.Short Tubes by External Pressure.
II. Illustrative of Dr. W. Marshall Watts's Paper on the Spectra
given by Carbon and some of its Compounds, and, in
particular, the " Swan " Spectrum.
III. & IV. Illustrative of Messrs. C. E. Magnusson and H. C. Stevens's
Paper on Visual Sensations caused bv a Magnetic Field.
V. Illustrative of Sir Ernest Rutherford" and Dr. E. N. da C.
Andrade's Paper on the Spectrum of the Penetrating y Rays
from Radium B and Radium C.
VI. Illustrative of Prof. J. C. McLennan's Paper on the
Absorption Spectrum of Zinc Vapour.
VII. Illustrative of Mr. L. Isserlis's Paper on the Application of
Solid Hypergeometrical Series to Frequency Distributions
in Space.
VIII. XI. Illustrative of Dr. T. Martin Lowry's Paper on an Oxidizable
Variety of Nitrogen.
XII. Illustrative of Sir Ernest Rutherford and Mr. H. Robinson'e
Paper on the Mass and Velocities of the a Particles from
Radioactive Substances.
XIII. Illustrative of Prof. W. M. Thornton's Paper on the Least
Energy required to start a Gaseous Explosion.
XIV. Illustrative of Mr. Ivar Maimer's Paper on the High
Frequency Spectra of the Elements.
ERRATA.
P. 291, line 16, for " (i1i2)lh " read " ftrl)/*, " or " 1— I//,."
„ lines 1921, for " the value of I ... . measure of it " read "the
value of I will be proportional to 1  R and I may be taken as
an inverse measure of R."
„ lines 29, 30, for " The value R'=I/t\  i3 will be greater than R,"
read " The value R' = l  I/fo i3) will be different from R."
P. 382, line 4 from foot of page should read : —
xy^ki, =XnXoiXio+X.
d*dy
00
P. 615. In the denominator of equation (12), for (lhy ££) read
P. 684, line 8, the first integral inside the bracket should be
£a!& *•*"*£,
p4w
(1+*V)
[This is printed correctly in the majority of copies of the November
number, but in some of the later "pulls" the :J appears to have been
broken off.— W. F.]
THE
LONDON, EDINBURGH and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[SIXTH SERIES.]
JUL Y 1914.
I. Notes on Hydrodynamics.
By Professor A. Gray, F.R.S*
I. Equations of Motion for any Axes. Theorems of Cir
culation. Proof of Constancy of Moment of Vortex
Filament.
1. 4 T time t the equation of motion along a streamline
JTJL in a perfect fluid is
oq t 07 , dV 1 op n /1X
Ot OS OS p OS
where p is a function of p.
For the first two terms are the acceleration along the stream
line at the point considered, P say, and the third and fourth,
with their signs changed, make up the force per unit volume
due to the field of force for which the potential at P is V,
and to the gradient of pressure op/os.
Integrating along the streamline from a chosen initial
point P0 to P as a final point, we obtain
Ru+i( sin 6
is a maximum. Equation (6) thus shows how we can pass,
in any direction, from the value of ^ at a point P on one
streamline to the value of % at an adjacent point P' on
another. Thus along any surface about the normals to
which at every point there is zero rotation &v, the value of
Prof. A. Gray : Notes on Hydrodynamics. 3
X is, in the case of steady motion, constant from streamline
to streamline. If there is no elemental rotation anywhere
throughout any finite portion of
the fluid in steady motion the value
of x is the same for all streamlines
in that portion.
4. To prove that cossr, defined
as above, has the value given b}r
the equation
~ds "?>"
2o}Ss' sin 6
51 (7)
we may proceed thus. Consider
the parallelogram (see figure) of
which adjacent sides are PP',
PQ, that is ds', ds. The average
velocities of the fluid along the
four sides PQ, QR, PP', P'P are
13?
2 3*
Q+ s^ds, q'+^ds+c
3*
13 be the mean angular velocity about P" for the
particles of fluid on the periphery at the instant considered
we get, since J p dc is twice the area of the parallelogram,
and this is also 2 ds ds' sin 6,
■bq
3*''
Thus (7) is proved.
B2
)„, sin 0=~
ds
4 Prof. A. Gray : Notes on Hydrodynamics.
It will be noticed that this result is independent of the
position of the point P" with respect to which the angular
velocities of the particles have been taken. A small spherical
portion of the fluid with its centre within the infinitesimal
parallelogram, has a component of angular momentum about
the normal to the plane of dsds', of amount
8 5
Y* ?rpr0 cogs',
where r is the radius of the radius and p the density of the
fluid. Thus we call cosst the component of elemental angular
velocity of the fluid about the normal considered. It is an
affair of an element of the fluid mass, not of a particle.
5. It is easy to extend the process by which (7) has been
established to show that the circulation round any closed
path drawn in the fluid is equal to twice the surface integral
of rotation of the fluid about the normals to the elements of
any surface of which the path is the bounding edge, and
indeed to prove Stokes's theorem connecting the line integral
of a directed quantity taken round the boundary of a surface,
with integral of the curl of the same quantity taken over
the surface. In what follows we shall assume the theorem
of circulation (not, however, of the constancy of circulation
for a closed path moving with the fluid).
6. We may interpret the result stated in (5) and (6)
above in the following manner. First integrate along the
path of which PP' is an element from an initial point A to a
final point B. We obtain from (6)
XA~ —% 1 cogsq sin 6 ds — 1
w
XBXA= * \ cogs/qsm6ds \ ^ds'. . . (8)
AB AB
Now as q is the resultant velocity of the fluid at any point
P on the path AB, q' ( = qcos0) is the component at P
along the tangent drawn there to AB, while q sin 6 is the
velocity with which each particle P on the path AB is being
carried by the motion towards the right (see fig. 2) in the
plane of the diagram. The product q sin 0 dsf is therefore
the rate at which an area of which ds' is an element of
boundary, and which is situated to the left of AB, is in
creasing (or if the area is situated to the right of AB, is
diminishing) in consequence of the motion of ds' as a whole, at
right angles to itself, in the plane of the diagram. We see,
therefore, that the first term on the left is twice the rate at
which the surface integral of elemental rotation, taken over
Fior. 3.
Prof. A. Gray : Notes on Hydrodynamics. 5
any area of which AB is part of the boundary, is changing
in consequence of the fact that each element ds' of AB is
being carried towards the right by the motion of the fluid.
Fig. 3 shows the effect of this
motion for a closed path of inte
gration. The area between two
streamline elements ds, and the
two positions of the connecting ds,
is evidently dsds' sin 6.
The second term on the right is
the rate at which as time passes
the flow along ^B~4schanging
apart from the mo£ion>Qf the
elements ds' . ,
If the path AB be
lefthand side of (8) vanishes and we hAte^
where (j) denotes integration round the
7. Now since the line integral xq'ds' taken rounc
closed path is twice the surfaceintegral of elemental rotation
about normals drawn to the elements into which any surface
bounded by the path may be divided, if we calculate the
whole change of rate of flow along AB due to the various
causes, we snail obtain a result which, extended to the whole
circuit, will give the exact rate of increase of the surface
integral of elemental rotation for any surface of which the
path of integration is the bounding edge.
Now the surface integral is increasing at rate
2 (J ) coss' q sin 6 ds' ,
in consequence of the motion of the elements ds1 at right
an iiles to themselves, and also at rate
; q S11T
closeo
IncT the
$16"
in consequence of the variation of q with time. There are
two other causes of variation due to the motion. Each
element ds' is being displaced bodily in the direction of its
length, and moreover the end of the clement nearer B is
undergoing displacement with respect to the end nearer A.
It can be proved that each of these displacements gives a
rate of increase of flow alono the element of amount
6 Prof. A. Gray: Notes on Hydrodynamics.
q'dq/'ds' .ds'. Thus the whole rate of diminution of flow
along the unclosed path AB is
 U j1®,.' q sin dds'+2 (V <**' + f ^' ss< j sin 6 =
(14)
which has been obtained as (6) above. From this the whole
motion of the fluid can be derived.
8. As has been noticed, if we take ds' in the direction of
the streamline we obtain, since # = 0,
(15)
which is the equation of motion for the streamline direction,
and yields at once the socalled theorem of Bernoulli.
If we take ds1 in the direction of the axis of spin, that
is along what has been called a vortexline, we obtain again
(16)
Thus Bernoulli's theorem is true also along a vortexline.
It is thus possible, in the case of steady motion, to draw
through each point of the fluid a surface on which lie inter
secting streamlines and vortexlines, and for every point of
which ^ has the same value.
Let a normal be drawn to such a surface at any point,
and dn denote a short step from the point along the normal.
Then clearly we have, by (14),
~dn
+ 2<
■ = 0.
(17)
If (j> be the angle between the streamline and the vortex
line which intersect at the foot of the normal, and o> be
the resultant regular velocity at that point, we have
G>sin cj) = a)sn, and therefore
OY
^ +2(oq sin eb
on * T
(17')
This equation is
in Lamb's ' Hydrodynamics/ § 164.
It is the particular case of (14) in which 6= ^7r, wsn — (o sin $,
and 'dq'lot = 0. The theorem of (14) is quite general.
8 Prof. A. Gray: Notes on Hydrodynamics.
9. If in (14) we take the step dsf parallel to the three
axes Ox, Oy, Oz in succession, and denote q' for these direc
tions by u, v, w, we obtain the three equations of motion
+2a,sz9sm^ = ^
g+2^,sin^=^
^— + 2G>s^sin6^=
Ot
(18)
It is to be noticed that these three axes may be inclined at
any angles, so that (18) are equations of motion for a system
of any three axes.
10. If we assume that the axes are rectangular, and write,
according to the usual notation,
2£=
2V.
"da
B<^
9^=
2f
5y'
(19)
axes
same
and regard 2f, 2t), 2£ as vectors associated with the
Ox, Oy, Oz respectively, the vector associated in the
way with any axis the direction cosines of which are I, m, n
is
2{Zf + mr}\n£).
If now I, m, n be the directioncosines of the normal to the
plane ds, ds', drawn outwards from the diagram (fig. 2), this
vector is a)ss' . Thus we can write (14) in the form
^+2q(lg + mV+n£)sm0=l^r . . (20)
(jt Q S
This form is much less compact than (14), but from it we
obtain at once the usual form of equations (18) for the case of
rectangular axes. Putting 1 = 0, we get q' = u, qn sin 0=—v,
qn sin 0 = iv. Similar results are obtained by putting m = 0,
ii = 0, in succession. Thus (18) become for rectangular
axes
**+.**— g
2w£+2u£ =
_ &
9.'/
2ur) + 2vrj = —
_ S%
d* J
(21)
These equations are usually derived from the Eulerian equa
tions of motion for the rectangular axes. The forms, however.
Prof. A. Gray: Notes on Hydrodynamics. 9
of (18) are more compact and more general, since they are
applicable to any system of three axes. Moreover, they can
be written down at once from (14) which it is easy to
remember.
11. From (18) we might deduce v. Helmholtz^s vortex
motion equations, but they are perhaps most easily obtained
in the manner indicated below. They are introduced here
as the discussion will be found to lead to what seems a new
proof of the constancy of the moment (product of angular
velocity by crosssection) of a vortexfilament.
By (1) above the equations of motion for the axes Oy, 0~
are
Bv ~dv _ BV 1 Bp 1
'dt y~ds ~ dy p~dy
~div , "div BV 1 "dp
i
I
(22)
These also hold whether the axes are at right angles or not.
Taking the axes at right angles differentiate the first equa
tion with respect to z, and the second with respect to ?/, and
subtract the first result from the second. We obtain easily
bB^' oy B~
dt
(23)
here © =
3" 3^
Be
the " expansion" (timerate of dilatation per unit of volume).
Two similar equations hold for 77, f and can be written down
at once.
It may be remarked that the righthand side of (23) can
be written as in the alternative equation
at a *B# ox dJ
Similar equations hold of course for the other axes.
From the equation of continuity
dtf+pS=0
at r
we obtain by substitution for @ in (23) and (23')
(23')
 (S)=
dt \pj
?B" vbu ?3"
PO<<' pou poz
I B" vo> Zoic
p 0^ p 3^' p 3
;
(24)
10 Prof. A. Gray: Notes on Hydrodynamics.
The first of these is typical of the three equations of
v. Helmholtz generalized for the case of a fluid of varying
density. The alternative forms here shown are sometimes
convenient.
If we multiply the first of the equations typified by (23)
[or (23') ] by £, the second by r), and the third by f, and add,
we get, since .
The equation just found may be written
^+6>e=^+^+?l^. . . (26)
dt co oo" co oo" co o oo"
by inclusion of the terms
do \coJ da\coJ 0(T\coJ
The expression on the right of (25) is thus the unital rate
of elongation in the direction of the tangent to the vortexline
at the point considered. It is thus a component part of
the dilatation of the fluid at that point, not following the
fluid as it moves. But the dilatation 0 is the volume dila
tation also at a fixed point in space, and might be expressed
equally well as the sum of a unital timerate of elongation
along the instantaneous direction of da, and a unital time
rate of expansion of area at right angles to that direction.
As will be proved below, that areal expansion will, to the
Prof. A. Gray: Notes on Hydrodynamics. 11
first order at least of small quantities, be unaltered by asso
ciation with the fluid as it moves, since, to the degree of
approximation stated, the effect of the displacement of the
fluid will be simply to turn the area round through a small
angle.
Supposing then ® to be expressed as has been suggested,
we subtract the timerate of unital elongation in the direction
of the tangent to the vortex filament from both sides of (25),
and obtain
Ir+^=0, (27)
where 2 is the unital timerate of increase of area. If S be
the crosssectional area of the vortexfilament at the point
considered, we have S = S2, and therefore also
&>Sa>S = 0,
that is o)S = constant.
The moment of the vortexfilament thus remains unaltered
as the fluid moves.
I have not hitherto seen this important result derived
directly from equation (22). Moreover, the proof seems
free from the objections brought by Stokes to the proof
given by Cauchy, objections to which the proof given by
v. Helmholtz is also open.
13. As to the point referred to above regarding the areal
expansion at right angles to the tangent at P to the vortex
filament, let rectangular axes be so chosen at P that the axis
of the filament is along Fz, and consider the expansion in
the plane which at the beginning of an interval dt is at righr
angles to P. Take a distance APB = J.i extending from
— \d.r, to 4Jfito, and another CPD extending from — \dy to
+ \dy. At A and B the component velocities of the fluid
are
_ U» 7 _1^, _ 1 ~dio .
2 ox 2 0
Further, it can be shown that to the degree of approxi
mation specified the angle between the new lines AB, CD
(which intersect in the new position of P) is given by
cos
mi+rK
It follows that the former area dx dy has in its new position
the value
^{i+(g4;>}
and the unital timerate of areal expansion is
~du ~dv
O^x ~dy
following the fluid, that is the statement made in § 12 is
justified.
Some further notes regarding vortexmotion, dealing
principally with surface and volume integrals, are reserved
for another communication.
[ 13 ]
II. Notes on Hydrodynamics.
By Professor A. Gray, F.R.S.*
II. Determination of Translational Velocity of a Vortex Ring
of Small Crosssection.
1. A VALUE for the velocity of displacement of a vortex
J\. ring in an unlimited fluid was given without proof
in a note by Lord Kelvin appended to Professor Tait's trans
lation of Helmholtz's celebrated paper (Phil. Mag. 1867).
A demonstration of the result was promised in the note, but,
so far as I know, Lord Kelvin never published it. Several
investigations have since been published with results differing
slightly from Lord Kelvin's value, which, however, has been
confirmed by Hicks (Phil. Trans, 176) and by Lamb
(Hydrodynamics, 3rd ed. p. 227). The following direct and
elementary proof (in which no use is made of elliptic in
tegrals as such) may possibly be of interest.
It is well known that the velocity at any point P in a
frictionless incompressible fluid, in which exists a single
vortex filament of any form, may be found in the following
manner. Let k be the strength of the vortex, that is double
the (constant) product of the elemental angular velocity
in the filament at any point by the area of crosssection
there, r the distance of the point P considered from
an element E, of the filament, of length ds, and 6 the
complement of the angle between the directions of the
element E and the line EP. The velocity Bq at P due to
the element E may then be taken as given by the equation
~ x ds . cos 0
s*=i^ — " d)
The direction of Bq is at right angles to the plane determined
by the element E and the line EP, and the flow is towards
the side of the plane specified by the rule given below.
This rule applied to all the elements of the filament, gives
the velocity at P as the resultant of all the vectors Sq given
by the elements composing the complete filament. Of
course there might be added to the right hand side of (1)
any term of: proper dimensions for which the integration
round the closed filament gives a zero vector.
The theorem here stated is the analogue of that by which
in electromagnetism the magnetic force at any point due to
a linear circuit may be found. There, if y be the current in
* Communicated bv the Author.
14 Prof. A. Gray: Notes on Hydrodynamics.
the circuit of which an element; E has length ds, and dl the
fieldintensity,
81=7.00.0$, (2)
and 6T is at right angles to the plane determined by E and P.
The resultant intensity at P is the resultant of the complex
of vectors SI given by the elements composing the circuit.
The Amperean rule for the direction of SI is well known :
that for the direction of Sq is simpler. Imagine a closed
path drawn round the element E, in such a manner that the
projection of the path on any plane dues not cross itself, and
at E let the path lie along the normal specified. Let a point
move round the path in the direction of the rotation at E :
the direction of motion at P is the direction of Sq.
2. Now imagine a circular vortexfilament of infinitesimal
crosssection to exist alone in an unlimited incompressible
fluid. If a point P be taken in the plane of the filament, the
velocity (q) there at right angles to the plane is given by
d?
cos0^, (3)
J
T
where r is the distance of P from the element E, of length ds,
and 6 is the complement of the angle
between ds and the line EP. From
this we can obtain an approximate
evaluation of the surface integral of
flow through a circle coaxial with
the filament and differing only slightly
in radius. Let the outer circle, of
radius a in the diagram, represent
the filament, and the inner circle, of
radius a — x, represent the coaxial
circle, in the same plane. The angle CEB is the angle 6 as
defined above, and so for the flow, d% say, through an
element of area rdO dr (EP = r) at P, due to the element E
of length ds, we obtain
dX=K~^dsd0dr. ..... (4)
Thus, if we suppose 0 to vary from 03 when EB is along EC,
the upper limit is sin1 {{a — a?) /a}. "We call this 6X. The
limits of r are the two roots of the equation
i*2ar cos 0 + a2(ax)2 = O. ... (5)
Approximately, these roots are 2a cos 6, .r/cos 6 ; a closer
Prof. A. Gray: Notes on Hydrodynamics. 15
approximation gives 2a cos 6 — x/cos 0, tf/cos 6. If we confine
ourselves to the rougher approximation we get easily from
the equation
%=£2f^'f ^, . . (6)
for the total flow, the result
X=«a(]ogia2). ..... (7)
The more exact value of the larger root of (.5) might have
been used without added difficulty, but a more accurate
solution can easily be derived from (7) in another way.
The order of approximation so far adopted takes that part
of the flow through the filament, which escapes passing
through the coaxial circle, as equal to that which might be
computed by taking the filament as straight. Of course, if
the filament is infinitely thin, this part is infinite, but the
infinity is avoided in any actual case by taking the cross
section as finite though very small. The flow which passes
outside the smaller circle may then be written ica log (#/e) ,
where e is a very small quantity. The term ica log e also
appears in the flow through the circle of radius a, so that
^ for the smaller circle has the value stated in (7).
Now let the smaller circle be moved out of the plane of
the larger through a small distance y while remaining
coaxial with the latter. If, then, c be the shortest distance
( = ^/x2 + y'2) between two points, one on the filament, the
other on the circle, the additional flow which escapes passing
through the circle of radius a— x is /ca(log c — log x). Hence,
to the same order of approximation as before
x=««(log^2) (8)
3. Of course c involves x, but any attempt to calculate from
(8) the axial component of velocity at the circumference of
the circle of radius a — x would lead to an erroneous result,
in consequence of our having neglected quantities of the
first order in x. We can now obtain, however, a closer
approximation to x by writing, as was done by Maxwell
(Electricity and Magnetism, § 705) for the electromagnetic
analogue,
X=*(Alog^ + B), (9)
and then determining A and B from the physical fact that
16 Prof. A. Gray: Notes on Hydrodynamics.
the flow through the circle of radius a~x due to a vortex
filament of strength and coinciding with the larger circle, is
equal to the flow through the latter circle due to a vortex of
the same strength coinciding with the smaller circle. We
assume therefore
A = a + Axx + , B= — 2a + B1x+..... . (10)
Since the vortex filament fas it was in the case considered
by Lord Kelvin) is to be taken of small though finite cross
section, we need not carry the calculation beyond terms
involving the first power of the ratio xla. No first power
of y, or indeed any odd power, can enter, since the value of %
cannot be altered by changing the sign of y.
We substitute then in (9), with the values of A and B
from (10), a — x for a and — x for x, and obtain
K1 1 x\, Sa 1 x } ,.,.,.
4. We now consider a vortex rino of small circular cross
section (radius r) made up of thin coaxial vortex filaments,
each of the same strength per unit area of section, and
calculate the flow through the circle, of radius a, which
forms the circular axis of the anchor ring surface of the
assemblage of filaments. Thus for the different filaments
c varies from 0 to r.
To find this axial flow, we calculate the rate of change of
^ when the ciicle through which the flow is taken is
widened, while the filament producing it remains fixed.
That is, we have to differentiate ^ with respect to x while
H = a — x remains unchanged. Thus, substituting Rf^ for a
in (11) we get
x=«{(R + i*)log®^)2B§*} . (12)
and therefore
!«')— (i+?)+i?
The axial component Sv of velocity at any point of the circle
of radius a is this divided by lira. Thus
5 k /' 8a \ k (I 2a\ k x" ,10,
Prof. A. Gray: Notes on Hydrodynamics. 17
If now We write fc = 2co cdcdO, and integrate from # = 0 to
B = 2ir (remembering that cos0 — .v/c), and £romc=0 to c = r,
so as to find the effect of all the filaments in the ring, we obtain
without difficulty
2ft)7rr2 , 8a ,^ ..
v== —. log — (14)
47ra e r K '
Now, for certain reasons which we do not discuss here,
the circular axis of the vortex ring is a coaxial circle of
radius R0 given by
Ro=~~ (15)
and of distance jjf0, from any chosen point on the axis of the
system given by
&=^E' (16)
The distance f0 i>s thus equal to the axial distance of the
circular axis of the anchor ring from the same chosen origin,
that is the two circular axes specified lie in the same plane.
The value of R0 is equal to the radius of gyration of a
uniform circular lamina of radius r about an axis in its plane
and at distance a from its centre. We have therefore
R0 = v/a2+jr*
=*+g£ (16)
approximately.
In consequence of the spin co the rate of advance of the
circular axis of the vortex ring is loss than the value of v
above obtained by ^wr2ja» Thus we obtain finally, writing /c
for the strength 2©7rr* of the whole ring,
d£o k /. Sa 1\ „_.
l = 4^(l0«T4> • • • • <17>
the value given by Lord Kelvin.
5. Any other point in the phme of the circular axis of the
vortex and near that axis might have been taken instead of
a point on the circular axis of the anchorring surface for
the specification of the circle at which the axial velocity is
calculated. In this case, however, it will be found more
convenient for the sake of the integrations to calculate first
the whole flow through the circle chosen due to the complete
vortex ring, and then determine the velocity sought by
differentiation.
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. G
18 Prof. W. M. Thornton on the
If £ be the radius of the circle chosen % can be expressed
in terms of a, f , and r. Differentiation with respect to (• and
division by 27r£ give, to the degree of approximation adopted
in this investigation, for the axial velocity
cor
€•«?+«>
where h = a — %. If h — r, so that the circle chosen is the
smallest, parallel to the circular axis, that can be taken on
the anchorring, this becomes
: log [cor.
vrra r
From this result the same equation (17) as before for the
velocity of the circular axis of the vortex ring can be obtained.
Innellan, April, 1914.
III. The Lost Pressure in Gaseous Explosions. By Prof.
W. M. Thornton, D.Sc, D.Eng., Armstrong College,
NewcastleonTyne *.
(1) The " suppression of heat " caused by the forces of
cohesion in molecular formation.
WHEN a quantity of heat Q is given to unit mass of a
gaseous mixture by slow external heating at con
stant volume, the rise of pressure is proportional to the
increase in the absolute temperature and can therefore be
calculated when the appropriate specific heat (Jv is known.
If, however, the heat is derived from the explosion of the
mixture the maximum pressure obtained experimentally is
about half of that calculated from the heat developed,
allowing for change of volume in combustion and with
values of Q and Cv derived from experiments under steady
conditions.
Four suggestions have been made to account for this
difference — (1) that there is rapid cooling by the cylinder
walls (Hirn) ; (2) that there is dissociation of the gases
formed (Bunsen) ; (3) that the specific heats rise greatly
with the temperature (Mallard and Le Chatelier) ; (4) that
there is " afterburning " (Clerk).
Evidence has been given that the first three do not explain
all the observed difference. Clerk's argument is largely
based on the fact that in certain cases the pressure in the
* Communicated by the Author.
Lost Pressure in Gaseous Explosions. 19
explosion cylinder remains constant for long intervals. This
persistence of pressure can, however, be possibly explained,
as shown later, by consideration o£ the transfer of energy in
the molecules of the products of combustion, after the act of
coherence by which new molecules are formed.
In slow combustion a certain portion of the total energy
of combustion is given to the products as heat, the amount
depending eventually upon the equipartition of energy. In
almost all physical processes into which consideration of
equipartition enters, the transfer of energy is from translation
to rotation or vibration. But in the act of combustion
forces (of cohesion) are suddenly introduced which materially
influence the method of transfer of molecular energy from
one form to another. There cannot be a rise in the number
of degrees of freedom sufficient to account for the deficiency
in translational energy. In several cases a rise from 7 to
12 degrees each sharing the energy equally is necessary,
and this does not occur, for it has been shown by Hopkinson*
and David t that the energy radiated up to the point of
reaching the maximum pressure is only 3 per cent, of the
total energy of combustion, and further, the increase in the
possible degrees of rotation is at most three.
(2) Simple case leading to a mean efficiency of
explosion of \.
The mechanics of two colliding and cohering spheres show,
however, that in such a case there may be a rise in the
rotational molecular energy at the expense of the trans
lational, and this of an order which may account in part
for the lost pressure.
In the explosion of a hydrocarbon gas carbon monoxide is
probably first formed J, and the atomic weight of carbon is
12 and of oxygen 16. Take for simplicity of treatment two
spheres of equal mass moving into union with equal velocity
v, having angular velocity &>, and having equal translational
and rotational energies before collision. If their approach
is in opposite directions in the same straight line, the trans
lational energy is all converted into vibrational energy at
the moment of collision, the total rotational energy either
remaining the same as before contact, or if cheeked also
adding to the latter.
* Proc. Roy. Soc, A. vol. lxxix. p. 138.
t Phil. Trans. Roy. Soc, A. vol. ccxi. p. 375.
X H. B. Dixon, Phil. Trans, clxxxiv. p. 97 (1893).
02
20 Prof. W. M. Thornton on the
If they approach out of line in parallel paths and cohere,
the whole of their translational energy is converted into
rotational energy (an irreversible process in combustion),
and the molecule spins about the point of contact. In the
case of oblique incidence we may resolve into two com
ponents and consider the energies before and after contact.
Fig. 1.
Before contact the total energy of each sphere is \{mv2 + Io>2),
I being its moment of inertia. The translational energy of
A parallel to B is J mv 2 sin2 6 ; so that in union each sphere
loses by this amount, for this is the component causing
rotation about the point of cohesion. The total rotational
energy of the two before collision is Io>2? after collision it
may reach Lo2 + mv2 sin2 6. The mean value of sin2# is J ;
thus, since Ia>2 was taken to be equal to mv2, the rotational
energy of the doublet formed is at most 1^ times the rota
tional energy of its component before collision.
The translational energy is before collision mv2; afterwards
it is ^mv2(l — sin2 0)+^mv2cos20, for B loses a part of its
translational energy in combining with A and the vertical
component of the latter is unchanged. This expression is
equal to mv2cos20, and in the mean to \mv2. Thus the
ratio of the total translational energy after collision (to
which the pressure of the gas is proportional), to that
immediately before collision is J; in other words, the resulting
pressure is only one half of that which would be obtained if
no new molecules were formed. The argument holds for com
bining atoms of unequal mass, but with the same energy of
translation, as for example hydrogen and oxygen in complete
mixture.
This excess of rotational over translation energy is not
permanent and is more or less rapidly equalized to pressure
and radiation ; but there is no experimental evidence other
than that derived from the velocity of an explosion wave, to
show its actual duration. The conditions in a travelling
explosion wave are somewhat different from the explosion
at constant volume under consideration. Dixon has shown
Lost Pressure in Gaseous Explosions.
21
that the velocity of a travelling wave corresponds to a
temperature in the wavefront twice that derived from the
heat liberated by slow combustion. Chapman has calculated
the specific heats of the products from the wave velocity, and
has obtained values nearly twice as high as those at low
temperatures. Berthelot has remarked that the pressure
indicated by any moving mechanism is probably not that
in the wavefront but on i corresponding to ordinary com
bustion.
If, however, the translational energy is suppressed for a
moment by one half in the act of explosion, as shown above,
the observed maximum pressures are the theoretical maxima
and the difficulty of reconciling theory and observation is
removed.
(3) Experimental values of the Explosion Pressure
ratio approximate to \,
The ratio of the observed to the theoretical maxima,
calculated on the assumption that all the energy of slow
combustion is converted into translational energy in explosion,
closely approaches one half.
In the case of hydrogen and air the observed and cal
culated values are as follows * : —
Table I.
Percentage of Gras
in Air.
Maximum Pressure of Explosion in lbs.
per square inch.
Observed.
Calculated.
Ratio.
143
41
68
80
883
124
176
•465
•549
•455
200
285
Mean...
489
In two cases of coalgas and air Dr. Clerk obtained the
following values f.
* P. Clerk, ■ The Gas, Petrol and Oil Engine,' vol. i. p. 105.
t Loc. cit. p. 136.
22
Prof. W. M. Thornton on the
Table IT.
Percentage of Gas
in Air.
Maximum Pressure.
Observed.
Calculated.
Eatio. r\.
67
I.
40
515
60
61
78
87
90
II.
55
67
75
76
93
102
105
I.
895
96
103
112
134
168
192
II.
105
110
118
127
149
183
209
I.
•447
•535
•582
•545
•572
•517
•468
II.
•523
•602
•645
•598
•625
•557
•505
71
77
83
100
125
142
Mean...
•523
•579
I. Glasgow Gas.
II. Oldham Gas
These are the only recorded values at different percentages
of gas and air readily available.
Berthelot & Vielle (Annales de Chimie et de Physique,
6e serie, torn. iv. pp. 390) obtained the following maximum
pressures of explosion (Table III.). Berthelot had previously
calculated* the theoretical maxima with values of the specific
heats of the products which aro certainly low, and therefore
give too high a calculated maximum. The mixtures here are
in every case those for perfect combustion.
From an extended series of measurements on acetylene,
Grover found ratios of observed to calculated maximum
between 0'42 and 0*73. Berthelot's ratio for this gas is
probably low, but retaining it the mean of all the ratios in
Tables L, II and III. is 0502.
(4) Relation of efficiency of explosion to percentage of
combustible gas in mixture.
The simple case considered cannot do more than suggest
the chief process in gaseous explosion, for there is the
addition of a second atom of oxygen to form carbon dioxide.
It is to be noted, however, that this addition does not affect
the ratio between translational and rotational energies so
much as the first combination. Further, in Table II. the
* M. Berthelot, ' Explosives and their power. Trans, by Hake and
MacNab, pp. 387 and 543.
Lost Pressure in Gaseous Explosions.
Table III.
23
Combustible Gas
in Oxjgen.
Maximum Pressure in atmospheres.
Observed.
Calculated.
Eatio.
Hydrogen
Carbon monoxide
98
101
210
153
161
161
163
20
24
51
455
42
38
34
049
042
041
034
038
042
048
Acetylene
Ethylene
Ethane
Mean ...
042
Fiff. 2.
s
\
\
\ N
\
\
\\
\
^1
jS
I
\
^
;
V
\
\
i
>
O 5 IO 15 20 25 30
PE.R CENT OF GAS IN AIR
more inflammable mixtures have the higher ratios, and the
variation of the efficiency of explosion with percentage of
gas appears to be in a regular manner. In fig. 2 the ratios
in Table II. are set down graphically, and the curves are
soon to have the same shape throughout, I. being for
Glasgow gas, II. for Oldham gas.
24 Prof. W. M. Thornton on the
When two atoms combine to form a molecule their energy
of combination is shared with surrounding inert molecules,
and the greater the proportion of inert gas the more com
pletely the observed maximum should approach a maximum
calculated on an energy basis only, since whatever form the
energy of combination may for the moment take, it must be
shared equally with the neighbouring molecules and in the
end the translational or pressure energy is equalized. Thus
the ratio of observed to the calculated maximum pressure
would approach unity in the weakest mixtures if combination
were not checked by the insufficiency of energy, set free by
the molecules which are formed, to raise the whole mass of
gas to a temperature such that all the combustible portions
may unite.
At the upper and lower limits of inflammability the ratio
expressing the " efficiency " of explosion is of course zero.
For coalgas these limits are usually 29 and 6 per cent.
Straight lines drawn from the upper limit through the
curves cut the vertical axis at 0'94 and 0'98. [In the case
of hydrogen, for which only three points are available, a line
from the upper limit at 70 per cent, through two of the
points cuts the axis at 0'78.] From this it is seen that the
efficiency of explosion of the richer mixtures decreases to
zero at a uniform rate as the percentage of gas is raised.
In weaker mixtures, diluted with air, every atom of com
bustible gas enters into combination and shares its energy
with the residue of the air. Thus the ratio of rotational to
translational energy is not at any moment abnormally great
for the whole volume, and the percentage of pressure
developed is higher than if there were no nitrogen present.
On the other hand, in richer mixtures with excess of com
bustible gas, all the oxygen is taken up but less combustible
gas, and at the upper limit none of the latter. The efficiency
of explosion should, therefore, always diminish in mixtures
above the point of perfect combustion.
Let N be the number of combustible gas molecules
entering into combination in unit volume of mixture, n of
" inert " molecules. The ratio rj of Table II. and the figure
is that of the translational energy of the whole mixture to the
energy of combustion in it. The translational energy is
directly proportional both to N, the combustion of which
gives rise to the heat, and to n which helps to transfer it as
pressure to the walls, and therefore to their product. As the
percentage is varied the sum N + w = M say remains constant,
and Nw = N(M — N). Thus we may write the translational
Lost Pressure in Gaseous Explosions. 25
energy in terms of N = aN — VN2 and the efficiency ratio
a^bW
V =
N?
where q is the heat of combustion of a single molecule.
We should then have
v = ABN,
and this is the equation of the line CD in the figure 2
obtained from the experimental results.
(5) General relation of maximum pressure of explosion
to percentages of combustible gas.
Let the percentage of combustible gas in a mixture be
represented by lengths from 0 to M, M being 100 per cent.
Set up vertically at M a line MR to represent to any scale
unit volume of combustible gas. At 0 set up OS to repre
sent unit volume of oxygen. Join OR and MS. The
percentage P in the mixture giving perfect combustion is at
that point at which the ratio of the ordinates PQ/PT is that
of the volumes required for perfect combustion. P being a
fixed point the lengths of OS and RM are decided by taking
PQ and PT to a convenient scale in the correct ratio, and
producing MQ to S, OT to R.
Fig. 3.
R
In the case of methane for example, which requires twice
its volume of oxygen for perfect combustion, PQ = 2PT.
Join OQ, then the ordinates of this line represent to a
certain scale the number of combustible units * in the
mixture below the point of perfect combustion, the ordinates
of QM of those above this point. Let U and L be the upper
and lower limits of inflammability. At the lower limit there
are FL combustible units, and they fail to ignite the mixture
because of the cooling influence of the mass of inert gas ; at
* The combustible or explosive unit is defined as the aggregate of one
molecule of combustible gas and the oxj^gen molecules required for its
complete combustion.
26 Prof. W. M. Thornton on the
the upper limit there are GU units, but the cooling or
retarding influence of the gas is greater than at the lower
limits and ignition again fails. The ratios
VL_ TP_WU
FL~QP~ GU
are those of gas to air for perfect combustion.
Now the heat set free in combustion is directly propor
tional to the number of combustible units in the mixture,
except at the limits where the inflammation suddenly ceases.
We may then write, for rich mixtures, from P to M,
Q = Qc(laN),
where Qc is the heat of combustion of unit volume of the
perfect mixture. For the part LP,
Q = QC6N.
The maximum pressure calculated from the heat liberated
is
Pc=Pl.~ '
('♦£>
where p1 is the initial pressure, r0, i\ the volumes before and
after explosion, Q the total heat of combustion, and a the
coefficient of expansion of the gas. Substituting the above
values for Q we have, since for the same gas Vi/vQ is
constant,
above the point of perfect c Qc(l — aN)al
combustion P : ^ = ^ij 1+ — — q > ,
below P : pc = cpx (1 + ^ J.
By definition y = p obg. \pc, so that
(a) above P : p oba. = cpi??{lf &(1— aN)};
and r] has been found to be of the form A — BN,
therefore in rich mixtures
pobs> = ci?l(ABN){l + Z:(laN)},
which may be written
Lost Pressure in Gaseous Explosions.
27
This is the shape of curve 1 (fig. 4), found experimentally
by Grover for coalgas and acetylene, in weak mixtures with
Fig. 4.
t oba
initial compression *, that is, with the number of combustible
molecules per unit volume N increased mechanically in the
ratio of compression.
{}>) below P : ^ob8. =j9,(ABN)&(l + &N),
= Pl(A, + K2NC2W).
This agrees with the shape of curve 2 (fig. 4), obtained
in explosions at initial atmospheric pressure f. The point
P appears, however, to be passed through without a sudden
change in the curve.
The agreement shows that by the use of the expression
taken for the explosion efficiency rj = A— BN, and the
conception of the unit of combustion, the two types of
curve expressing the experimental relation of maximum
explosion pressure to percentage of combustible gas may be
explained.
The peculiar feature of the curves connecting the observed
velocities of explosion with the percentage of combustible
gas is that they consist, for the most part, of two straight
lines forming a triangle upon the axis of N with the vertex
at or near the percentage of complete combustion, and the
base coinciding with the range of inflammability. The
straightness of the rising and falling sides of this can be
explained if the velocity falls from a maximum in proportion
* Clerk, ' The Gas, Petrol and Oil Engine/ vol. i. p. 1G4, fig. 53.
t Loc. cit. figs. 46, 49.
28
Prof. W. M. Thornton on the
to the decrease in the number of combustible units from their
maximum at P, the point o£ perfect combustion.
Fig. 5.
PQ represents to a certain scale the maximum velocity of
explosion ; the ordinates of LQ, QU the velocities of
explosion as the percentage of gas is increased from
the lower to the upper limit of inflammability.
(6) Influence of initial compression on the efficiency
of explosion.
The expression for the efficiency may be also used to show
how compression improves the efficiency of combustion.
Other conditions remaining the same, doubling the
pressure doubles the number of combustible units in a given
volume. This is represented in fig. 6 by extending the
upper limit. OUi is the upper limit at atmospheric pressure,
OU2 at two atmospheres, OU3 at three, and so on. The
lines corresponding to CD in fig. 2 are CUi, CU2, CUs....
It is well known that mixtures which are so weak that they
cannot be ignited at atmospheric pressure are inflammable
when compressed, so that N at the lower limit does not
Lost Pressure in Gaseous Explosions. 29
advance in the same ratio as the upper limit. If the lower
limit advanced at the same rate as the upper the efficiency
would not be changed by compression.
The efficiency of explosion is seen to be raised by initial
compression, and it should continue to rise indefinitely, a
point of practical interest in internal combustion engines,
which has led to discussion.
(7) Summary.
The total action in a gaseous explosion is a multiple of
that which occurs in the formation of a single molecule.
From the consideration of a diatomic case a value of ^ was
obtained for the ratio of translational energies before and
after formation. Experiment shows that this ratio varies in
a definite manner and that the mean of its values over the
working range in coalgas, and in many other mixtures
giving perfect combustion, also approaches J.
The suggestion now made is that the "suppressed heat "
in gaseous explosions may be explained by the influence of
forces of cohesion which come into action at the moment
of " contact " of two combining atoms. It would explain
(1) the cause of the lost pressure ; (2) the variation of
the efficiency of combustion with strength of mixture ;
(3) the shape of the maximum pressurepercentage curves ;
(4) the differences between the explosion efficiency of gases
having different limits of inflammability ; (5) the influence
of initial compression in raising efficiency of explosion,
as a consequence of the limiting shape of the efficiency
curves.
After explosion the molecular translational energy of the
products of combustion in a closed vessel would fall more
slowly than in simple cooling of a hot gas in which equi
partition is already established. This would have the same
influence on the pressure as afterburning, and may be the
cause, at least in part, of the maintained pressure observed
by Clerk in rich mixtures (in which, as shown above, the
pressure efficiency should be lowest), and of the prolonged
radiation observed by Hopkinson after the flame stage has
ceased.
[ 30 ]
IV. Notes on the Motion of Viscous Liquids in Channels.
By J. Proudman *.
1. TN a recent communication to this Journal f Messrs.
X Deeley and Parr remark that the conditions of the
steady flow o£ a viscous liquid in a parabolic channel, under
a constant force parallel to the length of the channel, have
not yet been ascertained. It is implied that the results
might be of interest in connexion with the motion of
glaciers.
In the present communication the problem is solved for
the special case in which the free surface of the liquid passes
through the focus of the parabolic section, and also for a
particular triangular section. Some remarks are also added
in connexion with the mathematical expansions used.
The general problem % for a channel of any section may
be reduced to that of finding a function % which satisfies
2u" Br "' (
over the section, which vanishes over the sides of the section,
and for which ^^/^w = 0 over the free surface. Here #, y
are rectangular Cartesian coordinates in the plane of the
section, and "dfon denotes differentiation along the normal
to the free surface.
The velocity of the liquid, which is parallel to the length
of the channel, is given by F^/2/jl, where P is the pressure
gradient along the channel, and fi is the coefficient of
viscosity. In applications, the function
where the integral is taken over the area of the section, is
required.
Particular Parabolic Section.
2. For convenience, take the length of the latusrectum of
the parabola to be 47T2. Then if we take polar coordinates
r, 0} having for pole the focus S, and for initial line the axis
* Communicated bv the Author.
t "The Hintereis Glacier," Phil. Mag. (6) xxvii. p. 153 (1914).
X See Lamb, Hydrodynamics, 3rd ed., p. 545.
Motion of Viscous Liquids in Channels.
SA, the equation of the parabola will be
f* COS \d — 7T,
and that of the latusrectum 62 = \ir2.
31
\£
Fig. 1.
S
Ejr. 2.
Let us take
f=?4cosi^ 9 = rising,
so that we have a conformal transformation if f, v be regarded
as Cartesian coordinates in another plane, the parabola Vans
Eorming into f =7r, and the latusrectum into £2 = v2. The
correspondence is shown in figs. 1 and 2, where corresponding
points are similarly lettered. °
Since
m~i<^
m
the conditions to be satisfied by % become, with reference to
fig 2,
P+g = «(f+r), .... (3)
over the area L'SL ,,x=0 on £=*, dx/0P=dr3,; on f
and dx/o?= —ox/9'; on ?=?/.
32 Mr. J. Proud man on the
Instead of trying to solve this problem directly, let ns
determine a function which satisfies (3) over the area of the
square bounded by £= +77, tj= + 77, and which vanishes on
the sides of this square. The determination of such a func
tion is known to be unique, and from considerations of
symmetry we see that over the triangle L'SL it will be the
function we require.
Now
4(7^ + f )t7r9V)  2 A» cosh (n + i)f cos (n + J)*?, (4)
where Aw is a constant, satisfies (3) and vanishes over
7] = + 7T. We shall see that we can choose the constants A„
so that it will vanish on f = +7r.
From Fourier's theorem, or otherwise, we have
4 oa / J\rc
for — 7r <: r] <> it, by which we see that if we take
( — 1 )n
A„cosh (w + i)7r = 327r v
(« + *)•'
(4) will satisfy all the conditions for %.
Thus,
0.0 3 (~l)w cosh(?i + i)£ .
The value of % on SL, which gives the velocity on the free
surface, is obtained by putting ?? = £ in (5). Doing this, we
obtain
in which f is connected with the distance r from the focus,
by £ = r*/ V'2.
For the flux of liquid through the channel we require the
function
taken over the area of the triangle L'SL (fig. 2), or, again
from symmetry, taken over the area of the square SL. The
integration is straightforward, the series for ^ being uni
formly convergent over the area, and we obtain
F 7T6 » 1 S 1
+ 2 2 7ttw; 2tt 2 . , M5 tanh (n + *>,
128tt2 15 ~Z0 (n r if Zo (n + ±)«
Motion of Viscous Liquids in Channels.
33
or, since
^O + i)6 15'
iow 2= ir~27rS , — — rrytanh (n + 4)7r.
12b7r2 5 n=o(™ + i)
If now we take the latusrectum to be 4a instead of 47r2,
F will be multiplied by (a/7T2)4, so that
F _ 128 _ 256 
5tanh(n + i)7r. . . (7)
Particular Triangular Section,
3. The section is that in which one side of the channel is
vertical and the other inclined at an angle \tt to it (fig. 3).
Fig. 3.
The solution for this case can be derived from an expression
previously given *, but it is just as easy to verify directly
that all the conditions of the problem are satisfied by
9 00.
X=0+*/) ('**) 2
7T tt=0 0 + ir sinh (2m + 1)tt
X {sinh (n + ^)(27T— i + i)(>f y) sin (n + £)(#—#)}, .
(8)
the axes being as shown in fig. 3. The boundary conditions
are that % = 0 on x = it and on x=—y, and that d^/dy = 0
on y = 0 ; but instead of the latter we take ^ = 0 on x=y>
again appealing to symmetry. We have taken 0A = 7r for
convenience.
* Lond. Math. Soc, Records for March 13th, 1913.
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. D
34 Mr. J. Proudman on the
The value of % on OA, which gives the velocity at the
free surface, is
,2a sinh (ft + i)(7r — x) sin (n+ \)x
(9)
and the function F, when OA is taken to be a instead of tt, is
given by
^=A"^J0(^coth(n+i)7r • (10)
Remarks on the Expansions.
4. The normal derivatives of (5) and (8) must vanish over
SL and 0 A respectively. For (5) this gives us
elMHt 1
s tt „=0 {n + i)2 cosh (n + J)tt
X {cosh (n + i)f sin (n + i)f+ sinh (a + J)f cos (n + £)f}, (11)
for — 7r<^<7r, while for (8) it gives us an expansion which
is easily transformed into (11).
Again, alternative forms can be obtained for (5) and (8),
and on equating them respectively to the above forms,
identities are obtained. Identities of this nature were noticed
by Sir G. Stokes *, and remarked upon by Thomson and
Tait f Those mentioned by these authors were examined by
F. Purser J, who pointed out their connexion with Elliptic
Functions.
Two additional remarks, however, seem worth making.
The first is that the identities can be easily obtained by
taking a twodimensional harmonic in algebraic Cartesian
form, and then finding a series of twodimensional harmonics
in normal Cartesian forms (i, e. in terms of trigonometric
and hyperbolic functions), which satisfies the same conditions
at a certain boundary.
The second is that when the expansions of conjugate
functions are combined to form a series of functions of
a complex variable, the resulting forms appear to be
interesting.
For example, we can thus obtain the following expansions,
valid over a square whose corners are given by
* Math, and Phys. Papers, vol. i. p. 190 (1846).
t Natural Philosophy, part ii. p. 249, 1883 ed.
X Messenger of Math. vol. xi. (1882).
Motion of Viscous Liquids In Channels. 35
 = !+;, 1 + z, li, 1i:
TT
*= !J. (^W^¥T¥)^ { sinh {n+i)z+ sin ('l+i)* }'
s2=  2, t — rrrs r^ — r~i~ J cosh (n + l)— cos (n + i)j I,
tt^o (« + i)"cosh()i + ij7r I \ 2J j>
(12)
(13)
= ^2„ fr + ti'U^ + ifr {Si"h (" + «* Si" (" + «' }' (")
* 47r4=gl(n + i)4os1h)(n + ^ { C°5h ^ + ^ + C0S f»+« ^ } >
and so on. • • • (15)
If in (12) we write z—%\\% and then take the real part,
f being real, we reproduce the expansion (11).
Numerical Values.
5. The series (6), (7), and (9) have been examined
numerically by Mr. J. K. Maddrell, of Liverpool University,
who has very kindly supplied the following results : —
32
?f" *'
0 17*41374
1 1740685
4 1730824
9 1689301
16 1578149
25 1352752
36 973117
49 486775
64 000000
Here r and Xi refer to (6), while x and ^2 refer to (9),
results are shown graphically in figs. 4 and 5.
The maximum value of %2 is found to be about 11656, its
position being given by ,r/7r = "60819. This maximum has
an interest in connexion with several other physical problems
which are mentioned in the note referred to in § 3.
The series for F/a4 in (7) and (10) have the respective
values 9293, 02610.
In connexion with a mathematically related problem Saint
Venant * pointed out that if we write
2F = kA4/I,
whero A is the sectional area of the pipe formed by the sides
* Comptes Rendus, t. Ixxxviii. pp. 142147 (1879).
D 2
8
 X.
TT
X2
0..
. . ooooooo
1..
... 0208361
2..
... 0546561
3..
... 0870728
4..
... 1096033
5.
... 1163877
6..
... 1028783
7...
. . 0650613
8..
... 0000000
id V
2 refer to (9). The
36
Motion of Viscous Liquids in Channel*.
Fig. 4.
 Wr
\
/
/
I'M
\
/
\
/
/
\
\
/
\
/
b
\
/
\
/
\
0 2 4 6 8
Surfacevelocity curve for Parabolic Section.
Fig. 5.
It/^tH
0 • •* 2 3 4 5 6 7 8 9 I ^
Surfacevelocity curve for Triangular Section.
o£ the channel and their reflexion in the free surface, and
I is the moment of inertia about its centroid of this section,
then for the majority of simple cases k has a value which lies
between 0228 and *0260. In the two present instances we
have respectively
* = 0252, /e=0232.
[ 37 ]
V. On a New ContinuousBalance Method of Comparing an
Inductance with a Capacity. By John P. D ALTON, ALA.,
D.Sc*
§ 1. TT7"HEN alternating current is used for inductance
▼ t determinations by null methods, the condition
to be fulfilled in order to obtain a true balance is that the
potential difference across the terminals of the indicator —
telephone or vibration galvanometer — should vanish at every
instant, so that the potential differences across each pair of
arms from a common junction to the galvanometer must
have the same amplitude and they must also be in the same
phase. Of the laboratory methods at present commonly
employed for determining the ratio —;, Anderson's method
(Stroud's is essentially the same, being the conjugate arrange
ment) is the only one which fulfils the requirements for
continuous balance. The original Maxwell scheme might,
perhaps, be added, but it hardly ranks as a practical method
on account of its tedious successive approximations. In the
Rimington and Pirani methods continuous balance can be
obtained for one definite frequency only, depending upon
the inductance, capacity, and resistances used ; and, if the
current is not simply harmonic, continuous balance is not
possible at all.
Since the advent of the tuned telephone and the vibration
galvanometer, a.c. methods have come into more general use,
and it is as well to have as many independent methods as
possible which give a true balance. The object of the present
note is to describe a method of comparison which has proved
workable, convenient, and satisfactory, and which deter
mines a continuous balance for all frequencies (that is, of
course, as long as the resistance is not a function of the
frequency].'
§ 2. It may be of some assistance to look first into the
cause of the imperfectness of the balance given by aggregate
methods when tried with alternating current. In the Pirani
arrangement, for instance, we have, leaving the galvanometer
out of account for the present, two circuits in parallel — the
* Communicated by the Author (now of S. A. School of Mines,
Johannesburg).
38 Dr. J. P. Dalton on a New Continuous Balance
one a noninductive resistance R', and the other, of total
resistance say R", a coil of inductance L in series with a
Fig. 1.
R' > <
jVVVVVv/VVVVA/vAA/nAA>VVVVAA/S^S/
C 3
TTVUB aoau OVTSJ
noninductive resistance over part of which, r, a condenser
of capacity C is shunted. Suppose now an alternating e.m.f.
is applied between the points A and B : taking its initial
phase to be zero, we may write for it E0 e^K The current
amplitude in the upper circuit is then ^ ; it is real and
consequently it, as well as the potential difference between
any two points in that branch, is always in phase with the
applied e.m.f. But the current amplitude in the lower
branch is
E0(l + Cnp)
(R"+Lz»(l + CWp)r2CV ' * " W
and therefore is not, in general, in the same phase as the
applied e.m.f. and cannot possibly be so for all values of p.
The same applies to the potential difference across any part
of the noninductive resistance in that circuit. The phase
of the voltage across the coil is determined by
E0(l + Cn>)(Rc + Lfo)
(R" + Lt/0(l + O*»rsCtp' " " ' ' K '
where Rc is the coil resistance, and it, too, cannot possibly
vanish for all values of p. As long as the coil and con
denser are in series in the same branch, a true balance cannot
be found which is independent of the frequency*.
* Forsvthe's recent very elegant method of comparison (Phys. Rev.
i. ser. 2, p. 468 (1913) ; Science Abstracts, A. xvi. No. 1724 (1913)) gets
rid of the difficulty by placing the condenser in series with a resistance,
x say, across the coil, so that r now includes the coil. The current in
the lower branch is now
Eo tt~ ^ ; n ;•• • (*)
(R"',)(i+L8>+r+'')+(,'+L!>)(ci+;l')'
id p —r2 the phase of
is then continuous for any frequency
If x=r and ~ —r2 the phase of this vector is zero, and the balance
Method of Comparing Inductance with Capacity. 39
If the telephone is connected between two points P and Q,
then balance is attained if
Bap RAQ(1 + Cnp)
R'  (R" + Lip)(l + Crip)r2Cip> ' ' {)
This determines as the frequency at which a continuous
balance is possible :
9 1 Rap .R" — Raq • R' fA\
P=LGr K^ ' • • ; (4)
and we then obtain
L __ 2 r^ Raq R' — Rap . R " . ... (5)
C Rap Rap
If, as is usually done, a steady balance is first obtained,
then an inductive balance can occur only when p = 0, and
the method cannot be used with alternating current. When
p = 0, equations (4) and (5) give the usual conditions
Rap R' ((»\
AQ J^
and L f .
o=r (7>
§ o. In the MaxwellRimington arrangement the con
denser is removed to the upper branch circuit, and it is no
Fig. 2.
longer impossible to obtain two points, one in each circuit,
such that the potential differences between them and A or
B are always of the same magnitude and in the same phase,
whatever the frequency may be.
The current in the upper circuit is now
°R'(l+ryO)fSpC' * ' ' * K)
where R' is the total resistance of that branch, and r is that
40 Dr. J. P. Dalton on a New ContinuousBalance
part of it which is across the condenser terminals ; while in
the lower circuit the current is
E° (9)
n"+Li>
In Rimington's method the indicator is placed between Q
and a point X between A and P, so that the voltages which
must be equal in order to obtain a balance are
¥L'(l + ripQ)rHp(?
across AX Eop//, , JLn\ ^Ln? . . . . (1U)
and across AQ
nAQ.+~Lip nn
and there is, in general, but one (real) frequency for which
these are equal. For that we find
2 _ Raq . R; — Rax ■ R ' /1 o\
p~ LCr(RxBr) ^ }
And then L = _^_ {^aq_(Raq .U'Kax . r»)}. (i3)
Here again, i£ a steady balance is first arranged an in
ductive balance cannot be obtained with alternating current.
With/>=0 (12) and (13) become the usual
Rax R'
Raq R
sS7"' '
C " Rr { }
..... (14)
and L ^2Raq
kxb
In Maxwell's original method the indicator was placed
between Q and P ; the points X and P are then coincident,
and r = RXB Continuous balance is then possible at all
frequencies, and the second relation becomes
ti=RxbRaq. ..... (16)
§ 4. In the Rimington method the phase difference between
the voltages across the arms AX and AQ and the applied
e.m.f. are of the form tan"1, P ,,, where a, b, and c are
b + cp2
constants, so that, in general, these phases can be the same
for one particular frequency only. But let us look for a
moment at the conditions obtaining along the branch PB.
Method of Comparing Inductance with Capacity. 41
The current in that branch is
^o /17n
TL'(l + ripC)r2ipC' ^ J
and, as the resistance is noninductive, the potential difference
between any two points along PB is proportional and parallel
to the same vector. Hence the voltage across YB, say, lags
behind the applied e.m.f. by an angle
tan'^'^0 (lg)
At the same time the voltage across QB lags behind the
applied e.m.f. by
tan'^ (19)
Both of these tangents are simply directed proportional to
the frequency, and consequently if they are equal for one
frequency they are equal for all. If then we equalize
potentials, not, as in Rimington's method, between Q and a
point on AP, but between Q and a point on PB, we can
ensure that the balance obtained is continuous for all fre
quencies, while we still avoid the troublesome double adjust
ment of the Maxwell method.
Writing
R1 = r1 + r2 = AP + PY^
R2 = YB L . . . . (20)
R3 = AQ and R4 = QB
the condition for phase equality becomes
Lp (Rir,)(R8 + r,)j?C
R3 + R4 R1 + R2 ' * ' * { }
while the condition that voltage amplitudes are equal then
becomes
R!R4=R2R3 (22)
Hence, for a balance,
£=?r,(R2 + r2) (23)
The same result is easily obtained for the conjugate
position, where Y and Q are the current points and the
indicator is across AB.
Putting r2 = 0 we get Maxwell's case; while if the arms
of the bridge are all made equal, (23) becomes
5=r1(Rhr2), (24)
42 Dr. J. P. Dalton on a New ContinuousBala
nce
or simply
R2
(25)
where r is the resistance which, in series with the single
arm, shunts the condenser.
§ 5. It was thought worth while to test the foregoing as a
practical method of comparison. Unfortunately no standards
were available, so a very severe test could not be imposed,
but it is perhaps of even greater value to experiment with
ordinary laboratory equipment, and ascertain how far the
method would serve as one for everyday use.
A small solenoid which happened to be available was used
as a test coil. Its mean diameter was 3*9 cm., and it had
300 turns of fine copper wire wound in a single layer on a
length of 15'9 cm. The coil was not sufficiently well con
structed to justify a very refined calculation of the inductance.
Regarding the field as uniform over any crosssection, and
equal to its value at the centre of that section, the inductance
becomes
L = 47rVa2( V^H^"a), . . . . (2S)
where I is the length and a the mean radius of the solenoid,
and n is the number of turns per unit length. In the present
instance this gives L = 0'75 millihenry.
A more reliable value was obtained by direct comparison
against a capacity of one microfarad by Anderson's method :
this gave an inductance of 0*7 64 millihenry.
A balance was then arranged in the following fashion : —
Fig. 3.
R2 and R4 are the ratio arms of an ordinary dial P.O. box.
In series with the third set of resistances of the box are
Method of Comparing Inductance with Capacity. 43
placed the inductive coil L, and a small cloth rheostat, r7i,
the latter enabling one to maintain a steady balance in case
of fluctuations in the temperature of L. The remaining arm
of the balance, R1? consists of two resistanceboxes between
which is stretched a wire of known resistance per unit length.
The condenser — in this case a Muirhead divided microfarad
— is between the junction of the ratio arms and a sliding
contact on the stretched wire.
A Cohen vibrating wire interrupter was used in con
junction with a Campbell vibration galvanometer tuned to
resonance; the range of the galvanometer permitted the use
of only comparatively low frequencies. As the inductance
was known to be comparatively small, moderate values were
selected for the resistances in order to increase the sensitivity
of the arrangement. The resistance of the coil was first
roughly determined, and the arrangement was then set for a
"steady " balance, i\ and r2 being made equal. Alternating
current was then turned on, and i\ and r2 were altered
(keeping their total constant) until the deflexion became
very small ; as the point of balance was approached, a little
successive adjustment of the rheostat, rht and the sliding
contact enabled one to obtain a very sharp zero. The in
terrupter was first placed across AC and the galvanometer
across BD, and then these were interchanged. The method
worked satisfactorily in every way, and gave beautifully
sharp and welldefined balancing points. (A coil interrupter
and telephone were also tried ; they worked quite well, too,
but they were not so sensitive as the galvanometer.) The
results of a few typical measurements are here given.
Test coil : short solenoid.
Inductance (approximate calculation) 0*75 mh.
„ (measured by Anderson's method) . 0*764 „
New method : —
(i.) Interrupter BD; Galvanometer AC.
C = 0'2 microfarad ; R = 100 ohms.
r1 = 21,3 ohms. r2 = 78'7 ohms.
.. ^ = 100278'72 = 3807xl03.
,\ L = 0761 millihenry.
44 Dr. H. E. Watson and Mr. Gostabehari Pal on
(ii.) Interrupter AC ; Galvanometer BD.
C = 02 microfarad; R=100 ohms.
Balance undisturbed.
C = 0'4 microfarad; R = 100 ohms.
r1 = 10,0ohms. r2= 90*0 ohms.
^ = 1002902 = l'90xl03.
.*. L = 0*760 millihenry.
Victoria College, Stellenbosch, S.A.
VI. On the Radioactivity of the Rocks of the Kolar Gold
Fields. By Herbert Edmeston Watson, D.Sc. (Lond.),
and Gostabehari Pal, M.Sc. (Calcutta)*.
THE Kolar Gold Fields are situated on the Mysore
plateau in lat. 13° N., and 50 miles east of Bangalore.
The present workings extend along a line about 5 miles
long running north and south, and several of them are
carried down for 4000 feet, the greatest vertical depth so far
reached being about 3600 feet. The "country rock" in
which the quartz lies consists of schists, and is apparently of
a very uniform nature throughout the workings. Conse
quently it was thought that it might be interesting to carry
out some estimations of radium with a view to determining
whether its distribution in this homogeneous rock was
uniform.
A few estimations of the radium in some of the other
rocks from the same locality have also been made for the
sake of comparison.
The following is a description of the method used and the
results obtained : —
Experimental.
The method used for the estimation of radium was a modi
fication of Joly^s fusion method (Phil. Mag. xxii. 1911,
p. 134). It is perhaps not capable of quite general application,
but it proved very efficacious for all the rocks dealt with in
the present experiments. The principle consisted in the
fusion of the rocks with potassium hydroxide under reduced
pressure; and the results obtained appear to show that by
this means the radium emanation is liberated just as com
pletely as by the higher temperatures necessitated by fusion
* Communicated bv the Authors.
Radioactivity of Rocks of the Kolar GoldFields. 45
with alkaline carbonates. A thick copper flask A, of about
300 c.c. capacity, with a long neck upon which was soldered
an outer tube to form a waterjacket, was used to effect the
decomposition. 10 grams of the rock, powdered so as to
pass through a 120mesh sieve, were mixed with 2 grams of
anhydrous sodium carbonate and introduced into the flask,
and 50 grams of stick potassium hydroxide added. The last
was not fused to remove water previous to use, as it has been
shown that vigorous ebullition assists in the liberation of the
emanation. The flask was then connected to the rest of
the apparatus as shown, by a rubber stopper, and the whole
evacuated with a Fleuss pump to a pressure of about 1 cm.,
as shown by the small manometer K. The tap B was then
closed and the flask and contents heated with a large Teclu
burner for 15 minutes, a very rapid stream of water being
passed through the neck. A certain amount of gas was
usually liberated, and in the earlier experiments the pressure
sometimes rose to such an extent that the cork was blown out
of the flask. To avoid this, a safetyvalve was devised. The
tube E, slightly greater in length than the barometric height,
was attached to the apparatus, and bent round under the
wider tube C about 50 c.c. in capacity, the joint being under
mercury. If the pressure rose much above that of the
atmosphere, gas escaped into 0. The gas liberated was on
one occasion pumped off, and found to be almost perfectly
pure hydrogen.
After heating, the copper flask was cooled in water and
the pressure fell sufficiently to cause C to fill with mercury
when the tap D was opened. The taps leading to the
electroscope were then very cautiously opened and the gas
washed out of the flask and dryingtubes which contained
calcium chloride, fused potassium hydroxide, and phosphoric
46 Dr. H. E. Watson and Mr. Gostabehari Pal on
anhydride, by opening the tap F connected to a tube leading
nearly to the bottom of the flask. The electroscope itself
was similar to those used by Joly, but about twice as
sensitive. The moving leaf was of aluminium about
30 x 2 mm., and a small piece of quartz fibre attached to
the end and illuminated from the side, made it possible
to take readings to the tenth part of a scaledivision with
ease. The insulation was a quartz tube with a sulphur rod
shaded by black paper on the end. Charging was effected
by means of a wire movable through an airtight ground
glass joint H. A potential of 300 volts from a battery was
used, and the wire always replaced in the same position and
earthed after charging. The whole was contained in a
500 c.c. glass flask silvered on the inside except for two small
holes, and earthed.
The quartz fibre was observed through a small microscope
with a scale in the eyepiece. The constant was determined
roughly at first by weighing out 5 milligrams of Joachimsthal
pitchblende which was mixed with a little rock and fused up
in the usual way. This quantity was found to be much too
large, and so 10 milligrams were accurately weighed, well
digested with nitric acid, and the solution made up to 100 c.c.
1 c.c. of this was taken and evaporated to dryness in a small
glass capsule. After three weeks this was fused up with
25 grams of potassium hydroxide, and the resulting leak
determined. Two such determinations were made at intervals
of eight months with different portions of pitchblende, and
in the second case twice the quantity was taken.
These two experiments gave the quantity of radium
which produced a leak of 1 scaledivision per hour as
0310 xlO12 gram and 0'315xl0~12 gram respectively.
Another preliminary experiment with a different leaf gave
0*275 X 10"12 gram. The pitchblende was found to contain
60*0 per cent, of uranium ; and it was assumed that the
amount of radium in equilibrium with 1 gram of uranium
was 3*15 x 10~7 gram (Pirret & Soddy, Phil. Mag. [6] xxi.
1911, p. 652).
The natural leak was fairly constant. It varied from 5*5
to 6'2 scaledivisions per hour, and was determined at frequent
intervals.
The materials used for fusion appeared to be fairly free
from radioactive matter. Blank experiments showed that a
correction of about 1 scaledivision per hour for 50 grams was
necessary.
It was found at once that the quantity of radium in the
rocks under examination was very minute ; and in order to
Radioactivity of Rocks oj the Kolar GoldFields. 47
be quite sure that this was not due to faults of the method
or of the apparatus, a number of experiments were made.
No increase in the leak was observed if the heating was
continued for more than 15 minutes. The resulting melt
nearly all went into solution on treatment with water, and
then hydrochloric acid. To be certain, however, that the
emanation had been expelled, these solutions, which were
quite clear, were kept for three weeks and then boiled under
reduced pressure and the gas introduced into the apparatus,
following the original method of Soddy (Roy. Soc. Proc.
lxxvi. 1905, p. 88). The small insoluble residue was also
brought into solution by fusion with carbonates, and the
solutions added to the others. Only a very slight increase in
the natural leak was ever observed, although a control expe
riment with a solution to which 1 c.c. of the uranium nitrate
solution previously mentioned had been added, showed that
this method was not at fault, even though the results were
lower than those obtained by the other method.
Although other observers have shown that very little
emanation is given up by a mineral on reducing the pressure
of the air in the vessel containing it, an experiment was
carried out in order to test this, in which only the electro
scope was evacuated. No abnormal result was obtained.
The method of sweeping out the gas from the copper flask
was also found to remove at least 95 per cent, of the emanation;
and as all determinations were carried out in exactly the same
way, no error should arise from this cause.
In all normal cases electroscope readings were taken
half hourly, for about 3 hours, starting about \\ hours after
admission of the gas, and the rates of leak reduced to their
maximum value by means of the curve given by latterly
(Phil. Mag. xx. 1910, p. 2).
The decay curve after three to four hours corresponded
approximately with that of radium emanation, although an
accurate determination was difficult owing to the extremely
small quantities of gas.
After a considerable number of determinations had been
made, an important source of error was discovered owing to
one specimen of rock giving rise to a leak which was less
than the normal leak of the electroscope. This was found to
be due to the presence of a large quantity of hydrogen in the
evolved gas. As the available data on ionization in hydrogen
are very meagre, some experiments were carried out to
determine the magnitude of the effect. It was found that
the natural leak in hydrogen was 1*0 scaledivision an hour,
while in air it was G 0. Also in an experiment with uranium
48 Dr. H. E. Watson and Mr. Gostabehari Pal on
nitrate similar to the others, but in which the apparatus was
filled with hydrogen instead of air, a leak of 36*5 scale
divisions per hour was observed, while the same weight of the
salt produced a leak of 133*5 scaledivisions per hour in air.
Using these data, and assuming that the rate of leak is a
linear function of the quantity of hydrogen (a doubtful
assumption), it maybe calculated that if 100 c.c. of hydrogen
are introduced into an electroscope of 500 c.c. capacity, a
leak of 15 divisions per hour would be reduced to 12*7 and
one of 10 to 8*4. With a natural leak of 5 it will be seen
that the error in the latter case is nearly 100 per cent., and
100 c.c. is not at all an unusually large quantity of hydrogen
to be evolved.
As the actual leaks increase in magnitude the relative
error decreases, but becomes by no means negligible.
In consequence of this, all the experiments which had
been done were repeated with a tube of redhot copper oxide
inserted between the heating flask and the first tap. The
effects of this were very satisfactory, as the pressure never
rose above half an atmosphere during the fusion ; and after
cooling the flask, it was rarely 30 mm. more than the initial
pressure. Even if this were due entirely to hydrogen which
was not absorbed in its subsequent passage over the copper
oxide, the amount (10 c.c.) would be insufficient to influence
the results.
It was found best to keep the electroscope continuously
charged ; consequently at the end of each day's readings it
was evacuated to a pressure of about 10 mm.; and if the
rock which had been examined had contained more than the
usual excessively small quantity of radium, air was again
admitted, and pumped out. At this pressure the leak was
only 0*3 division per hour, and consequently the fibre would
remain on the scale for a long period. Immediately after
the electroscope had been filled with fresh emanation, it was
charged to a higher potential, so that the pointer reached a
spot about 15 divisions off the scale, and no readings were
taken for at least an hour (except in the case of large leaks).
The reason for this was that the leaf fell very rapidly at first
if charged up directly from zero potential owing to some
" soaking in " effect of uncertain duration.
The sensitiveness of the electroscope remained practically
constant over the whole scale, as might be expected from the
potential to which it was charged ( — 300 volts) ; but the final
values were as far as possible deduced from readings over the
same part of the scale owing to a slight variation in parts
which was evidently due to rigidity of the leaf.
Radioactivity of Rocks of the Kolar Gold Fields, 49
The Results.
Mr. H. M. A. Cooke of Kolar very kindly supplied us
with large specimens of " country rock " selected at varying
depths from each of the chief mines. These were fir si,
roughly crushed iu order to obtain fair samples.
The following table shows the amount of radium per gram
of rock in units of 10 ~14 gram.
The depths from which the samples were taken are only
approximate, and are measured along the incline of the shaft.
The actual vertical depths are usually considerably less, for
instance, 4000 feet on the incline in the Mysore mine is only
2664 feet below the surface. As, however, the shafts follow
the direction of the strata, a comparison made in this way is
probably preferable to one made at equal vertical depths.
The names of the mines are given as they occur geogra
phically, Balaghat being the most northerly.
Table I.
Name of Mine.
Depth in feet.
1000.
2000.
3000. 4000.
Balaghat
21
17
18
34
14
17
24
21
21
Nundidroog
14
91
]8 14
23 96
Mysore
It will at once be seen from the above that there is no
obvious relation between the radium content and the position
of the rock; and in fact, with two exceptions, the numbers are
almost the same within the limit of experimental error. It
thus appears that the distribution of radium is uniform.
With regard to the high value for the deepest sample from,
the Mysore mine, a second experiment showed the figure to
be correct; but it was found on subsequent examination that
the rock was of quite a different character to the others, being
probably a later formation.
A few other samples of rock, the geological history of
which was known to some extent, were kindly presented to
us by Dr. Smeeth of the Mysore Geological Department, and
the radium content was determined. As these rocks were of
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. E
50 Radioactivity of Rocks of the Kolar Gold Fields.
a different character to those previously dealt with, the melt
was in all cases treated with water and hydrochloric acid, and
any residue weighed to see if the decomposition had been
sufficiently complete.
Table II.
Eock.
Radium Content 10 12 grm.
1 Oore^um ^000 altered
018
082
128
134
144
690
6. Quartz containing tourmaline, same age as 5...
The first o£ these samples was picked out from the piece o£
rock used for the determination given in Table I. It differed
considerably from the rest of the rock, and it was thought
that the radium content of the two portions might not be the
same. As may be seen, however, the two were almost
identical. Samples 2, 3, and 4 were of approximately the
same age, and related geologically to the deepest sample
from the JVlysore mine. A comparison of 3 and 4 seems to
show that the quantity of radium does not vary greatly with
the gold content. The comparatively large amount of radium
in sample 6 did not appear to be due to the tourmaline, as a
determination made on a small quantity of the latter gave no
abnormally high result.
One sample of water from a large spring at the bottom of
the Balaghat mine was also examined, both by boiling some
of the water and bv fusing the residue from over 2 litres
after it had stood for three weeks. Any radium was present
in such small quantities that it was practically undetectable.
It is interesting to note that these rocks, which are pro
bably some of the oldest known, contain so little radium. In
fact, the quantity appears to be equal to the smallest quantity
hitherto estimated in any rock. Also, as may be seen from
Table II., there appears to be some slight tendency of the
radium content of the rocks of this locality to increase
as their age diminishes; but the number of experiments
is not sufficient to show that this is definitely the case.
Dr. Smeeth has promised to secure a number of additional
Collapse of Short lubes by External Pressure, 51
specimens, and any results obtained will be given in due
course.
In connexion with the small radium content of these rocks,
it is worthy o£ note that the temperature Gradient in the
Kolar mines is quite abnormally small ; and it is hoped that
a communication on this subject may be made at some future
date.
Summary.
1. The radium content of a number of similar schists from
different parts of the Kolar Gold Field has been examined
by means of a new method which involves the fusion of the
rock with potassium hydroxide under reduced pressure.
2. The quantity of radium in these schists appeared to be
approximately constant, and has a mean value of 0'19 x 10~12
gram per gram of rock.
3. A few other specimens of different rocks of later date
were found to contain larger quantities of radium varying
from 0"82 to 6'90x 10"12 gram per gram.
Indian Institute of Science,
Bangalore.
VII. The Collapse of Short Tubes by External Pressure.
By Gilbert Cook, M.Sc, Assoc.M.Inst.C.E*
[Plate L]
npHE subject of the resistance of tubes to collapse by
JL external pressure is one in which, considering its
important practical applications, comparatively little experi
mental work has been done ; moreover, the greater part of
this has referred to tubes whose length is verv great com
pared with the diameter, and the most important researches
have included little, if any, reference to the strength of short
tubesf.
The experimental work of Carman^ and Stewart § has
shown that the relation indicated by theory, namely, that the
collapsing pressure is proportional to the cube of the ratio of
thickness to diameter, is substantially true for long tubes,
* Communicated by Prof. J. E. Petavel, F.R.S.
t A full discussion, and bibliography, of the present state of know
ledge in regard to tube collapse by external pressure, is given by the
author in a report to the British Association Committee on Complex
Stress Distribution. See British Association Report, Birmingham,
1913.
X University of Illinois Bulletin, vol. hi. No. 17, June 1906.
§ Transactions of the American Society of Mechanical Engineers,
19051906, vol. xxvii. pp. 730822.
E 2
52 Mr. Gr. Cook on the Collapse of
provided that the ratio does not exceed *025^ The formula
given by them for solid drawn weldless steel tubes is
p = 50,200,000 (£) , (1)
and it is notable that the numerical term is some 30 per cent,
less than the theoretical value *.
It is well known that a short tube has a much greater
strength to resist collapse than a long one of the same thick
ness and diameter ; it is also known that as the length
increases, its influence upon the strength rapidly diminishes.
Carman and Stewart have inferred from their experiments
that when the length exceeds six diameters, which has been
termed the critical length, the collapsing pressure is not
affected by a further increase in the length, and that it
differs, therefore, from that of an infinite tube by a negligible
amount. In the light of the experiments here described, the
above value for the critical length appears to be an under
estimate, and there can be little doubt that not only the
diameter, but also the thickness of the tube, is a factor wrhich
must enter into any expression for its value. It is, indeed,
possible that the theoretical value suggested by Southwell f,
namely
= kVT>
may be true, though there is at present no means of ascer
taining the value of the constant k by mathematical
analysis.
In view of the vagueness which exists in regard to the
value of the critical length, an accurate prediction of the
strength for any shorter length is hardly to be looked for.
It is, indeed, a fact that there is no experimental information
available which would indicate the manner in which the
collapsing pressure of such tubes is related to the three
variables, thickness, diameter, and length. Southwell % has
investigated the problem mathematically, and has arrived at
an expression for the collapsing pressure which is exceed
ingly interesting as affording an explanation of the hitherto
unexplained phenomenon of the variable number of lobes
* In regard to the theories which have been put forward to account
for the discrepancy, papers by S. E. Slocura (Engineering, Jan. 8, 1909)
and R. V. Southwell (Phil. Mag. Sept. 1913) should be consulted.
f Phil. Trans. (A) vol. ccxiii". (1913), pp. 187244.
% Phil. Trans. (A) vol. ccxiii. (1913), pp. 187244 ; and Phil. Mag.
May, 1913.
Short Tubes by External Pressure. 53
into which a tube divides in collapsing, but contains factors
for which a numerical value cannot at present be found.
It has been customary to assume that the strength is inversely
proportional to the length, the assumption being based solely
upon the early experiments of Fairbairn * and a few tests
on small seamless brass tubes by Carman f , who proposed
that the critical length should be regarded as the upper
limit of this relation, so that the strength of a shorter tube
would be given by the equation
Pi=j.p, . , .... (2)
where I is the length of the tube, p the collapsing pressure,
of an infinite tube, given (for solid drawn weldless steel
tubes) by the formula (1), and L, the critical length, = 6d.
The inadequacy of the data in regard to short tubes sug
gested to the author the experiments which are described in
this paper. It has not been found possible, however, to
represent the results accurately in any formula involving all
the variables, but it will be seen that the actual collapsing
pressure is widely different from that obtained by calculation
from equation (2), if Carman's value of the critical length
be taken. Much more extensive experimental work, covering
a greater range of dimensions than has yet been undertaken,
would, however, be required before it would be desirable, or
even possible, to propose a new formula.
The tests were carried out on solid drawn steel tubes 3 in.
internal diameter. A tensile test of a specimen cut longi
tudinally from the tube gave the following results : —
Stress at yield point 1S'4 tons per sq. in.
Ultimate strength 25'9 „ ,, „
Elongation on G in 15 per cent.
Elongation on 2 in 21 per cent.
In order to ensure that the thickness should be as nearly
uniform as possible, the tubes were carefully machined to
the desired thickness after having been cut to the required
length. The cost of the preparation was very kindly defrayed
by the Vulcan Boiler and General Insurance Co., Ltd. The
thickness and length of the tubes were varied, but a constant
diameter of 3 in. was selected because a sufficiently low
value of "7 could thereby be obtained without the thickness
becoming undul} small, and therefore liable to considerable
* Phil. Trans. 1858, p. 389.
t Physical Review, vol. xxi. Dec. 1905, pp. 381387.
54
Mr. G. Cook on the Collapse of
variation in any particular tube. A greater diameter was
undesirable in view of the fact that the length was limited
by the testing appliances to about 13 inches. It was found,
by means of a micrometer gauge, that the greatest and least
diameters did not differ by more than \ per cent., and
although a variation of onethousandth of an inch in the
thickness is equivalent to 2 or 3 per cent., it was found that
the departure from uniformity rarely exceeded this amount.
The manner in which the ends of the tubes were closed
before testing is shown in fig. 1. Two castiron disks were
Eff. 1.
machined with circular slots, against which the inner surface
of the tube accurately fitted. Longitudinal stress in the
tube was avoided as far as possible by providing a central
strut against which the disks were screwed, the joint being
made tight by means of a thin lead washer, a small clearance
being left between the ends of the tube and the bottom of
the slots. The slots were then filled with lead wool ham
mered in with a caulking tool, and the joint thus made was
found to be perfectly tight even against pressures as high as
2000 lb. per sq. in. A small hole drilled through one of the
disks, and communicating with a tube leading to the
atmosphere, ensured that the internal pressure remained
atmospheric.
The enclosure in which the tubes were placed for testing
was a vessel used for various experimental purposes, and
designed to withstand an internal pressure of 3000 lb. per
sq. in. It was provided with three valves, to one of which
was connected a hydraulic forcepump, the other two serving
as connexions for pressuregauges, and as outlets for the
air when the vessel was filled with water at the commence
ment of the test. The connexion between. the interior of
the tube and the atmosphere consisted of thick copper tubing
passing through one end of the vessel and attached to a
Short Tubes by External Pressure. 55
Utube indicator, whereby any leakage could immediately
be detected.
In carrying out a test, all the air was first displaced from
the vessel, and then the pressure raised very slowly until
collapse occurred. This was indicated by a sharp ring,
accompanied by a sudden fall in the pressure. Although at
this moment a condition of instability probably obtained over
the whole circumference of the tube, an inspection showed
that in most cases collapse had occurred in only one place.
Occasionally, however, two adjacent areas collapsed simul
taneously. The deformation thus produced evidently affected
the whole tube, for when the pressure was raised a second
time, collapse occurred in another part at a lower pressure.
The complete results are tabulated below, and curves have
been plotted to show :
(1) The relation of collapsing pressure to  for different
lengths (PI. I. fig. :
d
(2) The relation of collapsing pressure to length for dif
ferent values of 7 (figs. 3 to 7).
In the last two columns of the table a slight modification has
been made in the collapsing pressure, owing to small un
avoidable differences in the ratio 7, and these modified values
a
have been used in plotting the curves shown in figs. 3 to 7.
The present work was limited in scope by the expense of
the apparatus involved, and is insufficient to enable a general
law of collapse to be determined ; it fixes with accuracy,
however, the strengths of certain definite sizes of tubes,
which are given in the table, and in the curves shown in
figs. 2 to 7. In figs. 3 to 7 are also plotted :
(1) Curves showing the collapsing pressure as deduced
from equation (2), taking Carman's value of the critical
length, viz. six times the diameter ;
(2) Curves obtained from the same equation but assuming
the critical length to be much greater, and inversely pro
portional to the square root of the thickness, as suggested
by Southwell's equation
L = Ic
v? »
It will be seen that the latter method of calculation gives
results much more consistent with the experimental values
56 Collapse oj Sho?
(1)
(2)
(3)
0^ + i)p (xn+i)p ^ o»+i)p •
(^+l^l)?==(l)«S(l>('^(^+l)^
therefore
If c+l)
2ic + l
7r(«+l)— 2 COS TTCi . _ , _
w v ; n , (l)a[l(— 1)"]
3?2 — 2 A' COS
2/f + l
+ 1
and
2n(a? + l) '
. . . (13)
cos 7r(a+lW log («r — 2,/' cos tt + 1)
+ 2 sin
21og(2sin^±l.)}
« i 2ra
•n — tan'
2*+l "I
''—COS 77 ^
sm
7T ' .J
+ (l)ll(l)']logK,+1),
(14)
To find the Integral (10), we write
ml T>
•c1
1 "A
L/ » . i \ = ^ '^  * m = (p — B — l)>i — a (15)
Multiplying both sides by #rK and letting a?=rw we have
A = A^Z*1 = A_J_1 __1r,+i {u;)
Multiplying both sides of (15) by xm, gives
1 » A 0'»» »'• — 1
(17)
Differentiating both sides /c times and letting x = 0, we
obtain
It follows that B0=l, B1=B2 = ....=Bn_1=0, Bn=1,
#2,1= +1, and in general BB(C=(— 1)\
B« =
60 Prof. Max Planck : New Paths
Now m= (p—ft— l)n — u and a.g9(^+l)+ *
X
=0 (/> — (3—K— l)w — a — 1
L__i[i_(i)^] (J
The required Integral (1) is obtained by combining (18)
(19) with (7) (due regard being paid to the limits).
University of Pennsylvania,
Philadelphia, Pa., U.S.A.
IX. New Paths of Physical Knowledge : being the Address
delivered on commencing the Pectorate of the Friedrich
Wilhelm University, Berlin, on October 15th, 1913. By
Dr. Max Planck, Professor of Theoretical Physics *.
The Rector began his Oration thus : —
Honoured Assembly, Esteemed Colleagues, Dear Comrades :
Called to the head of the Administration by the confidence of
the accredited representatives of our Corporation, I have under
taken as my first official duty the task of greeting the members
and friends of our Alma Mater at the commencement of the new
* Communicated after revision by Sir Oliver Lodge, on the basis of a
translation by Dr. Fournier d'Albe.
of Physical Knowledge . 6 1
Session in an Address concerning the beginning of the Course of
Studies, as the Statutes prescribe.
Rather special are the feelings with which on this occasion we,
Teachers and Taught, regard the problems facing us in the new
Term ; for while the year just passed appears bathed in a festive
glory, illumined and warmed over its whole course by the thought
of national ideas, of the heavy sacrifices brought and of glorious
resulting victories, — the last and greatest of which is to be cele
brated in these days by the whole German people, — the coming
Term appears, on the other hand, likely to bear an everyday
character and to be devoted to regular work.
The best we can derive from the Memorial festivities of the
past year is the fervent wish that our successors may at some
future time look up to us as we look up to those men who, one
hundred years ago, fought and suffered in word and deed for the
Fatherland. Let no one reject such a wish as entirely baseless,
on the ground that such high aims cannot be contemplated today.
For, in the first place, we must not forget that the forces which
were then gloriously displayed derived their real strength from
the quiet everyday work of the simpler times preceding, which
may not have been so conscious of their high import but may have
been all the more intense and creative ; and in the second place,
none of us can know with what standard coming generations
may approach the estimate of our presentday performances. But
what we can assert with certainty, under all circumstances, is
that our generation can only hope to be judged honourably by
posterity if it endeavours to solve its own problems according to
its lights, in strict fulfilment of its duties : each one in the place
to which his calling and his fortune have led him.
So then may I be allowed today to lav before you a section of
the special work going on within the Science I represent, by a
survey of the progressive development of Physical Knowledge and
an endeavour to sketch those new paths which it has trodden
since the beginning of this century.
NEVER, probably, has experimental physical investi
gation experienced so strenuous an advance as during
the last generation, and never probably has the perception of
it significance for human culture penetrated into wider
circles than it does today. The Waves of Wireless Telegraphy,
Electrons, Rontgen Rays, the Phenomena of Radioactivity,
appeal more or less to the interesf of everyone. But if we
face the larger question in what respect have these new ami
brilliant discoveries influenced and advanced our under
standing of Nature and her laws, the outlook, at first glance,
does not appear at all correspondingly brilliant. On the
contrary, whoever endeavours to judge of the state of
present Physical theories from a higher point of view, may
62 Prof. Max Planck : New Paths
readily get the impression that theoretical investigation has
been rather confused by so many new experimental dis
coveries, which have been for the most part entirely
unexpected, and that it is now in a profitless period of blind
groping, which contrasts strongly with the clear calm and
security marking the theoretical epoch just passed, — an epoch
which, with some justification, can be described as the
Classical Epoch. Everywhere old and firmlyrooted con
ceptions are attacked, universally recognized theories are
discarded and are replaced by new hypotheses, — some of them
of a boldness which makes almost intolerable demands on the
intelligence even of the scientifically educated, and, in any
case, does not appear calculated to strengthen confidence in
a steady and effective progress of Science. So the present
science of Theoretical Physics may give the impression of
an edifice, venerable indeed but fragile, in which one part
after another commences to crumble away, and whose
foundations even are threatened.
Yet nothing would be more incorrect than such an idea.
It is true that in the main structure of Physical Theories
fundamental changes are taking place. But on closer in
spection, we see that this is not a work of destruction but a
work of completion and amplification, that certain blocks
of the edifice are only removed from their place in order to
find a better and securer place elsewhere, and that the real
foundations of theory are still as firm and safe as they have
ever been. I shall endeavour to prove in some detail this
assertion.
First a general consideration : — The initial impulse to
wards revision and modification of a physical theory comes
almost aways from the discovery of one or more facts which
do not fit into the present framework of the theory. Facts
always furnish the Archimedian point from which even the
most ponderous theory may be lifted off its hinges. In that
sense nothing is more interesting to the real Theoretician
than a fact which directly contradicts a hitherto universally
accepted theory, for it is just here that his real work
begins.
How what are we to do in a case like this ? Only one
thing is certain ; something must be changed in the accepted
theory, and in such a manner that it agrees with the new
fact, but it is often a difficult and complicated question at
what point of the theory the correction is to be applied, for
one fact is insufficient to furnish a theory. A theory, indeed,
consists as a rule of a whole series of theorems connected
with each other. It may be compared to a complicated
of Physical Knowledge. 63
organism, whose separate parts fit together so intimately that
any interference at one place is felt in various other places,
sometimes far removed. Wherefore, since every conclusion
of theory results from the cooperation of several theorems,
it follows that, as a rule, several theorems may be made
responsible for each failure of the theory, and there are
generally several possibilities of finding the way out.
Usually, the question is eventually reduced to a conflict
between two or three propositions which hitherto have found
a place in the theory, but of which one must be abandoned
in face of the new fact. The conflict lasts often for years or
decades ; and its final decision not only means the destruction
of the defeated theorem, but also quite naturally — and this
is specially important — a corresponding confirmation and
elevation of the victorious constituent theorems which
survive.
And now we must note the extremely important and
remarkable result that in all this war and conflict it is just
the great Physical Principles which have held the field, — such
as the Principle of the Conservation of Energy, the Principle
of the Conservation of Momentum, the Principle of Least
Action, and the chief laws of Thermodynamics. Their
importance has thus been considerably increased ; while, on
the other hand, the theorems which have succumbed in the
fight are those on which theoretical developments were based
tacitly, either because they seemed so selfevident that it was
not, as a rule, considered necessary to mention them, or because
they were forgotten. In general, then, one may assert that the
most recent development of theoretical physics is marked by
the victory of the great Physical Principles over certain
deeplyrooted and yet merely habitual assumptions and
conceptions.
To illustrate these statements, I may adduce some of
those theorems which have hitherto been used without any
hesitation as the selfevident foundations of any theory, but
which, in the light of new facts, have proved untenable, or
extremely doubtful, in face of the general principles of
Physics. I mention three : The Invariability of Chemical
Atoms ; The Mutual Independence of Space and Time ; and
The Continuity of all Dynamical Effects.
Of course it is not my intention here to quote all the
important arguments which tell against the Invariability of
Chemical Atoms. I shall only mention the single fact which
brought about an inevitable conflict between this assumption
— formerly always regarded as selfevident — and a general
physical principle. The fact is the constant evolution of
64 Prof. Max Planck : New Paths
heat by every compound of radium ; and the physical
principle is that of the Conservation of Energy. The conflict
ended finally in the complete victory of that principle, al
though voices were heard endeavouring to throw doubt upon
its complete validity.
A radium salt enclosed in a sufficiently thick envelope of
lead constantly evolves heat amounting for 1 gram of radium
to 135 calories per hour. It therefore constantly remains
warmer than its surroundings, just like a heated stove. The
Principle of Conservation of Energy asserts that this heat
cannot be evolved from nothing, but must have its cause in
some other change representing its equivalent. In the case
of the stove, it is the constant process of combustion, but as
no chemical process is going on in the case of the Radium
compound, we must assume a change in the Radium atom
itself ; and this hypothesis, which from the point of view of
previous chemical science is bold and unprecedented, has
been corroborated in every direction.
From a purely formal point of view, there is, no doubt, a
certain contradiction in the conception of a changeable
atom, since originally atoms were defined as the unchange
able constituents of all matter. Accordingly one might feel
bound to reserve the term " atom " for the really unchange
able elements, such perhaps as Electrons and Hydrogen.
But apart from the fact that we may perhaps never be able
to establish the existence of invariable elements in the ab
solute sense, such a change in terminology would produce a
wild confusion in literature. Indeed the modern chemical
atoms have long ceased to be the atoms of Democritus, they
can be numerically and accurately specified by a much sharper
definition, it is only they that are meant when we speak of a
Transmutation of the Atom, and any misunderstanding in
the direction indicated seems clearly excluded.
A thesis not less selfevident than the Invariability of
Atoms was, until recently, the Mutual Independence of
Space and Time. The question whether two occurrences
taking place at different points are simultaneous or are not,
had a definite physical meaning, without the need of any
enquiry about the observer who measured the time. Today,
that is altered. For a fact — so far invariably corroborated
by the most delicate optical and electrodynamic experi
ments— which is briefly and not quite clearly described as
the Relativity of all Motion, has brought that simple
conception into conflict with the socalled principle of the
Constancy of the Velocity of Light established by the
electrodynamics of Maxwell and Lorentz. This principle
of Physical Knowledge. 65
asserts that the velocity of the propagation of light in open
space is independent of the motion of the source of light.
If, therefore, we assume that Relativity is experimentally
established, we must sacrifice either the principle of the
Constancy of the Velocity of Light or the Mutual Inde
pendence of Space and Time.
For let us consider a simple example. Let a time signal
be given out by Wireless Telegraphy from a central station, —
say the Eiffel Tower, — as provided by the International Time
Service already projected. Then all stations in the vicinity
which are at the same distance from the central station
receive the signal at the same time, and can set their clocks
accordingly. But this kind of time regulation becomes
theoretically faulty, if accepting the relativity of all motion
we transfer our standpoint from the earth to the sun, whence
we must regard the earth as moving. For, according to the
principle of the Constancy of the Velocity of Light, it is
clear that those stations which, seen from the central
station, lie in the direction of the earth's motion, receive the
signal later than those lying in the opposite direction, be
cause the former stations are moving on in advance of the
light waves which they have to receive, and must be
overtaken by them, whereas the latter stations travel to meet
the waves. Thus the principle of the Constancy of the
Velocity of Light renders impossible an absolute determin
ation of time which shall be independent of the motion of the
observer. The two are incompatible. So far as the conflict
has proceeded, the principle of the Constancy of the Velocity
of Light has been decidedly victorious, and in spite of many
doubts which have latterly been raised, it is not at all
probable that any abandonment of that position will occur.
The third of the above theories concerns the Continuity
of all Dynamical Effects. This was formerly taken for
granted as the basis of all physical theories, and, in close
correspondence with Aristotle, was condensed into the well
known dogma — Natura non facit saltus. But even in this
venerable stronghold of Physical Science presentday in
vestigation has made a considerable breach. This time it is
the principles of Thermodynamics with which that theorem
has been brought into collision by new facts, and unless all
signs are misleading, the days of its validity are numbered.
Nature does indeed seem to make jumps — and very extra
ordinary ones. As an illustration, let me make an instructive
comparison : —
Let us imagine a sheet of water in which strong winds
have produced high waves. Even after the total cessation
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. F
66 Prof. Max Planck : New Paths
of the wind, the waves will be maintained for some time and
will pass from one shore to the other. But there will be a
certain characteristic change in them. During their impact
on the shore, or on other solid obstacles, the energy of
motion of the longer and coarser waves is converted to an
ever greater extent into the energy of motion of shorter and
slighter waves ; and this process will continue until at last
the waves have become so small and their motion so slight
that they are quite lost to view. That is the familiar
transmutation of visible motion into heat, of molar into
molecular, of ordered into disordered motion ; for in ordered
motion many neighbouring molecules have a common
velocity, whilst in disordered motion every molecule has its
separate and separately directed velocity.
This process of disintegration or subdivision does not
proceed indefinitely, but finds its natural limit in the size
of the atoms. For the motion of a single atom by itself is
always an ordered one, since the separate parts of an atom
all move with the same common velocity. The larger the
atoms, the less can the total energy of motion be subdivided.
So far, everything is perfectly clear, and the Classical
Theory is in excellent agreement with experience.
But now let us take another and quite analogous process,
not dealing with water waves but with waves of light and
heat. Let us assume that rays emitted by a brightly glowing
body are collected by suitable mirrors into a completely
enclosed hollow space, and that they are continually thrown
to and fro between the reflecting walls of that space. Here
also there will be a gradual transmutation of the energy of
radiation from longer waves to shorter waves, from ordered
radiation to disordered radiation. The longer and coarser
waves correspond to the infrared rays, and the shorter and
slighter waves correspond to the ultraviolet rays of the
spectrum. Hence, according to the Classical Theory, we
must expect the total energy of radiation to concentrate
itself upon the ultraviolet portion of the spectrum ; or, in
other words, we must expect the infrared and the visible
rays to disappear gradually and convert themselves ulti
mately into invisible ultraviolet or chemical rays.
But of such a phenomenon no trace can be discovered in
Nature. The conversion sooner or later attains a perfectly
definite and assignable limit, and after that, the radiation
conditions remain stable in every respect.
In order to reconcile this fact with the Classical Theory
the most varied experiments have already been made, but
the result has always been that the contradiction went too
oj Physical Knowledge. 67
deep into the roots of the Theory to leave them unhurt. So
again nothing remains but to reexamine the foundations of
the Theory. And again we must admit that the principles
of Thermodynamics have shown themselves to be unshakable.
For the only method so far found to promise a complete
solution of the riddle depends directly upon the two laws
of Thermodynamics; though it combines with them a new
and peculiar hypothesis, which, if we utilize the two illus
trations above mentioned, can be expressed somewhat as
follows : —
In the case of the Water waves, the disintegration of the
energy of motion is limited by the fact that the atoms hold
the energy together, in a way, each atom representing a
certain finite material Quantum which can only move as a
whole. In the same sort of way certain processes must be
at work in the case of light and heat rays, although they are
quite of an immaterial nature, which shall hold together the
energy of radiation in definite finite Quanta, and shall unite
it the more strongly the shorter the waves and the quicker
therefore the frequency of the oscillations.
In what way we are to conceive the nature of quanta of a
purely dynamical nature, we cannot yet say for certain.
Possibly such quanta might be accounted for if each source
of radiation can only emit energy when that energy attains
at least a certain minimum value ; just as a rubber pipe, into
which air is gradually compressed, bursts and scatters its
contents only when the elastic energy in it attains a certain
quantity.
In any case, the hypothesis of Quanta has led to the idea
that there are changes in Nature which do not occur con
tinuously but in an explosive manner. I need hardly remind
you that this view has become much more conceivable since
the discovery and investigation of Radio Active Phenomena.
Besides, all difficulties connected with detailed explanation
are at present overshadowed by the circumstance that the
Quantum Hypothesis has yielded results which are in closer
agreement with radiationmeasurements than are all previous
theories.
Moreover, if it is a good sign for a new hypothesis that it
is found applicable to regions for which it was not originally
devised, then the Quantum Hypothesis can surely claim a
favourable testimony. I shall only refer to a quite particu
larly striking point. Since we have succeeded in liquefying
Air, Hydrogen, and Helium, a new field of activity has been
opened for experimental investigation in the region of low
temperatures, and in this region a number of new and in some
F2
6$ Prof. Max Planck : New Paths
ways highly surprising results have been obtained. To heat
a piece of copper from —250° to —249°, i. e. by one degree,
we do not require the same quantity of heat as for heating
it from 0° to 1°, but a quantity about thirty times less. If
we took the original temperature of the copper still lower,
the corresponding quantity ol heat would turn out many
times smaller, without any assignable limit. This fact not
only runs counter to our habitual ideas, but also is out of
harmony with the demands of the Classical Theory. For
although for more than 100 years we have learnt to dis
tinguish between temperature and quantity of heat, yet we
were led by the Kinetic Theory of Matter to suppose that
these two quantities, even if not strictly proportional to each
other, preserved at all events a sensibly parallel course.
The Quantum Hypothesis has entirely cleared up this
difficulty, and in addition has yielded another result of high
importance, viz. that the forces controlling the thermal oscil
lations in a solid body are of the same kind as those which
control its elastic oscillations. With the help of the Quantum
Hypothesis, therefore, we can now calculate quantitatively
the thermal energy of a monatomic substance at various
temperature?, from its elastic properties, — a performance
which was far beyond the reach of the Classical Theory.
Hence arise a number of further questions which appear
verjr strange at first sight, — for instance, whether perhaps the
vibrations of a tuningfork are not absolutely continuous
but are broken up into quanta. It is true that in acoustic
vibrations the energy quanta will be extremely small, on
account of their relatively low frequency. Thus, in the
middle a, they would amount to only 3 quadrillionths of a
unit of work in absolute mechanical measure. It would be
just as little necessary to alter the Theory of Elasticity on
that account as on account of the quite analogous circumstance
that it treats matter as perfectly continuous, whereas it is
really constituted atomistically, i. e. according to quanta.
But fundamentally the revolutionary aspect of the new con
ception must be clear to everybody ; and although the nature
of dynamical quanta still remains somewhat puzzling, yet,
in view of the facts now known, it is difficult to doubt their
existence, in some form or other. For whatever we can
measure must exist.
Thus in the light of recent investigation, the Physical
representation of the Universe exhibits an ever more intimate
correspondence between its various features, and also mani
fests a certain peculiar structure whose refinement was
of Physical Knowledge. 69
hidden to the less trained eye and therefore remained con
cealed. But ever the question arises : What is the significance
o£ this progress in fundamental conceptions for the satis
faction of our thirst for knowledge ? Do we approach one
step nearer to a real knowledge of Nature by the refining
of our world image ? To this fundamental question let us
devote a brief consideration. It is not as if anything
essentially new could be said in this region, alread}
traversed by manifold and endless speculation, but that
while on this point modern views are often diametrically
opposed, yet everyone who takes a deep interest in the real
aims of Science must necessarily take up some position.
Thirtyfive years ago, Hermann von Helmholtz in this
very place expounded the view that our perceptions never
give us an image of, but at most a message from, the external
world. For every attempt fails to demonstrate any kind of
similarity between the nature of the external impression and
the nature of the corresponding sensation ; all conceptions
which we make for ourselves of the external world only
reflect our own sensations in the last resort. Ts there any
sense, therefore, in opposing to our consciousness an inde
pendent " intrinsic Nature w ? Are not indeed all socalled
" Laws of Nature u essentially but more or less effective
rules by means of which we summarize the temporal course
of our sensations as accurately and conveniently as possible ?
If that were so, then not only commonsense but exact
Science would have been fundamentally at fault from the
beginning. For it is impossible to deny that the whole
evolution of Physical knowledge up to now has aimed
towards the completest fundamental division between the
happenings of external Nature and the processes of human
perception. The way out of this embarrassing difficulty is
seen as soon as we go one step further along this line of
thought. Let us suppose that a Physical representation of
the Universe had been found which fulfils all our demands,
and therefore one that can completely and accurately repre
sent all laws of Nature empirically known ; still that that
image even remotely resembles " real " Nature, can in no way
be proven. But this assertion has another side to it, which
is generally too little emphasized : for, in exactly the same
sense, the much bolder assertion that the proposed image
represents real Nature in all points with absolute fidelity
cannot be in any way refuted. For the first step in such a
disproof would be the ability to assert anything with certainty
concerning real Nature, and that, as everybody agrees, is
absolutely excluded.
70 New Paths of Physical Knowledge.
We see that an immense gulf yawns here, into which no
Science can ever penetrate. The filling of this gulf is a
function not of pure reason, but of practical reason, — it is a
matter of commonsense.
Just as a given cosmic scheme cannot be scientifically
established, so we may also be assured that it will survive
every attack so long as it agrees with itself and with the
facts of experience. But we must not fall into the error of
supposing that it is possible to advance, even in the exactest
of all Sciences, without the help of any worldimage, i. e.
without any unprovable hypotheses. Even in Physics, the
phrase holds good that "There is no Salvation without
Faith," — at least a faith in a certain reality outside ourselves.
It is this confident faith which guides the advancing creative
impulse, this it is which gives the necessary support to the
groping imagination, this which alone can raise the spirit
depressed by failure and inspire it to new efforts. An
observer who does not allow himself to be led in his work by
any hypothesis, however cautious and provisional, renounces
beforehand all deeper understanding of his own results.
Whoever rejects faith in the reality of atoms and electrons,
or the electromagnetic nature of Lightwaves, or the identity
of Heat and Motion, can never be found guilty of a logical
or empirical contradiction, but he will find it difficult from
his standpoint to advance Physical knowledge.
It is true that faith alone does nothing. As the history
of all Science shows, it is liable also to lead astray and to
issue in narrowness and fanaticism. If it is to be a re
liable guide, it must constantly be tested by the laws of
thought and by experience which in the last resort can only
be furnished by conscientious and often laborious selfdenying
solitary work. There is no Prince of Science who is not
willing, in case of necessity, to do menial work, whether in
the laboratory, the library, in the open air. or at the
writingdesk. It is just this hard struggle which ripens and
purifies the cosmic view. Only he who has in his own body
gone through the process can fully realize its meaning and
importance.
And the Rector of the University concluded his Address
as follows : —
Thus I address myself, finally, particularly to you, dear
Comrades, who are about to cross the threshhold of a new term
of studies. The gates of our University are open, shortly you
will fill the Lecture Halls, and many a grain of seed will be
sown afresh, many a plant will approach fruition, fed and nurtured
Quantumtheory and RotationEnergy of Molecules. 71
by the treasures of infinitely manifold mental work transmitted to
you by your teachers. But do not believe that all that is offered
to you from the Chair is the last word in wisdom. So long as
there is progress in Science, so long it will be subject to temporary
error. He who has got so far that he never errs, has ceased to
work. If, therefore, doubts and difficulties occur to you in your
studies, do not consider them something unpleasant or forbidden
which must be shaken off or suppressed, but seek out their
meaning carefully, and go to your Teachers (who are your guides),
trust in their riper experience, and cling to the hope of attaining
a gradually increasing understanding of dark and difficult
questions by means of conscientious and sustained effort ; and
so secure the fullest and truest scientific advancement.
And if your honest efforts, verified by many tests, decisively
indicate to you new paths differing from the old, then — follow
your own conviction be}rond any other. That is and must remain
your highest, your most precious possession; for just as training
for scientific independence is the highest aim of academic in
struction, so does a scientific conviction acquired by honest work
give a firm anchorage for holding fast to a moral conception of the
universe in face of all the vicissitudes of life.
The noblest among all the moral fruits of science, and that
which is peculiarly its own, is Truthfulness : that truthfulness
which leads through the sense of personal responsibility to inner
Freedom, and whose estimation in our present public and private
life should be much higher than it is. In whatever degree our
younger generation takes part in the fight to win for Truth an ever
fuller recognition, to that extent it may feel at one with those
heroes who, a hundred years ago, sealed the genuineness of their
love for the Fatherland with their hearts' blood. With such
memories and such thoughts, let us enter upon the work of the
new Session.
X. On the Quantumtheory and the Rotation Energy of
Molecules. By Eva von Bahr, Dr. phil. Uppsala*.'
HPHE application of the quantumtheory to phenomena
JL connected with the rotationenergy of molecules
appears still more difficult to justify than in the case of the
vibrationenergy of molecules or electrons. In the latter case
we have a fixed frequency, but this cannot be assumed for a
freely rotating molecule, which we have at least been ac
customed to look upon as continuously varying in its rotation
velocity, as well as in its translationvelocity.
However, as early as 1911 Nernst f points out that, in
dealing with molecular rotation, we must probably introduce
* Communicated bv the Author.
t W. Nernst, Zeitschr. f. Elektroch. xvii. p. 265 (1911).
72 Dr. Eva von Bahr on the Quantumtheory and
the quantumtheory, since the rotation of a molecule — with
charged atoms — causes radiation, and that, even with in
finitely thick gaslayers, we do not get emission o£ shorter
wavelengths.
In the hope of finding this quantum effect, the disciples
of Nernst have since made investigations into the specific
heat of gases at low temperatures. The results obtained with
hydrogen are particularly interesting. For this gas A.
Eucken* has shown that, with a falling temperature, the
specific heat rapidly declines, at about 70° abs. asymptotically
approaching the value for the monatomic gases. Practically
this value is reached at 60° abs. Attempts to find — by the
aid of the quantum theory — a formula that represents the
results arrived at have been made by Einstein and Stern t,
as well as by Ehrenfest %. The latter proceeds from the
assumption that the energy of rotation of the molecules is
distributed according to quanta (Jiv), putting \{2ttv)2 .l = n~^>
where n is a whole number, h Planck's constant — and that
consequently the frequencies of rotation (y) of the molecules
do not vary continuously, but can only have certain fixed
values.
Strong support for the assumption of discontinuity in
the distribution of the rotationfrequencies is afforded by
investigations into the absorption of the ultrared rays by
the gases. It is proposed here to give a summary of the
most important results of these investigations, with special
reference to their significance for the quantumtheory.
Since Rubens and v. Wartenberg § found that certain
gases show strong absorption even in the case of rays of very
long wavelength (100350/a), it has been generally presumed
that this absorption is caused by the rotation of the gas
molecules, while, on the other hand, the absorption in the
case of shorter wavelengths (220 fi) is caused by the
vibrations of the atoms within the molecule. N. Bjerrum  ,
however, has drawn attention to the fact that absorption in
the case of short wavelengths is also probably dependent on
the rotation — to a certain extent.
If the vibrationfrequency of the atoms is v0, and at the
same time the molecules rotate with a frequency. of v, then,
* A. Eucken, JBerl. Akad. Ber. 1912, p. 141.
f A. Einstein u. O. Stern, Ann. de Phys. xl. p. 551 (1913).
% P. Ehrenfest, Verli. d. Deutsch. phys. Ges. xv. p. 451 (1913).
§ II. Rubens u. H. v. Wartenberg, Verh. d. Deutsch. phys. Ges. xiii.
p. 796 (1911).
 N. Bjerrum, Nernst Festschrift, Halle a. S. 1912, p. 90.
the RotationEnergy of Molecules, 7&
according to Bjerrum, emission and absorption may be
expected not only for v0 and v, but also for v0 + v. If, as
has been assumed hitherto, the rotationfrequencies of the
molecules are distributed according to Maxwell's law of
distribution, then the shortwave, ultrared absorptionbands
might be expected to consist of a sharp maximum for v0 and
two extended absorptionareas with maxima for v0 + v, when
z7 is the most probable rotationfrequency. For the very long
wavelengths the strongest absorption can be expected in the
case of the wavelength that corresponds to v, and this long
wave absorption can thus be calculated as soon as the two
maxima v0±v are known.
The latest investigations into the subject of ultrared
absorption have corroborated this view of Bjerrum's. It is
true that the three maxima anticipated by Bjerrum have not
been found, as it has been shown that, in general, the bands
have a tendency to dissolve into double bands *, but this is a
natural consequence of the circumstance that in this part of
the spectrum it has not been possible to work with sufficiently
great dispersion, but at the same time we always measure a
relatively considerable part of the spectrum — in the most
favourable case 1020/jl/jl* If the absorption for v0 consists
of a narrow line it can, on that account, only contribute to a
very small extent to the absorptionvalue found. Besides, it
must be considered as very uncertain — at least for diatomic
gases — whether there is any absorption at all for v0.
Fig. 1 1 gives the strongest absorptionbands for carbon
monoxide, nitrous oxide, and carbon dioxide, and similar
absorptioncurves have been found for a whole series of other
gases. If we calculate from these bands the absorption in
the case of long wavelengths, we shall find fairly close
correspondence with the results obtained directly by Rubens
and v. Wartenberg J.
Strong support for Bjerrum's view as to the origin of the
ultrared absorptionband is afforded, too, by the influence of
temperature. According to the kinetic theory of gases the
most probable rotationfrequency ought to increase pro
portionally with the square root of the absolute temperature
(T) ; hence the frequencydifference between the band's two
maxima also ought to increase proportionally with v^T, and
\/T . .? A 2 ought to remain constant, when \j and \2 are
the wavelengths for the two maxima.
* W. Burmeister, Verh. d. D.phys. Ges. xv. p. 589 (1913).
t E. v. Bahr, Verh. d. D.phys. Ges. xv. p. 710 (1913).
1 L.c.
74
Dr. Eva von Bahr on the Quantumtheory and
Fig. I.
4,8 4,7 4,6 4,5 4,4 4,3 4,2m
320 340 360 380 400 420 440 460 480
*■ Position cf Prism
In Table I.* are inserted the valnes I have found for
carbon monoxide, as well as the values I have calculated from
Table I.
Gas.
t°C.
X1X2
CO.
15
145
310
308
H20.
17
100
600
1000
1470
117
119
114
108
120
* E. v. Bahr, Verh. d. D.phys. Ges. xv. pp. 710 & 731 (1913).
the RotationEnergy of Molecules. 75
the measurements of the watervapour band at 6fi made by
Paschen * at various temperatures.
The constancy in the last column can be considered as
quite satisfactory.
In accordance with what has been said above, it must be
held probable that Bjerrum's view as to the influence of
rotation on the ultrared absorption is right in the main.
From this follows that, if the rotation frequencies are dis
tributed in accordance with MaxwelPs law of distribution,
the absorptionbands cannot be resolved into more than the
said maxima, and these must be continuous. This is, how
ever, far from being the case. By his remarkable investi
gations on the longwave spectrum Rubens f has shown that
the absorption of watervapour by no means forms a con
tinuous area here, but on the contrary a series of well
defined bands ; and by an indirect method I have shown that,
e. a. the absorptionbands of carbon monoxide and nitrous
oxide (see fig. 1) are not continuous. Finally, I have also
succeeded in directly measuring a whole series of separate
maxima in the watervapour band at 6*3//, and the hydro
chloric acid band at 3*5yu, J.
Thus fig. 2, curve (fr), shows an energycurve for rays from
a Nernst lamp which have passed through 2 m. of undried
air. The curve (a) is an energycurve measured by Paschen §,
given by rays which have been transmitted through 7 cm. of
watervapour at 100 degrees. As may be seen, some of the
separate maxima appear even in this curve. According to
Bjerrum every separate maximum that occurs in fig. 2 ought
to correspond to an absorption maximum in the longwave
spectrum. If the positions of these latter maxima are
calculated, then the values inserted in Table II. cols. 1 and 2
tire obtained. The first series is calculated from the right
hand part of the band, the other from the lefthand part.
The brackets round the values 69 and 64, 50 and 48 denote
that these absorptionbands so merge into one another (see
fig. 2) that, in the case of direct measurement according to
Rubens' method, they must presumably present themselves
as one single absorptionband. As is to be noticed, the two
series are, on the whole, in agreement, and they also corre
spond well with the values found by Rubens (col. 3). The
values between 10 and 20//, inserted in the same column will
he referred to below.
* F. Paschen, Wied. Ann. liii. p. 334 (1894).
t H. Rubens, Bed. Akad. Ber. 1913, p. 513.
J E. v. Bahr, Verh. d. D.phjs. Ges. xv. pp. 710,731 & 1150 (1913).
5 F. Paschen, Wied. Ann. Hi. p. 215 (1894).
FiR. 2.
160C
Quantumtheory and RotationEnergy of Molecules. 11
Table II.
\
(Rubens).
I
X (Eucken)
\(v.
Bahr).
385
n.
A = 173x1012
n.
A =075x1012
398
1
400
240
1 250
2
200
172
170
1
173
124
I 128
3
133
109
109
103
1
4
100
917
915
2
87
790
817
79
i
5
800
690 ]
70<>
087
66
6
666
637 J
615
540
559
58
3
578
7
572
497 1
493
50
8
500
482 j
474
452
447
9
445
415
419
4
433
39 3
38 6
10
400
358
—
11
365
333
—
5
347
12
333
304
—
13
307
234

6
239
14
285
27'2
—
15
267
253
175
157
143
133
124
116
7
8
9
10
11
12
13
14
15
248
216
193
173
158
144
133
124
11*6
16
250
109
u;
108
Fig. 3 snows energycurves for rays that have passed
through a 30 cm. long tube filled with air or hydrochloric
acid at pressures of 380 and 7b'0 mm. respectively. Fig. 4
gives the absorptioncurves calculated from fig. 3. The
continuous curves have been obtained by use of a quartz
prism ; the dotted curve is taken from Burmeister's work*
and is obtained with a fluorite prism, which in this part of
the spectrum has about onethird the dispersion of quartz.
As may be seen, this curve has quite the same generai
character as, e. g., the carbon monoxide curve in fio. 1.
The rotation wavelengths (V) calculated from fio. '4 are
inserted in Table III. as are also the correspondino
frequencies. °
* W. Burmeister, /. c.
Dr. Eva von Bahr on the Quantumtheoryland
Table III.
n.
v. 10n
x in (a.
n
1
75
74
403
745
2
146
133
215
698
3
203
20 0
149
672
4
261
263
115
655
5
314
326
94
640
6
368
—
82
613
7
408
—
74
583
The numbers appearing in the first column give the order
o£ occurrence reckoned from the midpoint of the band. For
hydrochloric acid no absorptionbands in the longwave
spectrum have yet been measured directly, but from the
determinations of Rubens and v. Wartenberg* it appears
that between 100 and 350 yu hydrochloric acid shows strong
absorption, which is stronger between 100 and 200yit than in
the case of longer wavelengths.
Figs. 24 show as clearly as can be desired that, if we
wish to maintain Bjerrum's view as to the origin of the
ultrared absorption, we must assume that the rotation
frequencies of the molecules (at least of the absorbing mole
cules) do not vary continuously according to Maxwell's or
any similar law of distribution. Hence it readily suggests
itself to try with hypotheses from the quantumtheory, and
this Bjerrum has in fact done in his abovementioned work.
He starts — as Ehrenfest does later on — from a distribution
of the rotationenergy according to quanta and puts
~(27TZ/)2.I = 71. ll.V,
(1)
where I is the molecules' moment of inertia, n a whole
number,, and h Planck's constant.
From this we get v — n^ 2j ; that is to say that when I
is constant v can only have certain fixed values, which form
an arithmetical progression with the constant difference =■
h
27T2f
* L.c.
the Rotation Energy of Molecules.
79
Bjerrum found, too. that the watervapour bands between
10 and 20 //, determined by Rubens and Aschkinass* (Table II.
eol. 3) form such an arithmetical progression, and he assumed,
therefore, that these bands also are caused directly by the
molecules' rotation.
Fig. 4.
1600
1700
But on referring to the curve in fig. 2, or to the values
calculated from this curve in Table II. (cols. 1 and 2), we find
at once that a simple arithmetical progression is out of the
question. The positions of the determined maxima seem to
be very irregularly distributed. A. Bucken f, however, has
attempted to arrange these calculated values in two arith
metical progressions, of which the one is a continuation of
the abovementioned series with a constant frequency
difference A = l'73xl012, and the other a series with the
difference 0*75 x 1012. According to Eucken these two series
may be explained by the lack of symmetry of the molecules,
which, he says, may result in the rotationenergy's being
divided between two chief moments of inertia.
* Rubens u. Aschkinass, Wied. Ann. lxiv. p. 584 (1898).
t A. Eucken, Verh. d, D.p/ujs. Ges. xv. p. 1159 (1913).
SO Dr. Eva von Bahr on the Quantumtheory and
The wavelengths and the accompanying number, repre
senting order in occurrence as calculated by A. Eucken, are
given in Table II. cols. 47. Their agreement with my values
is not particularly close for the longer wavelengths, and for
some of my maxima there are no corresponding values in
Eucken's series. However, ii must be borne in mind that
my values rtannot be very exact just in the case of the longer
wavelengths, and that it is not impossible that one or two
maxima in fig. 2 do not belong to the water vapour band
at 6'26 /jb. Judging from the fact that absorption is stronger
in the longwave part of the band, it is even probable that
we are here in the presence of another, weaker absorption
band, lying over the principal band. Another possibility is
that in reality there are three series, but that the third series
presents itself so feebly that it is noticeable only in the
neighbourhood of a rotation wavelength of about 4060 /j,.
The most serious objection to Eucken's series is that they
postulate an absorptionband at 87 fi} where Rubens has even
found great transparency for watervapour.
In the case of the diatomic hydrochloric acid there is
reason to anticipate a simpler composition of the absorption
band than in the case of watervapour, and to judge from the
curves given in fig. 4 such is also the case. As appears from
Table III. col. 5,  is not, however, constant, as it ought to be
•according to Bjerrum's formula, but decreases with increasing
frequencies. It is true that the determinations are not very
exact, and we can hardly claim that the divergencies of the
particular values from the mean value lie outside the margin
of error, but the steadiness of the decrease indicates that
here we have probably an actual divergence. It is possible,
however, that this divergence does not depend on the falsity
of the formula, but — as Eucken suggests — upon the moment
of inertia of the molecules (I) increasing with increased
velocity of rotation.
If, as first approximation, we assume that the rotation
frequencies of the hydrochloricacid molecules, as well as
those of the watervapour molecules, form arithmetical
series, as anticipated by formula (1), then it still remains to
enquire whether the formula still holds quantitatively. The
possibility of such a test is afforded by the calculation of the
moment of inertia, on the one hand from formula (1), and on
the other from the Kinetic theory of Gases. Similar calcula
tions have already been made for watervapour by Bjerrum,*
who found correspondence in respect of order of greatness.
* N. Bjerrum, I. c.
tlie Rotation Energy of Molecules. 81
As, however, he could only take into consideration the one
arithmetic series, and, besides, had no true value for v, it
seems to me to be of interest to repeat the calculations. Of
special interest, too, will be the investigation in the case of
hydrochloric acid, from which, on account of the simpler
chemical composition, more exact values may be expected.
If, in formula (1), we insert the frequency v = 20' 15 x 1011,
calculated from the maximum of the absorptionband, and
observe that the maximum coincides with the third of the
lesser maxima (see fig. 4) — ?z = 3, therefore — we get for
hydrochloric acid,
nh 3x6548x10" 5.lxl0„
27T2v2 7r2x2&15xl0n °1X1U '
According to the Kinetic theory of Gases the mean rota
tion energy of a diatomic molecule is kT, where A=« = 1*346
X 10~16. If now for the calculation we employ the value (v),
obtained from the maximum of the absorptionband, which is
approximately correct, we get
T_ JLL. 1346 x10»x 290 .0x1().4O>
i(2irv)2 i{27rx 2015 x 101
As we see, the correspondence is very satisfactory. It
must, however, be pointed out that the value of v inserted in
the last equation is very uncertain. Were the absorption
proportional to the number of the absorbing molecules, inde
pendently of the frequency, the mean of the squares of the
frequencies ought to be greater than (20' 15 x 1011)2, but
probably the intensity of the absorption increases with the
frequency. If we insert in the formula? a pvalue=JS8x 1011,
which in fig. 4 has a corresponding frequency, lying between
the fourth and the fifth maximum, we get a moment of
inertia half as large, or 2*5 X 10~40 — a value that agrees with
Ehrenfest's formula
V = JttU ^ Page )'
For watervapour we get the two moments of inertia 1*9
and 44xl0~4t) from formula (1), using the abovementioned
frequencydifferences. From the kinetic formula E=j&T,
we can naturally only calculate the mean moment of inertia,
and the calculation is very inexact, too, since from the
absorption measurements we can scarcely conclude more than
Fhil. Mag. S. 6. Vol. 28. No. 163. July 1914. G
82 Quantumtheory and RotationEnergy of Molecules.
that v lies between 20 and 40 x 1011. The mean value 30 x 1011
gives a mean moment of inertia = 3*3x 10~40.
For gases other than hydrochloric acid and watervapour
there are not yet any experimental determinations of the
frequency difference. On the other hand, we have more or
less exact values of v, as noticed above,, and by means of
the kinetic formula we can approximately calculate the
moments of inertia for these gases also. In Table IV. are
brought together — besides the values worked out above —
the moments of inertia (I) for several di and triatomic gases.
Table IV.
Gas.
v.1011.
I.1040(k.t.)
1 . 10io (q. t.)
A.108.
HC1
2015
167
83
30
38
25
49
67
29
33
205
480
51
V9; 44
18
23
16
HBr
CO
H20
N20
co2
It is striking that the moment of inertia of carbon monoxide
is more than four times as large as that of hydrobromic acid
although the molecular weight is three times less. But if we
assume the Rutherford atommodel, and, by means of the
moments of inertia given in the table, calculate the distance
(A) between the atoms, we get the values given in the fifth
column of the table, which are well reconcilable with mole
cular sizes determined by other methods.
According to formula (1) the frequency differences (^— ^ )
are inversely proportional to the moments of inertia. If the
formula holds, the separate lines in the absorptionband of
e. g. the carbonmonoxide, must lie six times as close together
as those of the hydrochloricacid band. The polyatomic
gases generally have moments of inertia much greater than
those of carbon monoxide, and, further, as they— like water
vapour — probably have at least two frequency series, there is
good reason to believe that their absorptionlines lie very
close together. From this would be explained the fact that
the influence of the pressure on the absorption generally
ceases the sooner, the greater the molecules, and that it has
not been possible to demonstrate any pressureeffect at ail in
Potentials required to Maintain Currents. 83
the case of very large gasmolecules, such as those of ether
and benzinevapour *. For the influence of pressure depends —
at least to a great extent — on the circumstance that with the
increase of pressure the single absorptionlines gradually
broaden, to merge finally into one continuous band f , and it
is likely that this final result is reached the more easily in
proportion as the lines lie nearer together. The real cause
of this pressureeffect, however, still remains unexplained.
If the formula (1) held absolutely, we should have to seek
the cause in a change of the moment of inertia brought about
by the impacts of the molecules. This could be caused by the
momentary effect of the impacts being that the molecules
have various rotationaxes, or that they suffer a temporary
deformation. But both these assumptions offer considerable
theoretical difficulties. Not less difficult to justify, however,
seems to me the assumption of Perrin t, that immediately
after impact the molecule has a given energy of rotation (E) ,
which, however, through some sort of friction or radiation,
rapidly decreases, until the remaining rotationenergy
satisfies the formula E = n . h . v.
Uppsala, April 1914.
XL The Potentials required to Maintain Currents between
Coaxial Cylinders. By John S. Townsend, Wykeham
Professor of Physics, Oxford §.
SEVERAL investigations have been made of the potential
required to maintain a current between a wire and a
large coaxial cylinder when a glowdischarge is produced by
a high electric force. A certain definite potential V0 is
required to start the discharge, and the rise of potential
V — V0 required to maintain a given current has been deter
mined experimentally.
When the current i is small the rise of potential V — V0 is
proportional to j, and in this case a simple formula  con
necting t and V — V0 may be obtained on the hypothesis that
the ions are produced from molecules of the gas by collisions
within a short distance of the wire where the electric force
is large.
«v
* E. v. Bahr, Ann. d. Phys. xxxiii. p. 585 (1010).
t E. v. Bahr, Verh. d. D.phy*. Ges. xv. p. 710 (1913).
X J. Perrin, < Die Atorae,' 1914, p. 146.
§ Communicated by the Author.
 ' Electrician/ June 6, 1913.
G2
84 Prof. J. S. Townsend on the Potentials required
For larger currents dY/di, the rate of increase of the
potential with the current, diminishes as i increases, and the
relation between V and i may be obtained on the same prin
ciples, when the pressure of the gas between the wire and
the cylinder is not very low, and ionization by collision takes
place only at points whose distances from the wire are small
compared with the radius of the cylinder.
The condition that the forces near the wire should be
sufficiently large to maintain a current is represented by the
equation
< dv,
'■
where a. and /3 have their usual signification, a being the
radius of the wire and c the distance from the axis at which
the force becomes so small that a and j3 may be considered
to vanish. This condition is independent of the intensity of
the current, so that the forces corresponding to the starting
potential V0, when i is infinitely small, will be sufficient to
maintain any current.
The current affects the field of force owing to the electric
charge produced by the separation of the ions in the gas, the
principal action being due to ions of the same sign as the
charge on the wire, which give rise to a volume distribution
of electricity p in the space between the outer cylinder and
the region near the wire where the ions are generated. The
distribution p' due to ions of the same sign as the charge on
the wire that move through a short distance of the order
c — a has such a small effect compared with the distribution
p that it may be neglected.
When a current is flowing the distribution p tends to
diminish the force near the surface of the wire, and in order
to maintain the current it is necessary to counteract this
effect by increasing the potential difference between the wire
and the cylinder. The relation between V and i may be
obtained on the hypothesis that the force at the surface of
the wire is equal to Xx ( = V0/alog bja), the force correspond
ing to the starting potential V0. For simplicity it may be
assumed that the velocities of the ions are proportional to
the electric force, and although this is by no means true for
the negative ions moving in gases at low pressures, the
formula obtained on this hypothesis, when compared with
the experimental results, shows very clearly many properties
of the negative ions.
to Maintain Currents between Coaxial Cylinders. 85
Let i be the current per unit length of the wire, k the
velocity of the ions due to unit force, p the charge per cubic
centimetre of the gas, and <£ the potential at the distance r
from the axis. At the surface of the wire r=a, cf> = 0; and
at the surface of the cylinder r=b, <£ = V.
If all the electrical quantities are expressed in electrostatic
units the following relations hold between cf>} p, and i : —
i= — 2iTprk^
dr
1 d ( d\ .
TdrVdr)=^
when p is eliminated the relation between dfyjdr and r
becomes
The constant of integration C is obtained from the condition
jr =A1=1 jj . when r — a.
dr a log bja
Hence
(£ )'«.•¥ ♦?
When 2z/£ is negligible compared with Xj2 this equation
becomes
, . v A , 2ir2 \i dr
which may be integrated by changing the variable r to
The total potential fall V thus obtained is
V=aX1{(l + 0)*l + Iog2&loga(l+(lf0)*)},
2ib2
where 0 = ^^ 2, 2i being small compared with kX^.
Hence V— V0l b „ , 9
86 Prof. J. S. Townsend on the Potentials required
When 6 is small this equation reduces to
»^2log bja
vv0=
%k\\
which shows that V— Y0 is proportional to i for small currents.
Y — Y 2ib2
The general equation connecting — ^ — °log bja and , 2^ 2
may be solved by means of a curve having these quantities
as ordinates, so that when V — V0 is determined experimentally
2ib2
for a current i the value of y 2V may be determined, and
lid 2v.
the value of k may therefore be found.
The theory may be tested by finding the values of k by
this method and comparing the results with the values ob
tained by more direct methods, or if the theory is accepted
the experiments provide a simple method of rinding the
velocities for a large range of pressures of the gas.
The following are the results of some experiments made
to investigate the values of kx and k2 for positive and negative
ions in air at different pressures. The cylinder was 7*49 cm.
radius and the coaxial wire *0268 cm. radius. The cylinder
was provided with two small side tubes, one at each end, and
a stream of dry air that had passed through tubes of phos
phorus pentoxide was drawn through the cylinder. A
series of experiments were made before all the moisture was
expelled by heating the cylinder, and the results obtained
with positive and negative discharges in air at three different
pressures are given in Tables I. and II. The pressures p
are given in millimetres of mercury, and the potentials V
and V0 and the currents i in electrostatic units.
Table I.
Velocities ^ of positive ions under unit electrostatic force.
p.
v0.
Y.
i.
321
*xxio 3.
k} p
760"
176
156
173
146
339
176
156
1853
660
158
. 365
84
985
1095
306
340
376
84
985
1173
642
365
404
30
575
635
305
108
420
30
575
672
560
120
470
1
to Maintain Currents between Coaxial Cylinders. 87
Table II.
Velocities k2 of negative ions under unit electrostatic force.
p
v..
Y.
i,
£2xl0 3.
760'
176
]83
192
317
270
63 (
176
183
199
655
270
630
84
115
12 02
350
78
860
84
115
1243
750
89
980
30
622
639
305
405
1600
30
622
650
570
432
1700
The cylinder was then heated and filled with dry air and
exhausted, the process having been repeated several times.
The experiments were again made with the drier air, and the
results are given in Tables III. and IV.
Table III. — Velocities kx of positive ions under unit
electrostatic force (after the cylinder had been dried).
p
Vo
v.
i.
1^X10 3.
*j P .
76U
400
430
176
176
1550
1550
1682
1828
290
680
173
186
84
84
983
983
1078
1133
337
570
445
442
492
490
30
30
565
565
607
653
305
687
160
153
630
610
Table IV. — Velocities k2 of negative ions under unit
electrostatic force (after the cylinder had been dried).
p
v0.
V.
i.
A..X10 3.
Kp
760 '
176
1837
1903
358
395
915
176
1837
1943
580
387
900
84
1168
1200
345
131
1450
84
1168
1228
660
123
1360
30
644
649
300
140
30
644
655
540
120
88 Prof. J. S. Townsend on the Potentials required
It will be observed that the rise of potential V — V0
for a given current is diminished by drying the gas.
The velocities are therefore reduced by the presence of
water vapour when the ions move away from the strong
field in which they are generated. The mean value of
^1^/760 = 415, deduced from the experiments in dry air at
176 mm. pressure, is in good agreement with the velocities
408 and 427 obtained by Zeleny* and Langevinf by direct
methods.
The mean values 491 and 620 obtained at the pressures of
84 and 30 mm. respectively, show that the velocity of positive
ions increases more rapidly than the inverse of the pressure.
Langevin, whose determinations were made with smaller
forces, found that kxp was practically constant for pressures
between 75 millimetres and 1420 millimetres.
The numbers given in Table IV. show that even at the
pressure 176 mm. the negative ions move much faster than
the positive ions, the velocity being eight times greater at
the pressure 30 millimetres. This is a well known effect,
and is due to the tendency of the negative ions to assume
the electronic state when the value of X/p increases. The
phenomenon only occurs at the higher pressures when the
intensity X is large, and has not been observed at the higher
pressure by the direct methods of finding the velocities.
The theory also gives satisfactory results when applied
to the experiments made by Watson J and Schaffers§ on the
discharges at higher pressures. Watson determined the
potential required to maintain currents from a wire *35
millimetres radius and a coaxial cylinder 10'2 centimetres
radius. The values of Jci and k2 for positive and negative
ions deduced from these experiments are given in Tables V.
and VI., the current i per unit length of the wire and the
potential V being in electrostatic units.
There is a good agreement between the values of the
velocities deduced from experiments at the same pressure
with different currents. The mean values are rather low for
dry air. This may be due to the oxides of nitrogen which
are formed by the discharge as they would condense on the
ions and diminish the velocities, an effect which may be
appreciable with the larger currents when the air is im
perfectly dried.
* J. Zeleny, Phil. Trans. A. cxcv. p. 193 (1900).
t P. Langevin, Annates de Chemie et de Physique, [7] xxviii. p. 289
(1903).
J E. Watson, * Electrician,' February 11, 1910. ■
§ V. Schaffers, Pht/s. Zeits. xv. (1914).
to Maintain Currents between Coaxial Cylinders, 89
Table V.
Values of ki deduced from Watson's experiments
with air at 748 mm. pressure.
V.
i.
K
7H0*
513
0
580
600
380
374
635
1200
425
418
725
2400
360
354
798
3600
340
334
Table VI.
Values of: k2 deduced from Watson's experiments
with air at various pressures p.
p
V.
i.
fa.
K n
760
528
0
588
600
446
440
748
635
1200
446
440
710
2400
422
415
760
3600
450
443
430
0
470
600
745
542
552
502
1200
790
575
553
2400
785
570
602
36U0
740
539
304
0
343
600
1580
740
356
364
1200
1660
780
405
2400
1520
712
437
3600
1490
700
In Schaffer's experiments the air was at atmospheric
pressure, nnd the variation of the potential with the current
was determined for wires in cylinders of different diameters.
The results show that the rate of change of the potential
with the current, dV/di, is small with the smaller cylinders,
but increases rapidly with the radius of the cylinder. It is
difficult to form an accurate estimate of the values of V — V0
for the smaller cylinders, but with the larger cylinders of
radii 3*85 and 5*85 centimetres, the rise of potential with the
current may be found to a sufficient degree of accuracy, to
calculate the values of k.
90
Prof. J. W. Nicholson on Atomic Structure
Table VII. gives the results of some of these experiments.
The values of k2 are in good agreement with the numbers
obtained from Watson's experiments, but hx is somewhat
larger.
Table VII.
Values of l\ and k2 for air at atmospheric pressures deduced
from Schaffer's experiments, b being the radius of the
cylinder and a the radius of the wire in centimetres.
XII. Atomic Structure and the Spectrum of Helium. By
J. W. Nicholson, M.A., D.Sc, Professor of Mathematics
in the University of London* .
AN earlier paperf has indicated that the crucial test of
Bohr's theory of spectra is to be found in its appli
cation to the ordinary spectrum of helium. For the theory
gives a precise specification of the constituents of a helium
atom, — a specification which is in accord with the results
obtained by a combination of the atomic number hypothesis
of Van den Broek and the experiments of Moseley, and with
those to which Rutherford and others have been led by some
other lines of study. The necessity for this accordance,
in fact, dictated the particular model which Bohr has
used. His corresponding models for lithium and the heavier
elements have been shown to be at fault, in that they involve
distributions of electrons which cannot exist by virtue of the
assumptions on which the theory is founded, in spite of the
fact that these assumptions are of a very general character.
The success of the theory therefore rests on its application
to the spectra of hydrogen and helium, — leaving out of
account, for the moment, the controversial question of its
application to Xray spectra, which was partially dealt with
in the earlier paper, — in so far as that application has been
made. The theory must therefore stand or fall according to
* Communicated by the Author,
t Phil. Mag. April* 1914.
and the Spectrum of Helium. 91
its capacity to take account more completely of the spectra
of: these two elements, to which the analysis of the preceding
paper does not apply.
The idea that series spectra are only an illustration of
Planck's law, by the energy which is radiated during the
passage of an atom from one state to another, is. at first sight,
very promising, for it takes account of the fact that all
frequencies in series spectra are differences of two functions
of whole numbers. Leaving aside, in order to avoid compli
cation, the existence of double and triple lines, satellites, and
such series as those known by the name of Bergmann, we may
say that for all ordinary elements whose spectra can be put
into series of Rydberg's approximate type, there are three
important series to be explained, — the Diffuse, Sharp, and
Principal series, whose mutual relations are exhibited by the
following formulas for their frequencies, where B is Ryd berg's
constant,
Diffuse, v = B{^)2(^},
Sharp, v = 13 '  — ? — — „ > ,
( (1 + */ {m + py )
nncipaJ
Bf_i ±—\
\ a+p)2 (m+*)2/'
where (p, s, d) are definite constants for the elements."
It is apparent that these series alone require, on Bohr's
theory, the existence of three different types of stationary
states, to which the constants m, $, p are individually peculiar,
and that the spectral series are formed during the passage of
the atom not between different states or configurations of the
same type, but between different types of states. If a helium
atom only contains two electrons and a comparatively very
inert nucleus, difficulties are at once apparent, for two
electrons have not the necessary amount of freedom, in
presence of each other, to take up three different types of
relative configurations. Moreover, the spectrum of helium
contains six such series, with six apparently independent
constants. The atom of helium, not only according to the
theory of Bohr, but in actual experiments such as those of
Sir J. J. Thomson, will not take up an extra electron, so that
the neutral atom and the atom which only retains one electron
are alone left to explain the entire spectrum.
It was shown in a recent paper"', that on the present form
* Monthly Notices of R. A. S. March 1014,
92 Prof. J. W. Nicholson on Atomic Structure
of Bohr's theory, the spectrum of helium as ordinarily mani
fested cannot be obtained. Moreover, it was proved that none
of the possible steady states of the two electrons leads to a
formula approaching that of Rydberg. The most important
steady states give formulae like Balmer's which do not pre
serve the Rydberg constant B. Tne reason for the latter
property is that Bohr's deduction of the universality of B in
all spectra depends on the assumption that steady states
exist in which one electron is at a great distance from the
others, whose joint effect, with that of the nucleus, is approxi
mately equivalent to that of a hydrogen nucleus The paper
in the Philosophical Magazine shows that this is at least
impossible for three electrons in all, and therefore for lithium,
which, as a matter of experience, does retain the Bydberg
constant in its spectrum. The paper in the ' Monthly Notices '
shows that it is equally impossible for helium. Stationary
states, derivable from the ordinary electrostatic forces, there
fore are very limited, and if Bohr's theory is to proceed
further, some change must be made.
When we consider the essentials of the theory, the limits
of possible change become apparent at once. The derivation
of the hydrogen formula requires that the angular momentum
in the atom should be a multiple of ^ — when there is only
one electron present. The same supposition is involved in
obtaining the Pickering series of lines, where again there is
only one electron concerned. This multiple property may,
however, change when there is more than one electron.
The second necessity which is vital for the hydrogen
formula is that the law of attraction of an electron to the
nucleus is that of the inverse square. This can easily be
seen by trying the law rn. We do not get even the form
of Balmer's formula unless n=— 2. As Bohr's theory has
been shown to be quite unsuccessful when there is more than
one electron, we must combine the following hypotheses in
any attempt to develop it further: —
(1) Nuclei attract bound electrons according to the inverse
square law.
(2) Bound electrons do not repel each other according to
this law.
(3) The angular momentum of an electron may cease to be
Til
^— , where t is an integer, if there are other electrons
present.
As we shall see, (2) and (3) are not alternatives, but are
and the Spectrum of Helium.
93
both necessary for further progress. The first hypothesis is
required, even for the Balmer series.
The Lithium Atom.
The conclusion of the preceding paper, that it is not possible
for three electrons and a nucleus to form a lithium atom with
a unit valency, after the manner of Bohr's model, was arrived
at without the second hypothesis. The electrons were regarded
as repelling each other under the law of inverse square. We
may now consider the possible laws of force between electrons
which can admit such a configuration, in which two electrons
form an inner ring, with one outside, while each electron has
a constant angular momentum. We shall not require to
consider whether the angular momenta of the electrons are
equal, or what may be the relation between their angular
velocities.
Let/(r) be the law of force between two bound electrons
at distance r apart, and consider the three electrons at the
instant when they are at the corners of a triangle ABC.
Since the nucleus 0 is very heavy, they are practically
describing orbits about it, whose radii at the instant may be
called (t\, r2, r3).
If AG, BO meet the opposite sides in L and M, we can at
once find CL, CM. For the forces on the electron A are
/(b) and /(c) in the lines CA, BA, due to the other electrons,
and the force along OA from the nucleus. The angular
momentum of A does not change if the components of f(J>)
and f(c) perpendicular to AO counterbalance, or
m
M
sinBAO
BL
LC
Thus
an<
sin CAO
BL :LC::cf(b):bf(c),
pTj _ abAc)
Similarly
CM =
ahf(c)
94 Prof. J. W. Nicholson on Atomic Structure
If the line CB is taken as an axis of x in a Cartesian system
with C as origin, the equations of AL, BM become
_&sinCQCL)
V fcoosCCL '
_ CMsinC(a?fl)
y CM cos C a '
and the coordinate .2? of the point 0 becomes on solution,
gCL.(CM6)ft . CMcosC(aCL)
CL.CM^
If r2 = r3, so that two of! the electrons are in a ring, x is ^a,
and this must be true approximately if the deviations of the
two electrons from their ring exist, but are not large.
This condition reduces to
a"b + a2GL.CM2a2b.CL = CK(aGh)(b2ja2c2);
and substituting for CL and CM, the law of force f(r) must
be consistent with
« { V(e) + cf(b) } { cf(a) + af(c) } = a%f{c) + 2abcf(a)/(c)
+ c(b* + a*c>)f(b)f(c)
for all values of a, b, and c. No law which is a power of r
will ordinarily serve, for writing f(r)=\rn, we have
a{bcn + cbn) {can + acn) = a2bc2n + 2abc . ancn + c(^2 + «2 — c2)bncn,
or on reduction,
(52_c2)c»i6»i=fl»+i(^icn1).
The case b = c is a solution for any such law. If b is not
equal to c, any case of n negative (and the case w = 0) makes
one side of the equation essentially positive and the other
essentially negative, so that it cannot be satisfied. The case
n = l compels b and c to be equal. n = 2 gives a= [5c(6 + c)]i,
unless b = c, and in fact, laws of direct distance beyond the
first are formally possible. But they cannot be permissible
from physical considerations, for the repulsion between
electrons cannot be thought of as increasing with their
distance apart, when that distance is comparable with the
atomic radius, as in Bohr's rings of electrons.
If the distance apart were comparable with the radius
of an electron, of order 10 ~13, such a possibility might be
admitted, and in a subsequent paper it will be found to lead
to valuable conclusions. But here we are not concerned with
possible nuclear structure, of which this is a question, but
and the Spectrum of Helium. 95
with rings of more ordinary atomic size, for which the only
conclusion is that h = c. This signifies that the electron A is
always equidistant from the other two which form a ring.
It therefore rotates with the same angular velocity, and from
this point the reader can easily supply the proof that all
three electrons must be in one ring. The conclusion extends
not only to the law Xrn for the force between two electrons,
but to any law f(r) which can be expressed in a power series,
— such a law, for example, as a mixed inverse square and cube,
of the form
We must finally suppose that the law of foroe between
electrons bound in an atom which can admit Bohr's lithium
model is either —
(1) No force at all, which evidently constitutes a
solution, or
(2) Any law whatever, provided that the three electrons
are all in one ring.
This last alternative cannot take account of the valency of
lithium, as in the last paper, whose investigation can be
extended.
But, in addition, direct distance laws are formally possible
for electrons which are actually on the confines of the
nucleus. These, however, are not rings of electrons in the
sense required by Bohr's theory, but are ^particles. They
would be too close to the nucleus to be capable of behaving
with it otherwise than as neutral doublets. We do not need
to consider their possibility any further at present. For we
cannot effect a compromise, such as would be suggested by
putting two electrons close to the nucleus (within a distance
1013) under this law, and the third as an outer valency
electron of lithium at a distance 10 ~8, acted on by the other
two according to the inverse square law. For such an
arrangement would give lithium, on Bohr's theory, a spectrum
which could not be distinguished from that of hydrogen.
Goplanar rings in Bohr's sense are therefore only possible if
bound electrons exert no force on each other, — for the case
of lithium at least. We can generalize this result from
lithium to any other element, but it is not necessary to give
the details of the argument. If the possibility of zero force
is admitted, there is no limit to the number of such rings
which are possible, or to the number of electrons which each
can contain. The atom becomes remarkably indefinite, for
any electron is only acted upon by the nucleus, and its
96 Prof. J. W. Nicbolson on Atomic Structure
behaviour is as simple as that of the electron in a hydrogen
atom. The consequences of such an hypothesis in connexion
with spectra will appear later. At present it has only been
shown that it is the only hypothesis which will, on Bohr's
theory, allow a coplanarring structure of the ordinary
elements.
We may notice that this hypothesis is a step towards
Sir J. J. Thomson's conception of tubes of force in the
atom. The relation between the two points of view is as
follows. If an electron when free has tubes of force, with
some material significance, radiating from it, it may when
bound have these tubes diverted to pass into the nucleus,
and all the electrons in an atom being in this connexion with
the nucleus, cannot exert force on each other. An accelerated
electron might radiate if its tubes extended to a distance, but
not when they all passed to the nucleus. This connexion
between two points of view is interesting, but may have no
special significance.
Tlie Spectrum of Helium.
On the supposition that a helium atom, when neutral,
contains only two electrons and a nucleus 2e, and that the
spectrum of the atom when one electron has disappeared is
that calculated by Bohr, and previously believed to be due
to hydrogen, we are compelled to suppose that the more
normal helium spectrum is produced by a neutral atom. For
a helium atom, on Bohr's theory and according to Sir J. J.
Thomson's experiments, will not take up a third electron.
The neutral atom must therefore possess six types of stationary
states, as there are six types of function to which the appli
cation of the combination principle gives the helium spectrum.
We shall attempt to arrive at any one of the^e functions by
Bohr's method in the most general manner which is yet
consistent with the theory of the spectra of the hydrogen atom
and the positively charged helium atom. The best series is
evidently the sharp or second subordinate series of parhelium,
given by the very accurate Hicks' formula
71=27174917  1097260 / fm + 861181  '°0^809 J *,
where n is the wave number. For the Rydberg constant is
in this case 109726*0, which is almost precisely Bohr's
estimate 109725 instead of the usual 109675 of hydrogen.
If the theory is correct, 109725 is the theoretical value for
elements heavier than hydrogen. We deduce that one of
the types of stationary states concerned in the production
and the Spectrum of Helium. 97
of this series is formed by the radiation, during the collection
o£ the atom from infinite dispersion, of an amount of energy
W=^L/(m + 8C1181^9)l
Now if there is any force whatever between the two
electrons, they can only have stationary states when they
are continually in a Jine with the nucleus, provided that
their orbits are in the same plane. This can be proved
immediately, on the supposition that they move with constant
angular momenta. When they are in this line with the
nucleus continually, their angular velocities must be equal ;
■©■
and since their angular momenta are specified, the radii must
be in a constant ratio, which is one of equality if the angular
momenta are equal. Let the rapulsion between them, when
at distance r apart, be — . Then their orbits are given by
d2ux
_{v_ _\ *> 1
ul=~.
d62 + ?'2 I r22 (r, + r2)» J h2 %* ?'2 "" r2'
In the ' Monthly Notices '* it is proved that these orbits do
not exist unless they are identical, under the inverse square
law. For other laws, we may proceed as follows. Let the
angular momenta mh^ and mh2 be given by h1 = oi2hi h2 = fi2h,
a2 and /32 being in a ratio of integers on Bohr's view, but
more generally, perhaps, in other constant ratios. Then if co
is the angular velocity,
and
= «2A, r22a>=£2A;
/3 a.
ri, u2=u^
* March 1914.
Phil Mag. S. 6. Vol. 28. No. 163. July 1914. H
98 Prof. J. W. Nicholson on Atomic Structure
so that
^i ,w_ Lu 2_x?, » «n I x
and therefore for a// values of w1? by division,
1 J 2 ^ » «" l 1 S °? 2 * w "" 1
Excluding w = 2, already dealt with in the other paper*,
this compels a to be equal to /3, so that the angular momenta
are equal. Then r1=r2, and the electrons are continually
equidistant from the nucleus.
For orbits in one plane, therefore, under any law of
electronic repulsion, the electrons are equidistant from the
nucleus ; and thus they move in the same circular or elliptic
path, or keep pace with each other in two equal paths with
the same major axis, as in the diagram given in the Phil Mag.
paper j*.
By analysis already given, and capable of extension J,
the same conclusion can be inferred for noncoplanar paths,
of which the figure gives an illustration, showing such possible
paths of each electron and of the nucleus round the axis of
ff
/?xz$
the atom. It is evident that it applies to any law of force
whatever which can be expressed in a power series. The
details of the analysis can easily be filled in by the reader.
We conclude, finally, that for any law of force between
electrons, the two in a helium atom must be equidistant from
the nucleus throughout any of Bohr's steady states.
* Phil. Mag. April 1914. t Page 657.
J ' Monthly Notices,' March 1914.
and the Spectrum of Helium. 99
Bohr has noted * the difficulty of any other supposition in
the case of two electrons, although apparently he has not con
sidered the example in the figure. A more general one can be
obtained by rotating the whole atom about an axis perpen
dicular to the axis of the atom, but as stated in the ' Monthly
Notices 'f, even this set of configurations does not give
anything resembling the helium spectrum when the law of
force between the electrons is the inverse square. In order,
therefore, to explain the helium spectrum, this law must be
abandoned.
Let us now follow out the consequences of the law of the
inverse nth power, as a preliminary to any other law which
can be imagined. We may suppose the path of the electrons
to be circular, as under the inverse square law, without a real
loss of generality. Taking the case in which the orbits are
coplanar, we see from the above reasoning that the electrons
are equidistant from the nucleus, and their circular orbits are
therefore identical. If the force between the electrons is —
where r is the distance apart, the angular velocity and radius
OL
are given by
2*a
mato2= — —
a 7h
ma 00= z—,
(*»)•
where the second is the condition of angular momentum.
The sum of the kinetic and potential energies is, if n is not
unity,
C + maW  — + * . .,
a n—l(2a)711
where C is the energy in a state of infinite dispersion. This
becomes
o^:
(>i3)\
(n— l)2naa1
The energy radiated in forming the atom it
W2g2 , Q3) A,
a (n— l)2»a*1'
where n is given by
V 2nan~2J
4i7r2m
Phil. Mag. July 1018. f L.
H 2
100 Prof. J. W. Nicholson on Atomic Structure
If n is not 2, X contains the (n — 2)th power of some constant
length, and it is very difficult to imagine what the length
could represent physically. For it is of the same order ;is
the diameter of the atom. This alone is perhaps sufficient
to remove the possibility of any law except the inverse
square. Nevertheless, we may give a brief account of some
other laws. For the inverse cube, n = 3, and
t2A2 9 2 X
4+w)(tvt*)=BT,
in which no assumption is made regarding ir as a function
of volume and temperature, and further vi denotes the
limiting volume " experimentally obtained,''' that is obtained
by extrapolating the CailletetMathias linear diameter to
absolute zero. As is well known, the b of van der Waals'
equation is onethird of the critical volume, whilst the
" actual " limiting volume is more nearly one fourth of the
critical volume. Berthelot (cf. Kuenen, Die Zustands
gleichung, p. 83) found that for oxygen, chlorine, carbon
dioxide, sulphur dioxide, carbon tetrachloride, and benzene,
the mean value of vi was 0*26 vc, where vc is the critical
volume. Guldberg some years previously had examined a
much larger number of substances (76 organic and 13 in
organic), and had found that yc=3*75 vi or vi = 0'27 vc, the
mean error not exceeding 4 per cent.
* Communicated by the Author.
• On the Internal Pressure of a Liquid. 105
Recently the internal pressure tt of a number of normal
liquids has been calculated by H. Davies (Phil. Mag. [6]
xxiv. p. 415, 1912). The calculations of tt carried out in
the present paper are of a similar nature to those of Davies,
in which the relation
RT
7T=
VVt
has been employed at 0° C, vi being taken to be 0*27 ve, the
values for vc being mainly those recently compiled by Young
(Proc. Roy. Dub. Soc.xii., 1910). It is clear that the results
obtained represent minimal values of tt0°, since one cannot
assume vi to be absolutely constant, and any possible change
is an increase with temperature. The values are, however,
of importance as fixing the inferior limit for tt for a con
siderable number of liquids. (It will be observed that the
values obtained are approximately one half of the values
given by the latent heat of vaporization of unit volume of
the liquid.) (Table I.)
In a paper in the Phil. Mag. [5J xxxii. p. 113 (1891)
E. Obach has pointed out that there is an approximate direct
proportionality between the latent heat of vaporization of
a liquid and its dielectric constant. The data, especially
those on the dielectric constant K, were and are meagre,
and further the quantities were not compared at the same
temperature. Comparison of substances of similar chemical
constitution leads, however, to a fairly constant proportionality
factor. Now, many years previously, Dupre had put forward
the view that tbe latent heat of vaporization of unit volume
of the liquid was a direct measure of the internal pressure tt,
and although this statement is not exact it is certainly safe
to say that a large latent heat accompanies large cohesion.
Hence one would expect that there should be a rough pro
portionality between the dielectric capacity of a liquid and
its internal pressure. This has been stated to be the case by
Walden (Zeitsch. phys. Chem. Ixvi. p. 407, 1909). Walden
inferred the relationship after having observed that solvents
with large cohesion possess likewise a marked dissociating
power upon dissolved substances, which property in turn
has been connected with the dielectric constant of the
solvent by the NernstThomson rule.
106 Prof. W. C. McC. Lewis on the Relation of the
Table I.
Minimal values for the Internal Pressure or Cohesion
of Liquids at 0° C.
[Values of vc marked (Y.) denote those given by Young,
those marked (Math. Wald.) are taken from a paper by
Walden (Zeitsch. phjsik. Chem. lxvi. p. 407, 1909).]
Substance.
Vc =
Molecular
critical
volume.
vl=
027 vc.
v0 =
Molecular
volume
at 0° C.
vQvi.
ET
7T0° C. =
in
atmospheres/cm. 2
Benzene
2561 (Y.)
3154 „
3068 „
2819 „
3913 .,
3450 „
2290 „
2276 „
286 0 „
1720 „
2845 .,
3864 „
3851 „
3079 „
3236 „
3509 „
2762 „
3073 „
2070 (Math.
Wald.^
166 (Wald.)
2707 (Y.)
4899 „
1176 „
1669 „
2195 „
1711 „
503(Wald.)
1905 „
3100 (Y.)
4271 „
4900 „
4819 „
3484 „
307 1 „
3518 „
2817 „
6715
8516
9906
7611
10565
9315
6183
6149
7722
4644
7682
10433
10398
8313
8737
9474
7457
8297
55 89
4482
7309
13227
3175
4506
5927
4636
1358
5144
837
12532
1323
13011
9407
8292
9499
7606
86 7(Y.)
1048 „
127 0 „
100 5 „
1387 „
1121 „
780 „
771 „
952 „
598 „
948 .,
1261 „
1273 „
9929 „
1032 „
1096 „
943 „
1126 „
7145
5878
91 7 (Y.)
1583 „
395 „
561 „
733 „
561
180
699
1116(Y.)
1428 ,,
1587 „
1605 „
1266 „
1054 „
1145 „
1087 „
2055
1964
2794
3439
3305
18^5
1617
1561
1798
1336
1798
2177
2332
1616
1583
1486
1973
2963
1556
1398
1861
2603
775
1104
1403
974
442
1346
2790
1748
2640
3039
3253
2248
1951
3264
1093
1144
804
653
679
1188
1389
1439
1249
1682
1249
1032
963
* 1398
1429
1512
1138
758
1443
1610
1207
863
2898
2035
1601
2307
5084
1218
805
1285
851
730
690
999
1135
688
Toluene
wHexane
Ethyl ether
Ethyl propionate
Propyl acetate
Ethyl formate
Methyl acetate
Ethyl acetate
Methyl formate
Propyl formate
Methyl butyrate
Methyl2butyrate ...
Chlorbenzene
Brombenzene
Iodobenzene
Carbon tetrachloride ..
ipentane
Acetone
Carbon disnlphide ...
Fluorbenzene
wOctane
Methyl alcohol
Ethyl alcohol
Propyl alcohol
Acetic acid
Water
Ethyl chloride
MPentane
ftHeptane
wOctane
Die'butyl
Dwpropyl
Hexamethy len e
Stannic chloride
Methyl propionate ...
Internal Pressure of a Liquid to its Dielectric Capacity.
The following are Walden's data : —
107
Substance.
7r (at the
boilingpoint).
K (at 20° C).
H.O
atmospheres/cm.2
10600
4000
3100
ca. 3800
„ 3600
„ 3000
2600
2600
1570
1220
1030
81
>80
ca. 95
46
35
HgSC^
HON
i (CH\0H)o
CH.OH
HCOOH
CH3CN
58
36
CH3N02
40
225
4G5
185
n.He
(CvH)20
n H "
Walden's values for it are doubtful, however, as be appears
to have calculated them by an erroneous application of the
Stefan principle regarding vaporization. In the following
table, therefore, the minimal values of it (already given) have
been compared with the dielectric capacity as far as data
could be found. The values of K are taken from the article
by Gr. Rudorff (Jahrbuch d. liadioakt. und Elektronik, vii.
p. 38, J 910), and also from Walden (Zeitsch. phys. Chem.
Ixx. p. 569, 1910).
Substance.
K.
Temperature to
which Iv refers.
7r0°C.
7T
K "
920
91
8 02
60
4315
546
2322
243
2246
2676
32
646
233
349
284
248
80
1°C.
81°
0°
14°
15°
12°
0°
0°
17°
0°
22°
20°
1°
0°
0°
0°
0°
1682
1389
1439
679
653
1429
1093
1144
1138
1610
1135
2307
1450
2898
2035
1601
6000
182
152
180
113
130
260
471
470
507
600
355
357
58
83
72
64
75
Ethyi formate
Ethyl propionate
Ethyl ether
Brombenzene
, Toluene
Carbon tetrachloride ..
Carbon disulphide ...
Stannic chloride
Acetic acid
1 Acetone
Methyl alcohol
j Ethyl alcohol
Propyl alcohol
Water
i
It is evident that there is no direct proportionality inde
pendent of the nature of the substance between cohesion and
dielectric capacity, though the substance with large cohesion
108 Prof. W. C. McC. Lewis on the Relation of the
has in general large dielectric capacity. The results point
to the existence of some relationship between tt and K of a
more complex nature than that embodied in Obach's or
Walden's generalization. A possible relation in this con
nexion will now be considered.
From the chemical standpoint the formation of molecular
compounds through the operation of "residual affinity/'
along with the applicability — at least approximately correct
— of the inverse fourth power law of attraction, as expressed
in the work of Sutherland and the equation of van der Waals,
have rendered it fairly certain that molecules possess some
thing of the nature of polarity. There seems to be no reason
why one should not identify cohesion with residual affinity.
In other words, cohesion is electromagnetic in origin. The
chemical significance of cohesion is thus apparent. Further,
this view of cohesion is in good accord with Baly's theory*
of residual affinity.
If we suppose that the force F between two molecules
is of electromagnetic origin, the molecules being regarded as
equivalent to small magnets, the attractive force may be
written
F= —
pi*'
where m is the magnetic pole strength, assumed the same
for all molecules, and jjl the permeability of the medium
itself. It follows, therefore, that the force of attraction (71)
across unit area of the liquid may be set proportional to
— g or proportional to — 2, where v is the specific or mole
cular volume of the liquid. Denoting the proportionality
factor by a, it follows that
a
7r= 2?
/JLV2
which only differs formally from the van der "Waals expres
sion by the introduction of the factor //,. Now if the liquid
possess a refractive index N (for very long waves) /a may be
N2 . .
written ^ , 2, where C0 is the velocity of light in vacuo and
K is the dielectric constant.
Hence
_aKC02 a0K
77 " tf2V2 N2V2*
* Balj and Kralla, Journ. Cliem. Soc. ci. 1912.
Internal Pressure of a Liquid to its Dielectric Capacity. 109
If we consider only those liquids whose molecular volumes
and refractive indices (for long waves) are of the same order
of magnitude, it is evident that in such cases it would be
directly proportional to K. This is suggested as the theo
retical basis for the ObachWalden generalization, the more
correct relation between it and K being given by the above
equation.
The ObachWalden relation is therefore of considerable
significance as regards the origin of cohesion between mole
cules. If cohesion were electrostatic in nature, one would
expect 7r to vary inversely as K ; the fact that there is ap
proximate direct proportionality appears to be explicable only
upon an electromagnetic basis of molecular attraction.
The significance of the K ternras employed above calls
for remark. It refers evidently to the dielectric capacity of
the medium separating the molecules. If the molecules were
very far apart (i. e. a dilute gas) the medium would be
practically unaffected by the presence of the molecules, and
K would be a constant having the value unity (E.S.U.). In
such a case the above expression becomes identical with
van der Waals* term (the numerical value being at the same
time very small compared to its value in the liquid state).
In the case of liquids with closely packed molecules, the
intervening space will be modified as regards dielectric
capacity, and the assumption involved above is that either
this quantity is actually evaluated when a dielectric constant
measurement is carried out, or at any rate that the dielectric
capacity of the medium between the molecules is proportional
to the observed dielectric capacity.
That cohesion between molecules is magnetic in origin has
been recently suggested by A. P. Mathews (Journ. Physical
Chem. xvii. p. 481, 1913), who writes: " In this view the
atoms [in a molecule] would be united by their electrostatic
affinities, and these same valencies and the other atomic
electrons by their magnetic effects produce the molecular
cohesion." Whether this is actually the relative mechanism
of the two kinds of valency or not is rather a matter of
speculation.
We may then consider the question of cohesion from the
standpoint of the electron theory as applied to liquid dielec
trics. The force — due to surrounding molecules — tending
to draw an electron out of a given molecule, is i7rne2.u, where
x is the distance through which the electron is displaced,
n the number of electrons per unit volume of the substance,
and e the charge upon an electron. If the force exerted upon
the electron by the molecule to which it belongs is fx the
110 Prof. W. 0. McO. Lewis on the Relation of the
whole force exerted upon it will be (/' — fyrne2)x. From
which it follows that
4W
The smaller /' is, i. e. the less tenacious the hold which a
molecule has upon one of its own electrons, the greater K
will be ; and this in turn should mean a large cohesion and
residual affinity. An interesting case in point is furnished
by liquid mercury. Its conducting power suggests that for
this substance /' is relatively small. Only a rough estimate
of it for mercury can be made. It is a very large quantity
however — possibly as great as 30,000 atmospheres/cm.2 —
which is thus in agreement with the considerations put
forward. A complete experimental study of the p, v, T
relations of liquid mercury and vapour should be of con
siderable importance from the standpoint of equations of
state. Further, attention may be drawn to the fact that the
conductivity of mercury (and other metals) increases as the
temperature falls, which possibly indicates a decrease in the
value of /', and consequently an increase in the value of K,
and therefore of the cohesion. That cohesion increases as
temperature falls is well known. As regards the intercon
nexion between dielectric capacity and the continuity of state,
reference may be made to the measurements of Fversheim
(Ann. der Physih, viii. p. 539, 1902; ibid. xiii. p. 503, 1904),
who observed (in the case of the liquids examined by him,
notably in the case of ether) that the dielectric constant of
the liquid diminished regularly as the temperature was
raised, there being a linear relation between the two until
the critical region was reached, at which the K decreased
abruptly within small temperature limits and then con
tinued to vary only slightly, i. e. it diminished linearly as
the temperature of the now gaseous system was still further
raised. (So sharp indeed is the change in K at the critical
temperature that the latter could be obtained from the KT
curve with very considerable accuracy.)
Numerical values for the permeability //, of a number of
liquids may be obtained from Pascal's* determinations of
the susceptibility k by employing the relation yu,= lf 47r£,
Pascal's values being given in electromagnetic units. As a
matter of fact, Pascal does not give the actual susceptibility
itself, but a quantity which he calls " the molecular suscepti
bility," i. e. the product of the molecular weight into the
* Pascal, Bull. Soc. Chim. France, [4] v. pp. 1060, 1110 (1909) ; ibid.
yii. pp. 17, 45 (1910).
Internal Pressure of a Liquid to its Dielectric Capacity. 1 1 1
" specific susceptibility," which is in turn the susceptibility
k divided by the density of the substance. I have carried
out the recalculation of jjl for a number o£ substances from
the data given by Pascal. If the expression it — — j is near
the truth, one may expect that for substances the molecular
volumes of which are not very different, a large value for it
should accompany a small value for /j,, and vice versa. The
following table contains the values of //, and it (the latter
being the minimal values already obtained).
Substance.
Molecular volume
atO° C.
ft.
ir (atmospheres
per cm.2).
1096 c.c.
1032
9929
1048
1005
108'7
0 9,89224
09,90124
09,91240
09,91852
09,92842
09,93628
1512
1429
1393
1144
Ethyl ether
Methyl propionate ...
653
688
The substances are arranged in order of ascending values
of fju. It is evident that the smaller the value for fju the
laroer the value for 7r, a result which is in agreement with
theory and is the complementary relation to that of Obach
and Walden. (The last two substances mentioned stand
indeed in inverted order, but the value of tt is practically
the same for both, and slight experimental error would have
large influence upon their relative positions.)
It will be observed at the same time that //, for these (and
other) substances is very nearly unity, so much so that as
far as numerical values for — 2 go, the expression is identical
with that of van der Waals, and since van der Waals' ': a ,s
is generally considered to vary with T and V, one may
anticipate a similar result in the above expression also.
Hence, although the foregoing considerations point to the
existence of electromagnetic attraction between molecules,
attraction of this nature does not completely account for the
phenomenon of cohesion, and in addition to the term
which appears as a correction term in an equation of state
of the van der Waals* type, another term or terms require to
be introduced as well — if we are to retain this type of
equation at all. To proceed in this direction it is necessary
to consider the magnetic properties of liquids from the
electron standpoint.
fir
112 Prof. W. C. McC. Lewis on the Relation of the
In the first place all ordinary liquids and vapours (with
the exception o£ oxygen and nitric oxide) are diamagnetic.
The susceptibility k for diamagnetic substances — bismuth
excepted — is taken to be independent of temperature (N.
Campbell, 'Modern Electrical Theory,' p. 118, 2nd ed.).
Pascal, I. c. states, however, as an experimental result*,
that the product kV, where V is the specific volume of the
diamagnetic substance considered, is constant, independent
of the temperature and of the physical state. Denoting this
constant by rft it follows that =— = ~s and since r0 is
a negative term ^r* is really positive, i. e. fi approximates
to unity as the volume increases.
As regards the temperature coefficient of the internal
pressure it. since it— —  it follows that
"dir _ a ~d/ji 2a ~dv
dT"~/^2§T~/u;3§T'
and since cV ^ / in d^ 2a ^v
55=0 (nearly), 5^=,^
Hence 1 Btt 2 ~dv 1 ^v
— ^tt = vT'=~a? where a =  == ,
7T Bl V qL V C)T
the coefficient of expansion of the liquid. Of course the
same result is obtained from van der Waals' original ex
pression. In connexion with the above relation, attention
may be called to the fact that Davies (Phil. Mag. [6] xxiv.
p. 421, 1912) finds just half the above value, which in turn
leads to the relation 7r=or7r = — . Davies employed
v fiv r J ' '
the general van der Waals* type of equation, viz.
(p + 7r)(v — b) =RT, his considerations being based mainly
upon the CailletetMathias law of the rectilinear diameter.
Davies is therefore dealing with a liquid Under the pressure
of its own saturated vapour, but this of course does not
account for the difference in the two values.
Returning to Pascal's relation, viz. kv = r0, it follows
that — 7^rnr = ~ =^7ii=«, a relation which could be tested
k ol v oi
* I am unable to judge the degree of accuracy of this remarkable
relation h V = constant, as Pascal simply states it in the above papers as a ,
fact without giving further details.
Internal Pressure of a Liquid to its Dielectric Capacity. 113
experimentally. Further, since fjL = l + 4;7rk, it follows that
oc= =£. The term ^777 can therefore not be zero,
1 — fi^l 01
provided Pascal's relation be true ; its value is, however,
extremely small. (It may be easily shown that if we no
longer neglect the ^r: term in the expression for ^7^, and
ascribe the discrepancy — between Davies' expression and
that obtained above for  ^rp, — 'to the term thus introduced,
7T Ol
we are led to the impossible result that a is a negative
quantity. The discrepancy is therefore not due to the /j,
term.)
Pascal's relation is of some importance from the stand
point of the electron theory. The susceptibility k of a
diamagnetic substance is given by the expression (Campbell,
loc. cit. p. 123)
n s^v
k= — 7 2 (electromagnetic units) ;
where n is the number of revolving electrons per unit
volume, e the charge, and m0 the mass of an electron, r the
mean radius of the orbit of the electron (presumably the
vaiency electrons of Stark, which give rise to the electro
magnetic field, and are the source of molecular attraction).
Pascal's relation may be stated, therefore, in the form
nVeV
4y??oC2 =a constant (r0),
or nVr2= J>—  =a constant.
But nV represents the number of revolving electrons
present in unit mass of the substance, and if we regard this
as constant (independent of T and V but naturally dependent
upon the chemical nature of the substance) it follows that r,
the mean radius of the orbit of the revolving electron, is
constant for any given substance. Mow the recent work of
Bohr (Phil. Mag. July, Sept., Nov. 1913) and Conway (Phil.
Mag. Dec. 1913) has shown, in agreement with a suggestion
first made by Nicholson, that the angular momentum of a
revolving electron in an atom is a universal constant, either
or r) where h is the Planck constant. If the electron is
Phil. Mag. S. 6. Vol. 26. No. 1G3. July 1914. I
111 Prof. W. C. McC. Lewis on the Relation of the
not revolving with a speed comparable with that of light, its
mass may be regarded as constant, and the constancy of
angular momentum leads to the relation that the frequency
of revolution x (orbit)2 is a constant independent of the
nature of the substance as well as of T and V. Now for
a given substance Pascal's relation leads to the conclusion
that the orbit r is a constant independent of T and V.
Hence it would follow that for any given substance the
frequency of revolution of the electron is constant inde
pendent of T and V. This refers in the first instance to
the steady revolution of an electron round the some orbit,
no radiant energy being emitted or absorbed during the time
considered. According to Bohr, however, there is a very
close connexion between this frequency of revolution and
the vibration frequency v of the light which the given sub
stance absorbs or emits. Bohr in fact assumes that the
frequency of emission is just half the rate of revolution.
Whether this simple relation exists or not, it is fairly certain
that constancy in the rate of revolution in any of the available
orbits will also lead to constancy in the frequency of the light
absorbed or emitted by the substance due to its revolving
electrons. That is, if Pascal's and Bohr's relations be simul
taneously true, one would expect the vibration frequency —
or frequencies — in the absorption or emission spectrum of a
given substance to be independent of temperature and pres
sure. Experiment shows that this is approximately but by
no means strictly true. The absorption band of a given
substance in the liquid state is nearly but not quite identical
in position with the band exhibited by the vaporized sub
stance. Thus the head of the absorption band for toluene
vapouris\ = 2610 (A.U.), for liquid toluene, \ = 2690 (A.U.).
For paraxylene vapour, A, = 272S (A.U.) for liquid,
X = 2750 (A.U.); the values are accurate to 57 A. units*.
If we take Pascal's relation as true, and assume at the same
time that nY is independent of T and P, then the variations
in X from liquid to vapour indicate that Bohr's relation
regarding constancy of angular momentum cannot be quite
true. Conversely, taking the X values to be approximately
independent of the physical state and assuming the con
stancy of angular momentum, we are led to a theoretical
basis for Pascal's relation. Actually, however, it would
seem that neither the constancy of A nor the constancy of
the angular momentum is to be regarded as strictly true.
* It has been shown also that emission spectra are only slightly
affected by pressure. 1 am indebted to Prof. K 0. C. Baly, J\B,S., for
the above information respecting the numerical values.
Internal Pressure of a Liquid to its Dielectric Capacity. 115
Probably the most correct view to take is that the orbit
(radius r) in the case of a given substance is constant, i. e.
is independent of temperature, pressure, and change of state,
whilst the rate o£ revolution of the valency electrons is not
thus constant, that is the angular momentum is not constant.
The possible variation in the angular momentum with tem
perature and pressure should, however, be directly propor
tional to the variation of the observed absorption or emission
frequencies with temperature and pressure.
Returning now to the question of electromagnetic mole
cular attractions due to revolving valency electrons of con
stant orbit, since such attractions are exhibited by diamagnetic
substances it is necessary to assume that each molecule has
at least two valency electrons revolving in opposite directions.
There will thus be a tendency for the molecules to arrange
themselves into chains in which the electron revolving in a
given direction in one molecule approaches as closely as
possible to the electron undergoing similar motion in another.
Liquids may therefore not be so " structureless ,J as js usually
assumed*, and in this possibly is to be found the source of
optical activity exhibited by certain liquids. At the same
time, the kinetic energy of the molecules tends to break up
any regularity of arrangement, thereby diminishing; tho
molecular attraction if this be electromagnetic. In agree
ment with this it is known that the cohesion diminishes as
temperature rises. This effect of temperature in modifying
the orientation of the molecules introduces virtually a factor
dependent upon temperature into the attraction expression,
so that the attraction is apparently less than that given by
the inverse fourth power law, —% . In agreement with this,
mention may be made of the empirical attraction term
—3 suggested by Dieterici (Wied. Ann. lxix. p. 685 (1899) ;
Drude's Ann. v. p. 51 (1901)) in his first equation :
This term could be interpreted as indicating that the
attraction between molecules varied effectively as the inverse
cube of the distance. The above equation yields the relations
vc=4by and RTc/pcrc = 3*75, being thus in much better agree
ment with experiment than van der Waals' equation in
respect of these quantities.
* See a paper by Cotton and Mouton, Journ. de Physique, [5] i. p. 40
(1911).
I 2
116 On the Internal Pressure of a Liquid.
It has been shown, however, that between the critical
point and higher pressures the equation does not apply so
well as that of van der Waals. At very low temperatures
presumably the inverse fourth power law would be more
closely obeyed, but this cannot be tested directly down to
very low temperatures owing to solidification of the liquid —
unless solidification or crystallization is itself occasioned by
the molecules arranging themselves under the inverse fourth
power attraction into definite positions which give rise to
crystalline structure.
Summary.
1. The minimal values of the internal pressures of a large
number of liquids at 0° C. have been calculated employing
Young's data.
2. The ObachWalden relation regarding the approximate
direct proportionality between the internal pressure and the
dielectric constant has been shown to follow from the hypo
thesis that molecular attraction is electromagnetic, not electro
static in nature.
3. As a corollary to (2) it has been shown that the sub
stances with the larger internal pressures possess corre
spondingly smaller values for the permeability.
4. Pascal's relation, viz. that the product of the suscepti
bility into the specific volume is a constant for a given
substance, is shown to involve constancy of orbit for the
revolving electrons, and this taken in conjunction with Bohr's
assumption of constant angular momentum leads to the con
clusion that the vibration frequency of the absorption bands
in the visible and ultraviolet should be the same for both
liquid and vapour, a conclusion which is approximately but
not strictly in accordance with experiment.
5. The effect of temperature in partially destroying the
orientation of the molecules appears to alter the inverse
fourth power law to the inverse cube as expressed in Dieterici's
equation
(i>+^)0&)=RT.
In conclusion I wish to express my indebtedness to Mr.
James Rice, M.A., Lecturer in Physics in this University,
for his helpful criticism and suggestions in regard to this
paper.
The Muspratt Laboratory of Physical and Electrochemistry.
The University of Liverpool, February 1914.
[ 117 ]
XIV. On the Spectra given by Carbon and some of its
Compounds ; and, in particular, the " Swan " Spectrum.
By W. Marshall Watts, D.Sc*
[Plate II.]
ALTHOUGH more than a century has passed since
Wollaston f first observed the spectrum given by
the base o£ a candleflame, now known as the " Swan "
spectrum, no complete agreement as to its origin has yet
been reached. Swan J, who observed it in 1857, attributed
the spectrum to a hydrocarbon. He obtained the spectrum
only by combustion of hydrocarbons. Van der Willigen§
found that the sparkdischarge between carbon poles gave
the same spectrum as burning olefiant gas, and ascribes the
spectrum to carbon. Attfield obtained the spectrum not
only by the combustion of hydrocarbons, but also, most
brilliantly, from the flame of dry cyanogen in dry oxygen,
and also by the electric discharge in dry cyanogen, carbonic
oxide, and carbon disulphide vapour at atmospheric pressure.
Since these substances have only carbon in common, " unless
the experiments are vitiated by impurities, they prove un
doubtedly that this spectrum is due to the element carbon "1".
Dibbits** arrived at the same conclusion as Attfield. In
answer to the objection raised that carbon could not exist
as vapour in the flame of a candle or of a Bunsen burner,
Dibbits argues that carbon is combined with hydrogen
before the combustion, and after the combustion that carbon
is combined with oxygen, and during the combustion it may
have been in an uncombined condition : a flame of carbonic
oxide does not show the same spectrum because the carbon
is already combined with oxygen. In the ease of cyanogen
the carbon is at first combined with nitrogen, and after the
combustion it is combined with oxygen, so that the same
explanation appliesft
* Communicated by the Author.
+ Wollaston, Phil. Trans. 1802, p. 365.
X Swan, Phil. Trans. Edinb. xxi. p. 411 (1857).
§ Van der Willigen, Pogg. Ann. cvii. p. 473 (1859).
 Attfield, Phil. Trans, clii. p. 221 (1862) : Phil. Mag. xlix. p. 106
(1875).
1J Schuster, B. A. Keport, 1880.
** Dibbits, Pogg. Ann. cxxii. p. 497 (1864).
tt Note. In the case of cyanogen burning in oxygen the temperature
probably reaches the volatilizationpoint of carbon, or that of the electric
arc, viz. 3500° to 3700° C. " The temperature of individual molecules in
the respective flames of cyanogen and acetylene may reach a temperature
of from six to seven thousand degrees." (Liveingand Dewar, Proc. Kuy.
Soc. No. 223, 1882.)
118 Dr. W. Marshall Watts on the Spectra
Morren * undertook experiments to prove that Attfield
was wrong, but became convinced that he was right. He
obtained the spectrum bj burning cyanogen in oxygen, and
also by taking the spark in cyanogen or acetylene at atmo
spheric pressure.
Pliicker and Hittorf f, Huggins J, Wiillner§, and Salet 
all arrive at the same conclusion as Attfield. Schuster ^[ in
his Report o£ the Committee of the British Association
" On the Present State of our Knowledge of Spectrum
Analysis," wrote in 1880: — " On the whole it may be said that
from the publication of Attfield's paper until the year 1875
every spectroscopist, whether he was a chemist or a physicist,
who had set to work to decide the question, came to the
conclusion that the candle spectrum was a true spectrum
of carbon, and the question appeared to be settled. In the
year 1875, after Angstrom's death, Thalen published a
paper ** in which he describes some experiments made
jointly with Angstrom. In consequence of these expe
riments the authors expressed the opinion that the candle
spectrum was due to a hydrocarbon. The experiments
which they gave in support of their view were made by
taking the spark of carbon electrodes in various gases, and
examining the spectra of the ' aureole ' or ' glory ' as it
might be called. If the spark is taken in oxygen, the
undoubted spectrum of carbonic oxide appears; in hydrogen
the candlespectrum is seen; and in nitrogen some blue and
violet bands are added to the candlespectrum which appear
to be duo to a compound of carbon and nitrogen. As it is
known thato acetylene is formed when the spark is taken in
hydrogen, Angstrom and Thalen conclude that the spectrum
seen in the * glory ' is due to acetylene."
The theory of Angstrom and Thalen, that the " Swan "
spectrum is due to a hydrocarbon, was adopted and main
tained by Liveing and Dewarf. They examined the spectra
seen in the electric arc between carbon poles in air, hydrogen,
nitrogen, chlorine, carbonic acid, carbonic oxide, nitric oxide,
and ammonia, and found that the green and blue bands of
the " Swan " spectrum, well seen in hydrogen, less strong in
* Morren, Ann. Chim. Phys. iv. p. 305 (1865).
t Pliicker and Hittorf, Phil. Trans, civ. p. 1 (1865).
I Huogms, Phil. Trans, clviii. p. 558 (1868).
§ Wiillner, Po#£. Ann. cxliv. p. 481 (1872).
II Salet, Ann. Chim. Phys. xxviii. p. 60 (1873).
H Schuster, B. A. Rep. 1880.
** Angstrom and Thalen, JSova acta Reg. Soc be. Upsal. ix. (3) (1875).
tt Liveing and Dewar, Proc. Boy. Soc.'xxx. pp. 152, 494 (1880).
given by Carbon and some of its Compounds. 119
nitrogen or chlorine, were present in the arc, whatever the
atmosphere. Nevertheless, putting aside the natural con
clusion to be drawn from these experiments, they attribute the
spectrum to hydrocarbon, the hydrogen being supposed to
come from impurities. They also examined the spark in
various gases with every precaution to exclude impurities —
in carbon tetrachloride and trichloride, in naphthaline and
between carbon poles in nitrogen. " In all the^e expe
riments the bands which Angstrom and Thalen ascribe to
hydrocarbons were always more or less plainly seen"*.
Again, Liveing and Dewar put aside the obvious conclusion,
and attribute their results to the presence of hydrogen as an
impurity.
Later t, as the result o£ further experiments, Liveing and
Dewar abandon the view that the " Swan " spectrum depends
upon the presence of hydrogen. In these further expe
riments they found that " the spark between electrodes near
together in wide tubes filled with saturated vapour of carbon
disulphide or carbon tetrachloride dried with phosphoric
anhydride and deprived as completely as possible of air by
pumping or boiling out, shows the spectrum of the flame of
hydrocarbons brightly. The flame of cyanogen from liquid
cyanogen which had remained in contact with phosphoric
anhydride or sulphuric acid gave the green line of the Swan
spectrum together with the cyanogen bands; but when oxygen
was used to raise the temperature of the flame all the hydro
carbonflame sets appeared with marked brilliancy."
In another experiment the spark was observed in carbonic
oxide at various pressures. The carbonic oxide was prepared
by heating potassium oxalate and lime, and was dried over
phosphoric anhydride. At atmospheric pressure the spark
(without condenser) showed the spectrum of the hydrocarbon
flame. On exhausting, the spectrum of carbonic oxide makes
its appearance, superposed on the former ; and as the ex
haustion proceeds it increases in brilliance until it overpowers,
and at last entirely supersedes the flamespectrum. If the
pressure of the gas is increased above the atmospheric, the
hydrocarbonflame spectrum grows brighter. [The line
spectrum of carbon also appears at high pressures : the
effect of increasing pressure being thus similar to that of
introducing a condenser.] On letting down the pressure
the same phenomena occur in the reverse order.
* Dewar, Proc. Roy. Inst. June 10th, 1881.
+ Lireinp and Dewar, Proc. Boy. Roc. No. '22S (1882), xxxiv. pp. 123,
41°.
120 Dr. W. Marshall Watts on the Spectra
Kayser, in the fifth volume of his great work HandhucJi
der Spectroscopie (1910), after describing the experiments of
Angstrom and Thalen, says: "I believe that the authority
which, with perfect justice, was at that time attributed to
the views of Angstrom upon spectroscopic matters is the
reason that the Swan spectrum is by many still spoken of as
the spectrum of hydrocarbons or of acetylene ; although, in
my opinion, this conclusion has been proved to be untenable."
Other experimenters who expressed the opinion that the
" Swan " spectrum must be attributed to carbon were Secchi*,
Lecoq de Boisbaudran f, Ciamician J, Deslandres §, and
Eder  . Eder H photographed the spectrum of burning
hydrocarbons from the yellow into the ultraviolet as far as
2449: the five groups of bands in the red, yellow, green, blue,
and violet (the " Swan " spectrum) he assigns to carbon, and
the very strong bands in the ultraviolet he assigns to water
vapour.
In a paper on " The Spectra of Carbon Compounds " in
the Philosophical Magazine for 1901, Prof. Smithells **
advances a new view of the origin of the " Swan " spectrum
and of the spectrum of carbon monoxide or dioxide in a
Geissler tube, attributing the first to carbon monoxide and
the second to carbon dioxide. Prof. Smithells's defence of
his position is based, for the most part, upon a study of the
phenomena of flames. He seeks to set aside the natural
conclusion to be drawn from the repeated observations of the
•'Swan" spectrum in carbon compounds not containing
oxygen by the suggestion that it is impossible to free these
substances from water, and other impurities containing
oxygen, and points to the difficulty of removing films of air
or moisture from glass, the occlusion of gases by electrodes,
and the fact that glass itself contains oxygen as forming well
recognized difficulties, quite apart from the purely chemical
difficulties of obtaining pure materials.
I have now to describe experiments in which the " Swan "
spectrum was observed in the absence of oxygen as far as it
* Secchi, C. P. lxxvii. p. 173 (1873).
+ Lecoq de Boisbaudran, Spectres lumineux, p. 43 (1874).
% Ciamician, Szb. Wien, lxxii. (ii.) p. 425 (1880).
§ Deslandres, C. R. cvi. p. 842 (1888) ; Ann. Chim. Phys. xv. p. 5.
 Eder, Beitriiye zur Photochemie, 1890.
51 Eder's photograph of the spectrum of the Bunsen flame obtained
"by 36 hours' exposure on an Erythrosin plate shows Cy, C53.54.55.56.57],
? e3(2) O.F.= 2213865 +14355m2[m53.54(55)56.57],
? e4(3) O.F.= 2139613 +141293m2[m53.54(55)56.57].
There is little doubt that the older measurements of the
red lines are in error, and I have not found it possible to
combine Deslandres' five formulae into one. The following
is a more satisfactory formula, for which I am indebted to
my friend, Rev. P. H. Jones, M.A., R.N.
O.F.(myacMo) = 2407015196(i9 + 00309)2 + ll6422(m + 83429L)2
where ^=45, 44, 43, and m = 63, 64, 65, 66, 67, 68, 69.
The following table shows the agreement between the
calculated values and those observed.
Observed.
Calculated Values (in Air).
Observed.
Angstrom &
Tbalen.
Deslandres.
Jones.
Fowler.
61882
(53)618806
(45,67)619102
61911
61199
611798
612180
61218
60573
6056 01
605998
60599
60018
6996*19
600518
6005 1
5954 5
594903
5967 02
59582
Kayser & Kunge.
Komp. (I. A.)
5635 43
(54) 563540
(45, 68) 5635715
5635 504
558550
558548
5565500
5585493
554086
554095
5541181
5540890
54^917
5502364
5501914
546655
6468906
5572670
Leinen.
516530
(55) 516527
(45, 69) 5165592
5165473
512936
513042
5129578
5129579
509901
5098134
507085
5071254
Hindrichs.
473718
(56) 473728
(44, 69) 4736131
4737147
471531
471586
4715523
4715453
469757
469809
4697 632
4697001
468494
468494
4683415
4083 491
4671791
4672979
Eder & Valeuta.
438193
(57) 438066
(43,69)438165
438091
437131
437090
437085
437114
436501
436427
436295
436427
126 Dr. W. Marshall Watts on the Spectra
It may be concluded, I think, that the three lines farm a
part o£ the " Swan " spectrum; but the shaded band /upon
which they are superposed probably does not belong to the
same spectrum, though always seen in the combustion of
hydrocarbons, in the discharge in liquids containing car on
and hvdrogen, or carbon, hydrogen, and oxygen, and occa
sionally in vacuumtubes. There is, I believe, no evidence
that it is due to carbon alone ; but there is evidence that
hydrogen is necessary for its production, and therefore it
may, provisionally at least, be termed the hydrocarbon band.
The group/ is seen in Vogel's photograph of the spectrum
of the cyanogen flame {Berliner Berichte, xxi. 1888).
In my early observations of the direct discharge in carbon
monoxide at atmospheric pressure which shows the " Swan "
spectrum, I sometimes observed /and sometimes the cyanogen
bands £ and 6 instead of/, a slight alteration of the electrical
conditions producing the change. It is now clear that the
gas used in these experiments contained both watervapour
and nitrogen as impurities. The " Swan " spectrum in
carbonic oxide usually shows /, unless the gas is very
thoroughly dried ; but that it can be so completely dried that
/ disappears is shown by the photograph of the spark in
the gas at atmospheric pressure (PL II. fig. 6), for which
I am indebted to my friend Mr. E. E. Brooks. The two
outside strips are the spectrum of the Bunsen flame, and the
central strip that of dry carbonic oxide together with the iron
arc. It can be seen that /has disappeared, but that " the
three" lines 4381, 4371, and 4364 remain.
Fig. 2 shows Mr. Brooks's spectrum of magnesium in a
coalgas vacuum*, and in fig. 3 portions of the Bunsenflame
spectrum are fitted on to this to show the presence o£ the
groups / and G6 in the electric spectrum.
Figs. 4 and 5 are photographs for which I am indebted to
my friend Mr. 0. W. Raftety, the first of the /group from
a Meeker burner, and the second of a hydrogen vacuumtube
in which the group /is seen amongst the lines of the secondary
hydrogen spectrum.
We have thus evidence of the production of the /group in
carbonic oxide with a trace of hydrogen, and in hydrogen with
a trace of carbonic oxide.
The u Swan" spectrum predominates at the negative pole
both in vacuumtubes and in the arc, whereas the spectrum of
carbonic oxide is seen most brightly at the positive pole,
In 1902 I observed in a vacuumtube containing coalgas
* See P. R S. Ixxx. p. 218 (1968), and Kayser, Hdb. Sped. v. p. 232.
given by Carbon and some of its Compounds. 127
at a pressure of 100 mm. the " Swan " spectrum and the
carbonicoxide spectrum simultaneously. As the pressure
was decreased the " Swan" disappeared and the carbonic
oxide spectrum remained ; but in the intermediate condition
the two ends of the capillary were of different colours, — the
negative end decidedly blue showed the " Swan" brightest,
whereas at the positive end the carbonicoxide spectrum was
the brightest.
Liveing and Dewar*, who observed the " Swan " spectrum
in an incandescent carbon lamp at the moment at which the
filament gave way, also observed a sort of flame giving the
carbonicoxide spectrum at the point where the filament
joined the positive wire.
Baldwin! (in " a photographic study of arcspectra ")
says, " The central arc is violet, outside this a blue sheath
strongest at the negative carbon and surrounded by an outer
yellow sheath shading into orange at the outside. The flame
which starts at the negative carbon extends some distance
up the positive carbon."
HagenbachJ, describing the arc between copper poles in
an atmosphere of carbon dioxide, says: — " The positive elec
trode was always hotter than the negative electrode ; with
decreasing pressure this difference became more marked, and
at 10 to 20 mm. the negative electrode never became redhot.
The bands of carbon oxide 66220, 6078"0, 5607*5, 5197*0,
48365, 45090, 4394'0, 4209*0, and 41300 were seen espe
cially at the positive pole. The * Swan ' spectrum 6188,
5634, 5164, 4736, 4375 to 4325, 4315, and 3890 to 3873
became more intense and sharper with decreasing pressure,
but was absent from the positive pole."
A study of the results obtained by Sir J. J. Thomson §
leads to the conclusion that the 'Swan" spectrum is due to
the negativelycharged carbon atom, and not to either carbon
monoxide or carbon dioxide.
The Swan spectrum is given by carbonic oxide at atmo
spheric pressure, but not by carbon dioxide. In carbon
monoxide as the pressure is reduced  the Swan dies away,
and at low pressures is completely replaced by the second
spectrum. This indicates that carbon monoxide contains
something not present in carbon dioxide, but that this
something gets smaller in amount as the pressure is reduced.
* Liveing and Dewar, Proc. Rot. Soc. xxxiii. p. 403 (1882).
t Baldwin, Phvs. Rev. iii. p. 370 (1895).
t Hageubach, Pkys. Zs. x. p. 649 (1910).
§ Thomson, 'Nature,' lxxxvi. p. 460 (1911).
ij Deslandres, C. 11. cvi. p. 842 (1888) ; Ann. Chim. Phys. xv. p. 72.
128
Mr. A. Ferguson on the Shape of the
Now Thomson finds that both carbon monoxide and carbon
dioxide contain C+ + , 0+ + , Cf , 0 + , C02 + , and C0 + ,
whereas carbon monoxide alone contains C — .
Further, Thomson shows that such negatively charged
atoms ought to be, and are, less numerous wheu the discharge
has difficulty in passing, i. e. at low pressures.
The spectrum given by carbon oxide at low pressures is
also generally seen in vacuumtubes enclosing hydrocarbons ;
but there is evidence that oxygen must be present. Whether
this spectrum is to be attributed to C0+ or C02+ there is,
at present, nothing to show, so far as I am aware.
XV. On the Shape of the Capillary Surface inside a Tube of
Small Radius, with other Allied Problems. By Allan
Ferguson, B.Sc> (Lond.), AssistantLecturer in Physics in
the University College of North Wales, Bangor*.
§ 1. rpHE present paper is devoted to the formation of a
JL closer approximation than usual to the outline of
the surface of a liquid contained in a vertical tube of small
bore, and to the application of the
formula developed to certain cases
of practical interest.
With axes as shown in fig. 1,
and symbols having the significance
there indicated, the differential equa
tion to the surface of the liquid will x'
be
Fig. 1.
gp(y+h)
HRi + lJ'
where Rx and R2 are the principal
radii of curvature at the point P. "
Substituting their values and putting
a25%, this becomes
a2x^+a2p(l+p2)=x{y + h){l+p2)l
Putting p= tan$? and s= sin<£, we readily find
■*"■*
dz z
dx~^~ x~
a2
(i.)
* Communicated by Prof. E. Taylor Jones.
Capillary Surface inside a Tube of Small Radius. 12^
Remembering that y is to be assumed small in comparison
with h, we obtain, after integrating (i.) twice, neglecting y
in comparison with h,
y = k \Zk*W*, (ii.)
the equation to a circle of radius h (k= 7). Putting this
approximate value of y in (i.) and integrating once, we have
Ji2 + 2a2 (4a4fe2)t 8a4 ... .
Z~ 2a2h X+ %a?hza> "dhV ' ' ^
or, putting lix — 2a2 sin 6,
. n 2a2 r3 sin2 d + 2 cos3 0 2n ,. N
,= sm0+_^ __ J. . (lv.)
O 2
Since 0= — — , and ^U is a small quantity, (iv.) may be
\Jl\p2 ,l
integrated in series, giving finally
_ 2a2 \/ia±K2x2 8a6 1 4,al
8a\ 2a2 + s/±a"li2x2 , .
+ 37?log Ia~2 ' ' W
as a second approximation to the Cartesian equation of the
meridional curve of the capillary surface inside a small
capillary tube.
§ 2. One of the constants of equation (v.) is A, the height
to which the liquid rises in the given capillary tube ; it is
convenient to obtain an expression for h in terms of the
radius of the tube, and the contactangle i of the liquid with
the tube.
In equation (iii.) z= sin c/>, and putting sin cf>= cos i when
x = r, (iii.) becomes
rh r (4a3 AV)* _8a*
~2a2 + A+ 3a2AV "~ 3/iV * (v1^
COS!
Hence, approximately
7 2a"
h= — cos 2,
r
which value, substituted in the small terms of (vi.), gives
more accurately
2a2 . . f . , 2 /sin2 i . cos e — 1\ I ....
*=— 00.1T.eo.^l+gl ^ )j. (vn.)
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. K
130 Mr. A. Ferguson on the Shape of the
For a liquid o£ zero contactangle (vii.) gives
2d2 r
^= — — 7v (viii.)
r 6
Equation (viii.) gives, to the order considered, the correction
for a liquid of zero contactangle due to the departure from
perfect sphericity of the liquid surface inside a capillary
tube. In most treatises dealing with the determination of
capillary constants one is told that in the case of a liquid of
zero contactangle a correction may be made for the weight
of the liquid raised above the plane X'OX (see fig. 1) by
adding ~ to h, which, as equation (viii.) shows, is perfectly
correct. But this result is invariably obtained by treating
the meniscus as hemispherical, a process which hardly seems
logically legitimate, inasmuch as the correction is to be made
precisely because the liquid surface is not hemispherical, as
is shown by the premisses from which (viii.) is deduced. In
fact, the meniscus cannot be treated as hemispherical unless
r be negligibly small compared with h (see equation (ii.)),
in which case, of course, the correction itself becomes
negligible.
Further, if we calculate the correction by the " weight "
method for a liquid of finite contactangle — that is, by treating
the meniscus as a segment of a sphere — we easily obtain
, 2d2 . . ,*. . .v r . 2 — sini— sin2^ i
h= — cos^— r sece 1sini) \ 1 ^ =. l ,
r I 3 cos2 1 J
which is not at all in agreement with (vii.), but becomes
identical with (viii.) when i becomes zero.
§ 3. It is of interest to obtain for a liquid of zero contact
angle the value of the radius of curvature (R) at the vertex
0, and of 77, the value of y corresponding to x=r ; the
elementary theory which assumes the meniscus to be hemi
spherical gives, of course,
r) = r, R = r.
But from first principles R is given accurately by
2a2
and therefore, substituting the value of h from (viii.), we
have
R
(l + £) ..... (ix.)
Capillary Surface inside a Tube of Small Radius. 131
Also, substituting in (v.) tbe value o£ li given bj (viii.)
and putting x = r, we have
v = r(l
With a change of sign similar formulae may be applied to
the case of a small pendent drop such as is shown in fig. 2.
In which case we have
6a .
(X.)
K.
■) 55 •>> T> ^2 = AL + tJ~ ,
where II is the atmospheric pressure, and Rx and R2 are
the principal radii of curvature at Ox and 02 respectively..
Henc^, subtracting
or
From fig. 2 b we similarly obtain
and therefore 7 , 2a2
But, substituting the value of R2 from (xi.), (xv.) becomes
i j 2a2 2a2
/ti— h2 — 5— = — +
r2
So that, finally, 9„2_a , * r22 .<
Za =(/ii — h2)r2—± (xvi.)
But this equation is identical with (xiv.). Thus we see
Capillary Surface inside a Tube of Small Radius. 133
that equation (xiv.), obt lined on the assumption that the
liquid surface is hemispherical, is precisely the equation
which does, to the first order, correct for deviations from
sphericity, and already embodies within itself the fact that
the drop is spheroidal in outline. The later refinement of
treating the drop as spheroidal and arguing by the "weight"
method, merely introduces a term of higher order into
(xiv.) or (xvi.),
§ 5. It is worth noting that direct measurements of lix and
rY afford a feasible method for the determination of contact
angles. M. Sentis' equation (xiv.) gives a value for a2 which
is quite independent of the angle of contact. Also from
(vii.) we have
1 cos i rY . f 2 sin2 z cos £—1 1 , .. .
77 = f ,r^sec l i *+ 7 2~ — r • (xvl1)
Y\x i\ la2 I 6 cos2 i J 7
Substituting in (xiv. a) the values of — and rr from (xvii.)
ill K2
and (xi.) respectively, we have
hi cos i , i\ . f. 2 sin2 £ cos t— 1 1 1 i\
2a* r, '2a* I ' 3 cos2i J T rQ ' 6a2'
cosz=V
2a r2
Approximately ._ / 7^
COS 2 — ?'i I ^
f i 2 sin2 z cos 2 — 11 1 ?'2
1 1 + o • r — +  + 7T
I 6 cos"* 2 J r2 ba'
or .... (xviii.)
ryr2 r? J  2 sin2 i cos 2—11
— ^r — o2 1 1+ 9 2^ — r sec*
ba zuS \. 6 cos 2 J
.... (xix.)
._ (K _L_!1\
~7l\2a2 r2 6a3/'
which value of 2, substituted in the small term of (xix.),
gives a still closer approximation to cos i.
If 2 = 0, (xix.) becomes
6a2(r1+r3)=r]r2(3/i1— r±— r2), . . . (xx.)
and a convenient test of a zero contactangle consists in
determining whether the value of a2 calculated from (xx.)
(which depends on the assumption that the contactangle is
zero) is in agreement with the value calculated from (xiv.),
which is independent of this assumption. If the difference
between the two results lies outside the limits of experi
mental error, equation (xix.) can be used to calculate the
value of 2.
§ 6. Before proceeding farther, it may be well to empha
size the limits within which these approximations can safely
be applied. Two assumptions have been made in order to
134
Mr. A. Ferguson on the Shape of the
effect the integrations: (1) that the radius of the tube is small
compared with the height to which the liquid rises in the
tube, and (2) that the ratio of r to a is such that any power
higher than its square is negligible in comparison with unity.
These assumptions are not independent of each other, but in
any given case it is simplest to test whether the conditions
are separately fulfilled. Thus in the case of water, the
approximations give very exact results in the case of a tube
\ mm. in diameter, and may be applied with fair accuracy
to a tube 1 mm. in diameter (for which r//i = *016, and
r3/a3 = *004). This latter value represents about the limit of
application of the formulae in the case of water.
§ 7. We now proceed to apply the equations of §§ 14 to
the method for the determination of surfacetension usually
known as Jaeger's method *. The practice of the method
consists in measuring the maximum pressure required to
liberate an airbubble from the end of a vertical capillary
tube plunged below the surface of the liquid under examina
tion. The apparatus needed is represented diagrammatically
in fig. 3 (which, for clearness of reference, is drawn out of
all true proportion).
Fiff.3.
Boyles Bottle:
V '
1
k
r
i
?'
^
_
V
J
I
With symbols having the meaning shown, the equation of
equilibrium of the bubble at any stage in its formation will
be
or
where
1_ 1 _H+y
Rx + K2 ~ a2 ' " *
H
Pi
JftiA'^
(xxi.)
* For «i different treatment see Cantor, Wied. Ann. xlii. p. 422 (1892),.
and Feustel, Ann. d. Phys. xvi. p. 61 (1905).
Capillary Surface inside a Tube of Small Radius. 135
Thus, as is shown by (xxi.), all the formulae already obtained
may be employed, substituting H for h.
Putting
p — ^ lix — li/ = r) + H, . . . (xxii.)
we require the conditions under which p is a maximum.
du
Let tan (f> = ■— , and let fa be the value of <£ for which
x = r. Then tj and H are functions of fa, and the condition
for a maximum or minimum value of p is given by
£=°
dfa
But, from (vii.)
2a2 . f 2 cos2 0! sin 0! — 11
xi= — sincpi— rcosecd)! i l + 0. • 9 ,  f
r ^' r I 3 snr^ i
and remembering that
the required condition for a maximum value of p gives
IT
fa=  , and therefore
H *"? r
LJmax. — „ • » •
r o
Hence the radius of curvature at the vertex under maximum
conditions is given by
RssiafrBJ' • • • (xxiii}
and the corresponding value of 7) is
A/3. , ■ s
'=r6T
The equation of equilibrium of the bubble under the con
dition of maximum pressure is
g ipjiip'(h'+v)] =g,
and, substituting for K and rj from (xxiii.) and (xxiv.), and
reducing, we have
where B is a known quantity.
136 Mr. A. Ferguson on the Shape of the
Approximately a2 = B; and therefore, more exactly.
a2 = B + f^JL, .... (xxvi.)
12v/B'
a very convenient equation from which to calculate a2.
The elementary theory which assumes the bubble to be
hemispherical gives
«2=iWip'/l') (xxvii.)
An idea o£ the relative importance of these correcting
terms will best be obtained from the consideration of the
results of an experiment made upon ethyl alcohol, using a
capillary tube slightly less than a millimetre in diameter
(such a tube is about the widest that can be safely employed).
Using water in the manometer, the heights hi and 1i were
measured by means of a travelling microscope. The follow
ing figures were obtained: —
Temp. = 12°5 0.
]h= 5072 cm., h' = 5059 cm.
pi=l gm. per c.c, p'= *795 gm. per c.c.
r= 04595 cm.
Erom which equation (xxvii.) gives
a2= 03038 sq. cm., or T = 23*69 dynes per cm.
The first approximation a2 = B gives
a2 = 02964 sq. cm., or T=23*12 dynes per cm.
The final approximation embodied in equation (xxvi.) gives
a2 = 02972 sq. cm., or T = 23'18 dynes per cm.
The final figures are in good agreement with the result
obtained by the writer by a totally different method * — the
measurement of the mechanical pull on a sphere of large
radius "just touching the liquid surface. This gave T15 = 23'll
dynes per cm.
§8. The formulae of §§ 14 may be used also in discussing
certain aspects of the classical experiments of Ramsay and
Shields f on the molecular complexity of liquids. In these
* Phil. Mag. Nov. 1913, p. 925.
t Phil. Trans. A. 1893, p. 647.
Capillary Surface inside a Tube of Small Radius. 137
experiments surfacetensions and their temperaturecoefficients
were measured by the capillaryrise method. It is of primary
importance to be perfectly sure that, when a liquid is in
contact with its vapour (as was the case in these experiments)
the contactangle with glass is zero. The experimental
test made to verify this condition consisted in forming a
bubble of the vapour of the liquid inside a capillary tube
(*65 mm. in diameter) containing the liquid under examina
tion. The two menisci so formed — which were, of course,
concave to each other — were then caused to approach as
nearly as possible. It was argued that, were the contact
angle zero, the capillary surface so formed would be spherical;
if the contactangle were finite and acute, the capillary
surface would be lenticular. Direct measurement failed to
show any deviation from the spherical form, and it was
therefore assumed that the contactangle was zero.
But even assuming that the capillary surface in such a
tube is a segment of a sphere — which is not the case — it
seems as if the method would fail to show the existence
of contactangles of a few degree^. Thus if we assume
a contactangle of 8° for which cos i is *99, this would
involve an error of 1 per cent, in the value of the surface
tension, as determined by the capillaryrise method. The
radius of the capillary tube (BA in fig. 1) being '033 cm.,
the value of OB would be '029 cm., the difference between
the two values being 4hundredths of a millimetre — a suf
ficiently small quantity on which to base so wide a general
ization.
But, assuming that quantities of this order could be ac
curately estimated, the deviation from sphericity of the
capillary surface should have been appreciable. Thus, to
take the case of ethyl alcohol, for which the contactangle
with glass is almost certainly zero, the value of BA being
•033 cm., the value of OB would be, from equation (x.),
about *031 cm., the difference therefore being about
2 hundredths of a millimetre*.
Taking into account the difficulties of measurement, the
test cannot be said to establish very satisfactorily the existence
of zero contactangles, and serves further to emphasize the
fact that, in spite of its wide usage, the capillaryrise method
is not very trustworthy for the determination of capillary
constants.
* For a discussion of certain other points connected with these
measurements, see Feustel, /. c.
138 Capillary Surface inside a Tube of Small Radius.
§ 9. Whilst discussing the results of capillary tube ex
periments, attention may be drawn to a recent paper on the
surfacetension of liquid sulphur*, in which, by a curious
series of elementary mistakes, a number of otherwise careful
measurements have lost much of their value. After measuring
by direct methods the contactangle between sulphur and
glass, and obtaining values of about 60° at 125° C, 43° at
190° C, and 27° at 257° C, the author proceeds to measure
the height to which the liquid sulphur rises in a tube of
which the diameter is expressly stated to be '103 cm. The
mean of a considerable number of readings — at a temperature
not stated — for the capillary rise was *246 cm. From which,
using " the approximate formula
1" 2
we get
T
103 x 1861 x 980 x 246 x *5
1156 nearly."
A truly extraordinary value, due to a confusion of diameter
with radius, and a misplacement of the term cos 6X. If we
transfer the peccant cos 01 to its usual place in the denomi
nator and substitute for r± the value "052 cm., we obtain
T = 23*12 dynes per cm.
If, on the other hand, we assume that "diameter" is a
misprint for " radius," and that the radius of the tube is
really '103 cm., we have
T = 46*24 dynes per cm.
This latter value is probably correct, as it is in fair agree
ment with the number (59 dynes per cm. at 160° C.) obtained
by Zickendraht using Jaeger's method.
University College of North Wales, Bangor,
April 1914.
* Proc. Camb. Phil. Soc. xvi. p. 55 (1910).
[ 139 ]
XVI. Higli frequency Spectra and the Periodic Table.
By Prof. W. M. Hicks, F.R.S*
TI1HE decisive importance in questions affecting the periodic
JL table of Moseley's law governing the highfrequency
spectra of the elements must be generally acknowledged, and
any discrepancy therefore between it and the ordinarily
accepted view of the constitution of the table requires
careful consideration. Such a discrepancy appears in the
interval between Ce and Yb. These elements have their
position in the table definitely settled by their chemical pro
perties, and the generally accepted constitution of the table
gives 16 spaces to be filled between them. The atomic
numbers, however, as determined by Moseley give 58 for Ce
and 71 for Yb, leaving therefore only 12 spaces. Or to put
it in another way, if the 16 spaces are accepted as actually
existing, the atomic numbers place Yb in the 01 group, Ta
amongst the Alkalis, and W between Cd and Hg, thus up
setting all chemical similarities. It is clear that both cannot
be correct. One way of explaining the difference would be
to suppose that the L series splits up into two at some
element between Nd and Yb, and that the new formula
depends on (N — 3)2 in place of (N — 7)2. In connexion with
this it is interesting to note that there appears to be some
disturbance in Moseley's curve in this region. But a better
explanation would appear to be that Moseley's rule is abso
lute, and that as a fact there are only 12 spaces between Ce
and Yb. Some strong independent evidence in favour of
this view is found in a recent discussion by J. R. Rydberg
on the system of the elements f He identifies each element
by the integral number oivino its place in the table, regarding
this as the "independent variable/' functions of which must
give all the properties of the atom. For a reason given
below this number is two units above that chosen by Moseley
for the " atomic number."
Starting from the acknowledged facts as to the two short
series or. 8 elements each, followed by two long series of 16
each, he supposes the table to be built up of long series con
taining 4p2 elements. Thus the two short series He through
Ne to A make one long of 16 elements (p — 2). The two
recognized long series A through Kr to Xe make one long
one of 06 elements (p = 3). He then supposes to succeed a
* Communicated by tue0Author.
t Lunds Universitets Arsskrift, N. F. Afd. 2, Bd. ix. Nr. 18, or
Kongl. Fysiografiska Sallskapets Handlingar, N. F. Bd. xxiv. Nr. 18.
140 Prof. W. M. Hicks on High frequency
long series of 64 elements (jd = 4), Xe through Ra Em to ?,
the end containing unknown elements. The first half of
this, like the others, begins at a rare gas (Kr) and ends at
one (RaEm), and this contains 32 elements in place of the
accepted 36. On the other hand, there should be a series
corresponding to^> = l of 4 elements. These are H (1), two
unknowns, and He (4), and at the beginning stands the rare
gas. Electron. And he suggests Coronium and Nebuliumfor
the unknowns. If we apply Nicholson's theory of the spectral
lines of these, Coronium has one positive charge in excess of
Xebulium. The latter should therefore be associated with
X = 2 and the former with N = 3. For evidence, reference
must be made to the original paper, but as it is not every
where accessible, the above general statement has been given
to render the present note intelligible. The accompanying
table, given by Rydberg, will also make the arrangement
clear.
As the table shows, Rydberg eliminates the old spaces
between Ra and Os, Rh and Ir, the space after Cs, and that
between Xe and Ra Em. He allows as two spaces for un
known elements (1) that of the next atomic number to Nd
(in the Mn group) as does Moseley, and that between Ce
and Th, the last of which Moseley 's numbers directly con
tradict. The observed highfrequency lines for Yb make it
come two places behind Ta, placing it, as the chemists have
done, in the La group. There is, moreover, spectroscopic
evidence putting La and Yb in the same group. This makes
Lu provisionally occupy the space between Yb and Ta.
Rydberg again places Sa between Pd and Pt, Eu between
Ag and Au, Gd between Cd and Hg. Bat the spectrum of
Eu is of the triplet, whilst Ag and Au are of the doublet
type, and they cannot therefore belong to the same group.
I have also given evidence* from spectral considerations,
which though not perhaps conclusive points to the suppo
sition that Eu comes between Cd and Hg. As Moseley has
determined the atomic numbers of Sa, Eu, Gd to be con
secutive, this would place Sa between Ag and Au and Gd
between In and Tl. Although the spectrum of Gd has not
been thoroughly investigated, I find indications that it is of
a doublet type with separations intermediate between those
of In and Tl. So far3 therefore, as any spectroscopic evidence
goes at present, it tends to placing Eu between Cd and Hg
and Gd between Ag and Au. A slight modification of
Rydberg's table would bring it into agreement with this
* Trans. Roy. Soc. A. ccxii. p. 58 (1912;, also A. ccxiii. p. 328 (1913).
Spectra and the Periodic Table.
141
Q
55
s>
53
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1 i rli I
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f 1 I 1 I 1 I 1 * =
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1 l 1 i 1 I 1 1 1 s I 1
1 1 J 1 £ I ^ 1 * 1 ^ ! t
To ~j en >f> to h* 1
" 5? 2 tJ 5?  5
1 I 1 I 1 1 1 1 1 1 1 5
0
i
1
+.
+ 1
O to
8
j
+
CO
+ 1
Cn CO
s § 1 1 1 1
I I T I 1 I 1 1 1
LJ
2?
LJ
3
+
+ 1
I I 1 I 1 1
1 I 1 1 1 I 1 I 1
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+
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ro
UN
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to
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1
142 Mr. J. A. Tomkins on the
arrangement. Instead o£ the carve in the diagram turning
in group viii. it would turn in group vii., each element Eu
to Er would be moved one to the left, so that Er comes into
the CI group, Tin comes under Cs as a new alkali metal, the
unknown metal (Moseley's Tm II.) where Rydberg has
wrongly placed Yb, and Yb and Lu go each one place to
the right. Another way would be to suppose the curve to
extend to the place between Rh and Ir. This would make
Sa and Eu come into group viii. and Gd into Rydberg's
group ix. This would not be absolutely contradicted by the
spectroscopic evidence, as we should expect triplet series in
an even group and doublets in an odd one — which Eu and
Gd respectively satisfy whilst Sa is unknown. But so far
as the evidence goes at present, it would seem that the first
suggested change is to be preferred.
XVII. The SlipCurves of an Amsler JPlanimeter.
To the Editors of the Philosojihical Magazine,
Gentlemen,
IN a paper on " The SlipCurves of an Amsler Plani
meter," in the April number of the Philosophical
Magazine, Mr. A. 0. Allen obtains the equation of the slip
curves, and explains their use in establishing the theory of
the instrument.
A similar proof, involving the use of these curves, is given
in Williamson's ' Integral Calculus/ where it is attributed
to Prof. Ball.
But it may be pointed out that this method of proof does
not require the use of the slipcurves as might at first sight
appear.
In a proof due, 1 believe, to Sir George Greenhill, the
element PQRS (p. 647) consists of the two circular arcs
PQ, US, having the pole A as centre, and two other arcs
QR, PS, described by the tracer about the fixed joint H as
centre. In this case the records of the wheel along QR and
PS cancel.
Further, it is obvious that QR and PS may be any two
similar curves whatever and the same method applied for
the records of the wheel, which will, in general, partly
slip and partly roll, along QR and PS will cancel and
the total record and area described will be the same in all
cases.
Slip Curves of an Amsler Planimeter. 143
The theory of the instrument having been established by
any one of the various methods when the pole is outside
the area to be measured, it can be applied at once to the
case in which the pole is inside the area by a simple
method which I have described in the 'Mathematical Gazette'
(July 1911).
This may be briefly indicated by reference to the figure,
in which, for simplicity, the curve is shown wholly outside
the basecircle.
Draw any arbitrary curve PS from the curve to the base
circle. Starting at P, take the tracer round the boundary
PQ RP, in the positive (clockwise) direction, along PlS,
round the basecircle SU TS in the positive (this time, anti
clockwise) direction, and finally back along SP.
The record is clearly the area of: the portion of the curve
outside the basecircle. But this record is merely that which
would be obtained by taking the tracer round the boundary
PQ RP, for nothing is recorded round the basecircle and
the records along PS and SP cancel. Hence by suppressing
these motions and taking the tracer once round the boundary
in the usual way, the instrument gives the excess of the
area of the curve over that of the basecircle.
Similar reasoning can be applied when the curve lies
wholly inside or partly outside and partly inside the base
circle.
Yours faithfully,
Technical College, Bradford. J. A. ToMKINS.
[ U4 ]
XVIII. The Ordinals of the Elements and the Highfrequency
Spectra. By J. R. Rydberg, Professor of Physics at the
University of Lund*.
AFTER having made use of integers of a similar kind
already in the year 1885 f in trying to find out the
laws of the atomic weights, in .1896 J I expressly emphasized
the great importance, or rather the necessity, of introducing
ior the qualities of the elements a true independent variable
instead of the atomic weight, which no doubt is not a simple
quality but of a most complex nature. As such an inde
pendent variable I proposed the ordincds of the elements
supposing, after the simple rule for the atomic weights of
the first elements from He to CI, that He had the number 2
and CI the number 17. Finally, having found in 1913 that
the system of the elements did not consist of any periods in
an ordinary sense, but of quadratic groups of 4p2 elements §
(p being the number of the group), I did not doubt that the
new numbers (which in the beginning of the series from He
onwards were 2 units greater than the old ones) must be the
true ordinals of the elements, and therefore ventured to
propose that they should be used as rational designations
besides the ordinary, as for instance Al (15), Co (29), Ni (30),
Ag (49), La (59), Ta (75), Au (81), U (94).
Now for me it has been of the utmost interest to see that
Mr. Moseley, in his excellent researches on the highfrequency
spectra , has found a simple relation between some of the
lines of these spectra and the ordinals of the elements. But
as there is a certain difference between my ordinals and Mr.
Moseley's numbers, I have allowed myself to calculate his
series of lines more completely and in a somewhat different
way.
As we see from the publications quoted, Mr. Moseley in
the examined spectra has distinguished several kinds of lines,
and of these he has published the wavelengths of six sepa
rate series which, referring to Barkla and Moseley, we will,
for the present, designate as K«, K/3, La, L/3, L<£, and L7.
* Communicated by the Author.
t " Die Gesetze der Atomgewichtszahlen," Bihang till Sv. Vetensk.
Akad. Htmdl. xi. No. 13 (1886).
X " Studien liber die Atomgewichtszahlen," Zeitschr. fur anorg. Chemie,
xiv. p. 66 (1897).
0 § " Untersuchungen liber das System der Grundstoffe/' Ltinds Univ..
Arsskrift, N. F. Afd. 2, Bd. ix. Nr. 18 (1913).
II Phil. Mag. xxvi. p. 1024 ; xxvii. p. 703.
for Ka,
108
A,
for La,
108
Ordinals of the Elements and Highfrequency Spectra. 145
Of these series Mr. Moseley has calculated only Ka and La
after formulse which we may write
Here v0 is my general constant (109720 instead of
109675?) and N the number of the element, supposing
N = 13for Al.
These numbers differ, as we see, from my ordinals with
2 units (Al in my system having the number 15), but the
order is the same from Al to Au including Co and Ni,
where the element of greater atomic weight precedes the
lower one. Only the element designated as Ho, which
according to Moseley has the number 66, corresponding to
my ordinal 68, must, according to my system, be
Dysprosium*.
In my calculation of the different series I have, in close
agreement with Mr. Moseley, made use of the general
formula
108
~ =^ = 109675. a2 (NC)2,
or _T „N / 108
a(N0)=/X/
109675 X
where we from the observations directly obtain the value of
the right member r.
In the left member we know that N varies by one unit
from one element to the next, and without making any
assumption regarding the true absolute values of N, we can
always write for two elements of a series
a(N1C) = r1
a(N2C) = r2,
where ru r2 and the integer difference N2Nx are known
for all lines.
* In the abstract of his second paper, which Mr. Moseley has been so
kind as to send me, I find that he has himself made the* same remark
and corrected Ho to Dy.
Phil. Mag. S. 6. Vol. 28. No. 163. July 1914. L
146 Prof. J. R. Rydberg on the Ordinals of the
Then we have
a(N2~N1)=r2r1,
and r2 — ri
and can in this way calculate a series of approximate values
of a, independent of the absolute values of N.
Having taken the mean of these values of a I have calcu
v
lated the values of  or N— C, and found that in all six
a
series these numbers, and consequently also the values of 0
(N always being an integer), approach very nearly to whole
or half units.
I have then assumed that the values of N— C end exactly
on *0 or *5, and on inverting the reckoning have calculated
vr p or a for every line in the spectra. If the formula of
Mr. Moseley is exact, we shall then find a constant value
of a in every series.
On using the N values of my system the values of C will
follow directly on taking for any one element the difference
N — (N — C). These values of C are given at the heads of
the columns a (N — C). As our N values are 2 units greater
than Moseley's, our Cvalues will also be greater, As we
shall see, the six lines form three pairs Ka and K/? ; La and
L/3; Ly and L<£, of which the Cvalue of the first line in
every pair contains 3, the second 3" 5 as a factor. I have
designated the corresponding avalues by aly bl9 a2, b2, and
a3, b3 in the three pairs.
In the following tables the abovementioned reckoning is
given for all the six series, with the exception of the later
part of L/3, where the lines seem not to belong to the same
series as the first ones, but to have been interchanged
in some way. The last 5 lines (Ta to An) , given by Moseley
for L/3, I have carried over to the series L<£, where their
wavelengths answer tolerably well. Probably there will
be more lines in the highfrequency spectra than those
hitherto measured or published.
As we see, the avalues in the different series are nearly
constant. Greater deviations occur only at the beginning
and at the end of the series, in Ka for the two first and for
the 3 (7) last elements; in K/3 for the two first only. In La
there are some greater values at the ends of the series, but
the differences are of no consequence. Ly and L(/> show
Elements and the Highfrequency Spectra,
147
nearly constant values throughout the whole series. 0£ L/3
I have already spoken. The results for this series are to be
regarded as rather uncertain.
First pair of Lines.
N.
Ea.
ax (N3).
K/3.
bx (N35).
Al(15)
8364
08701 . 12
7912
09335 .115
Si (16)
7142
08695 . 13
6729
09313 . 125
CI (19)
4750
08659 . 16
K(21)
3759
08652 . 18
3463
09272 . 17*5
Ca(22)
3368
08659 . 19
3094
09279 . 185
Ti(24)
2758
08659 . 21
2524
09272 . 205
V(25)
2519
08648 . 22
2297
2093
09267 . 215
09276 . 225
Cr(26) ....
2301
08655 . 23
Mn(27) ....
2111
08660 . 24
1918
09278 . 235
Fe (28)
1946
08658 25
1*765
0 9277 . 245
09278 . 255
Co (29)
1798
08661 . 26
1629
Ni(30)
1662
08675 . 27
1506
09285 . 265
Cu(31)
1549
08665 . 28
1402
09273 . 275
Zn(32)
1445
08662 . 29
1306
09271 . 286
Y(41)
0838
08681 . 38
__
Zr(42)
0794
08689 . 39
Nb(43)
0750
08717.40
_____
Mo (44) ....
0721
08674 . 41
Ru(46) ....
0638
08792 . 43
Pd(48)
0584
08781 . 45
Ag(49)
0560
08772 . 46
Second pair of Lines.
N. La.
Zr(42) 6091
Nb(43) 5749
Mo (44) 5423
fiu (46) 4861
Rh(47) 4622
Pd(48) 4385
Ag(49) 4170
Sn(52) 3619
Sb(53) 3458
La (59) 2676
Ce(60) 2567
Pr(61) 2471
Nd(62) 2382
Sa(64) 2208
Eu(65) 2130
Gd(66) 2057
Dy(68) 1914
Er(70) 1790
Ta(75) 1525
W(76) 1486
Os(78) 1397
Ir(79) 1354
Pt,(80) 1316
Au (81) 1287
a2(N9).
03708 . 33
03704 . 34
03705 . 35
03702 . 37
03696 . 38
03697 . 39
03697 . 40
03691 . 43
03690 . 44
03692 . 50
03695 . 51
03694 . 52
03092 . 53
03693 . 55
03695 . 56
03694 . 57
03699 . 59
03700 . 61
03705 . 66
03697 . 67
03703 . 69
03707 . 70
03707 . 71
03697 . 72
L2
L/3.
5507
5187
4660
4168
3245
2471
2360
2265
62(N105).
03959 . 325
03958 . 335
03940 . 355
03944 . 375
03944 . 425
03961.485
03971 . 495
03973 . 50fv
148 Ordinals of the Elements and Highfrequency Spectra.
Third pair of Lines.
N.
Ly.
a3(N12).
Pd(48)
3928
04232 . 36
La (59)
2313
04224 . 47
Ce(60)
2209
04233 . 48
Sa(64)
1893
04221 . 52
Eu(65)
1814
0 4230 . 53
Gd(66)
—
Er(70)
Ta(75)
1287
04225 . 63
Os(78)
1172
04226 . 66
Ir(79)
1138
1104
0 4225 . 67
Pt(80)
04226 . 68
Au (81)
1078
04215 . 69
I4.
63(N14).
2424
04310
45
2315
04314
46
1972
04301
50
1888
04309
51
1818
04307
52
1563
04313
56
1330
04292
61
1201
04305
64
1155
04323
65
1121
04321
66
1092
04313
67
On taking the means of the a and 6values for the
different series, we find : —
Series.
Constant.
Elements which enter in the mean.
Mean value.
Ka.
ax.
All except the three last (Ru, Pd,
Ag).
08671
»»
The constant middlepart from CI to Zn.
08660
Kj3.
bv
All measured by Moseley.
All except Al and Si.
09283
09275
La.
a.r
All.
The 12 constant, Eh to Gd.
03698
03694
L/3.
K
All here given in the table.
The six first only.
03956
03951
Ly.
a3.
All.
04226
L(&*,» 3l)=m,
representing the state of affairs contemplated by Archimedes
and the elementary treatises on hydrostatics.
(7) An interesting question is — what is the value of dx for
which Mx = M? Equating the righthand side of (vi.) to
zero we have
E*» = 4afR ^IL^n + ^V
1 L 3 \/2a + d1 2 3j 3 '
or approximately d±2 = 4
* L.c. p. 93].
Solid Sphere in contact with a Liquid Surface. 153
and the pull is therefore a maximum when the vertex o£ the
sphere is very slightly below the level of the " free" hori
zontal surface of the liquid.
It is obvious that equation (ix.) might be made the basis
of a practicable method for the determination of surface
tensions ; but in point of accuracy the method of which
equation (vii.) is the expression is distinctly preferable.
A simplified form of the experiment, involving the use of
a plate instead of a sphere, forms an interesting piece of
laboratory practice. Although very simple and straight
forward, I have not seen the experiment mentioned elsewhere,
and it may not be out of place to outline it briefly here.
A plate of glass is taken — a microscope slide does well —
and suspended by a thread from the underside of a balance
pan with its surface vertical and lower edge horizontal.
The balance being equilibrated and left free to move, a
basin containing the liquid to be examined is placed under
neath the plate on an adjustable table. The table is raised
till the liquid barely touches the plate. Weights are now
added to the balance to restore equilibrium, when we have
the wellknown equation
mg = 2(l + b)T, (xi.)
where m is the mass of the added weights, I the length and
b the breadth of the plate (the contactangle is of course
assumed to be zero).
Now remove the added weights, and, first noting the
position of the adjustable table, raise it until the balance
pointer is again in its zero position. If dY is the distance
through which the table has been raised, we have
%?r tubes 3 /n interna/ diameter
V thickness Rafro ±
■0097
V/fnenTct/ Curve.
".quafion (2) cxSSum/rrj Zi = 6c{ (Corman)
'quot/on (2.) assu/l/no L = J<^P (South rre/t)
k  I 73.
1 #
\£
«~— Sb"J. "'fnaJ?
// /
" /
vv
/
1 «*.«, /
I
z¥V
/•/•
77
X
// +
A^
k
i
/^.
/*
L'S'v"1
''>
XX
!'>'''*
V
^
(P
.+
"
*/° j'
SW, **»~ *iW. «^, »„„„
7 '
\
°z
£,„„„..?„/ C.L' "" "' ~
f
I
$'
i
f™ r,».*„ a) »„„„,„, i . i/pfs^M,,,!)
X
I
\
\
r
I
i.
*
111
<»
,„,« «,
«W
Fig.
4
? 'A
lZ"u,"nl"l't"''z,?",Z'Zi '".'""*
1
,.„«,,*««<„„,. *A.^.wf
W /"»» ^..*.„ i) .»»»»» i , AjS"f^m„J

\ V
*
a
•*=» T
«
~
"
— ._.
I
I
«•
V
sarr,ar,%^Sw
f
\>
.ST Am £■,„.*„, Ci) «,„„,,,, z . »^awHj
j
X
{
^
J
i"
I
U„,M (,
*,
Lj
Fio
Phil
Mag.
«. 6, Vol. 28. PI, I.
\
^ i
\\
I a,
;}■' '^ZSZT
i \
\
III
J '
\
J
i
c
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Fia
r~
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./»
r
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v
.„C,,,t„. ,..;,„„„,„, z.t^dMi
"V.
I

..Ill
Jtj
2
» • f!
8, PL II.
1
^■^H ■ S '■■♦ ■
Phil. Mag. Ser. 6. Vol. 28. PI. II.
■ ii^. 1
4737 Fio
pIG, 5._VacuiinilnW. H'limn un2#4 p0x2
(3)
Thus dy/ds is a function of x of known form, say X, and we
get for y in terms of x
as given by Beer.
f Xdx
y=±J •(iX')' W
Revolving Liquid under Capillary Force. 163
If, as in fig. I, the curve meets the axis, (3) must be
satisfied by x = 0, dy/ds = Q. The constant accordingly dis
appears, and we have the much simplified form
dy a(o2xz p0x ,.
37=~8T~+2T W
At the point A on the equator dyjds=l. If OA = a,
~8T~ + 2T ;
whence eliminating p0 and writing
^OH .... (io)
when we neglect higher powers of O than H2. Reverting
to a,', we find for the integral of (10)
±f(in+W(l#£(i#, (ii)
no constant being added since y = 0 when # = a.
If we stop at 12, we have
2 2
^ + a2(in)2=1 • : • ■ (12)
representing an ellipse whose minor axis OB is a (1 — XI).
When fl2 is retained,
OB = (ln + n2)a (13)
The approximation in powers of 12 could of course be con
tinued if desired.
M2
164 Lord Rayleigh on the Equilibrium of
So long as Q,
Here # = a corresponds to <£ = 7r, and ,v = 0 corresponds to
<£ = 0. Hence
OB= f2 sintytty (16)
The integrals in (16) may be expressed in terms of gamma
functions and we get
OB = avV.r($)7r(£)=4312a. . . (17)
When 12 > 1, the curvature at B is concave and p0 is
negative, as is quite permissible.
In order to trace the various curves we may calculate by
quadratures from (4) the position of a sufficient number of
points. This, as I understand, was the procedure adopted by
Beer. An alternative method is to trace the curves by
direct use of the radius of curvature at the point arrived aL
Starting from (7) we find
d?y [„Sw2 1Q,\dx
ds*
and thence
~ \ a* a J ds
a d2y ds2 3x2 1 n nQ.
~=a—/' =12— 0 + 1— 11. . . . (lb)
p ax/as or
From (18) we see at once that 12=0 makes p = a throughout,
and that when 12 = 1, ^ = 0 makes p = co .
In tracing a curve we start from the point A in a known
direction and with p = a/(2X2 + l), and at every point arrived
at we know with what curvature to proceed. If, as has
been assumed, the curve meets the axis, it must do so at right
angles, and a solution is then obtained.
The method is readily applied to the case 12 = 1 with the
advantage that we know where the curve should meet the
axis of y. From (18) with 12 = 1 and a = 5,
I24*2 ' (19)
Starting from #=5 we draw small portions of the curve
corresponding to decrements of x equal to "2, thus arriving
Revolving Liquid under Capillary Force.
165
in succession at the points for which o: = 4*8, 4*6, 4*4, &c.
For these portions we employ the mean curvatures corre
sponding to ^ = 4*9, 4*7, &c. calculated from (19). It is
convenient to use squared paper and fair results may be
obtained with the ordinary ruler and compasses. There is
no need actually to draw the normals. But for such work
the procedure recommended by Boys * offers great advan
tages. The ruler and compasses are replaced by a straight
scale divided upon a strip of semitransparent celluloid. At
one point on the scale a fine pencil point protrudes through
a small hole and describes the diminutive circular arc.
Another point of the scale at the required distance occupies
the centre of the circle and is held temporarily at rest with
the aid of a small brass tripod standing on sharp needle
points. After each step the celluloid is held firmly to the
paper and the tripod is moved to the point of the scale re
quired to give the next value of the curvature. The ordinates
of the curve so drawn are given in the second and fifth
columns of the annexed table. It will be seen that from
x—0 to x — 2 the curve is very flat.
±x.
±y
±y'>
±x.
±y.
±y'
00
216
ooo
26
206
075
02
216
ooi
28
203
083
04
216
003
30
199
090
06
216
006
32
195
095
08
210
010
34
189
099
10
215
0;14
36
181
101
12
215
020
38
172
102
14
215
027
40
161
100
16
215
034
42
149
098
T8
214
042
44
132
089
20
212
050
46
111
078
22
211
058
48
080
067
24
2 09
065
49
059
041
50
000
000
Another case of special interest is the last figure reaching
the axis of symmetry at all, which occurs at the point x = 0.
We do not know beforehand to what value of fl this corre
sponds, and curves must be drawn tentatively. It appears
that H = 2*4 approximately, and the values of y obtained from
this curve are given in columns 3 and 6 of the table. There
is a little difficulty in drawing the curve through the point
of zero curvature. I found it best to begin at both ends
(.r = 0, y = 0) and (#=5, y = 0) with an assumed value of O
and examine whether the two parts could be made to fit.
* Phil. Mag. vol. xxxvi. p. 75 (1893). I am much indebted to Mr. Boys
for the loan of suitable instruments. The use is easy after a little practice.
166 Lord Bayleigh on the Equilibrium of
When fl>2'4 and the curve does not meet the axis at all,
the constant in (3) must be retained, and the difficulty is
much increased. If we suppose that dyjds — + 1 when
x = a2 and dyjds =—1 when x = alt we can determine p0 as
well as the constant of integration, and (3) becomes
dy
(TO)'
xts 8T {*2'
■ ai2X^a2*) +
■axa2
(20)
a2—ai
We may imagine a curve to be traced by means of this
equation. We start from the point A where y = 0, x = a2 and
in the direction perpendicular to OA, and (as before) we
are told in what direction to proceed at any point reached.
When «£ = <*!, the tangent must again be parallel to the axis,
but there is nothing to ensure that this occurs when y = 0.
To secure this end and so obtain an annular form of equi
librium, ..„ /oox
£? = + — ^  ^(a + rcos^)2, . (23)
1 r a + r cos 6 21 v 7 v 7
in which p0 should be constant as 6 varies. In this
cos0 If r A 3r\ ' r 0/1 r2 ozl \
S+F^T* = «t ra+V1 + ^2)COS^^COs2^^2COS^N
(l+>s^=l£+JcoS^£2cos2^.
Thus approximately
a»o « r o.2a3/, r2 \ „/, 3r2 1 rdd*I + a + rcos0
a sin ^ 1 2a3 f )\ r% r02 1
"V ~2a"*~2r~0~2T0~ ~2T 1 a~ + a + 2a2 J '
or when we introduce the value of oy from (31)
3ju_,_P_2r3
»'o 4a r0 v y
170 Dr. C. Sheard on the Positive
The coefficient of cos 30 is in like manner
*£ + £+* U53)
r02 T 4a2 2r0 v y
These coefficients are annulled and ap0/T is rendered con
stant so far as the second order of r0/a inclusive, when we
take r4, r5, &c. equal to zero and
r2/r0 = r02/4a2, rjr0= 3r03/64a3. . . (34)
We may also suppose that ^ = 0.
The solution of the problem is accordingly that
r=rAi+icos2dMgcosS0} ■ • (35)
gives the figure of equilibrium, provided w be such that
=£&=!+.§•£ (36)
The form of a thin ring of equilibrium is thus determined ;
but it seems probable that the equilibrium would be unstable
for disturbances involving a departure from symmetry round
the axis of revolution.
XXIII. The Positive Ionization from Heated Platinum.
By Charles Sheard, Ph.D., Ohio State University *.
I. Introduction.
THIS paper will present the results of some investigations
made as to the dependence of the positive ionization
from heated platinum wires upon (1) temporary heatings of
the wire when earthed and at temperatures higher than
a chosen lower temperature at which the positive ther
mionic current was subsequently measured ; (2) temporary
heatings of the wire under negative potential and at
such temperatures as to give small negative currents, the
positive emission being investigated at an arbitrary, chosen
temperature ; and (3) heating the wire in a Bunsen flame
and in carbon dioxide, and also rewashing in nitric acid.
Some experiments carried out some time ago with a form of
apparatus somewhat different from that used in this work
and exhausted to a pressure of about 0*0001 mm., showed
that the positive currents at any given temperature could
be enhanced, by either heating the wire at a higher tempe
rature, the wire being connected to earth, or by discharging
* Communicated by the Author.
Ionization from Heated Platinum,
171
electrons to the cold electrode during a temporary heating.
After a wire, for example, giving a slow decay with time
current of 0*8 div. per sec. at +200 volts and 453° C, was
heated for an hour at 540° 0. the following currenttime
readings were taken, when the temperature of the wire was
again made 453° C. at + 200 volts.
Current in
divisions per
second
'.'J
199 156 129 65 42 2*8 1'8 1'
100
Time 1 Q
(minutes) J
10
25 35 45 60
85
Again, in a particular case the current, after continued
heating at 388° C, showed a nearly constant value of 0*7 div.
per sec. at + 400 volts. The w^ire was then heated for 30
minutes at 598° 0. and —400 volts, the negative emission
being measurable at this temperature. On returning to
388° C. and +400 volts, the initial current was found to be
equal to 14 div. per sec, or about twenty times its previous
value. This large current fell away fairly rapidly with
time at the lower temperature (388° C), as is shown by the
folio win 2 data.
Current in
divisions per
second
14 10 8 51 38 26 20 145 105 100
Time
(minutes)
}
10 25 35 45 60
85
100
We can explain these results if we suppose that the in
creased positive ionization, or, as we may call it, the
" induced leak," is due to some substance which gradually
decomposes under the influence of heat into at least one
positive ion. The reaction, assumed monomolecular, may
be represented by the equation P = ?>I4Q, where P repre
sents the unknown substance, n the number of positive ions
produced by each molecule of P, and Q represents other
products if existing. From the laws of chemical dynamics
we may write :
and hence
dt
P =A,** .
172 . Dr. C. Shear d on the Positive
If the " induced " ionization be denoted by X, then
X — n — — —Jr~P — Ir\ fi~kt — A />**
x~ndt~ Tt~k^kAie "A" '
so that logX=C— Jet.
In both sample sets o£ readings given above, if we plot
the logarithm of the difference between the actual current
and the slowdecay or " steady " current, a straight line is
obtained. Later in this paper, it will be shown that at least
two substances capable of producing ions under heating
must be assumed, and that the simple logarithmic law given
above is not generally applicable.
Complications arose, however, when fresh gas (air) was
admitted to the tube and allowed to remain there several
hours, the wire being cold during the interim. Upon re
pumping to a low pressure, it was found that the current
E.M.F. curves were either slightly convex to the voltage
axis or almost linear. With subsequent heating under
potential the usual saturationcurrent conditions obtained.
The positive thermionic current from the wire was found
to be initially about 15 to 20 times as great as the value
previous to the admission of air. This increased current
fell off rapidly with time and reached the slowdecay stage
in about 15 minutes. A preliminary account of some investi
gations along this line, made by Richardson and Sheard, is
to be found in the Physical Review, vol. xxxiv. p. 392 (1912).
Some experiments made recently by the writer of this paper
indicate that these effects are due in large part to small
quantities of water vapour. It was decided, therefore, to
carry out the present investigations in dry gas at atmospheric
pressure. This ensured the reproducibility of conditions
when the apparatus had been opened, the wire treated and
reinserted, and the thermionic currents again investigated.
Also any effects due to the contamination of the wire with
mercury or phosphorus vapour were eliminated.
II. Apparatus,
The main chamber used in these experiments consisted of
a cylinder of brass, 11*5 cm. long and 6 cm. diameter, closed
at the two ends with brass plates. Through the lower plate
two insulated heavy leads were passed, and across these was
fastened the platinum wire 0*01 cm. in diameter and about
3 cm. long. The receiving electrode was a hollow brass
cylinder, 4x1 cm., closed at both ends and attached to a micro
meterscrew passing through a threaded bushing of brass
Ionization from Heated Platinum.
173
which was insulated from the remainder o£ the apparatus.
The detecting electrode was kept cold by drawing air through
a system o£ tubes arranged within the electrode proper. Dry
air or other gas was filtered through calciumchloride towers
before admission to the apparatus. The ionization currents
were measured by means of a quadrant electrometer.
Throughout this paper one division per second represents
3'7xl0~13 ampere. For further details as to the apparatus
and methods of manipulation the reader is referred to the
Physical Review, series 2, vol, ii. p. 289.
III. The Increased Positive Thermionic Currents in Air
at a given temperature produced by heating
the wire at higher temperatures.
A measurable current (onehalf of one div. per second)
with +200 volts was obtainable at a temperature of 600° C.
The temperature was raised by several steps to 628° C.
before the ionization currents at 1200 volts showed any
detectable decay with time during twenty minutes testing.
At 632° C. a slow decay with time effect was found ; the
following is a sample set of data obtained.
Temperature.
628° C.
632° C.
Current
(div. per sec.)
14
15 14 133 135 14
31 To 21 22 20 19 18 17
Time
(minutes).
0
2 4 10 15 20
\ 0 2 4 6 8 10 15 20
The temperature chosen, therefore, as the datum or arbi
trary point at which to test the emission produced by increase
of temperature was 628° 0. Curve I. of fig. 1 shows the
initial current readings obtained after the wire had been
heated for 10 minutes respectively at each of the various
temperatures recorded, the wire being connected to earth
during the heating, the temperature then being reduced to
628° C. at 4 200 volts. The data for the curves of fig. 1 are
given in Table I.
Curve I. fig. 1 shows maxima after heating at G54° C.
and 756° C, the ionization currents being measured at
028° C. An attempt will be made in the following pages
to show that these maxima indicate the presence in the wire
of two substances capable of producing ions, the first beino
174
Dr. C. Sheard on the Positive
that operative in the lower temperature range, and the second
that which predominates at higher temperatures. Curves I.,
II., and III. of fig. 1 show two maxima. The first maximum
In all three curves is at 654° C. ; in Curve III., showing the
Fier. 1.
kOo° febo° txjo0 720* IScT ltd" »/o° ?40°C
Tern Ke rat u re
slowdecay values of the current, the second maximum
appears at 700° C. instead of 756° C. as indicated in Curves I.
and II. This shift is explicable in the light of what follows
later in this paper.
Moreover, experiments performed in another connexion,
while investigating the emission at f 200 volts, showed the
existence of a maximum at about 650° C, when the tem
perature was changed by short intervals from 630° C. to
900° C. The magnitude of this maximum decreased on
repetitions of the currenttemperature series and finally
practically disappeared. On allowing the wire to stand cold
for several days the increased ionization effects were again
obtained, the maximum occurring at the same low tempe
rature point. On the other hand, the positive currents
Ionization from Heated Platinum
Table I.
175
Applied potential = +200 volts. Wire at zero potential
during heating.
Curve 1(0). Initial positive current readings obtained at 628° C. after
heating in air at various temperatures.
Curve II (©). Slow decay currents, following Curve I.
Curve III (O). Initial current readings, as per method for Curve I.,
after first heating in Bunsen flame.
Curve IV ( A ). Steady current values following Curve III.
Curve V ( + ). Initial current readings after second heating in Bunsen
flame.
Minutes
heating
at V=0.
Air.
Bunsen Flame I.
Bunsen
Temp. cO.
Initial
Steady.
Initial Steady
Flame II.
(Curve I.).
! value
(Curve III.).
value
Initial
(Curve 11.).
(Curve IV.).
(Curve V.).
628
Fixed Point.
14
! "
24
31
644
8
502
135
654
10
1105
232
47
...
666
10
410
110
690
11
331
220
38
144
12
714
10
452
223
40
38
738
10
132
8
58
386
40
750
10
221
8
82
765
10
2006
6
89
112
83
774
10
101
41
786
...
*7l
*38
"i6
798
i'd
26
14
822
10
175
085
840
10
103
063
1
obtainable at higher temperatures (700° C. on) decreased
steadily with continued usage of the wire, and exhibited no
such effects as have just been described. These facts point
to the existence of a substance capable of producing ions
when heated at relatively low temperatures and that this
substance, when the wire is left cold, builds up ionizable
material with time as if it were radioactive. Sir J. J. Thom
son comments on the possibility of such a substance in his
' Conduction of Electricity through Gases,' page 214.
Referring again to Curve I., fig. 1, it will be seen that
the positive thermionic current, tested at the fiducial 628° C.
point, reached an extrapolated maximum value of about
260 div. per sec. when previously heated at 756° C. Heating
the wire at still higher temperatures gave currents, at 628° C,
falling off in value rapidly from 756° C. to 820° C.
176
Dr. C. Shear d on the Positive
Curve 1, fig. 2, or the data in Table III., shows the relation
between the negative currents and the temperature, and was
Fig. %
140
l
i
i '
l£0
4
i
!
go
6o
i'
/
/
/ i
£40
CO
i
1
II
1
C
O
750*
7?0*
81 o°
84^
S70°C
renr/i erasure
taken after Curves I. and II. of fig. 1. It will be noted that
the negative emission becomes measurable at 760° C. or ap
proximately the temperature at which the positive emission^
under the treatment described, begins to fall away from its
maximum value.
The quantity o£ positive ions, or of material capable of
producing such ions, decreases as the wire is heated from
756850° C, while the negative currents increase rapidly
within this same temperature range. It is a well known
fact that a heated metallic wire discharges positive electricity
at a lower temperature than it does the negative. The
negative electrification from hot wires and a considerable
number of salts has been found to be wholly electronic in
nature. In the case of certain salts containing the strongly
Ionization from Heated Platinum. 177
electronegative elements iodine, bromine, fluorine, it has
been shown by Richardson * and by Sheard f that the
negative constituent consists of negative ions or an admix
ture of negative ions and electrons. The following theory,
based upon experimental evidence of a general character
and upon the specific results outlined in the preceding para
graphs, will, I believe, account for these phenomena. Let
us assume a molecule consisting of two atoms, A positively
and B negatively charged. When the temperature is raised
decomposition of the molecule occurs i, and under the in
fluence of an applied electric potential the positively charged
atom A is expelled, for example, and the atom B, negatively
charged, is retained. At the still higher temperatures at
which the negative atom begins to discharge electrons, the
atom B loses a corpuscle. If the negative atom B is driven
off* before losing corpuscle?, as occurs in the case of some
halogen compounds, we should get an emission of negative
ions. If, however, the atom B loses an electron and this is
then discharged, B will be left without charge, and hence
will not normally recombine with the positive atom A. This
would explain the increased positive emission produced at
low temperatures by heating a wire under such conditions
as to give off negative electricity.
If the temperature is raised without the expulsion of ions
or electrons, then at low temperatures dissociation occurs
giving fAand — B. Some recombination doubtless occurs;
in the main, however, the net result would be to build up
positive material A to be later emitted as positive ions
under an applied electric force. At still higher temperature
B loses an electron ; this may find its Avay to the positively
electrified atom A, neutralizing its charge, so that A and B
are both without charge. Atom A, now having no charge
and being the electropositive element, can more readily lose
* Phil. Mag. Sept. 1913, pp. 452472.
t Phil. Mag. March 1913, pp. 370389.
^ The wire was heated electrically ; there was a drop of potential
across the wire varying from 0*75 to 1*2 volts within the range of
temperatures used in these experiments. The theory of electrolysis
is applicable, with some modifications, to liquids, gases, and solids ; in
the case of solids, however, there is no migration of the ions formed;
the potential difference across the wire in the heating circuit operates,
however, as an agent in accelerating molecular dissociation and in
preventing recombination. There is no reason to believe that the
r°sults given in this paper could not bo obtained by a nonelectrical
method of heating; there would probably be a diminution in the
magnitude of the effects at any given temperature, however. In this
connexion the recent work of Harker and Kaye (Proc. Roy. Soc, A.
vol. lxxxvi. pp. 379396 ; et al.) is of interest.
Phil. Mag. S. 6. Vol. 28. No. 164. Aug. 1914. N
178 Dr. 0. Sheard on the Positive
corpuscles than B ; hence A may again become positively
and B negatively charged. We should thus get a state,
dependent upon temperature, in which the number of
recombinations in unit time would approach equality to the
n amber o£ molecules dissociated in that time. The greater
the ease of getting the corpuscles out of B, the less the
amount of positive, or positive producing, material there
would be. This explanation would account for the marked
decrease in the positive ionization, subsequently obtainable
at lower temperatures, when ihe wire is heated from 756—
820° C.
The maximum positive thermionic emission obtained after
heating at 654° C. can be explained in an analogous manner
by assuming a substance capable of dissociation in this lower
temperature range into + C and — D ions. The maximum
positive emission would exhibit itself after heating at a tem
perature such that the negative ions begin to lose electrons.
Further increase of temperature would cause a greater
liberation of electrons, and the consequent neutralization of
the positive ions within the wire. No electronic emission is
detectable at these temperatures since the electrons do not
possess sufficient kinetic energy to enable them to escape
from the wire. Hence heating the wire under negative
potential at temperatures below 760° C. — the temperature at
which negative currents were detected in these experiments
would have no effect in increasing the positive emission
subsequently obtainable, at least until temperatures ap
proaching 760° C. are used. These latter effects are
discussed in Section VII.
This view does not preclude the possibility of secondary
chemical reactions within the wire or between a hot wire
and a salt at its surface. The proof seems conclusive that
potassium and sodium salts, present as impurities in metals,
constitute the source of positive emission. If so, sulphates
of these bases might be expected ; upon decomposition the
acid radicle may combine under temperature with the metal
of the wire ; hence corpuscular and not ionic emission would
occur.
Some observations were made on the decay of the positive
currents with time when the wire was earthed and heated for
10 minutes and the emission then tested at the temperature
previously employed. (See note.) The tabulation given in
Table II. shows the method of procedure. The data show
that heating at 765° C. for 10 minutes and reducing to
738° C. produced an increase of positive ionization ; heating
at 786° C., however, gives a much smaller current at 765° C.
Ionization from Heated Platinum. 179
than had been just previously obtained at this lower tem
perature. These results are in agreement with those given
in Curve I., fig. 1.
Table II.
Temp. = 738° C.
V = + 200 volts.
Heated 10 minutes
at 765° C. and
V=0; thermionic
currents read at
738° C.
Temp.= 765° C.
V = 4 200 volts.
Heated 10 minutes
at 786° O. and
Y= 0; thermionic
currents read at
765° C.
Time
(Min.).
Current.
Time
(Min.).
Current.
Time
(Min.).
Current.
Time
(Min.).
Current.
0
25
45
7
10
32
25
20
19
18
...
0
15
25
4
10
u
52
33
26
245
228
220
0
1
2
3
5
8
20
22
20
22
22
22
0
1
2
4
7
10
072
067
045
044
045
044
(Note. — The wire was in such a condition that very small
ionization currents were obtainable. The readings given in
Table II. were obtained after Curves I. and II., fig. 1.)
IV. Increased Emission produced by heating in a
Bunsen Burner.
Following Curves I. and II. of fig. 1 and before the heating
of the wire in the Bunsen burner, the following values of
the positive currents were obtained at the temperatures
shown : —
Temperature, ° C.
628
654
668
714 750 765
775 812 822
Current (div. ]
per sec.) at I
V= +200 volts. J
096
23
62
10 182 24
43 110 220
The wire was then removed and heated in the reducing
flame of the Bunsen burner for 10 minutes. The same
procedure was then followed as in obtaining Curve L, fig. 1,
N2
180 Dr. C. Sheard on the Positive
after the wire had been reinserted in the apparatus filled
with dried air. The values o£ the initial currents obtained
after heating at various temperatures, the wire being put to
earth, for 10 minutes in each case, are shown in Curve III. of
fig. 1, and the slowdecay values in Curve IY. Curve III.
is in form a repetition of Curve I. and shows maxima at about
the same temperatures. It is to be borne in mind that these
effects were obtained by heating in the gasflame following
the low current values given in the tabulation at the be
ginning of this paragraph ; hence the absorption of hydrogen
from the flame has in some manner produced a large increase
in the ionization subsequently obtainable. Under further
treatment the currenttemperature relations become com
parable with those given in the above tabulation. The wire
was again heated in the burner for 10 minutes. Curve V.
of fig. 1 and Table I. give data showing the effects produced
by the second heating. The thermionic currents were in
l creased but little in comparison with the effects produced by
the first heating in the Bunsen flame as shown in Curve III.y
fig. 1 ; the maximum effect at about 756° C. is, however,
clearly indicated.
The proof is conclusive that heating in hydrogen revivifies
I tne wire and produces an increased positive emission. The
role which such a gas plays, however, has been variously
explained. The recent work of Professor 0. W. Richardson *
has quite definitely settled this question in one of its aspects,
for he finds that there is no emission of gas ions and that the
values of e/m remain the same before and after heating in a
gas. The increased emission produced by hydrogen is pro
bably in part due to the removal of material, incapable of
producing ions, built up at the surface of the wire. This
removal may be occasioned by the mechanical effects of the
gas on diffusion into or evolution from the wire, or may be
due to its reducing action. The ionization effects following
the cleaning of an " old " wire in nitric acid t indicate the
existence of such a nonionizing substance. 0. W. Richardson
has shown that "merely straining the metal will cause a
revival of the positive emission," and that the renewal
of emission under various circumstances is probably mainly
a mechanical effect.
An important function played by hydrogen, however,
may be attributed to its affinity for electrons. Franck J has
* Proc. Royal Society, A. vol. lxxxix. pp. 507524.
t Sheard and Woodbury, Physical Review, vol. ii. pp. 288298 (1913).
% Verh. d. D. Phys. Gesell. xii. pp. 291 & 613 (1910).
Ionization from Heated Platinum.
181
recently pointed out that the affinity of gases for electrons is
in the following order : — Chlorine, nitric oxide, oxygen,
hydrogen, argon, and helium.
Curve 1, fig. 2 (data for curves of this figure are tabulated
in Table III.), shows the relation between the negative
currents and the temperatures and was taken after the
results given in Curves I. and III. of fig. 1. After heating
Table III.
Tenip. ° C.
After data
obtained in
Curves Land II.,
fig. 1. (See
Curve 1, fig. 2.)
After
first heating
in Bunsen fiame.
(See Curve 2,
fig. 2.)
After
second heating
in Bunsen flame.
(See Curve 3,
fig. 2.)
765
038
774
058
7S0
082
792
25
804
42
815
202
839
523
082
072
846
130
852
250
173
858
135
45
864
102
24
870
185
60
877
75
in the Bunsen flame and proceeding in the manner shown in
Curves III. and IV., fig. 1, the negative emission was that
shown in Curve 2, fig. 2. Following the second heating in
the gasflame Curve 3, fig. 2, was obtained. The relatively
large decrease in negative emission and the higher tempera
ture at which the negative currents were detectable in Curve 2
as compared with Curve 1, fig. 2, indicates the removal of a
considerable portion of the negative electrification between
the two stages. Curves 2 and 3 show by comparison a small
decrease in negative currents. The positive effects go hand in
hand with these changes ; an enormous increase in the positive
emission (Curve III., fig. 1) being accompanied by a subse
quent decrease in the negative emission (Curve 2, fig. 2).
In a similar manner the small increased positive effect
182
Dr. C. Sheard on the Positive
(Curve V., fig. 1) corresponds to a small decrease in nega
tive emission (Curve 3, fig. 2). The hydrogen presumably
accelerates the formation of positive ions, following out the
theory presented in the previous paragraphs, by inhibiting
the recombination of positive and negative ions produced by
heating. If this is so and if the gas has an affinity for
electrons, one might expect at high temperatures an escape of
hydrogen molecules carrying negative electrons ; these would
be held bound, in part at least, at the surface of the wire
under an opposing electric field. Recombination of positive
ions from the wire with the discharged negatively charged H
molecules would occur, and the positive thermionic currents
would decrease with increase of temperature above a certain
point.
The experimental results given in fig. 3 were obtained after
Fig. 3.
7gO° 8IO°C
Tem.fi.era+u.re
the treatment discussed with reference to Curve III., fig. 1.
The negative ionizationtemperature relations before heating
in the Bunsen flame are given in Curve 1 of fig. 2, and similar
Ionization from Heated Platinum.
183
relations following the positive ionizationtemperature effects
now being discussed are shown in Curve 2, fig. 2. About
15 minutes' time was taken to get the readings for each
curve of fig. 3 ; the curves were taken seriatim, proceeding
in each case from low to high temperatures. Curve 1 shows
a rapid dropping off of the positive thermionic current from
786° to 800° C. ; Curve 2 shows a less decided falling off
from 738° C. on ; while Curve 3 indicates that the cause of
this decrease in the two previous curves, whatever it may be,
has been removed. It is to be remarked that the highest
current readings recorded in these curves are approximately
in the ratio of 1 to 200 for the currents obtained in the
initial stages of heating the wire.
Y. Effects produced by Heating in Carbon dioxide.
When the wire was again in a condition of giving very
small positive ionization currents, it was heated in a dry
atmosphere of C02 and investigations carried out as described
under Section IV. No increase of thermionic currents was
obtainable in any case with this gas, although the wire
was heated for various periods of time and at different
temperatures. In fact, the positive currents were in each
instance lower than those obtained previous to heating in
this gas. The following is a simple set of data obtained
after heating the wire in C02 for 10 minutes at 750° C.
Temperature, ° C.
628
033
654 690 732 765 786
798 810 822
Current (arbi "J
trary units) 1
before heating [
in 0O2 J
032 031 ... 033 (M0
054 071 ...
Current (arbi ~\
trary units) 1
after heating (
inC0.2 J
027
025 025 023 025 025
031 040 083
These results were not anticipated in view of Horton's *
work, but would be on the view that carbon dioxide has little
affinity for electrons. Possibly, also, referring to Curve 3,
fig. 2, the wire should have been heated at higher tempera
tures than it was, since negative currents could not be
obtained in this stage of the history of the wire at a
temperature lower than 820° C. The latter explanation is
* Proc. Camb. Phil. Soc. vol. xvi. pp. 89 & 318 (1911).
184
Dr. C. Sheard on the Positive
probably the correct one, although other experiments using
a " fresher" wire are necessary before definitely settling this
point.
After the wire had been allowed to stand cold in C02 for
five days, a maximum current^ of 0*49 div. per sec. was ob
tained at 654° C, similar to the maximum effect at the same
temperature in air or hydrogen given in Curves I. and III.
of fig. 1.
YI. The CurrentTime Relations.
Fig. 4 shows the decay with time of the positive currents
at +200 volts : Curve 1 at 654° C, Curve 2 at 714° C,
Curve 3 at 738° C., and Curve 4 at 774° C. Fig. 5 shows
similar relations after the wire had been heated in the Bunsen
flame : Curve 1 at 628° C, Curve 2 at 690° C., Curve 3 at
738° C, and Curve 4 at 765° C. It will be seen that Curves 2
Fig. 4.
14 lb 18 20
Time  Minutes.
and 3 of fig. 4 and Curves 2 and 3 of fig. 5 show that the
decay effect is a composite one, indicating the presence of
one substance which decays initially from a maximum value
Ionization from Heated Platinum,
185
and of a second substance giving maximum ionization after
2 to 6 minutes' heating under potential depending upon
the temperature used. At the higher temperature (765° C,
fig. 5, and 774° C, fig. 4) this secondary rise of positive
currents with time is not much, if at all, in evidence. This
is to be expected since with increase of temperature the
decay constant of the second substance changes with
temperature ; the higher the temperature the more rapid
the rise of current to the intermediate maximum. This
second maximum would therefore be masked by the rela
tively large initial ionization currents due to the first source
Fig. 5.
Tint  /icau.T«v
producing ions. An analysis of such curves can be made
in a manner similar to that given by Rutherford in his
* Radioactive Substances and their Transformations,' p. 438.
The plate taken from the above reference is given in the
right hand portion of fig. 4. Such a curve as No. 3, fig. 4,
may be analysed into two currenttime relations, the first
being expressible by V = n^e~^yt and the second by the
equation
^•3 — *"2
Any attempts at details seems superfluous here, since the
decay constants \b X2, and \3 vary with the temperature.
Of course the interpretation to be given figs. 4 and 5 is not
the same as that given in radioactive analysis, since we are
presumably dealing here with two distinct sources of ions
and not with a succession of changes from A to B to C.
186 j Dr. C. Sheard on the Positive
VII. Increase of Positive Emission produced by previously
heating the wire under such conditions as to discharge
Negative Electricity.
In a general way the effects due to heating the wire for a
few minutes under —200 yolts and at such temperatures as
to give negative currents produced decided increases in the
positive currents subsequently obtainable. The decay with
time relations of the positive currents at various tempera
tures, following the above mentioned treatment, are almost
exactly the duplicates of those portrayed in figs. 4 and 5.
This was to be expected, since the expulsion of electrons
from the negatively charged atoms would build up positive
ions to be subsequently discharged under potential. The
ions thus formed would be characteristic of the substances
present as impurities in the wire just as shown in figs. 4
and 5. Since the latter mentioned effects have been dis
cussed in some detail in this paper, it is deemed sufficient to
simply call attention here to the similarity of results in the
two modes of treating the wire.
A sample set of data, chosen from some score or more
taken, may be of interest in showing how the positive
currenttemperature relations are affected by a preceding
shorttimed negative emission. The first row of figures
gives the current values obtainable at various temperatures
before heating the wire for 10 minutes at —200 volts and
768° C. The negative ionization current was 0*72 div. per
sec. The second row gives the positive current readings
then obtained.
Temperature, ° C.
628 640 654 670 690 714 732 750 765 782 792 810 825 840
Currents (before
negative emis
sion)
Currents (after
negative emis
sion)
1 09 26 62 166 10 152 182 24 31 43 110 220
20 56 38 71 166 123 143 17'5 25 32 46 190 134 185
It will be noted that the current at 628° C. is increased in
the ratio of 20 to 1, and furthermore that this increased current
condition falls away to smaller values as the temperatures are
raised, indicating clearly that the effect is to be attributed to
the production of positive ions under the treatment recorded.
As the wire aged, however, and higher temperatures had to
be employed before obtaining negative emission, it was found
Ionization from Heated Platinum. 187
that no increased positive emission could be gotten unless the
wire had been previously heated at a temperature such that
negative currents were obtainable.
Both rows of figures also show the maximum emission
under steady conditions at 690700° 0., in general agreement
with the curves of fig. 1.
These researches have naturally suggested the carrying
out of similar experiments with salts. Of particular interest
would be the effects obtained with salts of sodium and
potassium. (Such work is now in progress.
Conclusions.
As a brief summary of the paper, attention is called to : —
(1) Increase of positive thermionic currents produced in
high vacua (a) by heating a wire under zero potential at
higher temperatures than at which the ionization is subse
quently measured, and (7>) by heating the wire under negative
potential at such temperatures as to discharge negative
electricity. (Section I.)
(2) A detailed examination of the positive currents pro
duced by method (a) cited above in dry air at atmospheric
pressure. Maximum effects were obtained after heating at
654° C. and 756° 0. when tested at 628° 0. The intimate
connexion between the decrease of positive thermions pro
duced at temperatures above 750° 0. and the liberation of
electrons was pointed out and a general theory given to
account for the experimental results obtained. (Section III.)
(3) Similar results were obtained by heating the wire in
the reducing portion of the Bunsen flame. The role played
by hydrogen in producing an increased positive emission is
discussed and the theory advanced that hydrogen, through
its affinity for electrons, accelerates the production of positive
ions by (a) inhibiting recombination of positive and negative
ions, and (b) by removal of electrons. (Section IV.)
(4) A decreased rather than increased positive ionization
was occasioned by heating in carbon dioxide. A possible
explanation is offered. (Section Y.)
(5) An investigation of the decay with time of the posi
tive currents at different temperatures showed the existence
of at least two sources of ions ; the first giving decay effects
according to an exponential law, the second showing an
increase to a maximum followed by subsequent decay with
time. (Section VI.)
Physical Laboratory,
Ohio State University, Columbus, O.
March 1st, 1914.
[ 198 ]
XXIV. Visual Sensations caused by a Magnetic Field,
By 0. E. Magnusson and H. C. Stevens*.
[Plates III. & IV.]
IN a previously published paper t the authors verified and
extended the observations of S.P. Thompson J, Dunlap§,
nnd others on the visual sensations caused by a magnetic
field. It was shown that the effective condition of the
stimulus is the number of ampereturns in the coil, and not
a current of high amperage as was implied in the work of
Dunlap. It was further shown that the magnetic field pro
duced by a direct current at the moment of making or
breaking had a similar effect to that produced by an alter
nating current. The chief results of our former work are
here briefly summarized for the convenience of the reader.
Summary for Direct Currents.
" (a) Visual sensations were produced by an increasing or
decreasing field when the circuit was closed or opened. No
effect was noticed when the current had reached a constant
value.
" (b) The effect appeared as a narrow wave or band of
light in a horizontal plane and rapidly moving downwards
across the field of vision when the circuit is closed, and
upwards when the circuit is broken.
" (c) The direction of the lines of force in the field (up or
down) did not affect the direction of motion of the light
wave.
" (d) The wave observed on closing the circuit was brighter
and more definite in outline than the corresponding wave
when the circuit was broken. The rate of increase of field
strength on closing the circuit was also greater than the
corresponding decrease on opening the circuit. This tends
to prove that the intensity of the visual sensations depends
on the intensity and rate of change in the magnetic field."
Summary for Alternating Currents.
" (a) With the coil used, the threshold for current at
60 cycles was between 3000 and 4000 ampereturns."
" These values, no doubt, vary considerably with different
* Communicated by the Authors.
t Am. Journal Physiol, vol. xxix. p. 124 (1911).
X Proceedings of the Royal Society, B. lxxxii. {567) p. 396.
§ Science, n. s. vol. xxxiii. p. 68 (1911).
Visual Sensations caused by a Magnetic Field. 189
observers, and also with the physical condition of the observer
during the test. The observer rapidly becomes fatigued, and
after the excitation has continued for a while, larger values
are required to produce noticeable effects."
" (6) Increasing the ampereturns also increases the
intensity of the light sensations."
'; (c) The frequency of alternations has a marked effect on
the visual sensations. With the same strength of field the
effect appears greatest between 20 and 30 cycles per second."
" At frequencies below 15 per second the light pulsates
in a succession of flashes over the whole field. From 20 to
35 cycles the light appears as a network of standing waves
on which is superimposed a quivering flickering effect. The
size of the meshes is smaller for the higher frequency.
Above 40 cycles the light becomes more uniform and the
flicker more rapid. ';
" (d) The effect is the greatest in the temporal parts of
the field. In this connexion it is interesting to recall
Schoen's experiments upon the sensitivity to light intensities
of the nasal and temporal halves of the retina, in which he
found that the nasal retina was more sensitive than the
temporal retina/'
The cause of this phenomenon appears from fig. 10 (q.v.).
The nasal halves of the retina? are cut by more lines of force
than are the temporal halves of the retinaa.
The purpose of this paper is to present the results of
experiments which were aimed at the solution of a single
problem, viz., the dependence of the threshold of the light
sensation upon the frequency of the current. Knowing the
relatively simple law which W. Nernst and his pupils found
to exist between the minimal effective stimuli of alternating
currents of high frequency and intensity of current, we
hoped to discover a similar simple relationship between
frequency of alternation of the magnetic field and the
minimal effective intensity of current just sufficient to cause
the sensation of light. Nernst's law: — K= — r=* where K
V7
is a constant, t the minimal intensity of current just sufficient
to excite a motor nerve such as n. ischiadicus of the frog,
and / the frequency of the current, holds good for a large
number of results. It applies, as Nernst has shown, not
only to motor, but also to sensory nerves. If the mode of
excitation of a nerve by the alternating magnetic field is
* Pfliiger's Archiv, vol. cxxii. p. 293 (1908).
190 Messrs. Magnusson and Stevens on Visual
similar to the mode of excitation of a nerve by the alter
nating electrical current, one might reasonably expect the same
law to hold for both phenomena. Our work shows, however,
that such an expectation is not justified. Probably this
divergence from Nernst's law is due to the previously
noted greater brilliancy of the light sensations produced by
currents of frequencies near 25 cycles. Below 15 cycles the
light pulsates in a succession of flashes over the whole field.
The quivering network appearing between 20 and 40 cycles
is of greater brilliancy at 25 and 30 cycles. With increasing
frequency the meshes become smaller and change into steady
glow above 70 cycles. It is more difficult to determine the
threshold when the light comes as a phosphorescent glow
than for the pulsating flashes at the lower frequencies. With
the frequency above 100 cycles the light sensations remain
faint even from a field largely in excess of the threshold
value, while for frequencies below 40 cycles the intensity of
the sensations rapidly increase with the strength of field
applied.
Coil. — In the experiments described in this paper a coil
consisting of five sections connected in series was used. The
separate sections were kept in position by means of ropes
passed through eyelets, shown in figs. 1 and 2 (PI. III.).
In fig. 1, four sections are shown in position forming a
solenoid. The fifth section has been removed and placed
nearby so as to show the elliptical crosssection.
In fig. 2, the five sections are shown with the observer in
the position occupied while taking observations.
Data on Coil.
Length (five sections) 57*0 cm. (22*4 in.).
Majoraxis 23*8 „ (9*37in.).
Minoraxis 2P3 „ ( 8*37 in.).
Thickness of coil 1*14 „ ( #45 in.).
Area of crosssection (inside) 398*2 sq. (61*8 sq. in.).
No. of turns in series (five sections) 1075.
Size of wire #12 B & S, d.c.c. copper.
Insulation Bakelite.
Resistance of coil at 0° C 3*92 ohms.
Inductance of coil , *09 henry.
Reactance of coil for 60 cycles 34*0 ohms.
Strength of Field. — Considering the coil as equivalent to a
solenoid of equal length and circular crosssection of equal
area, the maximum strength of field, H, per sq. cm., is given
Sensations caused by a Magnetic Field, 191
by the expression
H=^(4uW1VV2), .... (2)
where N = total number of turns.
I' = maximum value of current in amperes.
1 = length of coil in cm.
Wi and W2 = tJ}e solid angles subtended by
the ends at any point inside the solenoid.
For any point p along the axis of the coil, Wj
and VV2 are given by the equations (2) and (3).
Using the values of a1 and a2 as shown in fig. 3,
Fiar. 3.
\7CM
57CM
^\Tl = 2 (1 cos ax)
W2 = 2 (lcosa2)
2NP
(3)
H = ^^ (cos ax+ cosa2). ... (5)
Since the ammeter gives effective values, the maximum
current can be obtained from Table I. by the factor \/~2.
F=V2l (6)
tt V2 NI ,
H= — ?j (cos a!— cosa2)« ... (7)
From this equation the value of H was calculated for a
series of positions along the axis of the coil ; and the results
were plotted, as shown in fig. 4 (PL IV.).
The zero of the abscissa? was taken at the lower end of the
coil and the distance shown in the curves of fig. 8 (PL III.")
extends to the middle of the coil. The amperes noted on
the several curves are the effective values as recorded in the
tables. H is the maximum value of field strength per sq. cm.
along the axis.
192 Messrs. Mapriusson and Stevens on Visual
o
The above calculations are for the field strength along the
axis and for sine waves. Since the optic nerve lies near this
axis (see fig. 10), and as the field strength at the position of
the head is fairly uniform, it will be assumed that the field
strength H, as found by equation (7), applies to the area
between the optic nerves.
With the observer in position, as shown in fig. 2, the eyes
and hence the optic nerves were at a fixed distance from the
end of the coil. This distance was measured and found to be
as follows: —
Observer. Distance from lower end of coil.
M . . . . 170 cm.
S . . . . 170 cm.
P . . . . 147 cm.
L . . . . 170 cm.
These positions are indicated in fig. 4 by the dotted lines.
Since all the observations recorded in this paper for the
threshold values were made with the observer and the coil
in the relative position shown in fig. 2, the value of H at the
distance of the eyes from the lower end of the coil is of chief
importance.
In order to ascertain conveniently the field strength in the
plane of the optic nerves of each observer, in terms of the
amperes as given in the tables, a set of curves were drawn
as shown in fig. 5.
Curve " C ,J gives H at the middle of the coil.
Curve "MSL" gives H at 17*0 cm. from lower end of
the coil.
Curve " P " gives H at 14*7 cm. from lower end of coil.
It is readily seen that the relation between the field
strength and current is a constant for each observer, as
follows : —
MSL 307
P 302
In Tables I., II., III., IV., H' is the product of the
respective average currents and these constants, and thus
gives the maximum number of lines of force per sq. cm.
along the axis of the coil and in the plane of the optic nerves
of the observer.
Generators and Wave Forms. — For frequencies of 11 to
67 cycles inclusive, the current was obtained from an
alternator having a slotted armature and with a normal full
load rating of 60 K.W., 60 cycles, 1100 volts. Power was
supplied by a d.c. motor, belted to the generator and
arranged for the required range of speedvariation. The
Sensations caused by a Magnetic Field. 193
slotted armature produced harmonics in the current and
voltage waves. In order to determine the nature of the
waveforms a series of oscillograms were taken, both of the
current passing through the coil and the voltage impressed
upon its terminals. Although the relative magnitude of the
fundamental and the harmonics, as well as their phase
relation, varied for the several speeds and loads, the wave
forms were fairly uniform. In fig. 6 is shown a typical
current wave : — Oscillogram L 31 : /= 30 cycles, E = 96 volts,
1 = 5 amperes. The wave was analysed, and its equation
found to be
Y= 3\L7 sin (# + 3° 15') +031 sin (3a; 26° 20')
+ 006sin(5.r48° 17').
Hence the 3rd harmonic is 9*8 per cent, and the 5th
harmonic 1*9 per cent, of the fundamental.
In fig. 7 is shown a typical voltage wave as impressed
upon the terminals of the coil : — Oscillogram L 27 :
/=58'5 cycles, E = 895 volts, 1 = 24*4 amperes. Its equa
tion was found to be
Y=224sin(#0°56')+063 sin (3^ + 175° 530
+ 012 sin (5tfll° 03').
Hence the 3rd harmonic is 28*1 per cent, and the 5th
harmonic 5'4 per cent, of the fundamental.
For frequencies above 67 cycles the current was obtained
from mi alternator having a smoothcore armature and a
normal fullload rating of 35 K.W., 137 cycles, 1100 volts.
Several oscillograms were taken of the current and voltage
waves, all showing approximately simple sine waves. In
fig. 8 is shown a typical oscillogram (L 55) for this machine :
/=102 cycles, V = 250 volts (upper curve), 1 = 4 amperes
(lower curve).
We have not determined what effect, if any, these dis
tortions of the impressed stimuli from the simple sine form
may have upon the threshold values. For current waves
similar to fig. 6 the maximum is somewhat less than
y/2 times the effective value. Hence the actual value of
H and H' would be correspondingly less than the values
#iven in the tables. For the voltage wave as in fig. 7 the
maximum is greater than for the equivalent simple sine wave.
The wave is also more peaked, and hence the duration of the
voltage above the average value is reduced. It seems likely
that the shortening of the time in which the induced voltage
is above a given value may, within limits, neutralize the
Phil. Mag. S. 6. Vol. 2S. No. 164. Aua. 1914. O
194 Messrs. Magnusson and Stevens on Visual
effect of a higher maximum value, and that the effective
instead of maximum values should be used in making com
parisons of threshold stimuli. Since in computing H and
H7 the constant y/2 was used, the values in the tables are
directly proportional to the effective flux values.
Lines of Force cutting the Retina and Optic Nerve. — An
alternating current produces an alternating field, and hence
the flux H moves radially toward the axis of the coil as the
current increases, and in the opposite direction when the
current decreases. Therefore, the flux appearing in any
area like oab in fig. 9, bounded by two radii ox and oy and
a nerve, represented by the line a, 6, cuts across the nerve,
ab four times for each cycle.
In order to show what part of the flux cuts the optic
nerve, fig. 10 was drawn showing the relative size and
position of the coil, the skull, the optic pathways, and field
of vision. In fig. 10 the coil is presented as circular. As a
matter of fact the coil was elliptical, the major and minor
axes of which were 23*8 and 21*3 cms. respectively. Since
the movements of the lines of force are at each point normal
to the ellipse, no flux will cross the major and minor axes.
The angles AOB and COD show the incidence of the lines
of force upon the right and left retinas. It is evident that
more lines of force cut the nasal halves of the retinae than
the temporal halves. It is also apparent that the optic nerves
lie almost parallel to the normal of the elliptic coil, and that
they therefore will be cut by comparatively few lines of
force.
The solenoid was made lightproof by fastening strips of
rubber insulator over the joints between the segments of the
coil and by covering the top with felt. The lower end into
which the head of the observer was placed was closed by
stuffing felt wadding around the neck of the observer.
Sensations caused by a Magnetic Field,
195
Inasmuch as the series of ten observations did not require
ordinarily more than ten minutes, the exigencies of fresh air
could safely be met. In determining the threshold, the field
was begun subliminally and gradually increased until the
observer signalled, by means of an electric light which he
operated with a button, that the visual sensation was just
apparent. Although the field was gradually increased in
strength by taking out resistance, the field was intermittently
made and broken by a switch which was operated by an
assistant. A pendulum of suitable length enabled the
assistant to keep the switch closed for one second and open
for an equal length of time. The period of stimulation and
the intervening interval were thus kept tolerably constant.
Following the observation in which the stimulus was in
creased from subliminal to liminal, there occurred an
interval of a few seconds when the instruments were read
and recorded. Inasmuch as three persons cooperated in
this part of the experiment, the work was done rapidly. In
the second observation of the series, the stimulus was begun
far above the threshold and decreased slowly and inter
mittently, as described above, until the sensation of light
02
196 Messrs. Magnusson and Stevens on Visual
became visible, the observer signalling with his light when
this point was reached.
By thus alternating, a series of ten observations were
taken with their corresponding readings on the instruments.
In Table I. are shown the intensities of current in the coil
for each observer's ten observations for each frequency.
These intensities are the values of the stimulus which were
just sufficient to cause a sensation.
The abbreviations "Inc." and " Dec." stand for increasing
and decreasing, respectively. An increasing stimulus is one
which changes from subliminal to liminal. A decreasing
stimulus is one which changes from supraliminal to liminal.
Also the average (Av.), the mean variation (M.V.), and the
probable error (p. e.) of the ten readings are given. H' is
calculated by the formula
H' = K x i, , (8)
where K is the constant derived from fl^. 5 and i the average
threshold value of the current.
Observed Data and Tables.
The ammeter readings for threshold values are recorded
in Table I. The value of J37 is the product of the average
current and the constants obtained from fig. 5.
Table II. and curve in fig. 11 give the maximum strength
of field H' in lines per sq. cm. along the axis of the coil and
in the plane of the optic nerve.
Table III. and curve in fig. 12 give fx H', values pro
portional to the induced voltages. Assuming that the
sensation of light is caused by this induced electromotive
force, then the numbers in Table III. represent the relative
magnitude of the stimuli producing threshold effects for the
several frequencies.
If a second assumption be made, that in whatever nerve
circuits or paths this alternating E.M.F. may be induced,,
there exists a simple resistance only, then the resulting
current should also be proportional to the same series.
Hence the quantities in Table III. correspond to the term
" i " in Nernst's law (1) . Dividing these values by y/f we
obtain Table IV. If the above assumptions be correct, and
since the values given in Table IV. are not constant but
increase with the frequency, it is shown that Nernst's law
does not apply to this form of nerveexcitation.
Sensations caused by a Magnetic Field.
197
Table I.
Threshold values — effective amperes in coil.
/=11 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
12/19/12.
12/19/12.
12/19/12.
12/19/12.
Stimulus.
269
398
942
550
Inc.
2*55
504
1011
570
Dec.
346
594
1315
515
Inc.
377
670
840
550
Dec.
494
678
1546
540
Inc.
337
648
899
530
Dec.
420
895
1442
500
Inc.
451
460
930
435
Dec.
468
630
1080
525
Inc.
442
434
1090
421
Dec.
380
582
1110
514
Av.
•70
107
195
•37
M.V.
•20
•30
•55
•10
Pe
119
179
335
158
H'.
/=18 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
12/5/12.
425
12/5/12.
12/5/12.
12/5/12.
Stimulus.
425
1058
600
Inc.
773
667
1060
630
Dec.
909
935
1325
640
Inc.
828
746
1020
500
Dec.
999
858
1440
720
Inc.
855
688
960
663
Dec.
664
922
1242
650
Inc.
693
810
920
473
Dec.
705
1000
1580
645
Inc.
753
621
960
500
Dec.
631
760
757
1160
002
Av.
142
138
196
•67
M.V.
•40
•39
55
•19
Pe
232
232
350'
185
H'.
198 Messrs. Magnusson and Stevens on Visual
Table I. {continued)
/=25 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
12/23/12.
12/23/12.
12/23/12.
12/23/12.
Stimulus.
305
552
970
425
Inc.
310
567
1015
508
Dec.
482
808
860
700
Inc.
378
675
7'90
548
Dec.
8'48
860
860
618
Inc.
635
700
630
565
Dec.
925
808
890
865
Inc.
708
610
6'40
666
Dec.
650
603
1000
804
Inc.
7'47
610
650
555
Dec.
629
679
831
625
At.
208
•92
122
107
m.y.
•59
•26
•35
•30
pe
193
208
251
192
H'.
/=32 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
11/23/12.
11/23/12.
11/23/12.
11/23/12.
Stimulus. '
427
538
580
7'20
Inc.
434
530
538
590
Dec.
432
558
621
675
Inc.
436
407
542
538
Dec.
450
572
558
731
Inc.,
443
415
550
556
Dec.
439
438
559
744
Inc.
425
428
500
601
Dec.
559
544
549
670
Inc.
405
436
563
582
Dec.
445
487
556
646
At.
•24
•62
•20
•62
M.Y.
•07
•18
•06
•18
Pe
137
149
168
198
H\
Sensations caused by a Magnetic Field.
199
Table I. (continued),
f=. 39 c)rcles.
C.E.M.
H.C.S.
F.B.P.
MX.
Direction of j
11/28/12.
11/28/12.
11/28/12.
11/28/12.
Stimulus.
1
347
409
4 68
512
Inc.
213
326
424
394
Dec.
204
383
486
571
Inc.
204
384
334
419
Dec.
357
375
338
480
Inc.
210
382
301
423
Dec.
363
505
523
432
Inc.
210
355
400
402
Dec.
326
420
517
441
Inc.
210
370
348
410
Dec.
2 64
391
414
448
Av.
•67
•32
•49
•43
M.V.
•19
•09
14
•12
Pe
H'.
81
120
125
138
/=46 cj'cles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
11/28/12.
11/28/12.
11/28/12.
11/28/11.
Stimulus. :
314
396
517
550
Inc.
293
291
444
396
Dec.
28S
3'13
494
484
Inc.
302
271
388
388
Dec.
285
335
452
601
Inc.
304
280
274
420
Dec.
313
442
452
470
Inc.
348
376
305
372
Dec.
337
368
437
585
Inc.
2*68
285
337
358
Dec.
305
336
410
462
Av.
•19
•48
•67
•76
M.V.
•05
•14
19
•21
Pe.
94
103
124
142
H'.
200 Messrs. Magnusson and Stevens on Visual
Table I. [continued)
/=53 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of 
11/21/12.
11/21/12.
11/21/12.
11/21/12.
Stimulus. 
j
530
598
850
890
Inc.
545
630
740
705
Dec.
5m
600
870
950
Inc.
575
570
690
780
Dec.
575
545
800
760
Inc.
575
624
700
815
Dec.
550
562
800
740
Inc.
545
502
630
675
Dec.
585
5'90
1000
830
Inc.
592
538
690
750
Dec.
563
576
780
790
Av.
•17
•33
•90
•56
M.V.
•05
•09
•26
•16
Pe
173
177
235
243
H'.
/=60 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
11/21/12.
11/21/12.
11/21/12.
11/21/12.
Stimulus.
!
450
580
685
605
Inc.
400
5'70
590
430
Dec.
530
440
726
445
Inc.
475
480
467
460
Dec.
570
550
748
468
Inc.
450
500
545
452
Dec.
650
595
695
458
Inc.
500
515
596
465
Dec.
555
510
850
434
Inc.
475
600
527
450
Dec.
506
534
.643
4 67
Av.
•57
•45
•82
•28
M,V.
•16
•13
•23
•08
Pe
155
164
194
143
H'
.
Sensations caused by a Magnetic Field.
201
Table I. (continued).
/=67 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
13/14/12.
11/14/12.
11/14/12.
Stimulus.
502
760
745
Inc.
437
443
568
Dec.
601
601
555
Inc.
440
4 50
484
Dec.
644
672
528
Inc.
383
590
428
Dec.
544
792
558
Inc.
424
636
425
Dec.
596
1001
478
Inc.
408
760
419
Dec.
498
671
519
Av.
•80 140
•69
M,V.
•22 40
•20
p. e.
153' 203
159
H'.
/=67 cycles.
H.C.S.
Direction of ,
H.C.S.
Direction of
11/14/12.
Stimulus.
11/14/12.
Stimulus.
360
Inc.
620
Dec.
387
»
576
446
„
530
w
625
>»
496
660
444
660
Dec.
407
Observex rested
365
3 minutes.
400
Inc.
682
Inc.
430
6'68
Dec.
512
650
„ I
575
"
525
Av.
•90
M.V.
•17
Pe
161
H'.
202 Messrs. Magrmsson and Stevens on Visual
Table I. (continued),
f— 74 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
1
Direction of
5/14/13.
5/14/13.
5/27/13.
Stimulus.
400
820
780
Inc.
250
840
715
Dec.
415
910
647
Inc.
388
640
7 00
Dec.
428
780
605
Inc.
413
580
590
Dec.
513
780
550
Inc.
443
760
620
Dec.
600
840
715
Inc.
513
710
640
Dec.
436
760
656
Av.
•65 70
•57
M.Y.
•18 19
•16
Pe
134 230
202
H'.
/=81 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
5/14/13.
6/3/13.
5/27/13.
Stimulus.
580
970
731
Inc.
520
750
645
Dec.
550
920
7'45
Inc.
520
800
692
Dec.
630
960
753
Inc.
600
660
760
Dec.
600
870
654
Inc.
590
630
740
Dec.
590
950
7'00
Inc.
550
680
703
Dec.
570
820
712
Av.
•30 110
•33
M.V.
OS 31
•09
Pe
175 248
219
H' !
Sensations caused by a Magnetic Field.
203
Table I. [continued).
f=88 cycles.
C.E.M.
H.O.S.
F.B.P.
MX.
Direction of
5/31/13.
5/31/13.
5/31/13.
Stimulus.
540
1010
730
Inc.
670
760
640
Dec.
470
1010
760
Inc.
520
680
710
Dec.
470
960
740
Inc.
510
720
620
Dec.
480
1000
600
Inc.
460
760
670
Dec.
480
1040
740
Inc.
600
860
670
Dec.
520
880
690
Av.
•50 120
•50
M.V.
•16 102
•14
Pe.
160 266
212
H'.
/=95 cycles.
C.E.M.
H.C.S.
F.B.P.
MX.
Direction of
5/31/13.
750
5/31/13.
5/31/13.
Stimulus.
1000
715
Inc.
710
720
535
Dec.
650
1060
780
Inc.
680
1090
595
Dec.
590
810
755
Inc.
640
1080
650
Dec.
530
810
8 30
Inc.
530
......
800
610
Dec.
590
1230
725
Inc.
550
860
580
Dec.
620
940
677
Av.
•60 140
•73
M.V.
•17 "39
•21
Pe
190 266
208
H'.
204
Messrs. Magnusson and Stevens on Visual
Table I. (continued),
/=102 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
6/3/13.
6/3/13.
6/3/13.
Stimulus.
840
1160
970
Inc.
790
790
580
Dec.
790
1280
760
Inc.
670
800
620
Dec.
680
1300
780
Inc.
780
820
660
Dec.
660
1220
765
Inc.
720
830
630
Dec.
690
1200
680
Inc.
650
780
580
Dec.
720
1020
702
... .Av.
•50 210
•91
M.V.
•16 59
•26
Pe
222 308
216
H'.
/=109 cycles.
O.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
6/3/13.
6/3/13.
6/3/13.
Stimulus.
760
1240
925
Inc.
790
860
625
Dec.
730
1270
700
Inc.
710
770
605
Dec.
880
1270
830
Inc.
800
760
590
Dec.
7'15
1270
620
Inc.
715
890
560
Dec.
770
1270
690
Inc.
900
810
540
Dec.
777
941
668
...... Av.
•52 223
•94
... ..:m.v.
•14 63
•26
Pe.
239 285
205
H'.
Sensations caused by a Magnetic Field.
205
Table I. (continued),
/=116 cycles.
C.E.M.
H.C.S.
F.B.P.
M.L.
Direction of
6/6/13.
6/6/13.
6/6/13. '
Stimulus.
670
1130
680
Inc.
555
900
590
Dec.
515
1300
690
Inc.
555
730
525
Dec.
545
1150
650
Inc.
590
820
580
Dec.
580
1170
710
Inc.
630
780
580
Dec.
645
1170
665
Inc.
655
770
555
Dec.
592
990
621
Av.
•42 190
■bo
M.V.
•11
161
•15
Pe.
182
299
191
7=123 cycles.
C.E.M.
H.O.S.
F.B.P.
M.L.
Direction of
6/6/13.
6/6/13.
6/6/13.
Stimulus.
670
1100
690
Inc.
690
780
560
Dec.
650
1190
610
Inc.
740
920
580
Dec.
710
1200
640
Inc.
640
890
560
Dec.
580
1200
620
Inc.
630
890
560
Dec.
600
1240
620
Inc.
740
823
550
Dec.
660
1023
590
•40 162
•30
M.V.
11 46
•OS
Pe
203 309*
181
H\
206 Visual Sensations caused by a Magnetic Field,
Table II.
Threshold values of W.
/.
H'.
O.E.M. H.C.S.
F.B.P.
MX.
11
119
232
193
137
81
94
173
155
153
134
175'
160
190
222
239 »
182
203
179
232
208
149
120
103
177
164
161
335
350
251
168
125
124*
235
194
203
230
248*
266
266
308
285
299
309
158
185
192'
198
138
142
243
143
159
202
219
212'
208
216
205
191:
181
18
25
32
39
46
53
60
67
74
81
88
95
102
109
116
123
Table III.
Threshold values of" H' x f.
/.
H'x/.
C.E.M.
H.C.S.
F.B.P.
M.L.
11
18
25
32
39
46
53
60
67
74
81
88
95
102
109
116
123
1309
4176.
4825
4384
3159
4324
9169
9300
10251
9916
14175
14080
18050
22644
26051
21112
24969
1969
4176
5200
4768
4680
4738
9381
9840
10787
3685
6300
6275
5376
4875
5704
12455
11640
13601
17020
20088
23408
25270
31416
31065
34684
38007
1738
3330
4800
6336
5382
6532
12879
8580
. 10653
14948
17739'
18656
19760
22032
22345
22156
22263
Theory of the String Galvanometer of Einthoven. 207
Table IV.
Threshold values of W x n//.
H'X V/
C.E.M.
H.C.S.
F.B.r.
M.L.
11
395
970
965
775
509
638
1260
1200
1265
597'
970.
1040
845
750
699
1290
1273
1330
1120
1440
1260
950
782
842
1705
1508'
1680
1980
2230
2500
2600
3120
2990
3220
3410
525
765
960
WW*
863'
96L
1760'
1185
1318'
1740'
I960
1980
2040
2190
2140
2060
2000
18
25
32
39
46
53
00
67
74
81
88
95
102
109
116
1150
1573
1500
I860
2250
2500
1965
2230
123
Electrical Engineering Laboratories,
University of Washington,
Seattle, Washington, U.S.A.
July 10, 1913.
XXY. Theory of the String Galvanometer of Einthoven.
By Albert C. Orehoke, Ph.D.*
THE theory of the transverse vibration of stretched
strings is very fully treated in Lord Rayleigh's ' Theory
of Sound 'f. The string galvanometer of Einthoven requires,
however, special consideration because of the manner in
which the impressed force is applied to the string. It is not
applied at a single point of the string as is the case with a
plucked string, a pianoforte wire, or a violin string, but
throughout almost its entire length, and, since an electrical
current forced through the string supplies the motive power,
the nature of the circuit external to the galvanometer has an
influence upon the motion of the string which must be taken
into the account.
Imagine a stretched string, fixed at its two ends, having
* Communicated by the Author.
t Rayleigh, ' Theory of Sound,' vol. i. Chap. VI.
208 Dr. A. C. Crehore on the Theory of the
sufficient electrical conductivity to admit of carrying a cur
rent, immersed throughout almost its entire length in a
powerful magnetic field, and we have the essential features
of the string galvanometer. The direction of the field is, of
course, approximately perpendicular to that of the string,
and the transverse motion of each element of the string due
to the mutual action between the current in it and the
magnetic field is in a third direction approximately perpen
dicular to each of the other two directions.
In this paper the string will be treated as if it were
immersed in a magnetic field the lines of which are parallel,
but the force may vary in intensity from point to point along
the string. As particular cases two different field dis
tributions are considered, first, the uniform field which most
nearly represents the galvanometer as at present constructed,
and second, the best distribution for simplifying the resulting
motion of the string. The open space at the centre occupied
by the microscope and the ends of the string which project
beyond the field are not specifically considered, as it will bo
apparent how these irregularities may be allowed for in an
approximate way when the data are known.
Let PQ, fig. 1, be the element of any string of length els,
Fig. 1.
T(+^T(
and confined to a plane. The element is acted upon by the
tension Ta at P and the opposite tension Ti+dTx at Q, and
the resultant of all external forces Fds in the plane of the
string acting at any angle %.
If the element of the string is at rest these three forces are
in equilibrium, and resolving along the tangent, we find
5+FcosX = 0, (1)
and along the normal
^FsinX = 0, ...... (2)
ds
where r = jn = the radius of curvature at P.
du
String Galvanometer of JEinthoven.
209
The mechanical force exerted upon an element of the
string ds, due to the mutual action of the current I and the
field H, is always perpendicular both to the field and the
direction of the elementary current, that is, it is in the
direction of the normal to the curve assumed by the string,
IT
and its value is HIds. Hence in (1) and (2) ^ = ; HI
takes the place of F, and we have
/7T
— 1 = 0, and Tx = a constant throughout the (3)
as
and
length of the string,
r =
HI
or r cc
H
W
The curve assumed by the string carrying a steady direct
current when the field is uniform is, therefore, the arc of a
circle, and since the deflexions employed in practice are very
small compared with the length of the string, the radius of
curvature is correspondingly large. Let OL, figure 2, repre
sent the chord of the string which forms the arc OBL with
Fig. 2.
*
3
o^
> "
y^
^
c}
P
/ i
y
0'
centre at 0', showing a small deflexion y0 at the centre of
the length. With 0' as origin, and abscissae parallel with
OL, the equation of the circle is
Transferring the origin to 0, so that OL becomes the axis
of a, by putting
x » m  g
and jf = y + p,
where 0'C = p, the equation of the circle is
x2lv+y2 + 2p!, = 0 (5)
Phil Mag. S. 6. Vol. 28. No. 164. Aug. 1914. P
210 Dr. A. C. Crehore on the Tlwory of the
And from the properties of the circle,
Hence the radius
v_ , ;vo
^+f (6)
This expression for the radius together with that already
found in (4) gives the tension
\ $Vn 2 /
In practice the deflexions are small and the second term
may be omitted, giving
TI=™ (7)
This expression involves three quantities Z, I, and y0, which
may be accurately and easily measured. H and Tx are not as
readily measured directly. If, however, H is determined for
one string and one strength of field, it remains the same for
any other string. The tension cannot be measured directly
with a fine quartz fibre, but if a fine wire is substituted for
the fibre, the upper end being fastened to the same point as
the fibre, and the lower end passed through a Vgroove held
in the same position as the lower end of the fibre, weights
may be suspended from the wire and the tension thus
directly measured. By this device the field strength is found
from (7), and it has the advantage that the average value is
measured in just the same position as the fibre occupies,
so that when the fibre is replaced its tension is indirectly
measured by a knowledge of H.
The order of magnitude of the deflexion and the radius of
curvature may be obtained from an actual case. With a
magnification of 900, and a deflexion of the shadow of the
centre of the string of 3 cm., and length of string 14 cm., the
deflexion y0 = qaa = '00333 cm. ; and ^— = 7350 cm. The
radius of the arc formed by the string is therefore 73#5
metres.
Solving (5) for y and using the positive sign before the
radical, we find as the equation of the circular arc formed by
the string
y = — p f ( — #2 f fa f p2)*.
y
String Galvanometer of EintJioven. 211
Putting z2=— x2 + las, and expanding the radical into in
finite series, we find
i2 _ ** 1.3£6
y"2p 2Apz 2.4. 6>5
Since p is always a very large quantity compared with z, the
maximum value of which is ^when x = ^ , all terms after the
first in the series are of the second order of smallness and
may be omitted, giving
=!(*** 2) (8)
This is the equation of the parabola which most closely
approximates the arc of the circle throughout the length of
the string.
The equation (8) may be developed by Fourier's series *
into the following, which will be used as the equation of the
string when deflected by a steady direct current to determine
later the constants in the equations of motion,
^=,yo_sinr + 3i8inr+5,«n — + .... J, • (9)
where — p is written instead of x in the previous equation (8) .
The Differential Equations.
We will next form the differential equations applying to
the string under the conditions of the string galvanometer.
Let y denote the deflexion of an element of the string at the
distance x from the end and time t, and Y the impressed
force. The partial differential equation f of motion of the
string is then
g + ij.rfJJ+Y (10)
at* at d,i K
*^The developments of x and or by Fourier's series from which (9) is
directly derived are
x = 2( sin x — ^ sin 2x + ~ sin 3x  . sin 4r + . . . . J
a 2r/7T2 4\ . 7T2 . 0 , /7T2 4\ . 0 7T2 . ,
* = A\l ~ V) Smr~ 2 9m 2vH 3 ~ sO Sm 3*~ TSm4a:
+(fi)
■J'
t Rayleigh, « Theory of Sound,' 1894 Edition, page 192.
P2
212 Dr. A. C. Crehore on the Theory of the
where h is a constant damping factor, and
T
a2 = — = a constant. (11)
P '
Tx being the tension and p the mass of string per unit of
length.
Instead of using this form of the equation it is advanta
geous to express it in terms of the socalled normal *
coordinates, which gives the equation the form
2
(f>s + k(f>s + ns2s =]&* (12)
where k is the air damping factor, and
ng—— j; orn/=^ — , s being an integer. . (13)
The deflexion of the string, y, in terms of the normal co
ordinates is
, . irx , , . 2ttx , , . Zirx ,., ..
y = ^1sinr + $2sin — ^ l^gsin— ^— + (14)
and the velocity
. 7TtV • . 2tT# • . Zirx .. K.
y = cj>1 sinr f<£2sm — , f$3sm—  + (15)
The customary expression for <3>s in terms of the mechanical
force is
<&g = I p Y sin ^^ dta,
where pYdx represents the force upon an element of the
string. In our case this is replaced by Hidx, where i
denotes any variable current, and hence
, = if KsmS™dx (16)
Assuming that the string is in a uniform field throughout
its entire length, this integral is
3>s= — (1cosstt) ..... (17)
As 5 takes in succession all integral values from 1 on,
, vanishes for all even values, showing that the impressed
force due to a current in the string generates no even
harmonics.
* Thomson & Tait's ' Natural Philosophy/ first edition 1867, § 337.
String Galvanometer of Evnihoven, 213
If we write a constant hs to abbreviate the expression
^ = ™(lcos^), (18)
we have
Ip
and the equation (12) becomes
'(ps + Jtfis + ntys^hsi (19)
The Circuit Equation.
When the circuit of the string is completed outside of the
galvanometer, the motion of the string of itself generates a
current which opposes the motion, giving rise to the well
known electromagnetic damping. If R is the total resistance
and L the inductance of the circuit, and e' the back electro
motive force generated by the movement of the string, we.
have
e=Bi+Ljt+e'.
Since the back electromotive force is proportional to the
velocity we may write
e =h/(j)$}
where hj is a constant quantity. The rate at which
mechanical energy is supplied to the moving string is equal
to the rate at which the electrical energy is converted into
beat. The former is the impressed force, hi times the
velocity 8 with
the time
L ■■■ /R L,\v /R 7 L , h\. R 2, ,,,* m\
214 Dr. A. C. Crehore on the Theory of the
The solution of this equation might be obtained under
certain conditions, but it is not the present purpose to
examine the effects o£ inductance in the external circuit.
Assuming, therefore, L to be small and negligible (21) is
reduced to the following equation of the second order,
*•+(*.+ S^ + n»2*'=e=l/(«)=^' • • (22)
Upon comparing this with (19) it appears that the only
difference is in the coefficient of £ — €s) . . . (25)
where
sine=[(n/_j;r+,sW]J • • • • (26)
cose= 1. . . . (27)
[Ww2)2+Wp
tane=^2 (28)
The values of n, and hs are given in (13) and (18) above,
and 1= =5 , the maximum value of the harmonic current.
String Galvanometer of Eintlioven* 215
For the complete solution the complementary function
containing the arbitrary constants must be added to (25).
This takes three forms according to the relative values of
the physical constants ns and h„
When n8>%k» the transient portion takes the oscillatory
form
_h£
8=(A8+Bst)e 2 (35)
As the velocities corresponding to the three cases just
given are required for finding the arbitrary constants, they
are given below.
When n8>\k„
fk 1 M
(j)s= —AA scos (n/t**,) +W sin (n.'t — a,) > e 2 . (36)
When ns<ps
4.= A.'p,'J*'t+B.p.''Jt9"f (37)
When ns=\ks
kst
$=[b.(A. + B^]«~2 (38)
216 Dr. A. C. Crehore on the Theory of the
Having obtained the expressions for the normal co
ordinates, we may write out the complete values of the
deflexions, ?/, and the velocities, y, at any point x and time t
by the relations in (14) and (15) between the normal co
ordinates and these quantities. As a preliminary step we
will write out in full the values of he, ks, nS) and es as s takes
in succession the values 1, 2, 3 &c.
If 5 is odd, by (18)
(39)
7 4H • 1  hl • h  lh •
tip 3 5
h  1
. . fls —
S
If even
/,2=A4 = /,6 = &c. = 0.
If 5 is odd
h2 h 2
■■iksk+smP
(40)
If even
^2 = &4 == &9 = &C = k.
By (13), (SG^and (27)
7T /l\ na nx //M.
= TV7;B! = i; •••^=7 • • • • (41)
p
ks(o s2%
(42)
Slne«~ [(5v«2)+^>2]§; °oses~ [(sV»2)+y©2]L
The permanent periodic portion of the motion of any
point of the string in a uniform magnetic field is therefore
(s being odd), by (14), (25), and (42),
4HI f cos («£ — 6i) . 7T^ cos (art — e3) . 3ttx
j : ii sin — 4 — sin ■
irp Ll[(V«2)2 + ^iV]" l 3[(3V^2)2 + W]* l
. S7TX^\
COS ((Ot — €)
It is to be remarked that the coefficients of the terms in
this equation have particular numerical values because of the
original assumption that the string is immersed in a uniform
magnetic field. This causes the curve for constant current
to be the arc of a circle. At each and every point of the
string, corresponding to a single value of x, the motion of
that point is evidently a simple harmonic function of the time,
having a period agreeing with that of the impressed force,
but differing from it in phase, since all the simple harmonic
3>
String Galvanometer of Eintlioven. 217
waves represented by the separate terms have the same
period and may be combined into a single wave of this
period, and differing* in phase.
If the impressed force were not a simple sinewave but a
complex wave, a similar expression to (43) might be written
for each component wave in the complex impressed force.
There would then be a sinewave in the resulting motion of
the string corresponding to each component in the impressed
force, but each differing in phase from the impressed wave
by a definite amount depending upon the physical constants.
These phaseangles differ from each other among the waves
of different frequencies, and, as a result, the record cannot
be a strictly faithful reproduction of the impressed force ;
but the amount of departure from it may be small, and
differs according to circumstances.
If we do not introduce the supposition of a uniform
magnetic field, but suppose that H varies from point to
point along the length of the string, the value of 5 in (16) is
s = i(lB.sm~d.v (44)
Jo l
And if K denotes the integral, we have as the second
member of (12)
*=f «=« (45>
where h now stands for the more general expression
^=z Hsin^^. . . . (46)
If the impressed force is harmonic, as in (24), the general
solution applying to any field distribution for the periodic
part is
t f A . , A v . 7tx h2 . , . v . 2irx
y = L<  — sin 6, cos (coi — eijsinT + : — sin eo cos (g)£— e2) sin — —
Lh'iO) I fc2(o ' I
+ ~ sin e3 cos (tot — €3)5111—, + f . (47)
Some consideration will be given to the best form of
field distribution along the length of the string to give the
simplest results, that in which the motion best represents the
impressed force, and the force generates the least motion
which does not occur in the original electromotive force, but
this is deferred to a subsequent section.
Referring to the solution (43) for the caso of a uniform
field and harmonic impressed force, it is to be expected that
218 Dr. A. C. Crehore on the Theory of the
when the frequency of the forced motion is very small com
pared with the natural fundamental period, and the damping
is not excessive, the curve of the string, at the instant of
maximum electromotive force, should be the same as if a
direct current flowed in the string. The equation shows
this to be the case. Putting £ = 0, when the e.m.f. (24) is
a maximum, we have
_ 4HI r n^co2 . ttx 3V0)2 . Zttx
y ^/rL(ni2«2)2+^wsm i +3[(3w®2)2+^y]sm i
whence, making o> small compared with nx and neglecting
hs(o as compared with 7ij2, we have
4HI f . irx 1 . ?>irx 1 . 07TX \ ,._.
^=7r^?(smT + ysinr + psmr •"•••) • (48)
7rpiii
and by (7) and (13)
4HI 32
IT'
Hence the form of the string agrees with (9), which is the
approximate equation of the circular arc.
As the frequency of the impressed force increases, other
things remaining the same, the form of the string at its
maximum deflexion changes. For example, let us increase the
impressed frequency in (43) so far as to reach the first critical
IT
frequency, so that a> = rc1. Then sine1 = l and €j= ~, and
cos (cot — 6^ = sin cot, which becomes zero when the time is
zero and e.m.f. a maximum. The harmonics present are the
only elements which prevent the string from taking the form
of a straight line at this time. To find the curve at the
77"
maximum deflexion, put g>£= — . If the dampingfactor is
so small that ksco may be neglected in comparison with
Z2n^ — g>2, and hb £•
It is not necessary to multiply examples, and we pass to a
consideration of the transient portion of the motion given
above in (29), (31), and (35) in the normal coordinates.
When n8 >^4, and the magnetic field is uniform,
_i? , M 9
y = Aii&> then ns>ik, and the oscillatory case applies to all
the harmonics which occur. It is now necessary to find the
values of the constants A„ A2, . . . al3 «2, according to
the assumption that the initial position of the string is the
circular arc given by (9), and that the velocity of projection
is zero when * = 0. Putting t=0 in (50), we find
3/=A1cos«1sin j + A2 cos a2siD^ + . . . A8 cos u3 sin S~u
This is the same curve as (9) and the coefficients of
. TTX . 2TTX 0
sm~T' SmT"' may equated, giving
a 32 1
A1COSa1=^?/0
7T
1 39
A3cos«s=^^y0 y. .(51)
A2COS a2 = A4COS a4 = A6COSa6 = &C. =0. ^
By writing out the velocities from (36), when ks — k we
have
TTX
!1 I
V—{ —A! I  cos(V* — «i) + V sin (n/^ — ai) si
— A2 ^  cos (n2't — oc2) + n2r sin (n2't  «2) J sin^
A8£ . . . &c. . . .  e 2 . (52)
220 Dr. A. C. Crehore on the Theory of the
Making j/ = 0, when t = 0, we find
tan#s = ; = r . . (53)
The constants are now determined in terms of the funda
mental physical constants in (51) and (53). y0 is the
deflexion of the centre of the string, and, substituting the
constants in (50), we finally have the complete motion
I^t) L v1 t)j
1 r / ^2 \s
+  cos I (3V X etan
n . 3tt;
sin
H
(54)
It is evident that when £ = 0 the string takes the form of
the arc of a circle, as the equation then reduces to (9).
Since the galvanometer is arranged to observe the middle
point only of the string, we may put
. 7r.v ., . 3*7r"
A9=~? + W=ff>* (58>
i t~» 1 32 aJ
and Bs= ?v>w5zi*° (59)
When s is even As=B, = 0, and hence
32 r i / h't k'k . wx
+3W=/?7)l~^3 +/*3" Jsln~+ ••••r • (b0)
32
y
+ 3
222 Dr. A. C. Crehore on the Theory of the
The values of fij and fis" have been given in (33) and (34),
by which /jl/ — fisn = 2ns' ; hence
^°{l^'L\2 */ ; ""("2""1 / V JsmT
..... (61
This equation gives the complete motion of the string in
terms of the four physical constants Th p, Z, and k. If the
external circuit is open fe=k, and
As the tension of the string is increased the values of
nj become imaginary one after another, and the oscillatory
case applies when s is greater than a fixed value, while the
nonoscillatory solution applies to the fundamental value of
w
or the velocity ( = — =) of electromagnetic disturbance is
equal to a /' = that of the rotational wave in the elastic
medium.
This also satisfies the usual relations in elasticity and
electromagnetism, since
and
*• @g)
_/dY_B£\
and further implies the condition
£+■■ •+••■«.
since
d# By a*
P/«7. M«7. S. 6. Vol. 28. No. 164. i%. 1914. Q
226 Prof. D. N. Mallik on the
3. Further, the equation of small motion in an elastic
medium (n, v,)* = *>,y,*,0
7. The object of the present paper is to deduce the various
solutions that have been proposed of these equations, starting
from Poisson's solution of f = c\2% (which is reproduced
here for convenience of reference), in a synthetic manner.
8. We know that if f = c2v2fj we maJ put
i / x sinh (cW) ,
I = cosh (c*V)%+ ctv +>
X, ijr being arbitrary functions of xs y, z.
Suppose, initially,
where F, /are given functions of a, y, z.
Then the solution of the above equation may be put in the
form
where the arbitrary functions are replaced by known,
functions. For, obviously,
?o = X=/ and &> = ±V
(putting t = Q in the solution).
In order to interpret this symbolic solution, let a, /3, y,
be a point (P) on a sphere of radius r, centre 0(#, y, z), and
consider the integral
!•
.(■l.«l,«S)
where JS is an element of surface of the sphere,
Dynamical Theory of Diffraction. 229
On transformation, this can obviously be written
so that
,2
f<
B2
I£ now z = r cos 0, c?S = 27rr2 sin 0 d0,
the integral becomes
27rr2r/cos^vsin6>^
2sinhrv
= ±7rr .
Putting now r = cty and remembering that dS = r2JH,
where d£l is the solid angle subtended by dS at the centre,,
sinh c^v * ff a"^ , . Wo
= r 1 U d* +... + .. . hill.
... "S^p «&/£+•■•+■••) mb
c \7 4ttJ * y
Writing now u=lct, &c, we have the symbolic solution
replaced by
f = M f(* + ^ &c.)
then it can be shown that at P (whose distance from d$ is r)
the disturbance is
A f 1 . 0 ft r + r{\ . . ,a
sr I sm27r r^ — Hcosnr — cos nr*) do.
2\J rrx VT A. / v l
12. Remembering that the direction of displacement must
be always perpendicular to the direction of propagation, the
Dynamical Theory of Di fraction. 231
above result will be correct, only on the supposition that the
direction of vibration is perpendicular to both r and rx.
If this is not the case, we must resolve the displacement
(which is necessarily perpendicular to r^) in the direction
perpendicular to r (in the plane of r and the vibration). In
doing so, we observe that if u', t/, w' be the components of
the velocity perpendicular to r, and u, v, w those perpen
dicular to r1} then
v! = u — ql, &c,
where q is the component of u, v, w along r.
In the equation (4), then, u0 must be changed into uQ—q0l,
a corresponding change being effected in the second term.
The modified equation will then be
i„s=^=qidS + (^o_l^o\dS _ _ (
r \or or J r K J
where
and I, m, n the direction cosines of r.
Taking now the particular case of a displacement along
the axis of z, viz. f \t — ) (the direction of propagation
along the axis of x), we have, neglecting terms depending
on r~2, and remembering that the resolved part of the
displacement perpendicular to r is f\t') sin <£, where (j> is the
angle made by r with the axis of z, from (14) and (15)
if 9 is the angle between r and the direction of pro
pagation, both measured in the same direction.
In particular, if
,/ a?\ . . 2irc / a?\
„,. 1 A 2tto 1 2we(. r\
which is Stokes's result.
232 Prof. D. N. Mallik on the
13. Now, according to Rowland electric and magnetic
displacements being perpendicular to each other, and the
mean energies being equal, the total illumination must vary
as 1+ cos#.
He further argues that Stokes's solution is based on dis
placement and rate of displacement of his elastic medium,
but in an elastic medium there is not only displacement buf
rotation also, and the components of this rotation must also
satisfy the equation of continuity. But when a wave is
broken up at an orifice, the rotation is left discontinuous by
Stokes's solution. Now, the equation of propagation of a
rotation is the same as that of a displacement, and the two
are at rightangles to each other : both are important.
Hence, according to Rowland, on the elastic solid theory
as well as on the electromagnetic theory, the true solution
of diffraction will depend on the sum of two similar terms;
and it will follow, therefore, that diffraction cannot supply
the criterion whereby we may determine the relation between
displacement and polarization.
Glazebrook has, however, pointed out that the magnetic
displacement is in consequence of the electric, and if we
take account of the latter, we have done all that is necessary.
Rowland's results, moreover, contradict the results experi
mentally verified and theoretically obtained by Lord Rayleigh
on the blue of the sky.
14. In any case, for the production of diffraction effect
7T
we must have sin = ~ nearly. On this understanding we
can work out the case of oblique incidence at a narrow slit
as follows : —
Let x be the distance of a small element dx from the
centre of the slit. The disturbance reaching any point will
be [since the difference of phase due to disturbance from
the central element and that at a distance x is equal to
— (sin i — sin #)],
A
cosi + cos(i — 0) C 2 . ft 8 X , . . . ax\
= 2^ /*.sm2,r(^(sm*sm0)J
1 cos i fees (i — 0) .
. . — = a sm
47r sinismp
( (sin z — sin 0)j. sin 2ir(^—\
where 2771™—) is the incident disturbance, and I is
breadth of the slit.
Dynamical Theory of Diffraction. 233
dde of the zero position the d
cos i + cos (i + 0) . [I , . .
On either side of the zero position the disturbance will
therefore be
sin i + sin 6
sin /  (sin i ± sin 6) J,
which accounts for the want of symmetry observed by
Mr. 0. V. Raman (Phil. Mag. Jan. 1909) when a diffraction
band is produced at oblique incidence.
15. Finally, to solve
(&*?)+ = *(;**,*)
we have only to find the particular integral.
Proceeding in the usual way,
* = D^T?PF' where D^c4
= irJ— +±1*
21) LD + cA DcAj
= ~ \e~ctA £e*'AF'dt' + e°tA Vrt"AFW
= i fccosh c(tt')A}F'Jt' = ~ (cosh (tt^Fdt
attending to the meaning of the operation cosh {(.'(£ — Ov^}
[art. 8],
=ttt n?,,F0"Djn^' since °^~*
[art. 10]
2 1 dv,
17T
where <£t> is an element of volume, which is Lorentz's
result.
234 J
XXVII. The Discharge of Electricity from Points,
By P. J. Edmunds, B.A., Queen's College, Oxford*.
1. rTlHE potential necessary to cause an electric discharge
X to pass through air between a pointed conductor
and a plane has been measured under various conditions, and
empirical formulae have been given which fit the results
more or less accurately for particular sizes and shapes of the
pointed conductor.
On the other hand, most of the measurements were made
with the air at atmospheric pressure, and the results have
not been shown to satisfy any relation deduced from a theory
of the mechanism of the discharge.
On the hypothesis that the discharge through a gas is
determined by ionization of the gas by collision, it has been
shown f that, if V be the potential difference necessary to
cause a discharge between two conductors A and B through
a gas at pressure p, then the same potential V will also pro
duce a discharge between two conductors A' and B', obtained
from A and B by reducing the linear dimensions in the ratio
1/k, the gas being at pressure p' = kp. It has already been
found experimentally that this relation is true for the case
of cylindrical conductors, the gas being air. The present
investigation was primarily undertaken to test the theory
for discharges from the hemispherical ends of wires at right
angles to a conducting plate. This method of obtaining
" points " of a definite shape was first used by Zeleny J.
2. The discharge apparatus is shown in fig. 1 drawn
Fig. 1.
fljMP t( GuAGE •
DRYING /lf>Pfi*ATU6
roughly to the scale of 1:6. EE are the walls of a glass
cylinder. The bottom was closed by the brass plate Z, which
* Communicated by Prof. J. S. Townsend, F.R.S.
t Phil. Mag. May 1914, p. 789.
j J. Zeleny, Phys. Rev. vol. xxv. p. 317 (1907).
The Discharge of Electricity from Points.
235
also served as the plane to receive the discharge ; the upper
face of this plate was platinized. To the top of the glass
cylinder was cemented a brass plate D. This plate supported
the mechanism for holding and adjusting the plate, and was
insulated from it by the ebonite ring C. The plate D was
raised to the same potential as the point, and thus served as
a guardring ; any leakage over the glass came from the
plate D and did not pass through the galvanometer used for
detecting a current through the gas. A silver wire P served
as the discharging point. It was fixed in a brass rod Q ; on
this rod was cut a screwthread of one millimetre pitch which
worked in the brass supports F and F'. This rod could be
turned from outside the apparatus by the part A, and the
distance from the plane was measured on the scale S, divided
into centimetres, and the wheel W whose circumference was
divided into ten parts. This gave a direct reading to one
tenth of a millimetre, and the distance could easily be adjusted
to '002 cm. The reading for zero distance was obtained
from the point at which there was electric contact between
the wire and the plane. Wire gauze, connected to the plate
Z, was fitted round the inside of the lower part of the glass
to prevent distortion of the electric field in the neighbour
hood of the point due to unknown charges on the surface of
the glass.
The arrangement of the apparatns is shown in fig. 2. It
is the same as that used for determining the Sparking Poten
tials of concentric cylinders, except that there is an additional
conductor leading from the leydenjars to the plate D.
M is a Wimshurst machine driven by a small motor, V is an
electrostatic voltmeter, Gr is a mirror galvanometer capable
of detecting a current of 10~8 ampere.
The gas used was air. It was taken from the room and
dried and freed from dust in the same manner as that used
with cylinders.
3. In taking the readings the pressure was adjusted and
236 Mr. P. J. Edmund? on the
then the insulated system slowly charged till a discharge was
detected by the galvanometer. If this discharge was con
tinuous, there was usually no difficulty in determining the
potential necessaiw to produce a discharge. Occasionally
the potential dropped slightly on the commencement o£ a
discharge, but if the system was discharged and then charged
anew to the lower potential and maintained at that potential
for a short time, a current would commence, showing that
this potential was sufficient to start the discharge as well as
maintain it.
On the other hand, if the discharge was discontinuous and
consisted of a single spark, the procedure w^as as follows : —
the potential at which the first spark occurred was noted and
the system was charged to a somewhat lower potential and
maintained at that potential. It wras usually found that a
discharge would occur at this lower potential after waiting
for a short time. The process was then repeated until a
point was found at which no discharge could be obtained.
From test experiments it was found that it might be assumed
the potential was below the sparking potential if no discharge
occurred within about five minutes. The lowest potential at
which a discharge was detected was considered as the true
sparking potential.
This phenomenon of "lag" or retardation is well known.
In the present investigation it was most marked in the
case of the smallest wire used, diameter 0*5 millimetre.
With this wire the potential could sometimes be maintained
for some minutes, without any discharge passing, at a poten
tial fif tj per cent, higher than that ultimately determined as
the sparking potential. The discharge in such abnormal
cases was usually rather violent (as judged by the drop of
potential, throw of galvanometer, and appearance of dis
charge). The fact of a discharge having occurred a short
time previous by no means made it certain that the following
discharge would be free from lag. To illustrate the above
remarks the following set of figures are given. They were
obtained wdth a wire 0*5 mm. diameter, the point being
distant 0*25 cm. from the plane, with fresh air at 761 mm.
pressure. The wrire was slowly charged up positively till a
discharge occurred. This was repeated as rapidly as possible.
The potentials, in order, at which sparks passed were :
4950, 4750, 3950, 3950, 4000, 3750, 4500, 4050, 4150,
4250 volts.
In order to lessen the difficulties caused by the lag it was
decided to increase the initial ionization by an external
'Discharge of Electricity from Points. 237
agent. A small quantity of radium bromide, sealed up in a
small glass tube, was fixed to the support F' inside the glass
jar (the small tube is shown in fig. 1 marked Ra). The
ionization produced by this was very small, being insufficient
to discharge a goldleaf electroscope placed near it. It had
no appreciable effect on the insulation o£ the gas at potentials
lower than the sparking potential. It was sufficient, however,
to diminish the lag very considerably, though without sensibly
affecting the sparking potential. Thus, after putting in the
radium, working with a pressure of 763 mm., the wire and
distance being the same as above, small sparks were obtained
successively at 3760 volts.
The results obtained were quite regular apart from the
lag, sparking potentials obtained on different days, with and
without radium, being quite consistent. With the radium,
however, the sparking potentials could be obtained much
more easily and quickly, and so radium was used for all the
later determinations.
Great care was exercised to exclude dust and to prevent
any large quantities of electricity passing from the point.
It is probably due to this that observations repeated after
various intervals of time showed good agreement. Small
particles of dust undoubtedly have a considerable influence
on the sparking potential; and Zeleny has shown that con
siderable alterations in the shape of metal points may be
produced by the passage of currents for some time*. This,
and the deterioration of the gas, may account for the "ageing"
of points.
4. If the sparking potential is determined by ionization by
collision, a discharge should occur whenever the values of the
electric force along a line of force satisfy a certain relation.
It is assumed, however, that there is already a certain amount
of ionization in the gas. In the case of a gas unsubjected to
any external ionizing agent, the ionization is very small, and
the spark will not pass at the lowest possible potential unless
there is a favourable arrangement of ions along the line in
which the electric force is greatest. For a discharge between
concentric cylinders all radii are equally probable lines of
discharge, but in the case of points there is a small region in
which the force is greater than in any other. Hence if the
ionization is small the probability of there being a favourable
distribution of ions is much greater with cylinders than with
points, and is greater with large points than with small. If
the distribution of ions is unfavourable there will be a lag.
* J. Zeleny, Phys. Key. vol. xxvi. p. 137 (1008).
238 Mr. P. J. Edmunds on the
This is in general agreement with experience. In the ex
periments carried out with cylinders lag was practically
unnoticed.
5. The theory shows that, assuming some initial ionization,
the number of ions produced by collision becomes infinite if
a certain distribution of electric force is maintained in the
gas. But as soon as the current has commenced two in
fluences come into play which tend to alter the electric field.
In the first place, the current tends to discharge the system
and so lower the potential. In the second place, the current
constitutes a charge in the gas which changes the electric
force. When one of the electrodes is sharply curved and
the other nearly plane, the effective ionization all occurs
near the sharply curved electrode. The result is that the
current is carried through the greater part of the gas by ions
of one sign only, namely that of the sharp electrode. This
charge lowers the force near this electrode and diminishes
the ionization taking place near it without raising the force
in other parts of the field sufficiently to cause extra ionization.
The first of the two counteracting effects depends on the
capacity of the system and the rate of supply of electricity.
The second depends chiefly on the velocity of the ions. This
velocity becomes large at low pressures, and is greater for the
negative ions than for the positive. With cylinders it was
often found that with the lower pressures the nature of the
discharge could be changed from a violent spark, discharging
the system almost completely, to a rapid succession of small
sparks by reducing the capacity of the system as much as
possible.
Now a discharge will continue even after the potential
has dropped considerably below that necessary to start a
discharge, provided there are a sufficient number of ions in
the gas. This number of ions depends on the current that
has passed through the gas immediately before. Thus with
a small capacity the potential drops before enough ions have
been produced to carry on the discharge, and the current
ceases at a fairly high potential ; whereas with a large
capacity in the same case there might be a spark nearly
completely discharging the system.
In the case of point discharges it was found that the
external capacity did not have much effect on the nature of
the discharge. In this case the current is much more con
centrated than with cylinders, and it is probably necessary
fo make the capacity very small indeed to bring its effect
Into prominence.
In order that a truly continuous or glowdischarge should
Discharge of Electricity from Points, 239
take place it is necessary that the charge in the gas should
come into effect, and there must be a balance between its
effect and the tendency to increase of ionization. Even
when this state of equilibrium is attained it may be unstable;
that is to say, a small increase or decrease of potential may
cause the discharge to increase to a discharging spark or to
cease altogether. In the present investigation it was usually
found that in cases where there were glowdischarges
at high pressures and sparks at low pressures, there was
some intermediate pressure where it was possible to obtain
both glow or spark discharges at the same potential. Or
a glow discharge would suddenly terminate with a spark.
Tins theory also gives an explanation of those cases, noted
by various experimenters*, where it was found possible to
obtain steady currents above a certain magnitude but not
below.
It appears from the above considerations that it is not
possible to transform a discontinuous discharge into a con
tinuous discharge by merely reducing the external capacity.
In a discontinuous discharge the amount of electricity
passing on each occasion is reduced by reducing the capacity;
but the tendency to become discontinuous is independent of
the capacity. If the system is continuously supplied with
electricity, the interval of time between successive discharges
may be extremely small. Thus Zelenyf, working with
liquid points, found that it was possible to have discharges
consisting of small impulses which follow each other so
rapidly that even a telephone fails to detect the dis
continuity.
6. If a be the radius of a wire and d the distance between
its point and the plane, the sparking potential for a pressure
p is the same as that for a wire of radius y and distance T
with a pressure hp. Thus if  is kept constant the sparking
potentials for different values of a will be the same if the
pressure is adjusted to keep ap constant. In the present
investigation, three different silver wires were used for the
point : their diameters were 0*5, l'O, 1"5 millimetres respec
tively, and the ends were rounded approximately to hemi
spheres. Each of these wires was adjusted to give the three
values 10, 20, 30 to 9 and the sparking potentials were
measured for positive and negative discharges for various
* A. P. Chattock, Phil. Mag. [5] xxxii. p. 295 (1891).
t J. Zeleny, Phys. Rev. n. e. iii. p. 88, Feb. 1914.
240
Mr. P. J. Edmunds on the
pressures. From these results curves were drawn showing the
relation between Y the sparking potential, and p the pressure.
The values of V for a set of values of ap were taken from
these curves and are shown in the following table. Thus
for each pair of values of  and ap are shown three values
of V, obtained from different wires. It will be seen that
the agreement between the corresponding values of V is
quite good.
Table I.
Sparking Potentials in kilovolts.
for wires with hemispherical ends in air.
a = radius of wire in cm.
d= distance from plane in cm.
p = pressure of air in millimetres of mercury.
Wire charged positively.
ap.
25
50
100
200
300
370
d
et
d
= 10

=30
a
a
a
a =025 a = '05
a =075
a = 025
a =*05
«=075
a = 025
a ='05
T25
126
120
138
130
132
146
140
170
166
163
185
176
176
196
189
250
231
230
276
260
260
290
284
386
363
360
435
400
395
465
435
480
474
530
523
561
560
556
616
a=075
137
186
272
412
545
Wire charged negatively.
25
105
105
110
116
125
125
; 130
130
50
157
170
163
170
181
180
1 186
192
100
240
252
247
261
280
280
! 290
303
200
372
397
390
430
445
430
465
472
300
527
515
586
561
620
370
622
595
G80
127
186
290
451
600
There is considerable difficulty in finding an exact solution
of the electrostatic problem of a charged wire opposite a
plane, but an approximation for small points is obtained on
the assumption that the wire is shaped like a paraboloid of
revolution.
The potential for a paraboloid of revolution is
Alogr(l4 cos#),
where (r, 6) ar the polar coordinates referred to the axis
Discharge of Electricity from Points. 241
and focus. The effect o£ the plane may be represented by
adding the potential of the image in the plane of the
paraboloid.
The total potential is thus
V=Alogr(l + cos#)AlogV(l+ cos0').
The diameter of a small point may be taken as the breadth
of the wire at the point where the breadth is twice the
distance from the vertex*.
Hence the parabola whose latus rectum is 21 corresponds
to the point whose diameter 2a equals 4Z.
This parabola passes through the point
Along the axis
V=Alog^=Alog
I a
r= a = I
and A is given by V = V1, when r=a/4z.
.'. Y1 = Alog7/ approximately, since j is small.
If X is the force at any point along the axis
A A
X=
r
hta
where #, the distance from the vertex, is small.
4A
Xx (the force at the vertex) =■ .
It is now possible to find an approximate formula for the
sparking potential by the method used by Townsend for
cylinders!. It is known that practically no ionization by
collision takes place in air at atmospheric pressure if the
force is less than 30 kilovolts per centimetre. Also Bailie's
results t give the empirical formula V== 30 s + 1' 35 for the
sparking potential V in kilovolts through air at atmospheric
pressure when s, the length of sparkgap, is of the order of
a millimetre.
Let s be the distance along the axis from the vertex of the
* This was the method adopted by J. Zeleiry, Phys. Rev. vol. xxvi.
p. 130 (1908).
t J. S. Townsend, ' Electrician,' June 6, 1913.
X Bailie, Ann. de Chim. et Thys. [5] vol. xxv. p. 486 (1882).
Phil. Mag. S. 6. Vol. 2S. No. 164. Aug. 1914, R
242 Mr. P. J. Edmunds on the
wire to the point where the force is 30 kilovolts per centi
metre, then
4A .
30 =
As + a
The average force over this distance s is approximately
Xx + 30
Xi + 30=3Q j 135
9, .«
or Xx30 _T35
2 " s '
30 = — . — : — = . — — ; ^ij
4^_X130
a ~~ 30 '
/. (X!30)8_ 135x4
60 "" a '
18
X1 = 30 +
a log — . , o N
.. Y.A^— ^(l.^
This formula gives the sparking potential in kilovolts for air
at atmospheric pressure. / ^
Since Vi is a function of lap.  1, for other pressures
where P is the pressure in atmospheres.
The formula is obtained on the supposition that % and aP
are both small. For the smaller wires it is found that the
calculated values lie between those found experimentally for
positive and negative discharges. The agreement for the
larger points is not very good, though the formula gives
results of the correct order of magnitude for quite large
points.
A table has been made up showing the calculated and
Discharge of Electricity from Points. 243
observed values for a number of points and pressures. In
the first of the tables the experimental values are the means
of the results given in Table I. In the second table are
given a few of Zeleny's results for small wires.
Table II.
Comparison of Calculated Results and Experimental.
Sparking potentials in kilovolts.
ap.
d
 =10.
a
d
=20.
a
*=30.
a
Experimental.
+ 
Calc.
Experimental.
+ 
Calc.
Experimental.
+
Calc.
25
122
107
123
133
122
143
141
129
155
, 50
166
163
181
179
177
210
190
183
227 1
100
237
246
269
265
274
311
282
294
337 i
: 200
370
387
406
410
435
469
438
463
507 !
300
473
521
547
526
574
Cr 33
5'53
i
610
685
Experimental Results taken from paper by J. Zelenv.
Phys. Rev. xxv. p. 317 (1907).
j
a.
c/=l5.
d=\.
Experimental.
+
Calc. j
Experimental.
+ 
Calc.
•00244
1600
1125
152
0039
200
1475
181
1895
1465
181
•0091
245
1975
266
230
185
251
The pressure was atmospheric.
The value of the electric force Xx at the vertex of the
point has been shown above to be given by
IS
Xx = 30H  for atmospheric pressure.
\a
For very small values of a this tends to
JL8
or Xx \/a = l8.
112
X1 =
244 Dr. F. Horton on the
In this formula Xx is expressed in kilovolts per centimetre.
If /is the measure of the force in electrostatic units,
/V«=60.
Now Zeleny, working with liquid points, obtained the
empirical result *
/v/«=56*9.
The similarity of these two results both in form and
magnitude of the constant is very striking, especially when
the extent of the approximations is considered.
In conclusion, I should like to express my thanks to
Professor Townsend both for suggesting the research and
for kind help in carrying it out.
XXVIII. On the Action of a Wehnelt Cathode.
By Frank Horton, Sc.D.'t
SINCE the discovery by Wehnelt of the large electron
emission which takes place when a limecoated cathode
is heated in a vacuum, several theories have been put forward
to explain its action. One of the earliest of these, and one
which received the support of the discoverer of the effectf,
was that the electrons proceed, not from the glowing oxide,
but from the platinum strip or wire upon which it is heated,
and that the action of the lime is confined to a reduction of
the amount of energy required to liberate the electrons from
the metal. More recently the view has been put forward by
Fredenhagen§ that the electron emission from glowing lime
occurs as a result of the recombination of calcium and oxygen
which have been separated by electrolysis. A modification
of this theory has been suggested by Gehrts , who assumes
that the calcium and oxygen are separated, not by electro
lysis, but by thermal dissociation of the lime at the high
temperature. This view simplifies the explanation of the
manner in which the activity begins. This is supposed by
Fredenhagen to be due to the small thermionic current from
the platinum support electrolysing the lime in passing through
it, and thus giving a supply of calcium and oxygen to start
the greater emission from the lime itself.
* J. Zeleny, Phys. Rev. n. s. iii. p. 88, Feb. 1914.
f Communicated by the Author.
% See Wehnelt, Phil. Mag. [6] x. p. 80 (1905).
§ K. Fredenhagen, JBer. d. Kgl. Sachs. Ges. d. Wiss. lxv. p. 42 (1913).
 A, Gehrts, JBer. d. Deutsch. Phys. Ges. p. 1047 (1913).
Action of a Wehnelt Cathode. 245*
Fredenhagen's theory has been tested by experiments with
Nernst filaments, which give a convenient form of oxide
cathode without the complication of a metallic support ; and
a paper has recently been communicated to the Royal Society
in which it is shown that the emission from such an oxide
cathode is quite independent of the possibility of electrolysis,,
and tbat the emission from the material of the filament at a
given temperature is the same whether it is heated in the
usual manner by conducting an electric current, or is powdered
up and heated upon a platinum strip. Further experiments
have been made to test these theories of the origin of the
activity of a Wehnelt cathode, and two of these experiments
are described in the present paper. The first was made to
test whether the emission from lime depends upon the nature
of the material upon which it is heated ; the second experiment
is a test of the separation of calcium and oxygen by the passage
of a thermionic discharge from lime, and of the connexion
between the recombination of these elements and the electron
emission/
The dischargetube used in these experiments and the
method of heating the Nernst filament and of covering it
with lime are described in an earlier paper*. The tempera
tures were measured by means of a Fery optical pyrometer,
for the use of which I am indebted to Professor T. Mather,
of the City and Guilds College, London.
I. A comparison of the Electron emission from Lime heated
upon a Nernst filament with that from Lime heated upon
Platinum.
In a recent paper * the author has described an experiment
which shows that a large electronic emission is obtained from
lime when heated upon a Nernst filament, but attempts to
compare the emission with that given by an ordinary Wehnelt
cathode were unsuccessful on account of the difficulty of
maintaining the filament glowing in a vacuum at a tempe
rature low enough to enable measurements to be made in the
absence of luminous pencils of cathode rays. The appearance
of these luminous rays is always accompanied by great
overheating at the points of lime from which they start ; so
that it is impossible to ascertain to what temperature of the
cathode the measured current corresponds.
At temperatures rather lower than that at which these
luminous rays are seen, a much fainter luminosity occurs in
the dischargetube and, at the low pressures used in these
* Proc. Camb. Phil. Soc. xvii. p. 414 (1914).
246 Dr. F. Horton on the
experiments, seems to fill the whole bulb. This luminosity
appears quite gradually as the temperature of the cathode is
slowly raised, so that it is very difficult to say when it first
begins. It is not accompanied by any sudden increase in the
thermionic emission, showing that the increase of temperature
of the cathode produced by the luminous discharge is general,
and is not confined to a few points of its surface. Further
■experience of working with Nernst filaments has made it
possible to control the heating of a filament even at tempe
ratures as low as 1100° C, and a comparison of the electron
emission from lime heated upon a Nernst filament with that
from lime heated upon platinum has now been made. The
filament used was covered with pure lime, prepared from
marble, in the manner described in the paper already re
ferred to. It was heated by an alternating current, and the
thermionic emission was measured in a vacuum obtained
by the use of charcoal cooled in liquid air. During the
observations there was a faintly luminous discharge through
the residual gas; but at the lowest temperatures the lumi
nosity was very difficult to discern, and perhaps was sometimes
absent.
For measuring the emission from lime heated upon
platinum a fine platinum tube was fitted into the apparatus in
the place of the Nernst filament. This tube was covered with
lime and could be heated by an alternating current from a
transformer. The gas pressure was reduced as low as possible
by means of charcoal cooled in liquid air, and the thermionic
current was measured as the temperature of the limecovered
tube was gradually raised. A large negative emission occurred
immediately the cathode became luminous. The current soon
became steady; there was no gradual increase as had been
•observed in an earlier research with a platinum tube covered
with the material of a Nernst filament, and the first obser
vations of the emission at different temperatures were the
largest values obtained.
In comparing the emission from this limecovered platinum
with that from a similarly covered Nernst filament, a difficulty
arises on account of the unsaturated nature of the thermionic
current given by a lime cathode. Even at the very low
pressures and comparatively low temperatures used in these
experiments, any increase in the applied E.M.F. caused an
increase in the observed emission, and with the highest
voltages it was deemed safe to apply, there, was no sign of
saturation. Since a saturation current cannot be measured,
it is necessary, in order to compare the electron emissions in
the two cases, to measure the thermionic currents with the
Action of a Wehnelt Cathode.
24,7
same potential differences between Hhe electrodes. The
difficulty is to allow for the effect of the field due to the
heating current. In the case of the Nernst filament there
was an alternating potential difference of about 50 volts
between the two ends of the cathode, while with the platinum
tube about 16 volts were used — the exact value in each case
being different at different temperatures. It was finally
decided to apply between the terminals of the dischargetube
a potential difference of 207 volts in the case of the platinum
tube, and 168 volts in the case of the Nernst filament — these
being values easily obtained from the highpotential battery,
while the difference between them makes allowance for the
difference in the field due to the heating current in the two
cases. For both cathodes curves connecting the thermionic
current and the temperature were drawn, and from these
curves the following values of the emissions at different
temperatures are taken. The area of the cathode covered
by lime was approximately the same in the two cases. The
numbers given in the table are the largest values of the
Temperature,
centigrade.
Thermionic Current in 10 ° ampere.
Lime on Filament. 1 Lime on Platinum.
1100
1150'
1200
1230
129
202
340
462
41
102
230
339
thermionic current in both cases. On continued heating,
the emission from the limecovered filament decreased more
rapidly than that from the limecovered platinum, apparently
because the lime adhered more readily to the metal than to
the material of the filament. From both lime cathodes there
was a continued slight emission of gas which, if it were very
unequal in the two cases, would render a comparison of the
emissions impossible j for, with the strong electric fields used,
ionization by collisions would play an important part in
deciding the value of the measured thermionic current.
Throughout the experiments a carbon tube cooled in liquid
air was in connexion with the dischargetube so as to keep
the pressure low, and as nearly constant as possible.
From the table it will be seen that the numbers measuring
the electron emission from lime heated upon a Nernst
2±8 Dr. F. Horton on the
filament are rather larger than those for the case of lime
heated upon platinum ; but considering the difficulties in
making the comparison the differences are small, and we
are justified in concluding that, in these two cases, the
emission from lime does not depend on the nature of the
material upon which it is heated.
II. 1 he Liberation of Gas from a Wehnelt Cathode.
Observations have been made of the rate at which the gas
pressure in the apparatus increases when the limecovered
platinum tube is acting as a cathode, and when it is heated
but no thermionic current is allowed to pass. For this
purpose the gas pressure was taken down as low as possible
by means of a carbon tube cooled in liquid air ; this tube
was then shut off by a tap. The mercurypump was
separated from the dischargetube so that the volume into
which the evolved gas could diffuse was considerably
reduced, being now only that of the dischargetube, the
phosphorus pentoxide drying tube, the McLeod gauge, and
the connecting tubes — in all 550 c.c. Taking observations
at intervals of 15 minutes with the platinum tube at 1360° C,
alternately with no thermionic current passing and with a
current of an average value of 8 milliamperes, it was found
that the pressure continually increased during the first two
hours' heating, but at a rather greater rate when the discharge
was passing than when no discharge passed. The increase of
pressure per minute, however, in both cases became less as
the heating was continued, until after two hours there was
a small decrease of pressure in the absence of the discharge.
The largest observed increase of pressure in 15 minutes with
a thermionic current of 8 milliamperes was '0007 mm., the
mean increase during the periods with no discharge imme
diately preceding and following this being '0002 mm. If
the difference between these two increases of pressure is due
to oxygen gas liberated by electrolysis of the lime under the
action of the thermionic current passing through it, the
amount of oxyger; so liberated is less than onethousandth
of that which would be set free according to Faraday's law,
on the assumption that the conduction of the lime is entirely
electrolytic, and that no recombination of the separated
elements takes place. It was thought possible that part of
the difference between these two increases of pressure was
due to the extra heating of the residual gas by the passage of
the luminous discharge. Observations were therefore made
to test this; but it was found that the pressure alteration due
Action of a Welinelt Cathode. 249
to the heating o£ the gas by the glowing cathode itself was
not more than onetenth of the increase observed above, and
was hardly measurable.
After the limecovered platinum tube had been heated for
some hours it was always found that the passage of the
luminous discharge was accompanied by a liberation of gas,
and that subsequent heating with no thermionic current
passing decreased the pressure; the actual numbers were
not very regular, but the same general result was always
obtained.
The most obvious explanation of this result is that oxygen
is liberated by electrolysis of the lime, and that recom
bination of oxygen and calcium goes on when the cathode is
afterwards warmed in tbe liberated gas. Assuming that
this is the correct explanation, it follows that recombination
is also going on at the same time as the electrolysis, and the
question arises: Is this recombination the cause of the electron
emission ? To test this point observations of the pressure
were taken at intervals of 15 minutes with the high potential
continually applied to the terminals of the dischargetube.
During the first 15 minutes a temperature of about 1400° 0.
was maintained, and a thermionic current of about 4 milli
amperes passed. The temperature was then lowered to about
600° C, and kept at that value for 15 minutes, after which
it was raised again to 1400° C, several observations at these
two temperatures being taken alternately. With the cathode
at the lower temperature, no thermionic current could be
detected with a galvanometer giving L division deflexion for
a current of 1*94 x 10~9 ampere; but there was an average
decrease of pressure of '00063 mm. during these periods of
15 minutes. At the higher temperature, with the thermionic
current passing, there was an average increase of pressure of
about the same magnitude. Since the effect of the decreased
temperature of the cathode upon the gas pressure is almost
inappreciable, it is evident that there was an absorption of
gas taking place while the cathode was heated at 600° C.
This absorption is presumably due to the union of electro
lytically liberated oxygen and calcium, and it was not accom
panied by any detectable electron emission. It follows,
therefore, that the electron emission observed at the higher
temperature cannot be due simply to the recombination
of electrolytically separated calcium and oxygen ; at all
events the existence of the high temperature is an essential
condition.
A similar conclusion also follows from the results of some
experiments made by the author in 1906, when comparing
250 Dr. F. Horton on the
the electron emission from calcium with that from lime *.
In these experiments a platinum strip was covered with
calcium by sublimation in a vacuum ; an excess of oxygen
was let into the dischargetube and the calcium was oxidized
to lime. No detectable ionization occurred during this
process of oxidation in the cold, nor even at a temperature
of 500° or 600° C, under which conditions the oxidation
must have been very rapid. It was only when the lime
formed had been raised to 700° 0. that a measurable therm
ionic current was obtained.
In this part of the present paper the term "luminous
discharge " has so far been used to indicate that faint
luminosity of the residual gas which appears gradually as
the temperature of the cathode is raised, but which is not
accompanied by luminous pencils of cathode rays. At
higher temperatures, or with greater applied potential dif
ferences, the nature of the discharge alters; brightly luminous
pencils of cathode rays appear, starting from points of lime
which are visibly hotter than the rest of the cathode,
and at the same time there is a sudden increase in the
thermionic current. The alterations of pressure which occur
during the passage of such a discharge are generally quite
different from those recorded above. In the early stages of
heating a cathode newly coated with lime there is usually
a considerable evolution of gas ; but if a brightly luminous
discharge is sent from a cathode which has been used for
some time, through air at a low pressure, there is nearly
always a diminution of pressure produced. This effect was
observed by the author some time ago f, and a spectroscopic
examination was made to see if the passage of the discharge
alters the chemical constitution of the residual gas. Some
new lines in the red part of the spectrum were observed, but
these were shown to be due to mercury {.
During experiments with this type of discharge the current
passing is several times as large as that with no luminous
pencils of cathode rays ; and it is certainly remarkable that
a thermionic current of 4 milliamperes is accompanied by an
increase of gas pressure in the apparatus, whereas a diminution
of pressure is produced by a current of 20 milliamperes.
The effect of these large currents is, however, uncertain; on
some occasions, even after longcontinued heating, an increase
of pressure was produced. The diminution of pressure when
* Phil. Trans. A. ccvii. p. 149 (1907).
t Phil. Mag. [6] xi. p. 505 (1906).
X Proc. Camb. Phil. Soc. xiv. p. 501 (1908).
Action of a Wehnelt Cathode. 251
'the heavy discharge passes may perhaps be due to the high
temperature of the lime at the points from which the cathode
rays proceed ; and it is interesting to note in this connexion
that some time ago Sir J. J. Thomson * suggested that the
origin of the large electron emission from barium oxide was
connected with a chemical transformation from BaO to Ba02
■at the high temperature of the cathode. A peroxide of
calcium also exists, so that it is possible that a reversible
reaction of this nature goes on in the case of lime. It is,
however, very difficult to obtain accurate information about
chemical reactions which may be taking place at the high
temperatures and low pressures of these experiments.
Conclusion.
From experiments made by the author in 1906 it; appeared
that the activity of a Wehnelt cathode is not due to an
escape of electrons from the molecules of lime simply as a
result of an increase in their thermal energy; for on this
view we should expect that the presence of the electro
negative oxygen atom in the lime molecule would hinder the
escape of electrons, and that, in consequence, the electron
emission from lime would be less, at a given temperature,
than the emission from the same amount of calcium in the
metallic state. The experiments referred to shewed that
the reverse was the case, and that the emission from calcium
is less than that from lime. The results of the experiments
described in the present paper may be summarized as
follows : —
(1) The electron emission from a Wehnelt cathode has its
origin in the lime itself, and the lime does not merely serve
to help the electrons to escape from the metal.
(2) When an electric current passes through lime at a
high temperature, the amount of oxygen liberated is only
a very small fraction of what would be expected if the con
ductivity of the lime were entirely electrolytic. It has been
suggested that the conductivity is entirely electrolytic, and
that the products of electrolysis diffuse through the lime
niul recombine; but it appears to the author to be improbable
that such recombination should go on, so rapidly and com
pletely, through a laver of solid lime, especially ;is the
charged oxygen atoms would be liberated on the vacuum
* 'Conduction of Electricity through Gases,' p. 427. Camb. Univ.
Press, 190(3.
252 Dr. C V. Burton on the Possible Dependence of
side o£ the oxide layer, and with a strong electric field tending
to drag them away from the cathode.
(3) The chemical combination o£ calcium and oxygen does
not by itself give rise to any detectable electron emission.
From (2) and (3) it appears that neither the theory of
Fredenhagen nor the modification of that theory proposed,
by Gehrts can be accepted as an explanation of the activity,
of the Wehnelt cathode.
The Cavendish Laboratory,
Cambridge.
XXIX. The Possible Dependence of Gravitational Attraction
on Chemical Composition, and the Fluctuations of the Moons
Longitude which might result therefrom. By C. V. Burton,
JD.Sc*
1. HMO what degree of accuracy is the Newtonian coefficient
A. of gravitation a universal constant of matter inde
pendent of chemical and physical properties ? To decide
this question, Newton himself made experiments on a number
of substances; and later Besself, as the result of a lengthy
investigation, found that, writhin the limits of experimental,
error (one part in 60,000), the weights of bodies were pro
portional to their masses ; the substances examined including
brass, iron, zinc, lead, silver, gold, meteoric iron, meteoric
stone, marble, clay, and quartz. Bessel, like Newton, used a
pendulum method.
2. In modern determinations of the acceleration due io
gravity, if observations were repeated at a given station with
pendulums of different materials, far greater accuracy could
be attained ; and additional refinements could no doubt be
introduced where the comparative behaviour of different
pendulums was the sole subject of research. But experi
menters are not readily attracted to laborious tasks from
which only null results are expected.
3. The planets of the solar system, differing considerably
in mean densit}r amongst themselves, may be supposed to
differ correspondingly in composition, so that the exactitude
with which their orbits conform to Kepler's third law is
reasonably regarded as evidence of the close uniformity of
* Communicated by the Author.
t " Versuche liber die Kraft mit welcher die Erde Korper von
verschiederier Beschaffenheit anzieht." Berlin A bhandl. 1830, pp. 41102 ;
Pogg. Ann. xxv. 1832, pp. 401417, reprinted in Astron. Nachr. x..
1833, col. 97108.
Gravitational Attraction on Chemical Composition. 253
the Newtonian coefficient. But it is in the earthmoon
system that a departure from strict uniformity of coefficient
would be most readily detected ; it is shown below that, if
the gravitational indices (§ 4) for earth and moon differed
by one part in 20,000,000,, the fluctuations of the moon's
longitude would comprise a term of period one lunar month
and amplitude one second of arc. Hence it follows that the
earth and her satellite, widely as they differ in mean density,
must have, within one part in many millions, the same
gravitational index. It should be possible to speak more
definitely when the short period terms in the moon's lon
gitude have been adequately discussed ; and this question,
I gather, is engaging the attention of Prof. E. W. Brown,
who has lately * expressed his belief " that sensible fluc
tuations with periods comparable with a month also exist/'
I. When it is not assumed that the Newtonian coefficient
of gravitation is universally constant, the simplest form which
the law of attraction between homogeneous masses m, m' can
take is given by the expression
yy'mm'r~2 ... ... (1)
for the attracting force, where 7, y' depend on the nature
•of the bodies m, m' respectively, and ?*is the distance between
those bodies. (It is understood that the term mass as here
used is equivalent to inertia.) It will be convenient to call
7, 7' the gravitational indices of m, m', and 77' the Newtonian
■coefficient for that pair of substances. To revert to the strictly
Newtonian law of gravitation, we have only to suppose that
y2 = y'2=. . ., each of these quantities being then identical
with the Bewtonian constant, which has now become the
Newtonian coefficient for all pairs of substances.
0. Let the masses of" sun, earth, and moon be M, mlz m2
respectively, and their gravitational indices G, 7^ 70. Then
the mean index of the earthmoon system is
{mlyl + m2y2)/(ml + m2),
and the excesses of 7^ y2 respectively above this mean are
( + m2, — ml){yi — y.2)/(m1\m2). ... (2)
In lunar theory attention is ordinarily restricted to the
mean index, which is taken to be the same for earth and
moon ; that is to say, any possible distinction between yx and
72 is disregarded. Our special problem is to find how the
motion is modified when 71 — 72 does not vanish, so that the
* Month. Not. R. A. S. Ixxiii. 9 (Suppl. 1913) p. 694.
(yi72); .... (3)
254 Dr. C. V. Burton on the Possible Dependence of
ierms (2) have to be taken into account ; and in this case, ii>
addition to the strictly Newtonian forces of attraction, the
sun's gravitational field gives rise to forces on the earth and
moon which are nearly equal and opposite, the corresponding
accelerationterms, both measured from the sun, being for
the earth
MG m2
Rx2 ??i1 + ?ws
and for the moon
, MG nh
+ R7^^2(7l_72) (4>
Now 71 — 72, if not actually zero, is so small that the effects
to be looked for have not yet been definitely disentangled from
the available observations ; hence in the expressions (3) and
(4) it will suffice to identify Rx and R2 in direction and
magnitude with the radius R drawn from the sun to the
masscentre of the earthmoon system, and the acceleration
of the moon relatively to the earth has thus a component
MG(7172)R2 = A6R2, .... (5)
say, measured in the direction of R. Again, we may dis
regard the inclination (5° 9') of the moon's orbit to the
ecliptic without affecting, to a first order, our estimate of
the lunar longitudeterms arising from the acceleration (5).
A great simplification may also be effected by treating the
orbit of the earthmoon masscentre as circular and uniformly
described, the moon's orbital motion around the earth being
similarly assumed as circular and uniform, except for the*
small fluctuations to be investigated. This substitution of
mean for actual radial distances, &c, still leaves the calculations
abundantly accurate for comparison with the observations.
6. It is the motion of the moon relatively to the earth which
we have to consider. Through G, the masscentre of the
earthmoon system, draw the arbitrary initial line GL in.
Gravitational Attraction on Chemical Composition. 255
some fixed direction in the plane of the moon's orbit. At
time t let the radiusvector drawn from the sun's centre to G
make with GD an angle <£, and let the line earthtomoon
make with GL an angle 6 at the same instant. We assume
0 =(ot + $, where $ is small, j
= £lt + j3, where ft is constant, J '
& and O being constant angular velocities. Let r, the distance
from earth to moon, = a + p, where a is constant and p a very
small periodic term. The radial acceleration of the moon
relatively to the earth is
''**=£ +&«»(**), ... (7)
where A/a2 = o>2a. In this equation, substitute for r, 0, <£,
rejecting squares and products of jn, p, 3; we thus obtain
'pZ(o2p2awk=/AR*cos{((Dn)t(3'(. . . (8)
Similarly for the perpendicular component acceleration, in the
direction of 0 increasing, we have
2rd + r#=/*K8 sin (#),
which to our order of approximation is the same as
2cop + ad==fJLR2sm{{cDQ,)tj3}. . . (9)
7. We are only concerned here with that particular
integral of the equations (8), (9) which is directly dependent
on the value of yu., and which makes p and 3 strictly periodic
functions of the time, the period being evidently 2ir j '(© — O) ,
or one lunar month. Thus, for example, when (yj is inte
grated with respect to the time, no constant of integration
has to be introduced, and the values of p, S are readily
obtained. That for $ is
»=5Cn(2mfl)(a,fl)ssm{(mn>/3}
= n^g(2gixgi)'8m/>Q)<^; (10)
if we put yi~ 72 = «G, and q^co/Q, the number of lunar
months in a year. Here MG2/R2 is the strength of the sun's
gravitative field at the earthmoon system, and is thus equal
to n2R, so that finally (10) becomes
•'•rd&^ir*1^0** (11)
256 Mr. S. S. Richardson on
8. Now # = 36526»2732 = 13'37 andK/a = 389: so that
the amplitude of 3 is, in circular measure, /ex 100*9, or in
seconds of arc tcx2 x 107. In other words, if the gravi
tational indices of earth and moon differ by about one part
in 20 million, then the principal effect upon the moon's
longitude will be represented by a fluctuation with an ampli
tude of one second of arc, the period being a lunar month.
The maxima and minima of 3 occur at times of halfmoon ;
and if = —
!r
a2 tan 0 /x02 tan 6
..tan d=—J±
jjlq tan
(1)
From the polar equation of the ellipse we hav
=/^2+(/V/v)sin20 (2]
I. The Squareended Prism.
In fig. 2, PQ represents the end face and AB the plane of
Fiff. 2.
division (the airfilm). The incident wave is plane and
parallel to PQ.
(a) The extraordinary ray. For transmission a must have
a value exceeding that given by the equation
veloc. of ray in air _ sin (63° 45' +.0)
veloc. of ray in spar cos a '
or cosa=rsin(63°45' + 0). ... . . (3)
When /x0 and fie are known, equations (1), (2), and (3) enable
us; to calculate successively 0, r, and a. The values of the
indices of refraction of Iceland spar for a large range of
Polarizing Prisms for the Ultraviolet. 259
wavelengths in the ultraviolet have been determined by
Gifford. (Proc. Hoy. Soc. 1902, 1904). For pur present
purpose it is sufficient to consider a few typical values.
We shall select the values for A = 6708 and \ = 4047 near
the ends of the visible spectrum ; and \= 2144 as the limit
of the ultraviolet spectrum to be considered. The results
are shown in the following table : —
X.
/V
/V
f*o2
He2
9.
r.
a.
6708
16537
14843
2736
2202
21° 39'
•663
48° 38'
4047
16813
14969
2825
2240
21° 21'
•6565
49° 31'
2144
18459
15600
3408
2434
19° 24'
•627
51° 30'
(b) The ordinary ray. In the squareended prism the
ordinary ray enters without deviation. Hence the limiting
position of the dividing plane for total reflexion is determined
at once from the value of the critical angle (c).
Thus  = sinc and /8=(90°c).
Taking the same wavelengths as before : —
\ = 6708
X=4047
X = 2144.
c = 37°12'
c=36Q30'
c = 32°48'.
/3 = 52°48'
/5 = 53° 30'
0=57° 12'.
It will be seen that if the dividing plane is inclined to the
line of vision at an angle less than 51° 30' the extraordinary
ray will not be transmitted at X = 2144, whereas if the angle
be greater than 52° 48' the ordinary ray at \ = 6708 will be
transmitted and this part of the light will not be plane
polarized. If the plane of section be at 52° the whole beam
will be planepolarized, provided its angular diameter within
the crystal does not exceed the following values for the three
wavelengths under discussion :
X=6708.
4° 10' =(0° 48' +3° 22'
3°59'
X = 4047.
= (l°30'+2°29').
X = 2144.
I 5°42' = (5°12'+30').
(The figures within the brackets represent the deviations
on opposite sides of the two internal lines of vision.)
It will be seen, therefore, that it the visible as well as the
ultraviolet portions are to be polarized, the transmitted
pencils must be almost parallel.
S2
260 Mr. S. S. Richardson on
II. The Ordinary Foucault Prism.
Referring to fig. 3, OP represents the face of the crystal,
OY the optic axis, OR' the transmitted ray, and PQ the
transmitted wavefront. The coordinates of P are a?1? yx. _,,
Fig. 3.
(a) Extraordinary rays. Taking the equation of PQ in
the form
y = mx ± y/(a?m2 + b2) ,
we obtain
tancf)
_ #i,yi+ V b2x\* + o?y\ + a2b2
The coordinates of P are a,1 = OPcosB ; y1 = OPsinB.
A1S0 0P=^ = ^4.
sin A sin A
Hence by substitution,
+. a. — tl pef^o sin B cos B + sin A \/ \x2 cos2 B f //,02 sin2 B — sin2 A
an9_/V~ (/xe2cos2Bsin2A)
. . . (4)
the positive value of the root being taken since the larger
value of is obviously required.
Polarizing Prisms for the Ultraviolet. 261
If OP is the natural end of the crystal,
A=19°8'; B = 44°37'.
Hence from (4), (1), and (2) we can determine , 0, and r.
The limiting angle for the transmission of the extraordinary
ray is given by : —
velocity of light in air _ P'B/
velocity of ray in spar Q'B/
sin (0 + tfO .
1
i. e. 
sinClSO0^0^'^*)'
sin (0 + 0)
(5)
r sin (153° 45' <£ a)'
from which a may be found.
Substituting the numerical values we obtain the results
shown below : —
X.
/V
/v
t
e.
r. a.
6708
16537
14843
57r0'
27°37'0"
•6569
55° 54'
4047
16813
14969
56° 51' 30"
27° 21' 30"
•6503
56° 35'
2144
18459
15600
56° 11' 20"
25° 33' 15"
•6184
59° 40'
(h) Ordinary rays. The angle of incidence is 19° 8'. If
8 is the deviation of the ray from the line of vision and c the
critical angle, the limiting angle of inclination /3 of the
section plane is found from
0=8 + (9O°c).
Hence we obtain : —
X,= 6708
\ = 4047
X2144.
8=7° 43'
8= 7° 54'
£=60° 31'
/3=61°24'
S = 8° 55'. /3=r.W}V
The values of a and /3 show that the angular diameters of
the transmitted pencils must be small if the light is to be
completely polarized ; also, if the inclination of the section
plane is less than 56° 35; no ultraviolet light will be trans
mitted. Taking the obliquity of the airfilm as 59°, the
internal rays may diverge to the extent shown by the following
figures : —
A = 6708. I A=4047. i A = 2144.
.4° 37' = (1° 31' 4 3° 6') I 4°49' = (2°24/+2025').  no axial transmission.
With the dividing plane inclined at G0° the values are : —
4°37=(31' + 4°6').  4° 49' = (1° 24' +3° 25').  6o27' = (6°7'+20').
262 Polarizing Prisms for the Ultraviolet.
III. The GlanFoucault Prism.
This is a squareended prism cut so that the optic axis lies
in the plane of division. I£ this axis is taken at rightangles
to the paper the sections o£ the wavesurfaces are both circles
and the conditions for transmission may be determined at
once from the values of the critical angle corresponding to
/x0 and fjie. We have therefore for the extraordinary rays :
X=6708 c = 42°21' a = 47°39'
\=4047 c = 41°54' * = 48° 6'
\ = 2144. c=39°52'. a = 50°8/.
Also, for the ordinary rays as already determined under I. :
\ = 6708 A = 4047 \=2144
/3 = 52° 48'. /3=53°30'. £=57° 12'.
Taking the dividing plane at 50° 17' to the line of vision, we
obtain the following values for the internal diameter of the
beam : —
X = 6708. I X = 4047. i X = 2144.
5°9' = (203r+2°38').  5°24' = (3°13'+20ir).  7° 4' = (6° 55' +9').
Theoretically the values thus derived for the internal
angular diameter of the beam are not quite exact owing to a
small variation in the refractive index when the direction of
the ray varies through a few degrees ; also they refer only to
a plane containing the optic axis and perpendicular to the end
face, i. e.} to a principal plane of section. They are, however,
sufficiently exact for ordinary experimental adjustments.
The external angle allowable will be somewhat greater than
the internal angle, — approximately 1*56 times as great, and
hence if the external beam does not exceed in angular
diameter the values given for the internal beam there will be
a safe margin for complete polarization.
The angular diameter of the polarized beam can be much
increased by the use of a suitable cement for the two halves
of the prism. Canadabalsam is inadmissible in ultraviolet
work, as even in thin films it arrests all rays beyond \3400,
but liquids such as castoroil and glycerine can be used.
It is hoped to deal with the cemented prisms in a later
paper when the necessary experimental data have been
determined.
Spectrum of the Penetrating y Rays from Radium. 2&%
The numerical values for the angles of Iceland spar em
ployed in the above calculation are taken from Prof. S. P.
Thompson's interesting paper on the Xicol prism read at the
Optical Convention of 1905. In conclusion, the writer wishes
to express his thanks to Prof. Wilberforce for his kindness
in providing the prisms for the experimental tests, and to
Mr. J. Proudman, B.Sc, and Mr. J. Foote, who respectively
have kindly verified the algebraical and numerical com
putations.
The George Holt Physics Laboratory,
University of Liverpool.
XXXI. The Spectrum of the Penetrating y Rays from
Radium B and Radium C. By Sir Ernest Rutherford,
F.R.S., and E. X. da C. Andrade, B.Sc, Ph.D., John
Harling Felloiv, University of Manchester*.
[Plate V.]
IX a previous paper f, we have given the results of an
examination of the wavelengths of the soft y rays from
radium B, for angles of reflexion from rocksalt between 8°
and 16°. It was shown that the two strong lines at 10° and
12c correspond to the two characteristic lines always present
in the spectra of the " L " series for heavy elements. It was
deduced from the experiments ofiMoseley, that the spectrum
of radium B corresponded to an element of atomic number
or nucleus charge 82. Direct evidence was obtained that the
strong lines of the y ray spectrum of radium B were
identical with the corresponding lines in the Xray spectrum
of lead — thus confirming the hypothesis that radium B and
lead have in general identical physical and chemical pro
perties, although their atomic weights differ probably by
seven units.
In the present paper an account is given of further experi
ments to determine the 7ray spectra of the very penetrating
rays from radium B and radium C. The strong lines from
radium B, which are reflected from rocksalt at angles of 10°
and 12°, undoubtedly supply the greater part of the soft
radiation for which /x = 40 (cm.)"1 in aluminium. There still
remained the analysis of the frequency of the lines included in
the penetrating radiations from radium B, for which /x = 0'5,
and from radium 0, for which fi= 0*115. It may be
mentioned at once that there is undoubted evidence that a
* Communicated bv the Authors.
t Phil. Mag. May 1914, p. 854.
264 Sir E. Rutherford and Dr. E. N. da C. Andrade:
large part, if not all, of these penetrating radiations give
definite line spectra and correspond to groups of rays of very
high frequency ; but it has been a difficult task to determine
the wavelengths of the lines with the accuracy desired.
We have been much aided by the development of a new
method for finding the wavelength, which depends on the
measurement of absorption as well as of reflexion lines.
In our first experiments the same general method was
employed as in the previous work. A fine glass tube con
taining about 100 millicuries of emanation was used as a
source. The distances between the source and crystal and
between the crystal and the photographic plate were equals
and, as in the previous experiments, about 9 cm. A beam
of 7 rays passing through a narrow opening in a lead block
fell on the crystal, the arrangement being that shown in
fig. 1 of our previous paper. The width of the direct photo
graphic impression on the plate was in general about 3 mm.
It was important in these experiments with penetrating
Y rays to use a thin crystal, since the rays pass right through
the crystal and the exact plane of reflexion is consequently
uncertain. The crystal employed was a slip of rocksalt
about 3 cm. long, 2 cm. wide, and about 1 mm. thick. For a
radiation which is reflected at about 1° from the (100) planes
of rocksalt, the diffraction line is displaced on the photo
graphic plate little more than 3 mm. from the centre of the
dark band. The fact that some lines may be reflected from
near the front face of the crystal, and others from further
back, introduces a possible source of error in the determina
tion of the correct angle of reflexion. This difficulty could
have been avoided if the lines could have been measured in
fhe second order as well as in the first, but the second order
lines were too faint to pick out with certainty in the presence
of a number of other faint lines. A large number of photo
graphs were taken and measured up, and the mean of the
deflexions of the strongest and easily observed lines should
not be much in error.
The crystal was kept in slow rotation by the method
described in the previous paper. This prevented the appear
ance of apparent lines due to crystal imperfections. The
*' centre " of the crystal was in most cases obtained by
observing, on a special photograph taken for the purpose,
the position of the strong 10° and 12° lines. This was
checked also by obtaining the same line on both sides of the
zero by rotating the crystal through about 180°; as described
in the previous paper.
Spectrum of Penetrating 7 Bays from Radium B and C. 265
The following table gives the results of these experiments.
The wavelengths have been deduced from the data given in
the previous paper.
Table I.
Penetrating 7 rays from radium B and radium C.
Angle of reflexion
Wavelength
from rocksalt.
in cms.
44'
•72xl0"9
B 1° 0'
•99
1° 11'
116
1° 24'
137
ri° 37';*
A 11° 43' J
159
169
2° 0'
196
2° 28'
242
2° 40'
262
3° 0'
296
3° 18' t
324
40 0, +
393
4° 22'
428
* The lines at 1° 37' and 1° 43' may possibly be one broad line, having a
mean reflexion angle of 1° 40'. The evidence, however, seems to be in favour
of the double nature of the line, which was by far the strongest line on most
photographs. We were for some time in doubt as to the existence of a line
reflected at 44', but its presence has been confirmed in several reflexion and
transmission (see p. 266 et seq.) photographs.
t Possibly second orders.
Of these lines, those marked A and B showed up strongly
on the photographic plate, A being the more intense. The
other lines were comparatively faint. In addition to those
recorded, which have been observed on Severn 1 plates, a
number of very faint lines have also been observed, but it
has not been thought desirable to include them.
It is of importance to decide which of these lines belong
to radium B and which to radium C. To bring out this,
photographs taken with a screen of 6 mm. of lead between
the source and the crystal were compared with similar
photographs either with no absorbing screen, or with a
screen of aluminium 2 mm. thick between the source and
crystal. The latter absorbed the soft 7 rays from radium B
but did not stop more than a small percentage of the
penetrating rays. The lead plate, on the other hand,
practically cuts out the greater part of the radiations from
radium B, and no doubt also some of the softer possible
266 Sir E. Rutherford and Dr. E. N. da C. Andrade:
constituents o£ radium C*, but does not stop half of the very
penetrating rays from radium 0. With the lead plate the
strong line A disappeared, but the group of lines reflected
in the neighbourhood of B still remained. We may conse
quently conclude that the radiations with reflexion angles
greater than 1° 24' belong mainly to radium B, and the
smaller reflexion angles mainly to the penetrating rays from
radium C.
Transmission method of determining wavelengths.
While the reflexion method described above left no doubt
as to the approximate wavelengths of the penetrating radia
tions, the uncertainty in regard to the plane or point of
reflexion of each type of radiation of the crystal might lead
to a possible error of some minutes in the angle of reflexion
of a line. This is a considerable percentage error for ai*
angle of reflexion of 1° or 2°.
A number of experiments were made to test whether a
more reliable method could be devised for determining the
wavelengths of these very penetrating rays.
For this purpose, the aray tube R was placed behind a
rocksalt crystal C, and a photographic plate PP placed at D,
the whole apparatus being placed between the poles of the
electromagnet to get rid of the effect of /3 rays.
Fig. 1.
Suppose, for simplicity, that the source R emits a radiation
of one definite frequency which would be reflected at an
angle 6 with the surface (cleavage plane) of the crystal. If
RE> is the normal to one face (and hence parallel to another
* See Rutherford and Richardson, Phil. Mag. May 1913, p. 722.
Spectrum of Penetrating y Rays from Radium B and C. 2Q7
face of the cubic crystal), the ray RA making an angle 6
with the normal is in position to be strongly reflected in
passing through the cubic crystal ; similarly also the corre
sponding ray RA'. The reflected rays from each point of EF
cut the normal in the neighbourhood of 0. A photographic
plate placed at 0 normal to RD should thus show a narrow
dark band on the general background, while if placed at D
it should show dark bands at B and B'. If a consider
able part of the rays in the direction RA, RA' are reflected,
the lines A and A' should be positions of minimum intensity
of radiation, and thus appear as white lines on the general
dark background. In other words, reflexion lines appear at
B and B' and absorption lines at A and A' *.
These conclusions have been completely confirmed by
experiment. When the photographic plate is placed at 0, a
central dark band appears with two absorption lines sym
metrically placed on either side of it. An actual untouched
photograph of this kind is shown in fig. 3 (PL V.), and brings
out the main points quite clearly f.
If an emanationtube, with the emanation compressed into
a small length of the tube, is placed normal to the face of
the cubic crystal, it corresponds nearly to a point source.
Under such conditions, the photographs obtained show a
beautifully symmetrical pattern. The two sets of strong
absorption bands cut one another at right angles, while the
crystal planes at 45° also give bands cutting the main system
of bands at an angle of 45° and passing through their points
of intersection. The 45° bands are relatively less marked
than the bands from the 100 planes. The results obtained
are in exact agreement with those to be expected from the
geometry of a cubic crystal. Reproductions of two photo
graphs of this kind made with different crystals are shown
in figs. 1 and 2, Plate V., fig. 1 showing the 45° lines well.
The nearly monochromatic radiation producing the pattern
is the "A" line, or doublet; the lines due to the other
radiations are too faint to show in the reproduction.
Considering the fact that no attempt was made to cut off
the effect of the general )/Iwj where Iw and I20
are the values of I when the speed of the rays are Wand 20 volts
respectively. Z has no special significance, but it measures
with sufficient definiteness the form of the curve ; it was
found that, in the range shown in fig. 2, the curve for which
Z was less was always flatter, and lay completely below the
curve for which Z was greater. The greater is Z the greater
is the ionization caused in the platinum by rays with a speed
between W and 20 volts *. For curves A and B Z is 1*40
and 0'47 respectively.
* It would probably have been better to measure V0 rather than Z ;
a small Z corresponds to a large V0. But in some cases the value of
Y0 was inconveniently large.
294 Dr. Norman Campbell on the
8. The results for Rw/ and W may he dismissed shortly.
W was never changed by any treatment of the platinum ;
there was not the smallest evidence of any alteration in the
ionization potential. When Z was very small the precise
determination of: W was somewhat difficult, for the minimum
became very flat ; but in the most extreme case an error of
2 volts could not have been made, and in most cases it pro
bably did not exceed ^ volt. Rm;, on the other hand, decreased
regularly with Z ; the smallest value found, corresponding to
the smallest value of Z, to be noticed immediately, was 0*302.
ROT' was rather more variable than Z, probably because it
would be affected considerably by any small traces of gas.
However R„/, and the early part of the curve generally, is
not of much importance for our purpose. Other observers
have shown that it varies somewhat with the precise state of
polish of the surface; and in these experiments it was probably
also affected by the distortion of the platinum on heating. We
shall confine most of our attention to Z.
9. After the state B was obtained, it was found that Z
was increased and the platinum tended to revert to its
original condition, if W was heated. W had been heated
continuously with P, and further heating for 24 hours caused
the evolution of no gas which could be measured on the
gauge (sensitiveness 0*0001 mm.). However, it was found
that P still continued to evolve gas on heating (doubtless
because it constituted with its leads a much more massive
piece of metal), and it is probable that W was still evolving
some gas. After the value of Z had been increased by this
means, it could be reduced to 0'47 again by heating P to a
bright red for two or three minutes.
Further heating of P, both in a vacuum and in oxygen,
produced a further gradual decrease in Z to 0*29. At this
stage the heating of W still caused an increase in Z, which
was still immediately reversed by a short heating of P. The
effect of making P the cathode of a discharge carrying about
2 milliamperes in oxygen was then tried. This procedure
reduced Z still further, the lowest value which was noted
being 0*15; but the decrease was only temporary; Z increased
very rapidly if W were heated, so rapidly, in fact, that it was
difficult to measure Z, since measurements involved, of course,
the heating of W. In two minutes Z had returned to about 0*3.
This rapid variation made it impossible to determine W for
the platinum in this condition ; it also made it difficult to
determine whether Z changed when W was not heated, for
the time occupied by a single measurement of Z was almost
sufficient to restore the value 0*3. But it is probable that
Ionization of Platinum by Cathode Rays. 295
it did not so change, for at least twenty minutes necessarily
elapsed between the passing of the discharge and any readings,
the time being occupied by producing the necessary vacuum.
I think also that the low value of Z was maintained if W
was heated while the rays from it were not allowed to
reach P; but on this point the record of experiment is not
conclusive.
10. The heating of P in a vacuum and in oxygen, with
occasional spluttering in the discharge in oxygen, was con
tinued for a fortnight. The normal value of Z (that is the
value which could be quickly restored after Z had been
increased by heating W, and after Z had been decreased by
the discharge) fell still further. Ultimately a value 0'21 was
reached ; the corresponding curve is shown in fig. 2, C ; it
will be noticed that there is no evidence that W has increased ;
if the curve were drawn on a larger scale it would be seen
that W has not decreased notably. At this stage Z could no
longer be increased by heating "W ; the supply of gas (?)
from W was exhausted. The evolution of gas from P when
it was heated had decreased very greatly, but could still be
detected ; it is likely that much, if not all, of this gas came
not from P but from the brass cylinder surrounding it.
Z could be decreased further temporarily to about (J 15, as
before, by passing a discharge with P as cathode ; but it was
found (the experiment had not been tried before) that the
discharge with P as anode produced exactly the same effect.
(There was a valvetube in the discharge circuit, so that there
is not likely to have been any doubt as to the true direction
of the current.)
At this point, when P had been heated in all for some
350 hours, the attempts to reduce Z still further were
abandoned. There is, of course, no evidence that the limit
had been reached, but progress had become so slow that it
did not appear desirable to pursue the matter further for the
time. Attempts were now made to restore the original value
of Z (about 1*4) obtained before the operations were begun.
It has been noted that restoration by heating \V had become
impossible. Practically no effect was produced by admitting
hydrogen and either heating P in it or allowing it to remain
cool. Even when P was left for three days in a mixture of
hydrogen, air, and the vapours of water and tapgrease
scareelv any increase of Z took place: Z remained below 0*3.
About 001 mm. of hydrogen was then admitted, AY heated
and the rays from it allowed to fall on P with a speed of some
80 volts ; no change was produced. No permanent change
was noted after P had been made the anode of a discharge in
296 Dr. Norman Campbell on the
hydrogen carrying 2 milliamperes for two hours; the only
effect was to produce the temporary decrease of Z already
mentioned. At length, however, by allowing 200 volt rays
to fall on P from W (current about 10 "9 amp.) for 24 hours
in a pressure of about 0*01 of a mixture of hydrogen and
tapgrease Z was increased again to 1*2, very nearly its
original value. The oriomal state of the metal had not been
reproduced however; for while formerly three days of heating
had been necessary to reduce Z to 0*3, now Z fell to 0*23
after two minutes' heating, during which a large quantity of
gas was evolved.
11. In the foregoing account the history of only one piece
of platinum has been traced. Less complete observations
have been taken in slightly different forms of apparatus on
other specimens. All agreed very closely in giving curve A
initially, and all could be made to give curve B after a few
days of heating. An attempt was made to produce state B
from state A initially by spluttering the surface of A in a
discharge; in two hours the discharge produced no effect,
though the walls of the tube were completely blackened ; a
further continuance for \ hour brought the experiment
to an end by the covering of the insulation with the
deposit.
12. The facts which have been recorded may be briefly
summarized : — The ionization produced by cathode rays in
platinum can be reduced very greatly by continued heating
of the platinum in a vacuum and in oxygen. At first the
heating produces no effect, then it produces a large and
sudden change from state A to state B, followed by a very
slow and longcontinued change the end of which has not
been reached. It is quite possible, or even probable, that
ultimately a state would be reached in which the platinum
would not be ionized at all by rays of less than 40 volts speed.
The initial state A of the metal can be restored by allowing
cathode rays to fall on the metal with a speed of some
100 volts in a pressure of gas less than 0'01 mm. ; mere
access to the metal of gases and vapours does not restore
that condition quickly, if at all. And when the initial con
dition is restored by the action of cathode rays, it is much
less permanent than before, and state B can be regained by
a few minutes, instead of many hours, of heating. There is
no evidence that the passage of a luminous discharge produces
any permanent effect, whether the platinum is anode or
cathode ; but in either case it produces a decrease in the
ionization, which is reversed immediately cathode rays fall
on the metal.
C3
Ionization of Platinum by Cathode Rays. 297
13. Before discussing the significance o£ these observations,
some other experiments may be noted briefly.
A great deal of time was devoted to an attempt to de
termine whether the ionization of the platinum varied with
its temperature. No such variation is to be expected, for
everything tends to show that it is not the free electrons of
the metal which are ejected in ionization, while the work
of Richardson proves that the work which electrons have
to do in emerging from the surface does not vary with the
temperature. But as the initial experiments showed an
apparent variation, the matter had to be investigated
thoroughly.
P was disconnected from the electrometer and heated by
the transformer, while A was connected to the electrometer,
the differences of potential between the various conductors
being unchanged. With this arrangement the (negative)
current received by the electrometer increases with an
increase of the ionization of P, whereas in the other arrange
ment it decreases. If Iw', I20' are novv the currents flowin
when the speed of the rays is ~\V and 20 volts respectively
•and if Z' = 20T , Z' will measure the ionization, as Z did
iw
previously. If P is at a temperature of more than 1000°,
there will be a thermionic current, for which a correction can
easily be made, since it is independent of the action of the
rays *. All the experiments were made after state B had
been reached ; in this condition the thermionic current at
1000° (approximately) had been reduced to less than 10~6
of its value in state A; the total current from the 2 cm.2 of the
platinum at that temperature was less than 10"10 amp., and
was small compared with the current due to the rays.
It was found that heating P invariably increased Iw'
and I20', and almost always increased Z', indicating an
apparent increase in ionization. The effect cannot be attri
buted to the evolution of gas from P because (1) the presence
of gas, while increasing I', would decrease 7/, (2)eveui if the
pump was not going the values of I' and Z' were unaltered
when P was cooled again. However, it was found that the
increase in I' was only temporary; after half an hour (during
which the pump had to be kept, going, since P always evolved
* Of course the object of the experiment is to lind whether the
the thermionic current is independent of the rays. The statement in
the text is better expressed thus : — If it be assumed that the thermionic
current is independent of the rays and the measurements, interpreted on
that assumption, show that the ionization is independent of the tempe
rature, then it is really so independent.
298 Dr. Norman Campbell on the
some gas) I' returned to its original value and Z' to near its
original value. Usually increases but sometimes decreases
in T/J were observed. There appeared to be no relation between
the changes and the temperature of P ; no changes were
observed unless P was heated till it just emitted light, but
they did not increase consistently when the temperature was
increased to a full yellow heat. The inconsistency of the
results is disturbing, but the observations are not easy ; the
sagging of the foil when heated doubtless introduced com
plications. The matter is not complely elucidated, but I
think it is clear that there is no direct effect of temperature
on ionization.
14. In all the experiments which have been described so
far the speed of the cathode rays incident on the platinum
was less than 40 volts. It is obviously interesting to inquire
whether the difference between states A and B persists when
rays of much greater speed are used ; for example, would
the difference persist if the cathode rays corresponding to
hard X rays were used? If it does not persist for rays of
all speeds, then at some point the B curve must rise more
steeply than the A curve; is this rise initiated by a 'kink/
similar to that which occurs at 11 volts, indicating the
attainment of the ionization potential for the metal itself ?
These questions have not yet been investigated thoroughly,
but there is no indication of an approach of curve B to
curve A when the speed of the incident cathode rays is
increased to 100 volts, and below this speed there is no
evidence whatever of a second ' kink' which should iudicate
ionization of the metal.
15. Some observations were made of the speed of the rays*
electrons leaving P in an apparatus without the tube T ; the
information can be obtained by varying V3. It was pointed
out in § 1 that the passage of the incident rays through the
speed "W is marked, not only by an increase in P, but also
by a change in the distribution of the velocity of the electrons
" reflected." It was examined whether this change took
place when the platinum was in state B as well as when it
was in state A. The conclusion was affirmative ; the state
of the platinum does not seem to affect the velocity of the
electrons leaving P. Indeed the change in the speed curve
of those electrons is often a more delicate test of the position
of the ionization potential than direct measurements of P.
16. Now let us inquire what conclusions may be drawn
from the facts which have been described. An explanation)
of them, on the whole satisfactory, can be offered in terms of
Ionization of Platinum by Cathode Rays. 299
the views which are prevalent as to the contamination
of metal surfaces with gas unless they are specially purified.
The contamination (which we shall suppose to consist of
hydrogen, partly on the ground of the value of W, partly
on the ground of other work) probably is present in two forms.
First there is gas condensed on the surface of the metal,
covering it completely; second, there is gas dissolved or
entangled in someway in the body of the metal. The change
from state A to state B may be supposed to represent the
abolition of the surface layer, the " dissolved gas " being
practically untouched ; while the gradual decrease in Z after
state B is attained represents a diminution of the u dissolved
gas" and a diminution of the surface of it exposed to the
rays. If it be supposed, further, that 11 volt rays ionize only
the gas and not the metal, the main features of the observations
will be explained in a very simple and obvious way. In order
to explain the difficulty of restoring state A permanently and
the ease with which state B, once attained, may be regained,
we may imagine that in order to liberate the surface gas
some alteration in the mechanical structure of the surface
must be made; and that after it is made it is not easily
reversed. There is some evidence for such an idea in the
change of Rw;'. When state A was completely restored by
the method of § 10, the value of B.J (in the state B) was not
restored. When state B was first attained ROT' was 0'44,
the original value in state A being 0*50. Before state A
was restored, Rm ' had sunk to 0*30; when state A was
restored, it rose to 0*52, its original value ; but when state B
was established once more, it returned to 0*33 and not to the
value it had when state B was first attained.
17. The temporary changes produced by the electric dis
charge indicate/ probably the effect of the surface "double
layers," studiea recently by Seeliger*. According to his
results, P, except just after it has been made the cathode of a
discharge, will he covered with a double layer, the potential
in which is 3 volts with the positive side outwards.
If such slow rays as are used in these experiments are capable
of penetrating the double layer at all, its presence will, of
course, have no effect on the speed with which they reach the
metal. But the double layer will help electrons liberated at
its inner surface to escape. Researches on delta rays have
shown that these slow electrons are undoubtedly aided in
their escape from a solid surface by a favourable field. The
abolition of the layer by the discharge would remove the field
temporarily, fewer electrons would escape and Z would
* Seeliger, Phys. Zeit. xiv. p. 1273 (1913).
300 Dr. Norman Campbell on the
decrease; in the conditions of my work the layer would be
very quickly restored. It is true that I found the effect of
the discharge to be the same in whichever direction it passed;
but Seelioer does not mention that he tried the effect of
making his surface the anode.
However, there is one serious objection to this explanation.
The double layer is presumably material, and it is extremely
improbable that 11 volt rays could penetrate even a single
layer of atoms ; they would be absorbed. The ionization
would take place at the outer surface of the double layer
and be unaffected by the potential in it. Moreover, the
estimate of the minimum ionization potential would not
require a correction for the potential in the layer, so long as
it was made by the method of § 6. We may then suppose
that the effect of the discharge which removes the layer is
merely to allow the rays to act upon the metal underneath
instead of on the material forming the layer. This supposition
would account well enough for the consequent diminution
of Z, but it would make the explanation which has been
offered between states A and B somewhat doubtful.
If the effect of the discharge can be explained on the
supposition of a double layer, a similar explanation might
possibly be offered of the complicated effects of heat. We
might suppose that heat temporarily changed the potential
of the layer.
18. However this may be, the important result of the
research, to my mind, is the proof that the ionization pro
duced at a metal surface by cathode rays can be decreased
very markedly by the same treatment which reduces so
greatly the thermionic and photoelectric effects. It hns been
suggested freely of late that the theories of these two actions
which were once accepted are quite erroneous ; that the
thermionic current is not due to an increase in the thermal
energy of the free electrons, and that the photoelectric effect
is not the analogue of the emission of secondary cathode
radiation due to X rays; it has been suggested that both are
quite subsidiary actions due to some form of circular chemical
action. The suggestion was doubtless right in its application
to some thermionic currents; the experiments of Richardson*
on tungsten prove conclusively that it cannot be applied to all.
It is rather the application to the photoelectric effect that I
wish to combat. I have never seen how an " indirect"
theory could explain the intimate relation between the speed
of the liberated electrons and the frequency of the incident
light ; but the experiments which have been described seem
* Richardson, Phil. Mag. xxvi. p. 345 (1913).
Ionization of Platinum by Cathode Rays. 301
to afford new evidence against that theory. Whatever
evidence there is that the photoelectric effect is the result of
some indirect action in which the gas plays a part, is now
available to prove the same proposition o£ the process of
ionization ; it clearly behoves anyone who supports the theory
in one of its applications to support it in the other. No
one can seriously doubt the truth of the main features of
the theory of ionization by charged particles ; there is no
reason to have any more doubt of the main features of the older
theory of photoelectric action. It appears that a much greater
energy is needed to eject electrons from a " pure " than a
" contaminated " surface of a metal by the agency of cathode
rays ; if the surface were perfectly "pure," it appears doubtful
whether an energy of 100 volts would be sufficient. Surely
it is likely then that a greater energy will be needed to eject
them by the agency of electromagnetic vibrations. The
reason why the photoelectric effect ceases when the surface
is perfectly " pure " is not that the process which gives rise
to the effect ceases in the absence of "contamination," but
simply that the pure surface holds its electrons much more
firmly.
There is, of course, one objection. Towards the cathode
rays the " contaminated " surface behaves as if were a surface
of hydrogen ; it is characterized by the value of W for that
gas. But towards light it does not ; light which will cause
the photoelectric effect in a " contaminated " surface would
not cause it in hydrogen. But I have purposely written
" pure " and " contaminated " in inverted commas; it does
not seem at all certain that the words describe correctly the
relation between the two kinds of surface. It is true that on
the assumption that they do, the experiments which have
been described can be explained very simply; but there are
other possible assumptions. It may be thought indubitable
that the process which results in the change from state A
to state B consists in the removal of hydrogen from the
surface, but it does not follow that, because it is accompanied
by a decrease in ionization, the ionization takes place in the
hydrogen. The ionization may always take place in the
metal, the function of the hydrogen being only to facilitate
the escape of the liberated electrons. If this were so, state
A rather than state B should be termed "pure," for it would
be that state in which the properties of the metal were dis
played most prominently. It is hoped by the continuation
of the work of which this paper is a preliminary account
to throw more light on these questions.
302 Dr. G. Bruhat on the Theories of
Summary .
§§ 12. Statement of the problem.
§§ 35. The experimental arrangements.
§§ 611. The main experimental results which are sum
marized in § 12 (p. 296).
§§ 1315. Some subsidiary results.
§§ 1618. The interpretation of the results. It is possible
to explain the observations by the assumption that the ioni
zation observed at a platinum surface for rays with a speed
of less than 40 volts takes place, not in the platinum, but in
gas (hydrogen ?) attached to the metal ; but this is not the
only possible explanation. Some curious effects observed
may possibly be connected with the double layer described
by Seeliger.
It is urged that the results throw light on the experiments
which have been described illustrating the abolition of the
photoelectric effect when the surface of a metal is " cleaned/'
and that they destroy most of the reasons for holding that
the prevalent theory of that effect is entirely incorrect.
Leeds, June 1914.
XXXVI. On the Theories of the Rotational Optical Activity,
To the Editors of the Philosophical Magazine.
Gentlemen, —
IN several papers recently published in the Philosophical
Magazine, Mr. G, H. Livens * compares the theories of
the optical rotatory power given by I) rude and Lorentz.
For my partj, I have attempted to verify experimentally
these theories. May I be allowed to set forth for the readers
of your Journal the reasons why I prefer Drude's theory ?
A first test of any theory must be obtained by the experi
mental verification of the formulae it furnishes for the
rotatory dispersion. Mr. Livens, in order to explain Prof.
Cotton's experiments by the theory of Lorentz, assumes that
the absorption band which is taken into account arises from
a group of nonrotationally active electrons. This assump
tion cannot be correct : Cotton has indeed shown that there
are anomalies of the rotatory dispersion and circular dichroism
only when the same molecule is absorbing and optically
active J, and I have observed the same phenomena in a pure
* G. H. Livens, Phil. Mag. xxv. p. 817 ; xxvi. pp. 362, 535 ; xxvii.
pp. 468, 994.
t G. Bruhat, These de Doctorat, Paris, June 1914.
% A. Cotton, Ann. de Chim. et de Phys. (7) viii. p. 347 (1896).
the Rotational Optical Activity. 303
substance, diphenylbornylimidoxanthide, in a state of super
fusion, in the absence of any inactive molecule.
Lorentz's theory can, however, give an account of the
experimental results, for the values calculated by Mr. Livens
correspond to conditions widely different from those which
are realized. The diagrams, pages 1004 to 1006, vol. xxvii.,
have been obtained for the values 0*1, 1, and 10 of the con
stant x, which is approximately equal to the maximum value
of the extinction coefficient. They represent the expected
phenomena in the case of substances for which the absorp
tion would be of the same order or' magnitude as for cyanine
(a = 0'75) or for silver (a. near 1). The polarimetric study
of such substances seems unfortunately impracticable, on
account of the magnitude even of the absorption. In the
experiments of Cotton and of 01 instead *, the extinction
coefficient has never exceeded 2 . 10~5; for the most absorbing
compound which I could study, it is at the maximum 4.103.
If we give to a the corresponding values, the expected
anomaly, according to the formula? of Mr. Livens, becomes
negligible : it is, in the case of solutions, lower than A
/a) 1 \ 1UUU0
of the mean rotation ( — < ) f.
\<*i 10000/
If the calculations pointed out page 1007 are made in the
same way, by neglecting a term which, in the case of solu
tions, never attains fk™. of the expected anomaly, the same
formula? are found as in Drude's theory, multiplied by
(e2 — 1) (e2 + 2): the variation of this factor as the fre
quency of the light varies is never great enough to introduce
between the two theories any discrepancy experimentally
appreciable.
The two theories are then quite equivalent from the point
of view of the observable phenomena of anomalous rotatory
dispersion and circular dichroism, and their comparison can
be made only by the results expected for the dependence of
the rotation on the concentration and on the nature of the
solvent. Whereas Drude's theory points out that the rota
tory power is independent of these two factors, Lorentz's
* L. B. Olmstead. Phys. Rev. xxxv. p. 31 (1912).
t Similar remarks apply to the formulae obtained for the magnetic
rotatory dispersion. It is known indeed that the attempts with aniline
dyes have not given any result : those obtained by Schmauss are due to
experimental error (F. Bates, Ann. der Phys. xii. 1903, p. 1080). The
magnetic rotatory dispersion is moreover not necessarily similar to the
natural rotatory dispersion (E. Darmois, Ann. de Chim. et de Phi/s. (8)
xxii. p. 247, 1911).
301 Theories of the Rotational Optical Activity.
theory makes it depend on the mean index of refraction of
the active medium by the relation (p. 826,' vol. xxv.)
M =/,(62_l) [a(62_l) + 1].
This theory has therefore the disadvantage that it does,
not explain the existence of cases, as that o£ saccharose.
for which [a>] remains constant at 9™, whilst the factor
(e2 — 1) (e2 + 2) varies about  .
Mr. Livens gives no numerical example showing the ap
plication of this formula. In all cases in which the Physical
Tables give data permitting us to verify it, I have found it
inadequate. It is so with tartaric acid, since the rotator v
power for the D line decreases with the concentration, whilst
the index for this same line increases. The rotatory power
of the same compound in different solvents, in much diluted
solutions, should vary in the same way as the indices of
these solvents ; it is not so in aboat one case out of two, for
instance in the case of camphor, camphoric acid, cinchon
idine. Even the example cited by Mr. Livens, of ethyl
tartrate, is not favourable to his theory, because the curves
of the form represented page 82± (vol. xxv.) can be obtained
only by assuming for the coefficients of the general formulae
unacceptable values ; the changes of sign cannot be expected
by Lorentz's theory, since the quantity (e2— 1) [a(e2 — 1) + 1]
should change its sign, which is impossible.
Since I could not find any case of a variation of the rota
tory power with the concentration and the nature of the
solvent of which Lorentz's theory gave a precise account,
I prefer Drude's theory : this theory brings us to Biot's law,
the considerable interest of which has been shown by such
researches as those of Darmois ; we may complete it, when
Biot's law can no longer be applied, by hypotheses as to the
modifications of the active molecule * or of the properties of
the electrons which it contains.
Yours truly,
G. Bkuhat,
Docteur es Sciences,
Agregepreparateur a l'Ecole Normale Superieure.
Paris, 30 juin, 1914.
* Those modifications, to which Lorentz's theory will be driven to
have recourse more often than Drude's theory, have not, I think, so small
an action as Mr. Livens has asserted. His reasoning regarding this
effect (vol. xxvi. p. 537) is based only on the constancy of the factor X
for all salts of the same acid or base, and for this there is not sufficient
evidence.
Magutctsson" & Steyens.
Fig. 1.
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XXXVII. The Connexion betiveen the /3\q&.$ Raiyfepectra.
By Sir Ernest Rutherford, F.E.S., Profe^siffof Physics,
University of Manchester *.
THE problem of the nature of emission and absorption of
radiation has occupied a very prominent position in
modern Physics, both on account of its outstanding importance
and of the great difficulties involved. It is clear that the
question of the excitation of X rays and their conversion
into /3 rays, and also the spontaneous emission of {3 and
7 rays from radioactive substances, must be included in any
general theory of radiation. A study of the /3 and 7 rays
from radioactive matter is of especial interest in this con
nexion, since the/3 rays are expelled with a very high velocity
and a considerable fraction of the energy is emitted in the
form of 7 rays of very short wavelength. It is to be
anticipated that a close study of the emission of these
radiations from radioactive bodies should throw light of a
fundamental character on the radiation problem on the high
frequency side.
During the last few years a number of careful investi
gations have been made in this Laboratory bearing on this
problem, and it may prove of interest to discuss briefiv the
evidence that has so far been obtained and to indicate
the general conclusions that can be drawn from it. The
problem is much too large and involved to hope for an
* Conimuuicated by the Author.
Phil Mag. S. 6. Vol. 28. No. 1.65. Sept. 1914. X
306 Sir E. Rutherford on tie
immediate and definite solution, but the experimental results
are sufficiently complete to afford some data for drawing
some tentative conclusions.
In a paper published two years ago * I discussed the
possible connexion between ft and y rays emitted from radio
active substances and outlined a general theory in explanation
of the magnetic " spectrum " observed when the ft rays are
analysed by their passage through a magnetic field. It was
pointed out that the emission of a large number of groups of
the ft rays of definite velocity from a single substance could4
be most simply explained by supposing that it is a statistical
effect due to a large number of atoms each of which gave
rise to a few only of the groups of ft rays observed.
In a transformation where primary ft and y rays appear,
it was supposed that each atom broke up with the emission
of a ft particle of definite speed. The latter in passing
through the external electronic system set it into vibration,
and energy was abstracted from the ft particle in definite
integral units depending on the vibrating system. If, for
example, the ft particle passed through two distinct vibrating
systems A1 and A2, the final energy of the escaping ft particle
was given by E0 — (pE, + gE2) , where E0 was the initial
energy of the ft particle, p and q wmole numbers which
might have any values 0, 1, 2, 3, etc., and E^ E2 the units
of energy abstracted in passing through A and B respectively.
It was supposed that the energy pE1 + qE2 which was
abstracted from the ft particle appeared in the form of
p gamma rays each of energy E1? and of q gamma rays each
of energy E2. It was suggested that the 7 rays so excited
corresponded to one or more of the types of characteristic
radiations brought to light by the experiments of Prof. Barkla
on X rays. This theory has formed a starting point for a
number of subsequent researches. In the first place, in
order to test the theory, it was necessary to know the energy
of the ft particles comprising the different groups with the
greatest possible accuracy. The initial experiments made
by Baeyer, Hahn and Meitner, and by Danysz on the groups
of ft rays emitted from radium B and radium C were repeated
with great care by Mr. H. Robinson and myself J. By the
adoption of a modified method, the magnetic spectra due
to radium B and radium C were separately determined.
The spectrum of radium C was greatly extended and found
* Phil. Mag. xxiv. p. 453 (1912).
t The particular point of view of which this formula is an expression
has been modified subsequently.
X Rutherford & Robinson, Phil. Mag1, xxvi. p. 717 (1913).
Connexion between j3 and 7 Ray Spectra. 307
to consist of a great number of lines, about 50 of which were
measured. It was pointed out that there appeared to be
certain simple numerical relations between a number of the
groups of /3 rays from radium C. In the meantime, the
problem had been attacked from another direction. According
io the theory, the 7 rays emitted from a radioactive substance
should consist of types of characteristic X radiations which
should be exponentially absorbed by a light substance like
aluminium. This question has been examined in detail by
Mr. H. Hichardson and myself *, and the results obtained
have fully confirmed this point of view. The 7 rays from each
radioactive substance can be analysed into a number of
distinct groups. Some of these groups undoubtedly corre
spond to the characteristic radiations to be expected from
elements of their atomic weight; but attention was drawn
to the evidence of the existence of other types of character
istic radiation not previously observed by workers with
X rays. It was found that the different radioactive sub
stances showed great variety in the types of 7 rays emitted,
but they could be classified by their power of penetration as
belonging to certain general types of characteristic radiations.
Mr. H. Richardson has continued these investigations and
has recently obtained evidence of the excitation of character
istic radiations in a large number of elements when the
,/3 rays of active matter fall upon them.
The discovery of Laue of the diffraction of X rays and
the subsequent work of W. H. and W. L. Bragg and of
Moseleyand Darwin and others, have placed into our hands a
powerful and simple method for determining the wavelengths
of the X rays. If the 7 rays from radioactive matter con
sisted of groups of characteristic rays, it v^as to be anticipated
that the rays would show a line spectrum when reflected
from a crystal surface. This point of view has been com
pletely confirmed by subsequent researches of Dr. Andrade
and myself. In the first paper t we gave an account of the
examination of the spectrum of the solt 7 rays from radium B,
and adduced evidence that the strong lines of the spectrum
of this substance were identical with the characteristic " L "
spectrum of lead. In a subsequent research J we have
determined the wavelengths of the penetrating 7 rays from
radium B and radium C and verified the results by the
adoption of a new experimental method.
* Rutherford & Richardson, Phil. Mag. xxv. p. 722 ; xxvi. p. 324 ;
xxvi. p. 937 (1913). Richardson, Phil. Mag. xxvii. p. 252 (1914).
t Rutherford and Andrade, Phil. Mag. xxvii. p. 854 (1914).
J Rutherford and Andrade, Phil. Mag. August 1914.
X2
308 Sir E. Rutherford on the
Distribution of energy between ft and 7 rays.
It was initially supposed that a large fraction of the
ft radiation from substances like radium B and radium C,
which give a marked /3ray spectrum, appeared in the form of
the homogeneous groups of ft rays observed. J. Chadwick *
has shown, however, in a recent paper that even the intense
lines in the magnetic spectrum of radium B represent only
a small fraction of the total number of ft rays emitted.
This result was obtained by direct counting of the ft particles
and confirmed by showing that an increase of intensity of
only a few per cent, of the ft radiation falling at a given
part of the photographic plate gave the impression on
development of a strongly marked band.
We may conclude from these results that the magnetic
spectrum of the ft rays from radium B or radium C consists
of a continuous spectrum of ft rays on which is superimposed
a line spectrum corresponding to groups of rays expelled at
definite speeds. A satisfactory explanation of these results
and also of other marked differences in the distribution of ft
and 7 rays from radioactive substances can, I think, be given
on the following lines. Suppose — as seems probable — that
the disintegration of the atom leads to an expulsion of a high
speed ft particle from or near the nucleus. This ft particle
in passing through the outer distribution of electrons will,,
on the average, suffer several collisions of an ordinary type
with the electrons, and will share its energy "with them. As
a statistical result of a large number of atoms, the velocity of
the escaping ft particles will, on the average, be continuously
distributed within certain limits of velocity. This would
give rise to the continuous spectrum of ft rays which is most
typically illustrated by the ft ra}rs from radium E. Next
suppose that there are certain wrell~defined regions in the
electronic distribution which can be set into definite vibration
by the escaping ft particle. These regions are to be identified
as containing the particular structures which give rise to the
" characteristic " 7 radiations from the atom. If some of
the ft particles in escaping from the atom pass through one
or more of these regions, they give rise to a linespectrnm
of 7 rays and at the same time to one or more groups of
ft rays of definite speed. The connexion between the energy
of the 7 ray and of the ft ray will be discussed later.
On this view, the appearance of homogeneous groups of
ft rays and the linespectrum of 7 rays are to be ascribed to
certain definite regions of vibration within the atom. It is
* J. Chadwick, Ber. d. D. Phys. Ges. xvi. p. 383 Q9U).
Connexion between ft and y Ray Spectra. 309
to be anticipated that characteristic y rays will always
accompany a linespectrum of ft tsljs. This seems to be in
harmony with radioactive data. Radium E_, which gives
rise to a continuous /3ray spectrum, emits exceedingly little
7 radiation in comparison with typical ft and 7 ray products
like radium B and radium C, which give well marked
spectra for both ft and y rays.
While it would appear probable that the greater part of
the y radiation from radium B and radium C is composed
of several groups of rays of definite frequencies, no doubt a
small part of the y radiation gives a continuous spectrum.
Such a result is to be anticipated from analogy with X rays.
This general radiation probably has its origin in the electronic
collisions of an ordinary type when the ft particle is escaping
from the atom or to the passage of the ft particle close to the
nucleus.
Importance of direction of escape of a ft particle :
There is another very interesting point that arises in
consideration of this question. "Why does radium E, which
emits ft rays of great intensity and over a wide range of
velocity, not emit y rays at all, or at any rate in very small
amount compared with radium B or radium C ? There
appears to be no reason to suppose that radium E would not
give rise to characteristic radiations of a frequency corre
sponding to its atomic weight or atomic number when
bombarded by cathode rays of suitable speed. In order to
explain this anomaly, it appears necessary to assume that
the primary ft particle from a given radioeleinent is always
expelled in a fixed position with regard to the structure
of the atom itself. Considering the remarkably definite way
in which the atom of the same substance disintegrates, this
assumption does not seem improbable. On this view, the
absence of y rays from radium E is due to the fact that the
direction of escape of the ft particle does not pass near or
through the definite regions where characteristic radiations
are set up, and in consequence only a continuous spectrum of
ft rays is observed. An explanation may be given on similar
lines of many remarkable anomalies in the types and relative
intensities of 7 rays emitted from radioactive substances.
This is well illustrated by a comparison of the 7 rays from
radium B and radium C which are nearly of the same atomic
weight. Radium B emits a very soft radiation which is
almost entirely absent in radium C, while radium B does not
emit the very penetrating radiation observed from radium 0.
We must suppose that the ft particle from radium B passes
310 Sir E. Rutherford on the
through one or more distinct regions which give rise to the
corresponding characteristic radiations, while the ft particle
from radium C escapes in such a direction that it does not
pass through the corresponding regions but does pass through
a new region not involved in the case of the expulsion of the
ft particle from radium B.
The general evidence available indicates that radium B and
radium D have identical general physical and chemical pro
perties with those of lead, although differing from the latter
in atomic weight. If this be correct, we should anticipate
that these elements should give identical X ray spectra when
bombarded by cathode rays. On the other hand, from the
work of Rutherford and Richardson, radium B and radium D
are known to emit types of 7 rays which are widely different
in relative amount and penetrating power. Such results are,
however, at once intelligible if it be supposed that the
ft particles from these two elements are expelled in different
directions with regard to the atomic structure.
It is to be anticipated that some of the lines of the 7ray
spectra of these two radioactive elements should be coincident,
but some may be absent or very faint in one spectrum and
strong in another. In fact, the relative intensities of the
spectral lines of the 7 rays from radioactive substances may in
all cases be very different from those which would be observed
when the element is bombarded by cathode rays. In the
latter case, all types of characteristic radiation have a chance
of excitation — supposing of course account is taken of the
speed of the incident cathode rays — since the ft particles
enter the atom on an average equally in all directions.
Theory of the origin of the ft and 7 rays.
One fundamental fact that has to be taken into account in
considering the origin of the ft and 7 rays is the conversion
of the energy of a 7 ray into the form of a high speed electron
or ft ray, and vice versa. This point has been emphasized by
Bragg in his papers on the nature of X rays. He supposed
that the energy of a single X ray could be converted by its
passage through matter into the energy of a single ft ray of
appropriate speed, and that no loss of energy occurred in the
process. It would appear, however, necessary to generalize
this conception, for it will be seen later that there is consider
able evidence from a study of the ft and 7 rays from radio
active matter that a train of X rays of the same frequency
may be given out each of definite energy, and that the whole
energy of this train of waves may, under suitable conditions,
Connexion beticeen ft and 7 Ray Spectra. 311
appear in the form of a swift ft ray. There is now strong
evidence from a variety of directions that the energy emitted
by a source of radiation of frequency vis in definite quanta E
when E — hv, h being Planck's constant. The general idea
that the energy of the 7 rays is emitted in definite units or
quanta appears to be necessary to explain the origin of homo
geneous groups of ft rays expelled from radioactive matter.
We shall first assume that the energy in a single 7 ray of
frequency v is given by Planck's formula, and then discuss
how far this particular relation is supported by the experi
mental evidence.
We shall suppose certain regions in the atom are set in
vibration by the escape of the ft particle during the atomic
explosion. If vl9 v2... are the frequencies of vibration, the
energy emitted in the form of y rays is phvx, qliv2 . ... where
p and q may have any integral values. For example, one
atom may emit one 7 ray of frequency v and energy liv,
another may emit a train of two 7 rays of the same frequency
but of energy 2/tv, another three, and so on. There is no
method at present of deciding the most probable value of p
for a single atom, nor to fix an upper limit to its value. This
may depend on the intensity of the disturbance communicated
to a vibrating system by the escaping ft particle. The energy
of these 7 rays is supposed to be partially or wholly con
verted into the ft ray form in their escape from the radio
active atom. Unfortunately there is no evidence available
at present of how this conversion occurs or whether any
energy is ^absorbed in the process. If the conversion takes
place without loss of energy, a train of 7 rays of frequency Vj
will give rise to a ft particle of energy given by J&ssphv^
It is possible the conversion may occur in one of the
regions of the atom which give rise to characteristic radiations,
and is accompanied by the appearance of a new type of
7 rays of frequency v2 etc. In such a case, we should
anticipate that the energy of the escaping ft particle is given
by E=p/iV]— qhv2, where p and q are whole numbers which
may have all possible values consistent with pvx being greater
than qv2.
The value of E here refers to the energy of the ft particle
at the point in the atom where the conversion of energy
occurs. It seems possible that there may be a further change
of the energy of the ft particle in escaping from the atom.
In this case, the energy of the ft particle after it has escaped
from the atom is given by phv1 — qhv3— A, where A may
have a negative value and is at present indeterminate.
It will bo seen that the present theory of the origin of the
312 Sir E. Rutherford on the
ft rays differs somewhat from that advanced in the earlier
papers (loc. cit.) . I there supposed that homogeneous groups
of ft rajs were due to the decrease of energy in definite
units of the primary ft particle in exciting the vibrations in
the atom. The present theory supposes that the homogeneous
groups of ft rays arise from the conversion of the energy of
the 7 rays into the ft ray form. In other words, the primary
effect in the atom is the excitation of 7 rays by the escape of
the ft particle from the nucleus. The appearance of groups
of homogeneous ft rays is a secondary effect due to the
partial conversion of the 7 rays into ft rays in their passage
through the radioactive atom. On the other hand, the
continuous ft radiation is ascribed mainly to the effect of
the primary ft particles escaping from the nucleus which
have lost energy, though not in definite quanta, in setting
the electronic system of the atom into vibration.
Consideration of the experimental evidence.
We shall now consider briefly some recent experimental
•evidence which has thrown light on this question. Robinson,
Rawlinson, and the writer"* have shown that the ft rays
excited by the 7 rays in their passage through matter
consist of definite groups which, no doubt, would be
homogeneous if the layer of matter in which the ft rays
were excited was exceedingly thin. As far as experiment has
gone, the velocities of these groups of " excited " ft rays are
in close if not complete agreement with the velocities of the
stronger groups of primary ft particles from the source of
radiation. The velocities of the corresponding groups
appear to vary slightly when the ft rays are excited in
different metals, but the differences, though no doubt real,
are not very marked. We may conclude from these results
that the primary ft particles, for example from radium B,
like the ft rays excited by the 7 rays in traversing absorbing
material, are due to the conversion of 7 rays into ft rays in
their escape from the radioactive atom. The observed
variations of velocity between the primary groups of ft rays
from the radioactive atom and the ft rays excited by the
7 rays in different substances, may be due to the variation of
the part — (qhv2 + A) in the expression for the energy
'E=phv1 — qhv2 — A. The experiments, however, show that
the variation due to this cause is small for the swift groups
of ft rays excited by 7 rays in different kinds of matter, and
for simplicity it may be assumed as a first approximation
* Phil. Mag. August 1914.
Connexion between ft and 7 Bay Spectra.
313
that the value of qhv2 + A. is very small compared with pliv1
for the actual groups of swift ft rays under consideration.
We shall now consider whether the experimental evidence
supports the view that the energy of the homogeneous
groups of ft rays from radium B and radium C are connected
with the values of hv, where v is the frequency of one or
more of the stronger lines in the ft ray spectrum.
For convenience of discussion, the frequencies of the
stronger lines of the 7 ray spectrum of the penetrating
7 rays from radium B and radium C are included in the
following table. The value of hv is calculated from the
observed frequency, assuming h = 6*55 x 10~27, and expressed
in the same form as the energies of the groups of ft rays
from radium B and radium 0 in the previous paper of
Rutherford and Robinson. The value of e is taken as
4*69 x 10"10 e.s. units — the value deduced directly by Planck
from his theory of radiation. The energy hv of the strong
line, which is reflected at 1° from rocksalt, is 1*25 X 1013 . It was pointed out, however, that in
several cases the adjacent lines appeared to be closer together
than would be indicated by this difference relation. Since
one of the strong lines of the 7 ray spectrum of radium C,
reflected at 1°, has an energy of 1*25 xl013£ on Planck's
relation, it is to be anticipated on the theory that the energy
of a number of lines should show differences of this unit or
integral multiples of it. As a special case, the energy of the
ft particle itself might be expected to be integral multiples
314
Sir E. Rutherford on the
of this unit. The original difference unit '4284 x 1013 ey
when multiplied by three gives 1'285 xl013£, which does
not differ from 1*25 x 1013 e by more than the experimental
error. It will be seen in the table below that a number
of the lines, including some of the stronger lines, can be
expressed very closely as integral multiples of the unit
E = 1285 xlO13*.
The numbers of the lines refer to those given in the
previous paper by Rutherford and Robinson.
Index
number of
line.
Intensity.
M45
K38
32
G29
F26
23
D20
17
0 14
11
m.
s.
m.f.
m.s.
m.
f.
m.
f.
m.
f.
Velocity as
a fraction
of light.
•750
•868
•917
•946
•952
•959
•9643
•9687
•9724
•9754
Observed
energy
=1013e.
259
516
768
1031
1149
1282
1409
1542
1671
1796
Energy
•l285xi013e.
2015
4016
598
802
894
997
1096
1200
1301
1397
Integral
multiple.
2
4
6
8
9
10
11
12
13
14
Of the twelve strong lines in the table marked with index
letters, six are seen to be expressed very nearly as integral
multiples of E:. In picking out possible other units, it
appears significant that another series of lines can be
expressed as a multiple of E2 = '74x 1013 e. This is seen in
the following table.
Multiple ...
2 4 6 8
9 12
14
15
16
1184
18
19
22 23
1628 1702
Calculated
energy ■? 10l3e.
1481 296 444 592
676 888
1036
1110
1332
1406
Observed
energy ^1013 e.
149 296 448 5'94
674 8 91
1031
m.s.
1107
f.
1186
y.f.
1328
1409 1628 1711
Intensity ...
f. m. m.f. m.s.
m.f. m.f.
m.s.
m. f. f.
Only two of these lines, viz. of energies 10'31 and 14'09,
are included in the first table. These lines may possibly be
close doubles.
It has been pointed out in the paper of Rutherford and
Andrade that the strong y ray line reflected at 1° 40' is
Connexion betiveen j3 and 7 Ray Spectra. 315
probably a close double which it is difficult to separate under
the experimental conditions. One component is believed to
belong to radium B and the other to radium C. This appears
very probable, for Hichardson has found recently that
radium C emits some 7 radiation which has about the same
absorption in lead as the soft 7 rays from radium B. We
should consequently anticipate that radium C should give a
strong line near 1° 40' of energy about *75 x 1013 e — in good
agreement with the unit found above.
Only two strong lines remain which are not included as
multiples of these two units *74 and 1*29 x 1013 e. These are
the two lines A and B of energies 2T02 and 17*5 x 1013
University of Manchester *.
DURING the past year, the National Physical Laboratory
has obtained a radium preparation certified by the
7 ray method in terms of the International Radium Standard
preserved in the Bureau des poids et mesures at Sevres. It
has consequently been possible to compare the standards in
use for many years in the Laboratories in Montreal and
Manchester with the International Standard.
It may be of interest to mention briefly the history of the
preparation which for ten years has served as a laboratory
standard and in terms of which a number of important
radioactive magnitudes have been measured.
In connexion with the radioactive work in Montreal, it
became important in 1903 to adopt a radium standard in
which to express the results of various measurements. For
this purpose, Professor A. S. Eve weighed out for me
3'69 milligrams of a preparation of radium bromide bought
from Dr. Griesel, and enclosed it in a sealed tube. This
preparation has served as the primary laboratory standard,
and was assumed to contain 3*69 milligrams of pure radium
bromide or 2*16 milligrams of radium element. At the same
time about one milligram of the same material, calibrated in
terms of the larger quantity by the 7ray method, was
dissolved and a number of standard radium solutions were
prepared. Part of this standard solution was sent to Pro
fessor Boltwood in New Haven in order to make a direct
determination by the emanation method of the quantity of
radium in a mineral per gram of uranium. The preliminary
measurements of Professor Boltwood gave a result much
higher than that measured by Professor Eve by direct com
parison of the 7ray effect of a uranium mineral with the
primary standard. It thus appeared that some error had
crept in the preparation of the standard solution. An in
vestigation by Eve showed that the radium in the standard
solution had partly deposited out on the walls of the glass
vessels, and that the solutions were consequently unreliable.
Another standard solution was then prepared in which the
precaution was taken of adding a considerable quantity of
hydrochloric acid to keep the radium in solution. These
standard solutions are believed to have kept their strength
unaltered during the past ten years. With the aid of these
standard solutions, the quantity of radium per gram of
* Communicated by the Author.
Radium Constants on the International Standard. 321
■uranium was determined by Boltwood. The result was
expressed in terms of the 3*69 milligram standard,, which
has generally been referred to as the " RutherfordBolt
wood" standard.
A radium standard in use in the Laboratory in Manchester
was accurately compared for me by Professor Stefan Meyer,
Secretary of the International Radium Committee, in terms
of the Vienna standard, which has been set aside as a secon
dary International standard. At the same time, Mr. Chad
wick, in the Manchester Laboratory, compared by a balance
method the laboratory standards with a secondary standard
kindly lent to me by the Radium Institute of Vienna.
Recently Dr. Kaye has compared two of the Laboratory
standards with the radium standard of the National Physical
Laboratory. The numerous cross measurements at different
institutions have all been found to be in excellent agreement,
and bring out the reliability of the 7ray method of mea
surement.
The original RutherfordBoltwood standard was assumed
to contain 3'69 milligrams pure radium bromide. The actual
7ray activity of the standard as employed in actual use was
found to correspond to 3'51 mg. RaBr2 in terms of the
International Standard, and was thus 4*9 per cent, too low.
Considering the time of preparation of this standard, the
choice of material has turned out to be very fortunate. No
doubt part of the radium bromide from which the standard
was prepared had been converted before weighing into car
bonate by exposure to the air, and this would account for the
high radium content, considering that no allowance was
made for the water of crystallization. In any case, the result
brings out the purity of the radium preparations sold com
mercially by Dr. Giesel more than eleven years ago.
It has been suggested to me that it would be a convenience
to many workers if the various radioactive magnitudes deter
mined in the Laborator}' at Manchester were recalculated in
terms of the International Standard. In my work * Radio
active Substances and their Radiations,' most of the results
were expressed in terms of the laboratory standard, but the
heating effect, which was determined while the book was
passing through the press, was given in terms of the Vienna
Standard, which had been compared with the International
Standard. Confusion may consequently arise in regard
to the standard in which some of the results have been
expressed.
Phil. Mag. S. 6. Vol. 28. No. 165. Sept. 1914. Y
322
Sir E. Rutherford on Radium Constants
In the following table the various magnitudes involving
the radium standard are expressed in terms of the Inter
national Standard.
Original paper.
' Radio
active
Sub
stances.'
Quantity.
Rutherford &
Boltwood
Standard.
International
Radium
Standard.
Rutherford & Bolt
wood, Auier. Jour.
Sci.xxii.p. 1 (1906);
Boltwood, Arner.
Jour. Sci. xxv. p. 269
(1908).
pp. 16,
462.
Amount of radium
in equilibrium
with one gram of
uranium.
34 X 107 gr.Ra.
323x107
gr. Ra.
Boltwood & Ruther
ford, Wien. Ber. cxx.
p. 313 (1911) ; Phil.
Mag. xxii. p. 586
(1911).
p. 557.
Production of
helium per gram
of radium per year.
Calculated value ...
156 c.mms.
155 c.mms.
164 c.mms.
163 c.mms.
Rutherford & H.
Robinson, Wien.
Ber. Oct. 1912;
Phil. Mag. xxv. p.
312 (1913).
pp. 578
581.
Total heating effect
of one gram of
radium and its
products in equili
brium with it.
Radium alone
Emanation ,,
• Radium A „
Radium B "1
Radium C J "
134'7 gr. cals.
per hour.
25*1 gr. cals
per hour.
286 „ „
305 „ „
505 „ „
Rutherford, Phil.
i Mag. xvi. p. 300
1 (1908).
p. 480.
V olume of the ema
nation from one
gram of radium in
equilibrium.
Calculated value ...
•60 c.mm.
•59 „
*63 c.mm.
•62 c.mm.
Rutherford & Geiger,
Proc. Roy. Soc. A.
lxxxi. p. 141 (1908).
pp. 128
133.
Number of a par
tides expelled per
second per gram
of radium itself.
Number per second
from one gram of
radium in equili
brium.
34x1010
136x1010
357 X 1010
143x1010
Rutherford & Geiger,
Proc. Roy. Soc. A.
lxxxi. p. 162 (1908).
pp. 135
137.
Total charge car
ried by the « par
ticles per sec. from
one gram of ra
dium itself and
from each of its
products in equili
brium with it.
31*6 e.s. units.
105x109
e.m. units.
332 e.s. units.
111x109
e.m. units.
on the International Standard.
323
Original paper.
' Radio
active
Sub
stances.'
Quantity.
Rutherford &
Boltwood
j Standard.
International
Radium
Standard.
Geiger, Proc. Roy.
Soc. A. lxxxii. p. 486
(1909).
p. 502.
Total current due
to the a. rays from
one curie of ema
nation.
(1) by itself
(2) with its a. ray
products.
275 XlO6
e.s. units.
946 XlO6
e.s. units.
289 XlO6
e.s. units.
994 XlO6 „
H. Moseley, Proc.
Roy. Soc. A. lxxxvii.
p. 230 (1912).
p. 204.
Total charge carried
by the (3 particles
emitted per second
by radium B or
radium C in equi
librium with one
gram of radium.
17*4 e.s. units.
18*3 e.s. units.
p. 459.
Calculated half
value period of
transformation of
radium.
1780 years.
1690 years.
It may be of interest at this stage to discuss briefly in
some cases the agreement between the observed and calculated
values, and to consider the fundamental data involved in the
calculations.
In the first place, it can be easily shown that
the calculated values of (1) the production of helium by
radium, (2) the volume of the emanation, (3) the heating
effect, (4) the life of radium, all involve the quantity ?iE,
where n is the number of « particles expelled per second per
gram of radium itself, and E the charge carried by the
cc particle. In other words, the agreement of calculation with
experiment in these cases does not involve the accuracy of
the actual value of n or of E (which is twice the unit charge),
but the product of these numbers, which is a measure of
the charge carried by the a particles expelled per second per
gram of radium
The value wE was directly determined by Rutherford and
Geiger, and is given by tiE = l'llx 10"9 e.m. units on the
International Standard. This value has been used in making
the calculations of the four quantities mentioned above.
The value of this quantity deduced from the rate of pro
duction of helium by radium (viz. 164 c.mm. per gram per
year) is 1*12 x 10"9 — a close agreement with the direct
experimental value.
Y2
324 Sir E. Rutherford on Radium Constants
In a similar way, the value of nE deduced from the
observed volume of the emanation is in close accord with
the experimental value.
There is thus a satisfactory agreement between theory
and experiment in the above cases, but the agreement is not
nearly so good for the value wE deduced from the heating
effect and the life of radium. These points will consequently
be considered in more detail.
Heating Effect of Radium Emanation.
In a recent paper, Mr. H. Robinson and myself* have re
determined the velocity and value E/tw for the a, particles
expelled from radium. From these data we have compared
the calculated heating effect due to one curie of radium in
equilibrium due to a. rays alone, with the observed heating'
efiect due to these radiations. On the Internationa] Standard,,
the observed heating effect due to the a rays from one curie
of emanation is 99*2 gr. cals per hour, and the calculated is
92*4, assuming nE = 11*1 x 10~10 e. m. units, and adding 2
per cent, for the energy of the recoil atoms. The calcu
lated heating effect is thus 7 per cent, lower than the
observed. If all the heating effect of the a ray products is
due to the energy of the expelled a particles, it would follow
that the value of ??E is 7 per cent, too small. This seems
improbable, so we must look for an explanation of this
apparent discrepancy in another direction. The general
radioactive evidence indicates that the loss of an a. particle
from an atom lowers the positive charge of the atomic nucleus
by two units, and the expulsion of a /3 particle from the
nucleus raises it by one unit. Without entering into a dis
cussion of the possible distribution and velocities of the
electrons external to the nucleus, it is to be anticipated on
general grounds that the kinetic energy of the total electronic
distribution external to the nucleus should increase with
increase of charge on the nucleus. The expulsion of an u
particle should thus result in a lowering of the total kinetic
energy of the electrons, and the expulsion of a fi particle to
an increase. Suppose, for simplicity, that this change of
energy is proportional to the variation of charge on the
nucleus, and is the same for each a ray transformation.
If A be the energy per atom liberated from the electrons
resulting from the ^expulsion of an a particle, then A/2 is
the energy absorbed in consequence of the expulsion of a
/3 particle. If E1} E2, E3 be the kinetic energies of the
* Wien. Ber. exxii. Abt. II a, Not. 1913.
on
the International Standard. 325
expelled a particles from the three products, emanation,
radium A and radium C respectively, then the energy per
atom released during the transformation of one atom is
Ex + A for emanation, E2 + A for radium A, s for radium
A
B, E3 4 ^ for radium C, since the latter expels both an a
and /3 particle, and thus its nucleus charge is finally lowered
by one unit.
The resultant energy due to a transformation of one atom
of all of these substances is E1 + E2 + E3 f 2 A, and the ratio
of the energy liberated from radium B and radium G together
to the total is
E2 + E2 + E3+2A t1'
In this calculation, no account is taken of the energy ex
pelled in the form of primary /3 rays and y rays, but only of
the energy from the a rays and from the electronic distri
bution.
If the above point of view is correct, the ratio of the
heating effect of radium B and radium C together (sub
tracting the energy due to ft and y rays from these products),
should be less than the value calculated from the energy of
the u particles. Now this point was carefully examined by
Mr. Robinson and myself (loc. tit.) some years ago, and we
drew attention to the fact that the heating effect of radium
C was distinctly less, compared with that due to the emana
tion and radium A, than the theoretical ratio calculated from
the energy of the expelled « particles.
It was found that the observed heating effect due to
radium (B + C) together in equilibrium with one curie of
emanation was actually in very nearly the theoretical ratio
with the calculated heating effect due to the emanation and
its products, viz. *403. Of the observed heating effect,
however, 43 gr. cals. out of a total of 103*5 per curie of
emanation per hour were to be ascribed to the absorption of
part of the ft and y rays emitted by radium B and radium C.
From these data, the value of A (equation 1) can be
deduced, and is found to be about 3*2 gr. cals. per hour
corresponding to one curie of emanation. The heatiug effect
due to the a rays alone should be consequently 6*4 gr. cals.
less than the observed value 99*2, that is 92*8. The actual
calculated heating effect comes out 92*4, assuming nE =
11*1 xlO"10 e. m. units. "When this additional factor is
taken into account, the observed heating effect is thus in
326 Radium Constants on the International Standard.
complete accord with the value o£ nE deduced by direct
measurement and by determining the volume of helium pro
duced from radium.
We have in the above made no assumptions as to the form
in which the energy is emitted from the external distribu
tion of electrons, but have supposed that it ultimately appears
as heat. It is, however, implicitly assumed in the calcula
tion that this energy is not emitted in the form of radiations
so penetrating that they are able to escape through the
absorbing material employed in the actual measurements.
No doubt part of the energy may be emitted (as possibly in
the case of radium itself) in the form of slow ft rays and soft
7 rays, but no certain evidence is available on this point.
It follows from these considerations that the heating effect
of all ol ray products should be greater (on the average
about 10 per cent.) than the kinetic energy of the expelled
a particles. Similarly the heating effect due to a ft ray
transformation may in some cases be less than the value
calculated from the energy of the expelled ft particles, but it
is difficult to be certain how far the energy of the latter may
be affected by the change of nucleus charge.
Life of Radium,
The halfvalue period of transformation of radium can be
calculated at once from the value nE, without any assumption
of the actual value of n or E except that E is twice the
unit charge. On the International Standard, the halfvalue
period comes out to be 1690 years #, taking nE = 11*1 x 10~1(>
e. m. units. This is much less than the experimental value
found by Boltwood t, viz. about 2000 years, but is in better
accord with the value 1800 years given by KeetmanJ, and
1730 years found by Stefan Meyer §.
Unless the determinations of nE from the charge carried
by the u particles, the production of helium and the heating
effect, are all seriously overestimated and to the same extent,
the value 1690 years cannot be far from the truth. It is
desirable that this important constant should be redeter
mined, and I understand that this is being undertaken by
Professor Boltwood and Mile. Grleditsch.
It should be mentioned that the accuracy of the original
* By an oversight, the period of radium was calculated as 1850 years
instead of 1620 years in ' Kadioactive Substances ' p. 459. The error
arose in the correction in terms of the International Standard.
t Boltwood, Amer. Journ. Sci. xxv. p. 493 (1908).
% Keetman, Jahrb. d. Radioakt. vi. p. 265 (1909).
§ Wien. Ber. cxxii. p. 1086 (1913).
Ions produced by ft and y Radiations from Radium. 321
determination of the period by Boltwood is quite indepen
dent of the correctness of the radium standard, since it merely
involves the comparison of two quantities of emanation.
There is no radioactive method of checking the accuracy
of the value of n, the number of a particles expelled per
second per gram, except by comparison of the value of E,
which is deduced from the measurement of the total charge
rcE carried by a counted number of a particles with other
measurements of the unit charge. Taking the original
determination of Rutherford and Geiger, the electronic
charge comes out to be 4'65 x 10~10 e.s. units. If, however,
we substitute the recent value of e found by Millikan*, viz.
4*77 x 1010, the value of n reduces to 3*48 x 1010 instead of
357 x 1010.
An accurate redetermination of the value of n and of
?iE for radium is much to be desired ; for both of these
quantities are fundamental constants which should be known
with the greatest possible precision.
University of Manchester,
June 1914.
XXXIX. The Number of Ions produced by the ft and y Radia
tions from Radium. By H. G. J. Moseley, M.A., and
H. Robinson, M.ScA
RUTHERFORD aud Robinson J have shown that the
heat evolved in each of the first four transformations
of radium corresponds closely with the kinetic energy of the
a particles emitted. In the transformation of radium 0 part
of the energy is carried by the ft and y radiations. The heat
evolved in the absorption of tin.' y radiation alone could be
deduced from the measurements, but the heating effects of
the a and the ft particles could not be separated. It was
therefore not possible to find the energy of the ft particles
by this direct method. An approximate knowledge, how
ever, of the relative amounts of energy associated with the
three types of radiation may be gathered from a comparison
of the total numbers of ions which they produce in air. The
work described in this paper was done with this object in
view and at Professor Rutherford's suggestion. Our main
results were included in the paper by Rutherford and
Robinson, but no details have hitherto been published.
* Millikan, Physical Review, ii. p. 109 (1913).
{ Communicated bv Sir Ernest Rutherford.
X Rutherford and Robinson, Phil. Mag. xxy. p. 312 (1913).
328 Messrs. Moseley and Robinson on the Number of
The number of ions produced by the a particles from a
gram of radium has been accurately measured by Geiger *.
Eve f has devised an elegant method for finding the ioniza
tion by the j3 and 7 radiations. The ionization at different
distances in the air surrounding a known quantity of active
material is sampled inside a small chamber, with walls and
framework so light that they do not affect the radiation.
One gram of radium then produces in all,
N = I ±7rr2ndr (1)
pairs of ions per second ; where n, the ionization per cc,
is a function of r. In the case of the /3 radiation the value
of this function can be found experimentally for all distances
for which n remains appreciable. The effect of the 7 radia
tion is spread over far too large a volume for this to be
possible.
Fortunately the penetrating part of the 7 radiation, which
carries very much more energy than do the softer types, is
absorbed according to an exponential law, and so in this
case
« = 5'*r (2)
and
N
= 1 47rKe^r=^=: . . . . (3)
Jo A*
If the value of /x, the absorption coefficient in air, is
already known, N can be found by using the value of the
the constant K given by a single measurement of n. Eve
has used equation (3) to calculate the ionization produced
both by the /? and the 7 radiations. Since, however, the
absorption of the ft radiation does not even approximate to
an exponential law, great uncertainty attaches in this ease
to his result. Our chief object in repeating Eve's work was
to measure the /3 ray ionization directly by means of equa
tion (1), and so to include the effect of the softer types of
radiation, which in many of Eve's experiments were stopped
by the walls of the tube containing the active matter.
In the case of the 7 radiation, where these objections do
not apply, there is a serious discrepancy, which will be
discus. Hater, between our results and those of Eve.
Our scarce of radiation was from 1 to 50 millicuries of
* Geiger, Proc. Roy. Soc. A. lxxxii. p. 486 (1909).
t Eve, Phil. Mag. xxii. p. 551 (1911) ; xxvii. p. 394 (1914).
Ions produced by /3 and y Radiations from Radium, 329
radium emanation, which Mr. Chadwick very kindly com
pared with the laboratory radium standard by means of the
7ray balance f. The emanation was contained in a thin
glass tube, whose walls only stopped the a. particles by an
amount equivalent to 2 cm. of air> so that the absorption of
/3 radiation by the glass was quite slight. Measurements
were made of the ionization produced by the j3 and 7 radia
tions together at distances from 10 cm. to 9 metres, and in
many cases were repeated with 1 cm. of aluminium sur
rounding the source in order to find the effect of the 7
radiation alone. At short distances the rapid variation of
n necessitated the use of a very small ionization chamber,
while at long distances a large chamber was needed to obtain
a measurable ionization current. Three chambers, A, B, and
C, were therefore used in turn.
Chamber.
Kange of use.
Approximate
dimensions.
Walls Frame
made of work of
1
A
10 cm. to 50 cm.
30 cm. to 3 m.
1 m. onwards.
1x28x24 cm.
72x71 x7'15 cm.
206 X 206x20*5 om.
Al leaf. Fine wires.
B
Al foil
Fine wires.
0
•002 mm. 
Tissue Steel
paper lined
Al leaf.
knitting
needles.
The wire frameworks of A and B were stretched inside
cages of steel knittingneedles, and each chamber was fixed
on light supports at about 2 metres from the laboratory
floor. Eve f has shown that under similar conditions the
presence of the ionization chamber has no appreciable effect
on the radiation, so that a true measure of n is obtained
by dividing the ionization current by the volume of the
■chamber. Since the sides of A were somewhat irregular,
its volume could not be measured accurately, and it was
therefore calculated from that of B by comparing their
ionization currents when at the same distance from the
source. Each chamber was furnished with a very light
central electrode connected with a Dolezalek electrometer.
The walls of the chamber were kept at a potential sufficient
to ensure saturation, and the ionization current measured
with a standardized condenser by Townsend's balance
method %. The connecting wire was carefully f1 ^ed, for
* Rutherford and Chadwick, Proc. Phys. Soc, London, xxiv. p. 141
(1912).
t Eve, loc. cit.
% Townsend, Phil. Mag. vi. p. 603 (1903).
330 Messrs. Moseley and Robinson on the Number of
unless completely surrounded by earthed conductors it col
lected a large number of ions from the air, even when at a
considerable distance from the charged chamber. The
values found for n, after subtracting the effect of the
natural ionization, were corrected so as to give the values
in air at 76 cm. pressure and 15° C. The actual distance, r,
traversed by the radiation was, when considering the absorp
tion of the (3 rays, replaced by the equivalent distance r',
which was corrected to this standard density of the air and
included a small correction for the absorption by the glass
walls of the emanationtube.
The Ionization by the /3 rays from Radium B and
Radium C together.
The values found for nr2 with no aluminium screen are
plotted in fig. 1 against r' '. The ionization in the distance
10x8
k
Ffc. 1.
1
!
\
■
\
o cm
+ C/VA/
1BE/I B
\
X CHAA
18£f?C
/
a
i 2 3 Metres
up to 10 cm., in which measurements were not made, was
mainly due to the a particles, but excluding the effect of
Ions prodvced by (3 and 7 Radiations from Radium. 331
these, the extrapolation can be made with sufficient accuracy
from knowledge of the numbers and absorption coefficient of
the different types o£ (3 radiation emitted by radium B and
radium 0. The total ionization N was calculated by means
of equation (1) from the area included by the curve, so that
any error in the extrapolation had little effect on the final
result. The ionization by both /3 and 7 radiations up to
3 metres was in this way found to be 6*42 X 1013 X 47r. It
will be shown later that of this only 0*18 X 1013 x 47T was due
to the 7 radiation, leaving 6*24 x 1013x47r for the ionization
by the /3 particles. From 3 metres onwards the value of
nr2, after the 7 ray effect had been subtracted, was found
to fall off exponentially, the value at 3 m. being 6*0 x 10ia
and the absorption coefficient '0056 cm.1. From these data
the total /3 ray ionization from 3 m. onwards was found by
means of equation (3) to be 1*07 x 1013 X 47T, and adding this
to the ionization up to 3 m. the total value of N was
9'20xl014. This result has been calculated in terms of the
RutherfordBoltwood radium standard, which is now known
to contain 5 per cent, less radium than its nominal value.
The corrected value for N, the total /3ray ionization from
radium B and radium C, is 9'65xl014 pairs of ions per
second per gram of radium on the international standard.
The Effect of the ft rays from Radium C alone.
No attempt has been made in these experiments to separate
the yS ray effects of the two contributory transformations. It
is, however, possible to calculate approximately the relative
numbers of ions produced by radium B and by radium O.
An equal number of particles is emitted in the two cases *.
Those from radium B have an absorption coefficient of about
100 cm.1 in aluminium. Those from radium 0 can be
divided into two groups f, of which about 50 per cent, have
\=50 cm.1 and the rest \ = 13 cm.1. Further, the number
of ions produced per cm. in air is a known function of X i,
and allowing for this factor the values of N for the two
transformations should be in the proportion
160 1/120 80\. ... ...
IOO:2Uo + lT5)(::34:6(,)
From this we obtain the estimated value for the/3 radiation
from radium C alone G*ixl0u pairs of ions per gram per
second.
* Moseley, Proc. Rov. Soc. A. lxxxvii. p. 230 (1912).
t Ibid. p. 241.
t Ibid. p. 248.
332 Messrs. Moseley and Robinson on the Number of
The Ionization produced by the 7 rays of Radium C.
We desire to modify the value 1'3 x 1015 previously given *
for the total ionization produced by the 7 rays. This value
was obtained from measurements of the ionization when all
the primary /3 rays were stopped by an aluminium cylinder
0*95 cm. thick.
The results of measurements made under these conditions
are shown in Table I.
Table I.
7 ray ionization.
(Uncorrected for absorption by aluminium.)
Distance.
nr\
103 cm
584 X 109
564 X 109
528 X 109
531 X 109
542 X 109
545 X 109
554 X 109
559 X 109
568 X 109
203 cm
300 cm
400 cm
500 cm
600 cm
700 cm
800 cm
900 cm ...
(It will be seen from the table that the condition expressed
by equation (2), namely, that nr2 should be constant except
for the unimportant factor e~iJr, is only approximately satis
fied. The reason for this variation will be discussed later.)
The results were then multiplied by the factor 1*14 to
correct for the absorption by the aluminium, and the total
ionization calculated by means of equation (3), assuming
Chadwick's value t '000059 cm."1 for /*.
Through the kindness of Sir Ernest Rutherford and
Mr. H. Richardson, we have been permitted to examine
some unpublished absorption curves obtained in connexion
with their work on the " Analysis of the 7 rays from radium
* Rutherford, Kadioactive Substances, etc., p. 295, 1913.
t Chadwick, Proc. Phys. Soc. xxiv. p. 152 (1912).
Ions produced by ft and y Radiations from Radium. 333
B and radium C " *, and this enables us to arrive at a more
satisfactory estimate of the total 7 ray ionization.
For reasons which will be explained later, the value of nr2
at the distance of 3 metres (Table I.) has been made the basis
of our calculations. Assuming the effect of 3 metres of air
to be equivalent to that of *15 cm. of aluminium, this corre
sponds to the effect after absorption by 1/1 cm. of aluminium.
From the curves of Rutherford and Richardson it appears
that of the ionization current observed under these condi
tions 81 per cent, is due to radium C, and 19 per cent, to
radium B.
Taking the total ionization current at this point to be 100,
the effects corresponding to radium C and radium B with no
absorbing material are found from the curve to be 92*8 and
30*5 respectively. (This does not include the very soft
7 radiation of radium B, with absorption coefficient 40 cm.1
in aluminium f.)
Taking the value for nr2 at 3 metres — 5*28 x 109 — from
Table I., the value of the constant K for the 7 rays of
radium C comes out to '928 x 5*28 x 109
= 489 xlO9.
The corresponding value for radium B is *305 x 528 x 109
= 161 xlO9.
Before basing any calculations on these figures it will be
well to discuss the serious discrepancy between our results
and those of Eve J, who in three distinct series of experi
ments found the values 3'8xlOn, 3'74xl09, and 381 xlO9
for K. These are much lower than our value, which is
moreover taken from the lowest result in our table, and
if the 7 rays of radium B were not entirely suppressed in
the experiments of Eve, the discrepancy is still more serious :
it is far too large to have arisen in the measurement either
of the quantity of radium or of the ionization currents, and
the same radium standard (the RutherfordBoltwood) was
used in both cases. The chief point of difference lies in the
methods used to stop the primary ft radiation. In our
experiments the effect of the secondary ft radiation excited
in the aluminium screen was superimposed on that due to
the rays excited in the air alone. By placing the source
and screen in a magnetic field, this effect was measured, and
* Rutherford and Richardson, Phil. Mag. xxv. p. 722 (1913).
t Rutherford and Richardson, loc. cit.
% Eve, loc. cit.
■334 Messrs. Moseley and Robinson on the Number of
was found at a distance of 3 metres to contribute 5 per cent,
to the ionization. Since Eve always used a magnetic field
as well as a screen to stop the j3 rays, this accounts for a
small part of the divergence between our results. We pro
duced the magnetic field by means of a large electromagnet
with poles so far apart that a clear path was left for the
7 rays throughout a cone of more than 50°, but even so the
presence of the electromagnet when unmagnetized diminished
the ionization at 3 metres by 16 per cent. This result is
scarcely surprising, seeing that the 0 particles which give
rise to the ionization are themselves excited in the air all
round the ionization chamber and at a distance up to several
metres from it. It is difficult to arrange an electromagnet
so that the 7 radiation is unaffected throughout the region in
which the operative /3 particles are generated, and an explan
ation of the divergence between our results and those of Eve
may possibly lie in this direction.
Before using equation (3) to calculate N, it is necessary to
enquire whether equation (2) really holds in this case. It is
obviously not true at short distances ; for it gives very large
values of n in the neighbourhood of the source, a region in
which few /3 particles, except those generated in the alu
minium, will have been excited. As these few appear
initially to follow the direction of propagation of the 7
radiation, the value of nr2 excluding the effect from the
aluminium will here be small. In the extreme case in
which the secondary /3 particles continue precisely in this
direction,
nr2=Ke~*r {l—e~kr),
where X is the absorption coefficient of the secondary /3 radi
ation. Here nr2 vanishes in the neighbourhood of the source.
In practice these simple conditions are not satisfied, and it is
impossible to calculate the variation of nr2 at short distances.
At 3 metres, however, the effect cannot be large, and is
roughly balanced by the radiation from the aluminium,
which itself only accounts for 5 per cent, of the ionization.
At greater distances the effect of scattered 7 radiation from
the walls and floor of the laboratory may perhaps become
appreciable, so that, as already remarked, we have adopted
the value of nr2 at 3 metres in all our calculations.
We have seen that this value led to the results
K = 4'89 x 109 for the 7 rays of radium C, and K = 1*61 x 109
for the 7 rays of radium B. Now the value of yu, found by
Chadwick for air at 15° C. and 76 cm. pressure, after
the 7 rays had passed through 1 cm. of lead, was
000057 cm."1
Ions produced by j3 and 7 Radiations from Radium. 335
Equation (3) gives then for the total ionization produced
by the 7 rays of" radium C,
N=^**8?*.10' = 108x10"
5*7 x 10"°
pairs of ions per gram of radium per second.
We are without data for the absorption coefficient of the
y rays of radium B by air ; as the effect of these rays is in
any case very small, we can, however, make a sufficiently
close approximation by assuming fju to be 4*4 times the value
for the 7 rays of radium C (the ratio found in the case of
absorption by aluminium *). On this assumption we get for
the 7 rays of radium B,
4*4 x 5*7x10 °
80 that the total 7 ray ionization is 1*16 x 1015, which after
correcting in terms of the international radium standard
becomes 1*22 xlO16 pairs of ions per gram of radium per
second.
From the nature of the assumptions made in the course of
the calculations, this result can only be regarded as an
approximation to the truth. A more accurate value could
only be obtained as the result of a much more extended
investigation, in which all the data required would be deter
mined under specially suitable conditions. For example, a
possible source of error lies in the use of an absorption co
efficient for air, which was determined with an unsymmetrical
disposition of the absorbing material around the source.
In any case the separation of the effects of the very soft
y rays from that of the ft rays, and of the soft /3 rays from
the « rays would be attended with some uncertainty.
Discussion of the Results.
For the a particles emitted by radium C alone,
N = 8*46x20l51\ For the 7 rays from radium B and C,
N = 1'22 X 1015. These quantities are in the ratio 39*4 : 5*7.
The corresponding heating effects are 39*4 and 6'4 gram
calories per hour per gram of radium, respectively % ; so that
the amount of energy expended in making a pair of ions
appears to be approximately the same in the two cases.
* Rutherford and Richardson, loc. cit.
t Geiger, loc. cit. Geiger's result has been increased by 5 per cent, to
bring it to terms of the international standard.
X Rutherford and Robinson, loc. cit.
336 Ions produced by fi and 7 Radiations from Radium.
Further, since the process of ionization by 7 rays is sup
posed to be indirect, the bulk of the energy of the 7 rays
being transferred to the secondary /5 rays, which themselves
produce the ionization, it is reasonable also to suppose that
the total ionization produced by the primary /9 rays is an
approximate measure of their total energy.
The energy of the different types of rays calculated on this
assumption is given in Table II.
Table II.
N.
Heating
effect gr. cal.
per hour.
Average energy
emitted
per atom.
.y radiation from EaC
y radiation from EaB
(3 radiation from EaC
(3 radiation from EaB ...
M34xl0151
•084xl015j
•64 xlO15
•325 Xl015
596
64 obs.
•44
3'35 calc.
1*71 calc.
194x106 ergs.
014x106
109x106
•55x106
Total (3 and y radiation
from EaC
1774 xlO15
93 calc.
302x106
Total (3 and y radiation
from EaB
•41 XlO15
2*15 calc.
•70X106
Now according to Butherford*, a primary ft particle is
expelled from the nucleus of the radioactive atom with a
definite initial velocity, but loses a variable part of its energy
by exciting the characteristic types of radiation while passing
through the electronic system. The fact that each atom of
radium B or radium C emits on the average rather more
than one /3 particle t indicates that sometimes the energy
converted into 7 radiation is transferred back to a secondary
/3 particle before leaving the parent atom. In any case, the
combined energy of the /3 and 7 radiations should represent
the energy with which the original ft particle would have
escaped had it not excited the 7 radiation. The average
energy per atom is given in the last column of Table II.,
and the combined energy per atom for radium C, namely,
3*02 x 10 ~6 erg, should on this theory be not less than that
carried by the fastest of the /3 particles actually observed..
* Eutherford, Phil. Mag. xxiv. pp. 453, 893 (1912).
t Moseley, loc. cit.
Contact Difference of Potential of Distilled Metals. 337
Rutherford and Robinson * have found a few /3 particles
from radium C carrying as much as 3*9 x 10" 6 erg.
Similarly the fastest {3 ray observed from radium B has an
energy of *6 x 10 ~6 erg, while a very rough approximation
to the combined energy of the (3 and y radiations gives
•7xl0"5erg.
The results in the case of radium 0 are thus at variance
with the theory, but the discrepancy is hardly greater than
can be explained by the uncertainty in the exact value of the
heat produced by the y radiation, and in other factors in
the calculations.
We wish to thank Sir Ernest Rutherford, who suggested
the work, for his advice and interest during its progress.
Physical Laboratory,
University of Manchester.
XL. The Contact Difference of Potential of Distilled Metals.
By A. Ll. Hughes, B.A., D.Sc, Assistant Professor of
Physics, The Rice Institute, Houston, lexas'f.
THIS paper contains a short account of some experiments
on the contact difference of potential between metals
and particularly on its relation to the presence or absence
of occluded gases. ' The research was suggested by a
comparison of the photoelectric experiments of Richardson
and Compton with those carried out by the writer. A
discussion of the relations between photoelectricity and
contact difference of potential will be given before the actual
experiments are described.
One of the most important problems in photoelectricity is
the determination of the exact relation between the maximum
emission velocity of the photoelectrons and the frequency
of the light causing their emission. Experimental diffi
culties made it impossible, until recently, to determine the
exact way in which the velocity increased with the frequency.
The relation between the two quantities has been shown by
Richardson and Compton J, and by the writer § to be that
the velocity squared, or the energy, of the fastest photo
electrons is a linear function of the frequency. This is
generally expressed in the form
V = knV0, (1)
where V is the potential just necessary to stop the electron
* Rutherford and Robinson, Phil. Mag, xxvi. p. 717 (191,3).
f Communicated by the Author.
X Richardson and Compton, Phil. Mag. xxiv. p. 575 (1912).
§ Hughes, Phil. Trans. A. ccxii. p3 205 (1912).
Phil. Mag. S. 6. Vol. 2$. No. 165. Sept. 1914. Z
338 Prof. A. LI. Hughes on the Contact
and therefore proportional to its energy, n the frequency,
and k and V0 are constants. V0 varies from metal to metal,
while k is almost independent of the nature of the metal.
In a theoretical discussion * of the photoelectric effect,
Richardson has identified V0 with the " intrinsic potential "
of the metal. The meaning of this may be made clear in
the following way. If Y0(Cu) be the value of V0 for copper
and V0(Zn) be that for zinc, then V0(Zn)— V0(Cu) obtained
from photoelectric experiments should be identical with the
contact difference of potential between the metals.
Quite recently, some very interesting results, which may
.lead to a more definite knowledge of the nature of photo
electricity, have been obtained by a number of investigators.
Kiistner f found that no photoelectric effect is shown by
zinc when it has been scraped in a vacuum after extraordinary
precautions have been made to exclude gases, particularly
active ones. (The shortest wavelength available in these
experiments was probably X 1850.) Wiedmann and Hall
wachs X found that the removal of occluded gases from
potassium by repeated distillation in a very high vacuum
caused its photoelectric effect to disappear completely.
(Since the light had to pass through glass, the shortest wave
length was probably about X 3400.) These results were
confirmed by Fredenhagen §. On the other hand, Pohl and
Pringsheim  found that potassium behaved in practically the
same way whether it was freed from gases by repeated
distillation in a very good vacuum or prepared in the usual
way for photoelectric cells. We cannot therefore regard
the suppression of the photoelectric effect by the removal of
gases from the metallic surfaces as established beyond dispute.
Nevertheless, the results are of great interest. It is implied
in the papers of Wiedmann and Hallwachs, Fredenhagen,
and Kiistner, that the photoelectric effect does not exist when
occluded gases are completely removed from the metal. A
less violent departure from current views would be to regard
the experiments as indicating that V0 in equation (1) is
increased to such a value that light of the wavelengths
available in their experiments is no longer capable of exciting
the photoelectric effect. The law expressed by equation (1)
holds not only for the electrons released by light from metallic
* Richardson, Phil. Mag. xxiv. p. 570 (1912); Eichardson and
Compton, Phil. Mag. xxvi. p. 550 (1913).
t Kiistner, Phys. Zeits. p. 68 (1914).
X Wiedmann and Hallwachs, Verh. d. Deutsch. Phys. Ges. p. 107
(1914).
§ Fredenhagen, Verh. d. Deutsch. Phys. Ges, p. 201 (1914).
 Pohl and Pringsheim, Verh. d. Deutsch. Phys. Ges. p. 336 (1914).
Difference of Potential of Distilled Metals. 339
surfaces, but also for highspeed electrons released by Rontgen
rays. Consequently, it may be inferred that the processes of
emission of electrons from metals by light and by Rontgen
rays have much in common. There is not the slightest
evidence for believing that the emission of highspeed
electrons from metals struck by Rontgen rays, depends in
any way upon the presence of occluded gases, and conse
quently we should not expect the photoelectric effect to
depend, directly at least, upon the gases in the surface. The
value of Y0 may be a function of the amount of gas absorbed
in the surface, and on this view the complete removal of
occluded gases may cause V0 to increase to such a value that
the long wavelength limit of the photoelectric effect is
beyond the part of the spectrum used by the investigators.
Ever since contact difference of potential between metals
was first observed, two essentially different theories as to its
nature have been advanced by physicists and, as yet, no
experiments have been made which can be said to support
one and disprove the other. According to one theory, two
metals in contact are at different potentials ; on this theory
the contact potential is a specific property of the metal itself.
The other theory maintains that the difference of potential
observed is the difference of potential between the gaseous
films on the surfaces of the metals and not between the metals
themselves. The difference of potential between the film and
the metal depends upon the nature of the metal and on the gas,
and hence this theory may be called the chemical theory
of contact potential. The great difficulty of removing or
altering the exceedingly tenacious films of gas which adhere
to metallic surfaces as soon as they are exposed to air, has
been an obstacle in the way of obtaining definite information
as to the nature of contact potential.
The novel feature about the experiments described in this
paper is that the metals were prepared by distillation in vacuo,
and measurements of the contact potential were made before
the metals had been exposed to air at any appreciable
pressure. The apparatus used is shown in fig. 1. The
base is a brass plate about 35 cm. in diameter and 6 mm.
thick. Through the centre passes a vertical steel rod or
spindle, which is turned down to a point at the upper end.
A plate glass disk G with a circular hole in the middle is
supported by a brass cap C, which in turn rests on the steel
spindle in such a way that the cap and disk can rotate freely.
The position of the disk was controlled magnetically from
outside. An almost opaque film of platinum was deposited
on the under side of the glass disk from a platinum cathode.
Z2
340
Prof. A. LI. Ruolies on the Contact
(This was done in another apparatus.) The film was
generally exposed to the atmosphere for about two days
before contact potential experiments were made. Two quartz
furnaces, F, which could be heated electrically, are situated
TO £LfCTf?OMET£R
below the glass disk and in positions making an angle of
120° with one another and with the brass electrode E.
One contained bismuth and the other, zinc. The furnaces
with the metals in them were heated beforehand for a
considerable time in an auxiliary vacuum to get rid of
occluded gases as much as possible. The distilled metals
were deposited on the under side of the glass disk. The
partitions A, surrounding the furnaces and the electrode E,
served to confine the deposit to definite areas of the disk.
The distance between the disk and the electrode E was about
1 mm. This electrode was connected to a quadrant electro
meter whose sensitiveness was 700 divisions per volt. The
apparatus was exhausted by means of a Gaede molecular air
pump, to which it was joined by a short brass tube about
25 mm. wide. All joints were made airtight by a mixture
Difference of Potential of Distilled Metals. 341
of beeswax and resin which had been subjected to prolonged
heating to drive off vapours. The bulb B, provided with
electrodes, served to test the vacuum during the course of
the experiments. Its diameter was about 15 cm.
The method of experiment was as follows : — The electrode
E connected to the electrometer, previously earthed, was
insulated. Then the disk, connected to earth through the
•spindle, was rotated. Since the disk is always equidistant
from E, no change will take place in the deflexion of the
electrometer as the disk is rotated, provided that the portion
of the disk opposite E is always of the same metal. But if a
part of the disk covered by another metal comes opposite to
E, the deflexion of the electrometer will alter. By
applying a suitable potential to the disk (through the
spindle), the deflexion can be brought back to its original
value. The potential necessary to do this is the contact
difference of potential between the two metals on the disk.
The method was tested by electroplating a brass disk of the
same size as the glass disk with copper on one part and zinc
on the other, and measuring the contact difference of
potential between them. It was found to be *75 volt, which
agrees well with the values given by other methods.
Two typical sets of results will be given. The apparatus
was exhausted by means of the molecular pump. After
about 15 minutes, a patch of zinc was distilled on to a portion
of the disk. The contact potential between this patch and
the rest of the platinum surface was measured as described
above.
f 2 uihis. after Zn electropositive I #f>1 lf
(a)\ distillation. to Pt by j i voir.
[ 10 mins. later. „ '23 „
Air was then admitted to the " rough vacuum J' of the
molecular pump to a pressure of 1 cm.
( Zn electropositive  .601
to Pt by J
I f) mins. later. ,, '79 ,,
(b)i& „ .. „ "87 ,.
j 5 ,. „ „ '92 .,
! 5 .. „ „ 86 „
15 „ „ H '80 „
Rough vacuum pumped out.
5 mins later.
Zn electropositive"! 01 ...
toPtby jSlrolt.
•83
•87
342 Prof. A. LI. Hughes on the Contact
Air was admitted to the rough vacuum of the molecular
pump to a pressure of 2 cm.
{Zn electropositive ] .^n ,.
5 mins. later. „ "66 ,,
Rough vacuum pumped out as before.
I 5 mins. later.
Zn electropositive j '76 volt.
(e)\ toPtby /.80
Air admitted to the rough vacuum to a pressure of 10 cm.
and then pumped out.
5 mins. later. „ *46 .,
{
Air at atmospheric pressure admitted to apparatus.
(g) { 20 hours later. Zn electropositive j ^ ^
Apparatus pumped out completely.
[_ 50 mins. later. „ 39 ,,
Air at atmospheric pressure admitted to apparatus.
,.J Zn electropositive 1 on , ,
(*){ ~ toPtby f '39* volt
The contact difEerence of potential between the zinc and
the platinum immediately after the distillation of the zinc is
evidently quite small and increases slowly with the time (a).
When the pressure of the air in the apparatus is slightly
raised, a great increase in the contact potential occurs (b).
It is necessary to point out here how small the increase of
pressure in the vacuum really is. The " rough vacuum " of
the molecular pump was maintained by a Gaede boxpump
giving a vacuum of about *01 mm. On passing a discharge
through the bulb F, the alternative sparkgap was 8 cm.,
which indicates a good vacuum when one considers the size
of the bulb. When air at a pressure of 10 mm. was admitted
to the rough vacuum, the alternative sparkgap was reduced
* Note.— It was observed that a surface of distilled zinc after exposure
to air was always more electronegative than a surface of polished zinc.
Hence the structure of the surface has, directly or indirectly, a con
siderable effect on the contact difference of potential.
Difference of Potential of Distilled Metals. 34$
by less than 1 cm. Another way of considering the vacuum
is as follows : — According to the information supplied with
the pump, a pressure of 10 mm. in the rough vacuum means
a pressure of '0005 mm. in the high vacuum when the pump
makes 8000 revolutions per minute. As the speed in these
experiments was about 9600 revolutions per minute, it is
probable that the pressure in the high vacuum was below
*0005 mm. It is, therefore, clear that the introduction of a
very small quantity of air into the apparatus causes a great
change in the contact difference of potential when one of the
metals has never been exposed to air before. At this stage,
the contact difference of potential is sensitive to changes in
the pressure of the air. The new zinc surface is absorbing
gas slowly and, up to a certain point, the contact potential
difference increases with the amount of gas absorbed. The
absorption of more gas caused a reduction from *92 to
"80 volt (observation (b)). From observations (c) (and also*
(d), (e)9 and (f)), it may be inferred that the gas absorbed is
not yet held very firmly, for, on reducing the pressure again,
the contact difference of pressure increases. From obser
vations (g), (A), and (i), we see that when the distilled metal
has been in contact with air for several hours, its contact
potential relative to platinum is no longer altered by changing
the pressure of the gas around it, and probably the absorbed
gas has become firmly attached to the metal and is not
appreciably affected by the pressure of the gas in the
surrounding atmosphere.
The evidence that the change in the contact potential on
varying the pressure of the gas in the vacuum is localised at
the surface of the distilled metal is straightforward. The
deflexion of the electrometer on admitting air into the
vacuum was observed, (1) when the platinum surface of the
disk was opposite to the electrode E, and (2) when the newly
distilled deposit of zinc was opposite to E. No appreciable
alteration in the deflexion was observed in case (1), while
very considerable changes in the deflexion always occurred
in case (2).
Part of another series of observations made under similar
conditions is given below.
( 1 min. after Zn electronegative \.m lf
distillation. to Pt by J U
2 mins. later. ,, *00
, 0 Zn electropositive "I .n..
1 J » » to Pt by J UJ
1 2 „ „ ;, 06
2 „ „ „ 09
1 2 „ „ „ 11
344 Prof. A. LI. Hughes on the Contact
Air was then admitted to the re
molecular pump to a pressure of 1 cm.
Air was then admitted to the rough vacuum of the
f Zu electropositive
to Pt by
J 5 mins. later ,,
1 R
},
20 volt.
•61 „
I s ;; „ „ 64 „
15 „ „ „ 64 „
The admission of air at atmospheric pressure reduced the
contact potential to '30 volt.
About twelve separate experiments were made on the
contact difference of potential between platinum and newly
distilled zinc, and results of the same character were always
obtained. The newly distilled zinc was usually electro
positive to the platinum film by •■! or *2 volt, but on a few
occasions it was slightly electronegative. Even in the best
vacuum obtainable in these experiments, the electropositive
character of zinc invariably increased slowly, and this
increase was always greatly accelerated by the admission of
a trace of air. The electropositive character of zinc does
not increase indefinitely ; after a certain point, it decreases
and reaches a steady value at which it remains and after
wards is practically independent of the pressure of the air
about it. These results suggest that there is an intimate
relation between the electropositive character of zinc and the
amount of gas absorbed in the surface. At first, immediately
after distillation, the zinc surface is almost free from absorbed
gases and is strongly electronegative. Then, as it absorbs
gas, it becomes more and more electropositive up to a certain
point. Beyond this point, the absorption of still more gas is
accompanied by a decrease in the electropositive character
of the zinc until a fairly constant end value is finally obtained.
The principal features of the relation between the amount of
gas absorbed and the electropositive character of zinc
(relative to platinum) may be represented graphically as in
fig. 2.
There is a close similarity between the alteration in the
electropositive character of zinc on the one hand and the
shifting of the long wavelength limit of its photoelectric
effect on the other, as the amount of gas absorbed in the
surface increases. Kustner's result that a new zinc surface,
prepared in a vacuum from which all reacting gases have
been excluded with the utmost care, has no photoelectric
effect, has been interpreted as indicating that its long wave
length limit under these conditions is beyond X 1850. With
this result we may link an investigation by Pohl and
Difference of Potential of Distilled Metals. 345
Pringsheim, who did not take such elaborate precautions in
preparing their vacuum. They found that the long wave
length limit of freshly distilled zinc was in the ultraviolet,
Amount of gas absorbed
but as time went on it moved slowly into the violet and
towards the red. Presumably this was due to a slight
evolution of gas inside the apparatus and to its diffusion up
to the zinc. The admission of air caused the long wave
length limit to move back to the violet. We may therefore
conclude that both the long wavelength limit of the photo
electric effect of zinc and its electropositive character (as
measured by its contact potential) alter in the same way with
the amount of gas absorbed in the surface. Results of this
kind are to be expected from Richardson's interpretation of
V0 in equation (1). Any process which makes a metal more
electropositive (t. e. V0 less) should, according to equation (1),
be accompanied by a movement of the long wavelength
limit of the photoelectric effect towards the red end of the
spectrum and vice versa.
A number of experiments, similar to those already described,
were made using bismuth instead of zinc. The contact
difference of potential between the platinum film and the
newly distilled bismuth altered with the pressure of the
air in the vacuum in the same way as when zinc was used.
In some experiments the bismuth and zinc were distilled
simultaneously, and their contact difference of potential
relative to platinum and to each other could be studied under
identical conditions. The results of these experiments may
be summarized by saying that bismuth is initially electro
negative to zinc by about '1 volt, and that the change in it
346 Contact Difference of Potential of Distilled Metals.
contact potential on admission of a trace of air is not quite
so large as in the case of zinc.
It may be urged that the platinum in the form of a thin
film deposited from a cathode on to a glass disk is not a very
suitable standard for measuring the contact potential of
distilled metals. From a number of experiments in which
zinc was distilled on to a copper disk, there was no reason
for believing that the platinum, in the form of a film on glass,
was in any way abnormal. A platinum film on glass was
found to be rather more convenient than a metal disk.
It is generally agreed that for the production of the best
possible vacuum, the apparatus must be exhausted con
tinuously for several days and maintained at a high tem
perature for a considerable portion of that time. It was
inconvenient to run the molecular pump continuously for
more than a few hours and impossible to heat the apparatus
on account of the wax joints. Hence although the molecular
pump is an extraordinarily efficient instrument, it is probable
that still better vacua could be obtained than were used in
these experiments. A combination of cooled charcoal with
a Gaede pump appears to be the most efficient method of
completely exhausting an apparatus. However, liquid air
was not available when these experiments were made and so
the molecular pump alone was used. If the view of the
relation of the photoelectric effect to the contact potential
is correct, we should expect from Kustner's result, that zinc
completely freed from gases would be still more electro
negative than the distilled zinc used in these experiments.
It is unlikely that the zinc distilled in these experiments was
absolutely free from occluded gases, and so we may expect
that further precautions would result in zinc still more
electronegative in character. It was not possible to get
exactly the same contact difference of potential between newly
distilled zinc and platinum in different experiments made
under apparently similar conditions. This is probably due to
the fact that the vacuum obtained cannot be precisely the same
on each occasion, and we have seen how sensitive the contact
potential of new surfaces is to traces of air in the vacuum.
It seems clear from these experiments that the contact
potential between metals under ordinary conditions depends
to a very great extent upon the gas absorbed in the
surface. The results suggest several important and inter
esting questions. Is there any contact difference of potential
between metals completely freed from gases and, if so, how
will it differ from the accepted values ? Which of the gases
in the air is responsible for the change in the contact
Disintegration of the Aluminium Cathode. 347
potential of distilled zinc and bismuth ? It is not possible at
this stage to offer any explanation of the precise way in
which absorption of gases by the surface affects the contact
potential, and indeed whether it is due to mere absorption or
to something of a more definite chemical nature.
Summary*
An investigation has been made on the contact difference
of potential between zinc and bismuth, both distilled in
vacuo, and platinum. Initially, there is but little contact
difference of potential between the distilled metal and
platinum. The admission of a trace of air into the vacuum
causes a great increase in the contact difference of potential.
The change is located at the surface of the distilled metal,
which becomes more electropositive. A maximum is reached
after which the contact potential difference decreases to a
steady value which is scarcely affected by the presence or
absence of air.
A connexion between these results and some recent work
on the photoelectric effect has been pointed out.
The Rice Institute, Houston, Texas.
May 30th, 1914.
XLI. Disintegration of the Aluminium Cathode.
By L. L. Oampbkll *.
IN 1891 Crookes f made the first quantitative determi
nation of the rate of disintegration, or " electrical
evaporation, " as he termed it, of a number of metals used
as cathodes in dischargetubes containing rarefied air. He
found that the following series represented the relative rate
of disintegration of the cathode, in descending order : Pd,
Au, Ag, Pb, Sn, Pt, Cu, Cd, Ni, Ir, Fe. The disintegration
of the Al and Mg cathodes he found to be zero. This lack
of disintegration, or " spluttering," on the part of the
aluminium cathode has made its use in spectrum and other
dischargetubes very general.
Shortly after the discovery of the noble, or monatomic
gases, argon, helium, neon, xenon, krypton, it was found by
Orookes J, Travers §, Baly , Soddy and McKenzie II,
Claude **, and others, that the aluminium cathode
* Communicated bv Sir J. J. Thomson, CM., F.R.S.
t Proc. Roy. Soc. 1. p. 88 (1891).
% Zeitschr.f.2>hys. Chemic, xvi p. 370 (1895).
§ Proc. Roy. Soc. lx. p. 449 (1897).
 Ibid, lxxii. p. 84 (1903).
«[ Ibid. lxxx. Ser. A, p. 9l> (1908).
** Compt. Rend, cliii. p. 713 (1911).
318 Mr. L. L. Campbell on
disintegrated badly when an electric discharge was sent
through tubes containing the above gases in a rarefied state.
c © ©
For the spectroscopic study o£ the monatoinic gases Baly *
has called attention to a special form of aluminium cathode
that claims to lessen the disintegration during the electric
discharge.
In gases other than the monatomic ones, the aluminium
cathode was supposed not to disintegrate, or the amount oi:
disintegration was considered to be negligible.
In the autumn of 1913, while bombarding AgCl with
cathode rays from an aluminium cathode, Prof. J. J. Thom
son found that the dischargetube became heavily coated
with a metallic mirror, which on examination proved to be
aluminium. It was supposed that the cathode rays had
decomposed the AgCl into Ag and CI, and that in some way
the CI had been instrumental in bringing about the dis
integration of the aluminium cathode.
©
The following preliminary investigation was undertaken
to see if the aluminium cathode did disintegrate in the
presence of chlorine, and of other gases not included under
the list of monatoinic gases.
©■
General Method of Procedure.
The dischargetubes first used were of the type shown in
fig. 1. The length of the tubes was about 30 cm., the
diameter of the spherical bulb G cm., the diameter of the
larger cylindrical portion 2*5 cm., while that of the smaller
end portions was 1 cm. The anode consisted of a small
fused mass of aluminium, while the cathode was made of
an aluminium wire 2 mm. in diameter and 4 cm. long, to one
end of which was fastened an aluminium disk 1*0 mm. thick
and 1*5 cm. in diameter. To the discharge tube wras con
nected an annex tube, X (fig. 1), in which could be placed
easily decomposable salts for supplying the desired gases to
the dischargetube. The dischargetube was connected to a
waterpump, to a mercury pump, McLeod gauge, drying
tube, and charcoal tube for liquid air. The tube was
exhausted by means of the pumps, and the electric discharge
from an induction coil was allowed to pass through the tube
for a considerable time to drive out the major portion of
the gases occluded in the electrodes. A pressure of about
O'Ol mm. was obtained with the mercury pump, then the
pressure was reduced to the Xray stage by the application
of liquid air to the charcoal tube. Then a small amount of
gas or vapour was let into the dischargetube from the annex
* Baly's ' Spectroscopy,' 2nd edition, p. 126 (1912).
Disintegration of the Aluminium Cathode. 349
tube. If too much gas got into the dischargetube, a portion
was frozen out by means of the charcoalliquidair tube.
Fin. 1.
Charcoal tube
In the early work an induction coil was used to excite
the discharge, while later the discharge was sent through
the tube by means of a battery of small accumulator cells
giving a voltage of about 2400 volts. Whenever the elec
trodes were contaminated in any way by use, new ones were
introduced in the tubes.
Disintegration in Chlorine.— The chlorine was prepared by
placing a small amount of gold chloride, Au013, in the annex
tube, and gently heating the salt by holding a lighted match
beneath the bulb of the tube. First the water of crystalliza
tion was driven off, and this was removed by the phosphorus
pentoxide and the charcoalliquidair tube. Gold chloride is
said to decompose at about 120° C, at atmospheric pressure,
while under the very low pressure of the dischargetube, it
must decompose at a still lower temperature. At any rate,
a very gentle heat was all that was required to set free the
small amount of chlorine needed. This amount of chlorine
did not entirely remove the green fluorescence of the Xray
vacuum staoe. The exact pressure was not measured, as
the letting in of chlorine into the McLeod gauge would have
contaminated the mercury. It seems fair to assume from
the °Teen fluorescence and the width of the cathode dark
space*, that the pressure in the tube was much below the
pressure of O'Ol mm.
350 Mr. L. L. Campbell on
The disintegration began from the large aluminium wire
back of the cathode disk, and the small portion of the cathode of
the tube (1) received the mirror deposit first. In about an
hour the large bulb B (fig. 1) was about half covered with a
bright opaque metallic mirror, showing interference colours.
During the disintegration process the cathode wire and
disk were covered with a fuzzy pink glow extending 1 to
2 mm. from the surface of the metal. No trace of chemical
action between the aluminium electrodes and the chlorine was
apparent to the eye. The dischargetube was removed and
opened, and the deposit was examined in the usual way with
caustic potash, and was found to consist of aluminium metal.
Disintegration in Bromine. — The bromine was prepared
by placing gold monobromide, AuBr, in the annex tube and
heating it gently. The bromide decomposes under normal
pressure at about 115° C. into Au and Br. The discharge
tube was exhausted as in the case of chlorine, and a small
amount of bromine let in. The disintegration began, as
before, behind the cathode disk, and proceeded much as in
the case of chlorine, except that at the end of an hour the
deposit seemed greater than the deposit formed when
chlorine was used. The metal of the electrodes remained
bright during the disintegration, and no evidence of
chemical action was apparent. The metallic deposit when
examined proved to be aluminium.
Disintegration in Iodine. — The iodine was prepared in the
annex tube from gold monoiodide, Aul, which decomposes, at
atmospheric pressure, at about 120° C. When the same
condition of pressure was obtained in the dischargetube as
was the case when chlorine was used, the disintegration
began and was very rapid. A white ring deposit was
formed in front of the metallic mirror. On examination
this was found to be aluminium iodide. This compound was
probably formed by the union of the metal and iodine after
the metallic particles had left the cathode.
Relative Rates of Disintegration in the Halogens.
The discharge was sent through two tubes connected in
series. The tubes were of the same dimensions, the one
contained chlorine and the other bromine. The cathode
darkspaces in the tubes were kept of the same width as far
as practicable. Thus the pressure and voltage, as well as
the current strength, were the same in each tube. The
bromine tube received its mirror more rapidly than did the
chlorine tube, and after an hour the deposit in the former
tube seemed about twice as great as in the latter.
Disintegration of the Aluminium Cathode. 351
In a similar manner, the discharge was sent through a
chlorine and an iodine tube in series, the pressures in the
tubes being kept practically the same. The disintegration
in the iodine tube was much more rapid than in the chlorine
tube, and after the discharge had been running for an hour,
the deposit in the iodine tube appeared to be three or four
times as great as that in the chlorine tube.
Thus it would seem that the rate of disintegration of the
aluminium cathode in the halogen gases increases in the
general order of their molecular weights. This is in accord
with the investigations of Kohlschiitter* with a platinum
cathode in hydrogen, helium, nitrogen, and argon. As
J. J. Thomson t has pointed out, the disintegration of the
cathode is probably brought about by the bombardment by
positive particles, which give to the atoms of the metal
sufficient energy to escape ; and that the heavier molecules
may more readily receive multiple charges, and that thus
their kinetic energy is increased, as shown by the relation,
Kinetic Energy = V.e, where V is the cathode fall in poten
tial and e the electric charge on the molecule.
In the above cases, the discharge was excited by an
ordinary inductioncoil. Later, a storage battery giving
2400 volts was employed. The total voltage between the
electrodes was measured by means of a Braun electrometer,
and the current strength employed was read on a railliammeter.
Tt was found difficult to maintain a discharge from the small
cells through the long tubes of fig. 1 type, so a shorter tube
.(fig. 3) was employed. The tube was 15 cm. long and
Fii
<
>u
y
=r* TO PUMP, ETC.
2*5 cm. in diameter. The cathode consisted of an aluminium
wire 2 mm. in diameter and 3 cm. long.
It was found that some disintegration occurred at lower
voltages, yet the spluttering was not rapid until the voltage
passed 1000 volts. In the case of iodine the cathode did not
disintegrate rapidly until the voltage had reached some
1500 volts. This increase in voltage may be due to a
* Zeitschr. f. Electroch. xii. p. 365 (1906).
t ' Rays of Positive Electricity/ p. 104 (1913).
352 Mr. L. L. Campbell on
hindering effect o£ the aluminium iodide formed. The
current strength employed was from 4 to 5 milliamperes.
The size and form of the tube and of the cathode seem to
influence the voltage required to bring about this disinte
gration of the cathode.
Disintegration in Cyanogen. — The cyanogen, (CN)2, was
prepared by gently heating silver cyanide, AgCN, in the
annex tube, X, fig. 1. Under the electric discharge the
(GN)2 changed to paracyanogen a?(ON), and other portions
were probably changed to C and N, while a portion of the
(CN)2 remained undecomposed. When the cathode dark
space had reached the walls of the tube opposite the cathode,
disintegration of the cathode set in, but was not so rapid as
was the case with any of the halogens. On portions of the
tube, not covered by the metallic mirror deposit, was found
a brownish deposit of carbon, with probably some para
cyanogen.
Disintegration in Pentane. — Pentane, C(CH3)4, was chosen
as a typical and fairly stable hydrocarbon. The vapour of
the pentane was supplied to the dischargetube by means of
an annex tube shown in fig. 2. When the cathode darkspace
had reached the walls of the tube, disintegration of the
cathode began, but the process seemed somewhat hindered by
decomposition products brought into existence by the dis
charge. A deposit of carbon was found on those parts of
the walls of the tube where there was no metallic mirror
formed.
Disintegration in Mercury Dimethyl. — The mercury di
methyl, Hg(CH3)2, was placed in an annex tube of type
shown in fig. 2. The vapour was soon partly decomposed by
the discharge, as was shown by the presence of mercury
spectrum lines. The disintegration of the cathode was quite
rapid, and was attended by a deposit of carbon on the tube,
as was the case when pentane was used. The mercury set
free from the dimethyl was probably the active agent that
brought about the disintegration of the cathode.
Disintegration in Mercury Vapour. — Vapour of mercury was
let in from the annex tube in small quantity, and the disinte
gration of the cathode in the presence of this vapour was
probably the most violent of any observed.
Disintegration in Cadmium Vapour. — Finely ground cad
mium metal was placed in an annex tube. Beneath the
bulb containing the metal was placed a small gasflame that
melted the cadmium and kept a small supply of the metallic
vapour in the dischargetube. In the presence of this vapour
the cathode disintegrated rapidly, and the rate of deposition
Disinter) rat ion of the Aluminium Catliode. 353
seemed not much below that brought about by the mercury
vapour.
Disintegration in Carbon Monoxide. — The carbon monoxide
was prepared from ferrocyanide o£ potassium and purified in
the usual way. Small portions of the gas were let in from
the annex tube of the type in fig. 2. Under the electric
discharge the cathode became covered in part with a brownish
coating, probably due to the separation of carbon from the
CO. As the discharge continued, the pressure fell rapidly,
and the CO seemed to be absorbed and to disappear. More
gas had to be let in several times, before any disintegration
was noticed, and after an hour or more only a small deposit
had been made. The presence of the carbon set free may
have retarded the rate of disintegration in this instance.
Behaviour of the Cathode in Nitrous Oxide. — The nitrous
oxide was prepared from ammonium nitrate and purified.
The gas was let into the dischargetube as in preceding cases.
"When the discharge was started the pressure fell rapidly, the
green fluorescence gradually disappeared, and in about ten
minutes the discharge refused to pass through the tube, and
jumped the gap at the spark terminals of the coil. A small
additional amount of gas was let in, and again the pressure
dropped, and in about ten or twelve minutes the discharge
refused to pass through the tube any longer. This procedure
was repeated some seven times, with the same result. No
disintegration took place, and it seemed useless to continue
the process. The cathode had become darkened near the
centre during the discharge, and some chemical action had
seemingly taken place between the cathode and the decom
position products of the N20. The N20 gas, or its
decomposition products, exhibited a very vivid afterglow
when the discharge was turned off. The glow was of a
yellowish pink colour, and lasted for some seconds. The
glow may have been due to the " active nitrogen " studied by
Strutt *.
Delayed Disintegration, — During the above study of the
disintegration of the aluminium cathode, sometimes the
deposition of the metallic mirror began almost as soon as
the discharge was started ; at other times the disintegration
began only after several hours of discharge. In one or two
instances, the discharge was sent through the tube for a long
time, with no deposition of a mirror apparent ; then the gas
in the tube was left exposed over night to the phosphorus
pentoxide tube, and in the morning the disintegration began
* Rov. Soc. Proc, Ser. A. lxxxv. p. 377 (1911) ; Ser. A. lxxxvi. p. 262
(1912). *
Phil Mag. S. 6. Vol. 28. No. 165. Sept. 1914. 2 A
354: Disintegration of the Aluminium Cathode,
as soon as the discharge was sent through the tube. Hence
it would seem that a very small amount of water vapour
present hinders or prevents the disintegration of the cathode.
The delay in other instances may be due to the fact, that the
thin and invisible layer of oxide or nitride on the surface of
the cathode must first be removed hj bombardment by the
positively charged particles.
In support of the necessity of this initial " cleaning " of
the cathode surface, may be mentioned the fact, that in the
cases where the disintegration was pronounced, the metal of
the cathode appeared bright after the discharge ceased, and
when the tube was opened to the air, the peculiar " fibrous
growth " took place on the surface of the cathode. This
growth of aluminium hydroxide is supposed to take place
only when a perfectly clean surface of the metal is exposed
to the air.
This ability of aluminium to long delay the process of
disintegration may be the reason it is so much used and yet
disintegrates so little.
General Considerations,
1. The aluminium cathode, which had been supposed not
to disintegrate to any marked extent, except in the monatomic
gases, does " splutter " violently in the presence of the
halogens, certain metallic vapours and other gases.
2. The disintegration of the cathode in question seems to
take place best when the pressure in the tube has reached the
ordinary Xray stage, and when the cathode darkspace has
reached the walls of the tube next the cathode.
3. The rate of disintegration seems to increase in general
with the molecular weight of the discharge gas.
4. It seems probable, that under proper conditions, the
aluminium cathode will disintegrate in all gases, more or
less rapidly. It is intended to investigate the behaviour of
this cathode in other gases, and to make quantitative deter
minations of the rate of the disintegration, the cathodefall,
and the current strength necessary to bring about this
phenomenon.
The writer wishes to express his gratitude to Prof.
Sir J. J. Thomson for suggesting and directing this pre
liminary investigation.
Cavendish Laboratory.
June, 1914.
[ 355 ]
XLII. The Crystalline Structure of Copper, By W. Law
eence Bragg, B.A., Allen Scholar of the University of
Cambridge *.
rriHE copper crystals used in this investigation were some
L natural specimens, for which I am indebted to
Mr. Hutchinson, of the Mineralogical Laboratory at Cam
bridge. In their natural state these specimens are obviously
rough crystals, and have some faces of large dimensions
(1 cm. each way), but these faces are very much warped and
distorted. An attempt was made to obtain an Xray reflexion
from various natural faces of such crystals, but it was not
successful. Apparently the outer surface of the crystal has
been so battered and distorted that little regular crystalline
arrangement is left. Any attempt to grind crystal faces
artificially also destroys the crystalline character of the
surface and so prevents the reflexion of X rays from the face.
It was observed, however, that when the crystal was placed
in nitric acid until the surface was eaten away to an extent
of perhaps \ millimetre, the faces were etched deeply into
numerous parallel facets, which all reflected the light simul
taneously in the usual way. This suggested that, internally
the crystal structure was perfect, and showed further that in
some cases the whole specimen was composed of a single
cryst.il. Moreover, in this case the surface layers are not
pulled about, and so are capable of reflecting the X rays
falling on them. This method of obtaining a crystal surface
was suggested by some previous experiments on natural zinc
oxide, zincite. Zincite occurs very rarely as crystals, and
the specimens used had merely a platy structure. However,
by partly dissolving a block of the mineral in hydrochloric
acid, the etched mass showed indications of crystalline
structure sufficient to serve as a guide in the preparation of
various faces. The faces reflected the X rays and led to the
determination of the arrangement of zinc and oxygen atoms.
Copper crystallizes in the cubic system, holohedral class.
The natural crystals of copper used in the experiments were
mostly of one type, being composed of two individuals
twinned about the plane (111). The faces of the simplest
crystals approximated to those of the rhombic dodecahedron
{110}. The face first investigated was that parallel to the
twin plane (111) of the best of the crystals. The crystal
appeared like two triangular pyramids joined base to base,
and the apex of one pyramid was ground down on a carbo
rundum wheel until a triangular face (111) was formed.
* Communicated by the Author.
2 A2
356 Mr. W. L. Bragg on the
This was roughly polished, treated with nitric acid to dissolve
away the outer layers, and then mounted in the usual way in
the Xray spectrometer *.
By assuming various arrangements for the copper atoms,
we can calculate the spacings o£ various planes of the crystal,
starting from the density of copper, 8*96, as a basis. For
instance, if the copper atoms are arranged at the corners of
cubes, so as to form a simple cubic lattice,, we have the
relation
(dcmy. 896 = 6357. 164. 10~24,
since we know that
1. Mass of a copper atom =63*57 . (Mass of a hydrogen atom)
= 63*57. 164. 10"24 gram,
2. The unit cube of the structure contains one copper atom,
This gives the relation
dam = 226. lO"8 cm.
The X rays from an anticathode of palladium, such as was
used for the purposes of this experiment, have a wavelength
of 576 . 10 s cm. This is reflected from the planes d(ioo) at
an angle given by the equation
\ = 2 . d(ioo) . sin 0.
Substituting the above values for \ and i(ioo), we find that
e=r 20'.
Suppose, on the other hand, that the copper atoms were on
a facecentred cubic lattice. This lattice has a point at each
corner of a set of cubes, and one at the centre of each cube
face. The volume (rf(ioo))3 now contains only onehalf a
copper atom. From this it can be calculated that ^(100) =
1*80 . 108 cm. So for the other faces ; the results are
summarized below.
Smoino of planes Glancing angle of
bpacing ot planes. reflexion, ?d rays.
Facecentred lattice :
d{m)=l'80 . lO"8 cm. 0(1Oo) = 9° 13'
tf(110) = l27 . lO"8 cm. 0(uo) = 13° 2'
iau)=208 • 10"8 cm. 0(ui) = 8° 0'
Simple cubic lattice :
Jao0) = 226 . lO"8 cm. 0(ioo) = 7° 20'
d(110) = l59 . 10"8 cm. 0(iio)= 10° 22'
d(111)=l30 . lO"8 cm. 0(iu)= 12° 50'
* For a description of the instrument and its manipulation, see Proc.
Roy. Soc. A. vol. lxxxviii. p. 428, and A. vol. lxxxix. p. 468.
Crystalline Structure oj Copper.
357
The face (ill) of the copper crystal, prepared as described
above, was first investigated. The chamber was set at 25° 40'
( = 2 X 12° 50'). The crystal face, being adjusted so that the
rays fell on it at a small glancing angle, was turned so that
this angle assumed in turn a series of values between 6° and
20°, in order to find if, in some position, it reflected the
X rays. This had to be done because the true orientation of
the plane (111) in the crystal was not known with any
exactness. At none of these angles was there a reflexion into
the chamber, the simple cubic arrangement of the copper
atoms being therefore ruled out.
On setting the ionization chamber at 16°, however, a
marked effect was found. Fig. 1 shows the current in the
chamber for a series of angles at which the crystal was set.
Fig. 1,
5° 10° 15'
Cl/iA'C/A'S /1A/Ci.£ Of S?ArS O/VCri^TAL FACC ...
Between 8° and 11° 30' there is a marked increase in the
ionization current, which rises to a maximum at 9° 30'.
Now if the crystal were perfect, the range of angles at which
it reflected the X rays would be limited to some 30' at most.
The fact that the crystal reflects over such a wide range of
angles shows that its planes are distorted to an extent of
several degrees, instead of being strictly parallel. As the
crystal is turned round, one set of planes after another comes
into the reflecting position and causes an ionization current
in the chamber. From the curve we deduce that when the
•crystal is set at 0° 30' the area of its face so oriented as to
reflect is larger than at any other angle.
A scries of readings are now taken with the chamber at
various angles " #," and the crystal in each case at the angle
Q
" 9+1° 30'/' in order always to make use of this larger
reflecting area. The results are shown in the curve for the face
358 Mr. W. L. Bragg on the
(111) of fig. 2. Here we have two peaks close together in
the curve, representing the two lines in the spectrum of
palladium. The curve shows a decided first and second
order spectrum, and even perhaps a third, though this last i&
Fig. 2.
Copper
(100)
* 18 50
k
38
(no)
c5 40'
S4?"40'
^
^
(m)
A
* 16
32 W
49*
30'
0° 10° 20°
Angle of C^.iM8£/t
60
somewhat doubtful. It must be noted that the range of
angles, over which the ionization chamber may be set so as
to receive the reflected beam, is confined to 1°, though the
crystal structure is so imperfect. If the crystal were perfect
it would be scarcely smaller, and the reason for this is clear.
Although there are a number of settings for the crvstal which
enable a set of planes somewhere on its distorted face to
receive the incident X rays at a glancing angle of 8° and
so to reflect them, the reflected rays all converge and are
received by the chamber when set exactly at 16°*. There
fore, when the chamber and crystal are moved simultaneously,,
the reflexion is only found when the chamber is in the neigh
bourhood of 16°.
Fig. 2 also showrs the curves for faces (110) and (100) of
copper. The angles at which the spectra are found are as
follows : —
0(ioo) = '163.
#(1]())=13018' sin(9(11o) = *230.
0(111)= 8°0' sin0(111) = 139.
Calculated angle 9° 13'
„ 13° 2'
„ 8° 0'
sin#(100):sin 6{
(110)
sin 6
(in)
= 1:141: '85 = 1: V^a/^.
This relation between the angles of reflexion from th&
* Cf. W. H. Bragg, Phil. Mag. May 1914, p.
Crystalline Structure of Copper. 359*
three principal faces of the crystal is that which would exist
for a facecentred cubic lattice. In such a lattice
^(100) 1^(110)1 ^(111) — 1
1 2
We have already seen that the absolute values of the angles.
at which reflexion occurs are those to be expected if the
copper atoms lie on a facecentred lattice. It is further to
be observed that the 1st, 2nd, and 3rd order peaks reflected
from the faces of the crystals are in every case quite normal,
the first order being greater than the second, and that
greater than the third. This, as has been shown in former
papers (Proc. Roy. Soc. A. vol. lxxxix. p. 472) implies a
regular arrangement of reflecting planes, equally spaced and
identical in nature. Lastly, as a check, a search was made
for spectra at half the angles at which the first spectra of
fig. 2 occur, in the case of the planes (100) and (110).
This search gave a negative result. Taking all this into
consideration, there can be little doubt that tlie atoms
of a copper crystal are arranged on a facecentred cubic
lattice.
The results are interesting in that they show that con
siderable accuracy of measurement can be obtained with the
Xray spectrometer, even when the crystal itself is highly
irregular. If the cubic symmetry of the copper crystal had
not rendered this unnecessary, it would have been possible
to measure the axial ratio of the crystal within 1 per cent, of
the truth, although the faces of the crystal were distorted by
many degrees. Since the crystal is so irregular, only a
fraction of its surface reflects at any one angle, and therefore
the electroscope had to be very sensitive when measuring the
small ionization current. This explains the very obvious
irregularities of the curves in fig. 2. The dots in this figure
represent a set of readings. All these curves were repeated
several times, some with different crystal faces. Some curves
were more irregular than others, but all agreed closely in the
positions of the spectra.
I wish to take this opportunity of again thanking Mr. Hut
chinson for his kind help, both in supplying material and in
aiding with his advice the preparation of the various crystal
faces. I wish to thank Professor Sir J. J. Thomson for his
kind interest in the experiments. I am indebted to the
Institut International de Physique Solvay for a grant with
the aid of which the apparatus used in these experiments was
purchased.
360 Prof. J. C. McLennan on the
Summary.
It was found possible, by treating with acid prepared
surfaces of a natural crystal of copper, to obtain crystal
faces which could be used as reflectors in the Xray spectro
meter.
The results of the investigations, thus rendered feasible,
showed that in a copper crystal the atoms are arranged on a
facecentred cubic lattice. This is the closepacked lattice,
to which attention has been drawn by Pope and Barlow.
The crystal structure is the most simple of any as yet
analysed.
The Cavendish Laboratory,
July 16th, 1914.
XLIII. On the Absorption Spectrum of Zinc Vapour. By
J. C. McLennan, Professor of Physics, University of
Toronto *.
[Plate VI.]
IN a paper by Wood and Guthrie f on the absorption
spectra of certain metallic vapours, the statement is
made that with zinc vapour they found no trace of any
absorption in the range of the spectrum investigated by
them. This region began at about X5200 and, from indi
cations in the paper, did not extend beyond X2150.
In view of the prediction made by Paschen % and subse
quently confirmed by Wolff §, that the emission spectrum of
zinc should include a series of single lines with the first
member of the series at X2139'33, it seemed to the writer
desirable to look for absorption by zinc vapour in the region
below X2150.
With the object of investigating this point, some pure zinc
was placed in a clear fused quartz tube 2 cm. in diameter,
which was then highly exhausted and sealed up. This tube
was placed before the slit of a quartz spectrograph and heated
with a Bunsen burner. When the spark between zinc
terminals in air was used as the source of light, it was found
that the vapour evolved when the zinc was melted and raised
to a red heat was sufficient to completely cut out the line
X2139#33. With this moderate heating the absorption band
was sharply edged and quite narrow. With stronger heating,
* Communicated by the Author.
f Wood and Guthrie, Astrophys. Jl. vol. xxix. no, 1, 1909, p. 211.
% Paschen, Ann. der Phys. 1909, vol. xxx. p. 746, and 1911, vol. xxxr.
p. 860.
§ Wolff, Ann. der Phys. 1913, vol. xlii. p. 825.
Absorption Spectrum of Zinc Vapour.
361
however, the band widened symmetrically, and with very
strong heating with a Meeker burner itcould0easily be made
to cover a range of from one to two hundred Angstrom units.
No trace of any other absorption was found either above
A.2139'33 or in the region of the spectrum below this line
down as far as X1840.
From the work of Wood *, Stark f, and others it is known
that the light corresponding to the lines \2536'72 and
X1849'6 is readily and easily absorbed by mercury vapour.
Wood and Guthrie % have also found that with cadmium
vapour there is strong absorption at \2288*78. With
cadmium vapour of high density these investigators have
found, too, that it is possible to obtain a narrow absorption
band at \3260*17 in addition to the one at \2288'79. From
the summary of wavelengths given in Table I., which are
taken from Paschen's paper, it will be seen that all of these
Table I.
Single line series... n — l'b, S — m, P.
m.
Mercury Cadmium
(X). " (X).
Zinc
(X).
o
18496 228879
140271 10693
1268*9 152673
12506 146935
etc. etc.
213933
1589 64
145704
137697
etc.
3
4
5
Combination series... // = 2, p2— m, S.
m.
Mercury Cadmium
(X). " (X).
Zinc
(X).
15
253672
4078 05
283707
256414
etc.
320017
441323
30828
27571
etc.
307599
42940
29818
20064
etc.
25
35
45
* Wood, Astrophys. Jl. 1907, vol. xxvi. p. 41.
t Stark, Ann. der Phys. 1913, vol. xlii. p. 239.
X Wood and Guthrie, lac. cit.
362 Absorption Spectrum of Zinc Vapour.
absorption bands correspond either to members of Paschen's*
single line series n=l*5, S — m, P, or to members of his
combination series ?* = 2, p2— mf S, in the emission spectra
of mercury, cadmium, and zinc.
If zinc vapour acts in a manner analogous to that of
mercury and cadmium, the numbers given in the table
indicate that one should expect to find absorption with zinc
vapour at \3075*99. In the experiments of the writer with
this vapour every effort was made by strong heating to see if
absorption occurred in the neighbourhood of the line
A3075'99, but no evidence of any absorption was obtained.
It is just possible, however, that the failure to get absorption
in this region was due to the fact that in these experiments
either vapour of sufficiently high density or a spectrograph
of great enough resolving power was not used.
Considering the character of the absorption band which
one obtains with mercury vapour at X2536*72, it may be of
interest to note that this band is, apart from its asymmetrical
nature which was first described by Wood *, in intensity
more like that which Wood and Guthrie f obtained with
cadmium vapour at \2288'79 and the writer with zinc vapour
at \2l39"33, than it is with the absorption which Wood and
Guthrie obtained with cadmium vapour at X326CK17. From
observations on such factors as intensity, line structure,.
Zeeman effect, and Doppler effect, Stark J has been led to
take the view that the line X253G*72 belongs to the mercury
single line series and that it is really the first member of that
series ; i. e., according to him the Hg line \2536'72 and not
the Ho line \1849*6 is homologous with the CdlineX2288*79
and the Zn line \2139'33. The considerations regarding
absorption given above, it will be seen, lend some support to
his view.
It would have been interesting to see if absorption also
occurred with mercury vapour at X1402*71, with cadmium
vapour at Xl669'3, and with zinc vapour at A.1589'64, as well
as at lines farther down in the single line series of these
metals, but to investigate this point it would have been
necessary to use an optical system of fluorite, and this
equipment the writer did not have at his disposal.
Since the evidence adduced so far indicates that one should
expect to find absorption by these three vapours at the wave
lengths mentioned above, it suggests that the lines of
Paschen's single line series in the emission spectra of
* Wood, Phil. Mag. Aug. 1909.
t Wood aud Guthrie, he. cit.
X Stark, Ann. der Phys. 1912, vol. xxxix. p. 1612; 1913, vol. xliu
pp. 238 & 243.
Secondary ?
RG£
0 /
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/
/
y
y r *
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iX^:
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15 20 ^5 30 36^,
Fisr. 2.
So
/?//?<: f rzv
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374 Mr. H. A. McTaggart on
from Traube (Journ. de Chim. et Pliys. iii. p. 587, 1905)
and Freundlich (Kapillarch.. p. 64). Tbe curves showing
the effect of concentration on tbe velocity, i. 5)
found for it a small but easily measurable electric osmosis
through certain porous diaphragms — the sign, however,
depending on the nature of the diaphragm. Besides this,
methyl and ethyl alcohols give rise to electrification in water
fall and in bubbling experiments, which might lead one to
expect some effect.
In methyl alcohol, bubbles of air of a size *10 mm. showed
no sign of motion under potentials of 100 volts/cm. The
application of the potential in this case had the curious
effect of hastening the absorption of the bubble.
In ethyl alcohol 125 volts/cm. gave no motion to an air
bubble, not even in a 50 per cent, aqueous solution.
In a 25 per cent, solution of propyl alcohol no motion
can be observed with 125 volts/cm. for airbubbles artificially
made.
In isobutyl alcohol no motion of an airbubble is
apparent.
It does not follow, of course, that a cataphoresis of other
particles than airbubbles does not occur in these alcohols, as
the case of benzine spheres in 25 per cent, propyl alcohol
shows as well as Baudouin's experiments.
The results for the first four fatty acids are given here.
The concentrations are necessarily small for the reason ex
plained in the last paper. A complete series of observations
for formic acid is given to show again the effect of size, these
acids being adsorbed like the alcohols and being very active
in reducing the surface tension. The concentrations are
given in ccs. of acid per c.c. of water.
The readings are not as regular as they should be — perhaps
because of some unavoidable impurity introduced with the
bubble by the pipette, or perhaps — where the final reading
for a bubble is large — because of some residue left by the
acid in the surface.
Electrification at LiquidGas Surfaces.
Formic acid and water.
375
Cone.
10'
10"
1/2
10 . 1/4
10"\l/8
10"6.1/16
10 . 1/3:
10 "6 . 1/64
Diam.
•16
•10
•06
•02
•14
•10
•08
•02
•10
•08
•00
•02
•12
•10
•04
•14
•10
•06
•04
•14
•08
•06
•04
•14
•10
•06
02
Time
22 sees
16^
15
14
24
16
121
14
14
13
18
15
13
12£
13
17
13
14
16
18
12
12
13^
14
13^
13*
18
Yolts/cm.
35
Acetic acid and water.
Cone.
Diam.
Time.
Volts/cm.
I
10"4
•12
23 sees.
35
10~4. 1/4
28
10_4.1/16
20
10 "4. 1/64
14
TO"4. 1/256
15
10~4. 1/1024
25
10 "4. 1/4096
14
Di9t. water
11
376
Mr. H. A. McTaggart on
Propionic acid and water.
Cone.
Diam.
Time.
Volts/cm.
104
•10
30se cs
35
10~4.l/2
23
10_4.l/4
23
10~4.l/8
18
10~4.1/16
17
10~4.l/32
15
10"4 . 1/64
16
10~4. J/128
13
10 "4. 1/256
13
Water
11
Butyric .
xcid and water.
Cone.
Diaua.
Time.
Volts/cm.
10~6
14
22 sees.
40
10~6.2
12
40
10"6
3
14
30
10"6
4
12
50
10"6
5
12
69
10"6
6
12
33
10"6
7
14
32
10~6
8
14
32
106
9
12
29
10~5
•12
30
In the last table the solutions were each made separately
and not by successive dilution as in the other cases.
On account of the difficulty of getting readings fairly
consistent, only the most general comparison of the effects of
these acids on the surface charge can be made. They all
lower the charge as do inorganic acids, formic and butyric
acting more vigorously than the others. They show parallel
activity in lowering the surface tension. They exhibit with
the alcohols the variation of charge with the size of the
bubble.
Discussion.
The addition of alcohols to water reduces the electrification
at the liquidgas surface, the effectiveness of anv alcohol
Electrification at LiquidGas Surfaces. 377
corresponding in this respect to its effectiveness in reducing
the surface tension of water. This reduction of surface tension
is caused by a positive adsorption of alcohol into a surface
layer of small thickness. In this layer there exists — on the
electric double layer theory — a double layer of ions, the
outer surface negative, the inner positive. The number of
ions, i. e.y the charge, will be influenced by the relative
proportions of alcohol and water in the layer. Consider a
bubble of air in a mixture of alcohol and water. It is
enclosed in a layer of small thickness which contains
relatively more alcohol than the liquid around it. Suppose
this surface to diminish in area by the gradual absorption of
the air into the liquid. If, as the diameter of the sphere gets
smaller and the curvature of the surface greater, the pro
portion of water in the surface layer increases, one would
expect the electric charge to approach more nearly to the
value it has in pure water — i. e., one would expect a bubble
to move faster under a given potential difference as it
becomes smaller. This is exactly what happens in the case
of: the alcohols and other substances which exhibit a positive
adsorption. It seems probable that this equalization of the
concentration in the surface layer with that in the solution
goes on as the bubble becomes smaller, until at the moment
of disappearance the surface layer has the same concentration
as the rest of the liquid. In other words, this effect indicates
a dependence of the degree of adsorption upon the curvature
of the surface — an effect analogous to the change of vapour
pressure in pure liquids with the curvature of the surface.
An instance of an effect the converse of that shown by
these organic substances was given in the last paper. In a
solution of thorium nitrate a bubble of air decreased in
velocity as it became smaller, and even reversed its direction
of motion before disappearing. This behaviour is explained
as before if the salt is negatively adsorbed — i. e., if the
surface layer at first contains less salt than the liquid.
It is rather unexpected to find that alcohol appears to show
no electrical surface charge in these experiments, while in
waterfall experiments or in bubbling experiments it does
give a small electrification. Prof. Thomson (Joe. cit.) in
waterfall experiments found alcohol to give an effect, but
much smaller than water. Bloch also (Ann. de Chimie,
pp.22 & 23, 1911) found it to be "active" in bubbling
experiments. The difference in behaviour is probably due
to the difference in the conditions of experiment. In cata
phoresis experiments there is no violent mechanical rupture
of the surface as in the other types of experiment. In
378 Electrification at Liquid Gas Surfaces.
waterfall experiments Lenard supposes this disturbance to
tear apart the negative from the positive electrification in a
double layer at the surface. If this is so, it cannot be the
same double layer which surrounds a bubble of air and is
active in a cataphoresis experiment. If it were, there should
be parallel effects observed in the two cases. For example,
Prof. J. J. Thomson has shown in waterfall experiments that
the merest trace of methyl violet in the water reverses the
electrification produced. The author has tried solutions of
methyl violet in cataphoresis experiments, but found no
reversal. Again, Prof. Thomson in his experiments showed
that water falling through its own vapour gained no charge,
and from similar results in other cases concluded that the
liquid and the gas in contact must be chemically different to
give any electrification. The author could not try a bubble
of water vapour in water in a cataphoresis experiment, but
found that particles of ice in water moved almost as fast as
bubbles of air. In this case, at least, the two phases in
contact need not be chemically different.
The mechanism which produces the electrification in water
fall experiments is different from that in cataphoresis
experiments, and is most probably and most simply explained
on Prof. Thomson's view mentioned in the last paper.
Alcohol, in these cases, can unite with a molecule of oxygen
for example, and when the combination is violently disturbed
ionization may take place, while in a cataphoresis experiment
an adsorption of the ions in the liquid is necessary. This
may not be possible in alcohol.
Summary.
Alcohols reduce the electric charge at the liquidair
surface in water, showing in this respect a parallelism with
their action on the surface tension of water.
There does not appear to be any cataphoresis of air bubbles
in pure alcohols.
The fatty acids reduce the charge at the gasliquid surface
in aqueous solutions, but no reversal was observed with the
concentrations used.
A large variation of velocity with size of air bubbles in
cataphoresis experiments is produced by the presence in
aqueous solution of substances which have a marked effect
on surface tension.
Evidence is given to show that the electrification in water
fall experiments is not due to the difference of potential
observed in the cataphoresis experiments.
[ 379 ]
XL VI. The Application of Solid Hij per geometrical Series ta
Frequency Distributions in Space. By L. Isserlis, B.A. *
[Plate VII.]
§1. r"^HE connexion between the frequency distribution of
JL one variable character, and the hypergeometrical
series is well known. It has been discussed exhaustively
by Professor Karl Pearson in a series of memoirs in the
Phil. Trans, from 1895 on, where a series of frequency
curves are studied which may be described as parallels to
the hypergeometrical series.
In a paper entitled " On certain properties of the hyper
geometrical series, and on the fitting of such series to
observation polygons in the theory of chance," Phil. Mag.
1899, pp. 2362i6, Pearson showed how to determine from
a given frequency distribution the constants of the corre
sponding hypergeometrical series. The method of moments
was employed, and the fitting of the series was equivalent to
the statement of a problem of chance with a theoretical
distribution of events similar to the actual distribution.
Although much progress has been made with the study
of the normal surface, no general theory of frequency
surfaces analogous to Pearson's Skew frequency curves
exists at present, and this paper is an attempt to make a
first step towards such a theory by solving the problem of
fitting a double hypergeometrical series to a frequency
distribution with two variable characters.
§ 2. The corresponding chance problem may be stated as
follows : —
A bag contains n balls of which pn are white and qn are
black ; r balls are drawn and not replaced ; a second draw
of r' balls is made. This is repeated N times. If N is a
large number, the theoretical frequency of s black balls in
the first draw and s' in the second is
{n)r+r,
We may denote this by Nc(s, /),
where
/h\_h(hl)(h2)...(hk + l)
w=~ p. —
and
(h)k = h(hl)(h2) . . . (A* + l).
Communicated by Prof. Karl Pearson, F.R.S.
380 Mr. L. Isserlis : Application of Solid Hypergeometrical
The " fitting " of a double hypergeometrical series to a set
of statistics, say the frequency of a certain age of husband
and a certain age of wife, in a large number of marriages
will involve among the other things the determination of
r, r', n, and q in the corresponding chance problem. The
value of q and the ratios of r and r' to n will be numbers
characteristic of the particular type of frequency distribution.
§ 3. Let
[h]k = h(h + l)(h + 2) . . . (A + &1),
then
("W [pnrr' + l]s+s,
If we put
— r = at, — r' = a, —qn = j3, pn — r — r' + l=zy,
then
^'*j~ (n)r+r, " il«'![7W '
Let F(a, a', /3, 7, #, ?/) denote the double hypergeometri
cal series
•7 s\s'\ [7J5+S, ' 'J ■
So that
22.0, 0= {r$~ F(«, *', A % l, 1),
\JI')r+r'
or say =AFi(«, «', 0, 7).
§ 4. We will begin by finding two partial differential
equations satisfied by F(a, a , /3, &, y) as a preliminary to
the calculation of the moments of the series.
The coefficient of tl^MIll' in F(^ a'? fr % Xy ^
is T W,[/3 + 5']s.i's „. , ,
' *1[7 + «V =*(«»£ + ' '? + «»«)
= Xs say.
Similarly write Ys = F(a', /3fs, 7 + 5, ?/).
Series to Frequency Distributions in Space. 381
Then F(«, a!, J3, 7, .«/)
Xo+Xll^7'/+X2^72 tTtTI)3' + • • • •
Y° + il lTV +ls i.2.7(7TTT' + ' • • '
orF=2Xs./ = 2W'.
Now F(a, /3, 7, ,r) satisfies the equation
ff(l*)g + (7_(a+£+l)*)g_«/32=0,
Multiply this by ?/'' and sum with respect to s' ; writing
z for F(«, a', /3, 7, .r, //) we obtain
X(l.v)^+(y(u + (3+l).v)^al3: + (l.v)ls'^:(X,f')
Oits'X,/' =0.
9y
But^5i.=2s'Xs,//,
(2)
d— )g+(7(«+^+D);^+(i)^)^=o;
or finally z satisfies the two partial differential equations
*(i)g+{7(+^+i>}+ya)d«y^=o • (i)
§ 5. 77*£ moments of Fx(a, «b /3, iy). We imagine an
ordinate of magnitude z(s, s') erected at the point
,r = cs, y = cs', where c, <"' are constants to be determined in
the fitting. LetV be the volume of the polyhedron of which
the tops of the ordinates are vertices, /, e. a polyhedron
made up of elementary prisms on base cc', and of height
z(s, s')j and letpw denote the (t, t')t\\ product moment about
the centroid vertical, the elements of volume of the polyhedron
being concentrated alonu the ordinates. Loty>'«' be the corre
sponding moment about the planes
ff=— c, !/=c'.
382 Mr. L. Isserlis : Application of Solid Hyper geometrical
Thus Y = cc'F1 = cc'22As,s say,
p'tt Y = Cc'%Xks, s'(s+l)Xs' + 1)W.
Let F(«, a', & % *, */) = 2£ASS. a/
= %oo>
and letx*,*' = ^'Xoo>
7S 7s
where 0(w) = ^— {xu) and t'{xy) = {s+iy(s' + iyxy. . . . (3)
We have then
y
Fi= (7Cao)x=9=1= —t
:and
p'tt,Y = cc'.cic'v(X t,t>)x=y=v
P
'* = *!?*(&*) (4)
V %oo 7^=1 v ;
^00 satisfies the differential equations (1) and (2) .
Similarly ^=Xoi%oo,
Solving these equations, we find
^2§^=X2o3%io+2xoo,
.3«
2,
005
W §^ =Xo2  3%oi + 2^00.
Multiply equations (1) and (2) by se, y respectively after
writing m1 = a + ^, m1/ = a/ + /5J m2 = a/3, m2' = «p, and
noting that 7 — a — «' — /3— l = n.
Series to Frequency D istributions in Space. 383
«
The equations (1), (2) lead to the following :
(i^2g~ + (ww^J, +(7'« must have
the same values as Pearson's vu v2, ... vt found for the single
iiypergeometrical series.
For let
fes = ASq + A^i f A.sg . . • + Ass'
_ H.D8J. . . M.[«WW ,
' »IW. + *!*'i[7]s+,
_ «l[7]. t +,"+ » ![*+•> "K'J
[«].[/3.] n(Y+«i)n(y«'/3i)
" s![7], n(7/3l)Ii(7«,l + *)'
384 Mr. L. Isserlis : Application of Solid Hyper geometrical
which reduces to
_ (wr)i H,[^], \__
a*(nrr')\ s! [y]s [y+s]r.
= („r)! 1 , [«].[?].
(n — r—r')\ [7],., * 5 ! \_y + r')s"
p',0V=oc'[S,/+S1(2cy+...]
or , _ ,/S04S1(2y + S2(3)< + ...\
^""l S0+S1+8S+.... 7
n 4 J^2^ 4. «("4DA/3+l)(3)' ■ "I
= C*L lfy + r') 1.2.(y+r')(7+r' + l) +"J.
i+ l.(7 + ,.')+
Now our a, = Pearson's a,
our /3 = Pearson's /3,
and our 7 +r; = Pearson's 7 ;
our //«) = Pearsons ^.
§ 6. To find the higher moments and product moments we
write equations (5) (6) in the form
(1^)[%2o + %n+%io(wi3)%oi(^ + l)+%ooKmi + ^ + 2)J
+ (n — r')xio + rjeoi+%oo(— r — ?n2 + / — w) = 0 . . (9)
(ly)[Xo2 + %n+%oi(V3)x10(/ + l)+Xoo(V^i, + / + 2)]
+ (^r)%oi + ^%io + %oo(^V + ^~w)=0, . . (10)
or (ltf)P + Q = 0, ..... (11)
(l?y)P+S = 0 (12)
Now 6(xu)=x(l + 6)ii,
6\xu) = x(l + 0)ku,
and .'. dk(lx)u=[6k(l + 0k]u+(lxXl + 6)ku.
Similarly **(ly)w= [**(! + ^)*>+ (l~y)(l + ^);fcM.
Hence
^[(lar)w] = [(^(l + ^)*]^/ + (l.p)(l+^)*^M
^*[(ly>] = [* (1 + £)*] 0*w + (ly)(l + 4>)k0hu,
[^(l.i'>],^i = U^(l + W^}i • (13)
[^(l^]^i={^(l + ^)^}!. . (14)
Series to Frequency Distributions in Space.
385
Operating on each of equations 11, 12 with 6 and and
putting x—y = l in the result, we obtain, since by (13), (14)
[0(l.fP + Q)]1 = [P + 0Q]
L*(lrrf, + Q]i = (*Q)i
[tf(lyB+S)]1=(«)1
[^(T^R + S)]1=[R+^S]1,
the following equations : —
— X20— %n— %iof^i — 3)+%oi(^ + l)— %ooO»2 — mt + r+2)
+ ("^')%2o + ^%ii + %io(— r + m.2 + r' — n) = 0, . . .
(n_ r')xii+^o2 + Xoi(~ *•— m2+rfn)=0, ....
%o2Xn%oi(^i'3)+Xio(r' + l)%oo(w2'Wi/ + ^+2)
+ 0' — »0Xo2+^Xn + Xoi(— ^'W + r — /i) = 0 ....
z=y=l
(15)
(16)
(17)
(18)
Solving these equations, we find after some reductions,
(g)i=^I[(l + '?)(l + '''?)»7(«'' + '' + »')l], • (19)
and these are the values of J~ and *fr.
CC cr
We can transfer these to the mean, since P2o=p'2Q—p'iQ2
and pn=p'n—p'oip/iO'
The results are
S)1%7^^1 + 3^ + r(rl)?*)l<,. + :>)?], . (20)
_cV9(l9)(nr)
*»— („_1) ' • •
_«Vrflg)(»Q
*■ („l) • •
ccVo/1— 0)
^■=(,17 ' •
As shown in § 5 we may use Pearson's v3 for
whence
_ cV(l g)(l 2g)(» )•)(» 2)Q
/>30 =
> 1)0.2)
/>03 =
_ C'V(l9)(l2?)(»r')(»2
'•')
(» 1)02)
• (21)
• (22>
• (23)
our Pm
• (24)
• (25)
Phil. Mag. S. 6. Vol. 28. No. 165. Sept. 1914. 2 C
386 Mr. L. Isserlis : Application of Solid Hyper geometrical
To find p2\ and p12 we again apply (13) and (14), and
obtain
[^(l^P + Q]! =[0P + 0;J0_ (ni)(n_2)
t.'*r'q(lq)(l2q){nr')(n^2r')
^03~ {nl){n2)
(40)
(41)
_ c*q(lq)(nr)
^(^1)^2)0.3)
c*g(lg)QiQ
*"(nl)(n2)(n3)
r>r(l + 3yl qv 2)
+ n@ql^r6^+l6r \ (±2)
,+6/2(l3^1^) )
1
n2(l + 3?l •' 2)
+ n^T^r'S / + 1  6r') I (43)
+ 6rB(l3?T^) J
2 C 2
388 Mr. L. Isserlis : Application of Solid Hypergeometrical
I£ the values for the moments be examined it will be seen
that a frequency distribution is not hypergeometrical in type
unless certain conditions are fulfilled. For example
P2i2^opo3 =PuPo2Pso • • • • (44)
P02 P20 P21 P12 = Pn poz Pso .... (45)
Other identities will appear later. This explains why we
have to use more moments than there are unknowns to be
determined.
§7. The solution. Let p denote the correlation so that
2 = _Pn_ (46)
P20P02 ;
and let X2 = ^^2' (47)
P20P21
From (21), (22), (23) we get
1 _P2offo2 (n—r)(n — rf) ,.„.
?— ^F= ^ • • • • («)
and using (38), (39)
x2=r nr/nJr^y
r' n—r'\n—2rj
Put^1 = £ f — 1=77, then
r r
&=p (50)
g^* («)
whence f = —7 — ~r\ , (&%)
p(p + \)
X being taken of the same sign as p.
Equations (50) and (52) give £ and 77.
Next from (21), (22), (40), and (41) we have
Pzl _ (I*??. *l (w2^)2 (53)
p032 _ (12?)2 n1 Qi2r')2 (54)
^ »— ^T^)(n2)»r'(nr')'
Series to Frequency Distributions in Space,
while from (38) and (39)
p212 _(l.2qf n1 (n2r)2r
P202Po2~ qQq) {n—2y'(nr)\n—r')
pil_ _ (l2q)2 n1 (n2r')2r
389
(55)
(56)
iW>20 q(lq) (n2)* (nr')2(nr)
These last four equations involve three more identities as
we have four values for the product
(l2g)'(nl)
2(12) (n2)2'
The best value of this quantity which plays an important
part in the numerical fitting is found from (55) and (56)
which involve the body of the table, and we shall take
(l2g)» (ii1) _ puftifV*
q.0— q) («2):
(57)
«rV8(£l)(i,l)'
which is obtained by multiplying equations (55) and (56)
and writing as is usual a and a' for \/ />20 and \/po2
If we remember that p=pn/acr' by (48), we can write
(57) in the form
(12?)* (n1) /,,/',,
gaS)(»2)!
0,say . (58)
PiiXSl)(vi)
These identities necessitate the use of higher moments for
the determination of n.
For this purpose we use (42), (43).
Following the usual notation we write
A  '':!"
/3/
P'2o I *20
P<>3 p ' _ P
(59)
60
that
e ??— r' ~2 r n—r'
= and — = — ,
n — r
r n — r
390 Mr, L. Isserlis : Application of Solid Hyper geometrical
Hence (21), (22), (40), (41), (42), (43) give
*2[fe(ra~fS~3)(3n6)] =^ + «3 + 6(.V»2~~0 • (60)
*&%rF=n4+ *(*i'»*.i) ........ (61)
*'[/8,'("^"~3) (3»6)] =n«.+«»+6(*»«V)
... (62)
2!'[ft'"»l!]="H4(:i'!""V) ...... (63)
Eliminating.^, ^'we obtain
But &' = ^VlZL is known from (50), (52) ;
z2 r n — r < s '
. V /r'V(n\2nr' r (1 + ?)2^ ,,,,
in fact — = ()() — , = ^t — T2> =7 say . (64)
z2 WW r' n—r (1 + V)s ' J
. 3A(n2)2ft(n3) +6n6 _ ( „
' " 3/31'(n2)2f32'(n3) + 6n6 7 ' * V •'
orw=6f Ttft^'+DM+D] . . (66>
l7(3A/2A' + 6)(3A2A + 6)i
n being known, q is given by (58) since
(l2# ,(712)'.
2(12) "
n
1
The
values
of
r and rr are given
by
n
r"f+i"'
/
71
Equations (21) and (22) will give c and c' and (7), (8) will
give the position of the mean.
Let N be the total number of observations, then to obtain
the various ordinates of the hypergeometrical frequency
polyhedron we must take the various terms of
which is equivalent to the double series AFX (a, a', 0, 7) of §3.
Series to Frequency Distributions in Space. 391
§8. This solution fails for symmetrical distributions.
If r=r' , it follows that f =tj and from (64) that y = l, so
that the value of n given by (66) becomes
Now for nearly symmetrical distributions ft would be
nearly = ft' and ft = ft', so that it would be the ratio of two
very small numbers and liable to an exceedingly high error;
in fact n is quite indeterminate in the exactly symmetrical
case.
In this case, however, the solution can be completed by
the use of marginal moments alone, and the formula given
by Pearson for n (eqn. 32 I. c.) can be adopted. In fact let
ft= IhojPzoPzo, ft' = PQ0/P02P03, we must have approximately
4ft'  10ft/ + 6ft' + 2 _ 4ft 10ft + 6ft + 2 _
ft'4ft' + 3ft' + 2 ■ ft 4ft + 3ft + 2 ~n
We may therefore use
B=V
(4/8,' 10/3/ + 6/8/ + 2)(4/8„ 10A + 6/9, + 2) ,,..,.
08,'4A' + 3/31' + 2)08,4/9, + 3/81 + 2) ' ^S;
The other constants can be determined as before, or if we
are at an early stage of the calculations convinced of the
symmetry of the distribution, the heavy work of calculating
the product moments may be omitted and the whole solution
carried out by Pearson's method.
It may be observed that the solution by marginal mo
ments only can be used even in unsymmetrical cases if (6S)
is approximately true. On the other hand, the higher
moments being subject to very high variations in individual
samples, a more accurate fit is to be expected from the use
of p2i*Pi2,Pu than, from ft, ft' which depend on psg, />ur>
and which replace these moments in the " marginal "
solution.
By whichever of the above methods n is determined, its
probable error, since it depends on the higher moments (the
fourth or fifth), is much greater than the probable errors
r v
of  , — in the determination of which no moment higher
n n
than the third is employed. The quantities g, c, c' are
derived from n and are subject to similar variations.
§9. Another solution involving no moments higher than
the third can be obtained for distributions of discrete
392 Mr. L. Isserlis : Application of Solid Hyper geometrical
variables where we know a priori that approximately
Let us assume that cc'=l. Multiplying equations (21),
(22) we have
(nl)>2Qpo2
1,
or g(l— q) = —jy w ,, , . . . (69)
■/(n — r)(n — r)rr
(nl)oo'
Now from (58) we have easily
n—L
Substituting fxT' "TXT ^or r* r' *n ^^ anc* eclua^ng
the two values of ^(1 — g) we find
[4(nl)+(9(n2)2]o(7' = nV^/(?+l)^ + l).
Or, since f rj — ^ , wis given by
^.D + ^V^^^. . (71)
NOW p = P\ijCF(Jf.
Hence if V* =/*(£+ l)(i + 1), • • • • (72)
n2(0) + <44<9)+4<94 = O. . . (73)
The rest of the solution follows as in §§7 and 8. It is to
be remarked that this solution applies to symmetrical dis
tributions as well. For convenience in numerical applications
a table is appended, giving the various constants in the order
of calculation.
§10. Numerical. — In numerical applications, we begin by
calculating the moments about some convenient origin and
transferring them to the mean by the formulae : —
^02 = P'()2—P'012,
p03 =^'o33pVoi + Vol3.
P0i = P'<)4— ±P'03P'01+$P'02P'012— 3/oiS
pot = v'mSp'oip'v + 1 Op Voi2 — 10p'02pV + 4/014,
Pii=p\i— p'oip'io,
P21 = p'2i—p'2oPfoi—2p'iip'io + VioVoi,
and similar formulas obtained by interchanging the suffixes.
Series to Frequency Distributions in Space*
For all three forms of solutions we calculate
<* = \/p2Q, d = V/Po2»
p = Pii/™',
\ = ),
393
pt
n is determined by the methods of §7, 8, and 9 by the group
of equations A, B, C, respectively,
A = P3o2lp2o\ Pi = PmVpw,
02 = P*0lP20\ @2 = iW/Wj
»_ 7(ft^ft,+i)(ftft+i)
6 " 7(3 A'  2ft' + 6)  (3 A  2 A + 6)
A = PsolpsoPso, fo' = Pto/p&tPoa,
2 _ (4ft 10A + 6A + 2)(4A'  10A' + 6A' + 2) KB)
"i
i/**u(e+i)0Hi),
n2{0) + >i(44<9) +4(94 = 0
(C)
l/?(W)=4 + 0(n2)7(nl)
r=«/(f+l)i r'^Vh + l).
c =
n
\/){nr)q(lq)
y/r\nr')q{lq)
Equations (C) will in general give the best value of n
when they are applicable, since the errors of the lower
moments are less, but they may fail to give a real value of n.
This is due to the fact that it is not always possible to fit an
arbitrary distribution with a hypergeometric series on the
assumption 20='64, i>30=032, ^4o=l1254857?
^02 = *56, ^03=_.()56, p04=8192,
^n=32, ^21=016, j912=032.
Whence we obtain
. ^=5345, ft = 00390625, #/ = 01785714,
/32 = 2'747768, &/ = 26122449.
The formulae of §10 give
\ = 2138, ? = 1'5, ^ = 23333, y=J;
so that, if we use
n = 6
we find
7(/31'/32' + l)(&/32 + l)
7(3A'2/32/ + 6)(3A2/32 + 6)?
16741074
1674106
10.
Series to Frequency Distributions in Space. 395
3
Hence r = 4, r' = 3, 6 = j^g,
n_2o)2 i
and .*. ±— — H =  so that <7 = '6 or '4.
We take q='Q for equation (41) shows that since \ ps0 is
negative q>p.
From (21) and (22) we find c=c' = l.
Finally
N 210 1
nOl)(n2) . . (nrr' + l) ~ 10*9 . . . 5'4 " 2880'
.. the given table is equivalent to the series
2^(t)(.?)(6>^>
Numerical Example 2.
The following table, based on the Registrar General's
report for 1910, gives the distribution according to ages of
the 235,252 bachelors and the same number of spinsters
who intermarried in 1910 and whose ages were stated.
Taking an arbitrary origin at 22*5 years of age for each
sex, we obtain the following values for the raw moments : —
p\o= '5227798 p'01 = 8860797 p'u=l*075566
p'20= 11860516 p'02 = 1888826 p'^ 2*7207547
y30= 28226710 y03 = 5543315 p'^ 31404579
p'40= 105105716 /0, = 2263546
Thus the mean age at marriage of spinsters is 25*11390y
and of bachelors 26*93040. "When transferring to the mean
we diminish p20 by ^ and pU) by ip^o—^io *° allow for the
grouping of the material (cf. W. F. Sheppard, Proc. Lond.
Math. Soc. vol. xxix. pp. 353380). The moments about the
mean are : —
p20= 8294195, p02=l'020356, />u = '012341, o = 910725,
^30=12482896, />03 = 19137522, p81= 1*0295785, 0i = 95141106, pl2 = 1067845.
The distribution is decidedly unsymmetrical. We will
first attempt a fit by moments of 3rd order.
The constants are as follows : —
^ = •673016, \=935105, £=1505454, 77 = 146650,
0 = 20308, (£ = 26427;
whence rc338532n+ 3*8532=0,
giving an imaginary n. Thus it is impossible to fit this
396 Mr. L. Isserlis : Application of Solid Hyper geometrical
table with a series whose terms are spaced equidistantly with
the entries of the table.
Ci OS CO <0
rjn O t CO
CO CI h Tt<
CO CO rH H
a o co
_
b
GO
T*
CO
o
CO
o
r^
oo
lO
OS
b
HH
CI
t<
CO
iO
1— 1
CO
o hh
— I CO O
rH CO 00
O —i CO rH OS rH
t OS CO CJ
00 CI
I— 1


^
Tf
rH
rH

CO
o
00
rH
CO

rH
CO
id
00
CO
CI
>
CO
o o
iO o
CO o
CI o
h CM io CO CO CI
O O CO rH
b o O kO
co^'^ioioocobir
'saopqo^g;
Series to Frequency Distributions in Space. 397
If we complete the solution by means of equations (A)
we find
A = 2730913 /3/ = 3*447G9 7 = 100513
ft = 8*77876 /3/ = 9*13828
92 = 14888
2= 0745
r = 59422
r' = 60362
c = 33259
c'=3'6793
The starting point of the series is at c(l + rq) to the left
and c\\\r'q) above the mean of our table, i.e., at age
12*7474 for spinsters and 12*7079 for bachelors.
Numerical Example 3.
The following table gives the actual distributions in 25,000
deals of the trumps in the first two hands, in whist with
ordinary shuffling *.
1
0
1
D
2
istribution of 25,000.
First Hand.
3 4 5 0
7
8
0
3
22
38 76
05
26
12
4
240
1
5
43
159
380 531
358
153
54
4
1687
2
20
183
740
1458
1590
920
323
49
7
5302
. 3
42
300
1451
2297
2059
934
222
34
4
7403
n
54
542
1588
2048 ! 1497
505
101
14
1
0350
cc 5
45
378
906
920
500
139
25
3
2928
6
34
166
312
255
90
21
2
880
7
10
47
69
34
16
2
178
8
5
2
9
4
20
215
1724
5202 7440 6371 2950
852
166
20
25,000
* Unpublished material of Professor Karl Pearson.
.'398 Mr. L. Isserlis : Application of Solid Hyper geometrical
Arbitrary origin at (3, 3)
p'10= 24944
y20 = l72352
y30 = l64960
/40 = 911360
p'50 = 1710464
p\n = 25048
p'Q2= 174592
^03=168592
p'u= 941008
p'Q5= 1764208
p'21 = 020404, yi2 = *02244, p'n= 49036.
Mean at ^=324944, y = 3*25048.
Transferring to mean
p20= 166130
pSQ= 39094
^0 = 80995
P50= 65004
^=•13242
p12= 13611
pn= 42788
p02 = l68318
Pos = 40545
p04=836635
^05 = 67460
^ = •03333 /3/ = '034462
^ = 29347 /32' = 29530
ft = 10009 /3/ = 98864
^=•2559.
\=l02152, f=38588, ^ = 39570, 7=9878.
Using the value of n for a symmetrical distribution
v
(4/3, 10/8, + 0/3, + 2)(4/33'  10/8,' + 6ft' + 2)
=49161.
(ft'4ft' + 3ft' + 2)(ft4ft + 3ft + 2)
we complete the solution by equations B and obtain
r = 10117,V = 9'9159, ^ = '25557, c=l'0319, c' = 10465.
If we determine n from moments of the first 3 orders by
equations (C), we obtain # = 027218, (£ = '097035, so that the
equation
(<9)?i2 + (446>>(44<9) = 0
becomes
giving
n255'737n + 55737 = 0
>i = 54723
q = 24485
r = ll263
r' = 11040
c = 99436
c' = 10090.
The starting point of this series is at ('487, 486).
Series to Frequency Distributions in Space, 399
The various terms o£ the double series are given by
/ ^_rt.oQ7/ll263\/ll04\(13'399)(12399)... 043995/)
in
lO
FFl
',
xft

N
3
£ Hi
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— '
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CM
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CT5
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1— 1
as
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CM
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CO
CO
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<*
t
CO
rH
rH
rH
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1—1
1—1
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iH
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i— 1
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rH
C7i
rH
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m
3
CM
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•pnBj£ puooag
Application of Solid Hyper geometrical Series. 401
Bv (9U „ _m(lq)(nqm)
B} (21) P*« q(nl)
and by (24)
_ m(l — q)(L — 2q)(nqm)(nq — 2m)
These equations lead to a quadratic for q,
L m p2o I L p2o m m l " p2o J
2^20
m1 = 0. . . . (75)
A similar quadratic can be obtained from the other
marginal values. If q. q' are the values of q obtained from
the quadratics we may adopt \Zqq( for the value in the
fitting ; denoting this by Q, we have
_m , _m
r~Q" '=Q"
and n is given by (23), since
_ rr>q(\q)
P"~~ ,<\ •
A simplified solution of this character in which a priori
values are assigned to certain constants has to be employed
with great care. In the case of the 25000 deals at whist
(ordinary shuffling) dealt with above, the two quadratics
are
m2. By
•(23) V/A = V/^rT(2e1n2)/(n~2) •£, but \/%=* ^1°*
n2
where a is the positive square root of //,2, .*. z± "^^ according
Hence (34) should be written
*=i(W^fe?>
the + sign occurring when /t3>*0 and the — sign when
/*3<0.
(35) becomes: — m2, eare roots of ^2—z1^{z2 = 0i e being the
greater root, and (37) becomes a, j3 are roots of f2— mL%+ m2 = 0,
where a is numerically less than /3.
As an illustration we may apply the formulae to the series
1 + 24+ 90 + 80 + 15,
which is really F(— 4, —6, 1, 1), i. e. we snould obtain
*=4, /3=6, y=l,^ = 04, 2 = 06, r=4, n=\0, c = l, d = 2'4.
We find
^ = 24, yu2 = 64, ^3=032^4=l1254f. /*5=21961f
^ = •00390625, /32=2747768, & = 1072321.
whence (32) gives
_4&10A + 6& + 2
n~ A4A + 3A + 2 "JU
From (33)
, *3("l) _576
^" 4(721) +2/32(n~2)/33(n4)°^'
By (34)
V n — 1
the minus sign corresponding to the negative value of y^3.
Thus ^= 48.
Surf acetensions of Liquids in contact with Gases. 403
By (35) m2 and e are the roots of
r2^i?+^2=0 or £248£+576 = 0,
•so that m2 = € = 24:.
Using (36)
_e — m2 — n2 _ Q
1— n '
.and a, /3 are given as roots o£
f27»1f+m2 = 0 or ?2 + 10?+24 = 0
by (37).
Since a is the numerically smaller root, a=— 4, /3=— 6.
The remaining constants are now given without any
.ambiguity by (38) to (41), viz.: y = nJo>,\ /3 + l = l,
. __/?_.£ y — a — 1 __
— =1, and 0= =2*4.
The Series
F(3, G, 2, 1) = 7 + 63 + 105 + 35
leads to 11 = 10, c2 = 504. The negative sign is required in
the ambiguity which occurs in the determination of Z\ since
yLt3<0. e and m are given by f2—46f+ 504 = 0 and e must
lie taken =28, m2 = 18 since e>m2. The equation lor a, /3 is
found to be f2 +"<)£+ 18 = 0, so that a=— 3 and y3=— 6 in
accordance with the rule a < /3 .
The effect of this note is to make the fitting in the paper
referred to determinate and unique.
The author wishes to express his gratitude for much valuable
advice ami kindly encouragement received from Prof. Pearson
daring the preparation of the present paper.
XLVIl. On the Surfacetensions of Liquids in contact with
different Gases. By Allan Ferguson, B.Sc. (Lond.),
AssistantLecturer in Physics in the University College of
North Wales, Bangor*.
TI^HE question of the effect of the nature of the super
1 incumbent gas on the surfacetension of the liquid
with which it is in contact does not appear to have been very
exhaustively investigated. The only experiments with which
* Communicated bv Prof. E. Taylor Jones.
2D2
404 Mr. A. Ferguson on the Surfacetensions of
the writer is acquainted are those o£ Forch and of Bonicke,
which were carried out by the capillar yrise method*.
In view of the fact that this method, despite its wide use,,
is by no means an ideal one for the determination of capillary
constants, it seemed very desirable to institute experiments
to investigate the point by a quite independent method.
The method chosen was that which consisted in deter
mining the surfacetension of the liquid under experiment
by observations of the maximum pressure required to release
a bubble of gas from the end of a vertical capillary tube
plunged below the surface of the liquid — usually known as
Jaeger's method.
In a recent communication j" the writer has developed a
formula which expresses, in convenient form, the surface
tension in terms of certain easily observed quantities. This
equation is
where r is the radius of the capillary employed, h± is the
maximum difference in level observed in the manometer,
pi the density of the liquid in the manometer, p' the density
of the liquid under experiment, h' the vertical depth of the
end of the capillary below the free horizontal surface of
T
the liquid, and a2, as usual, is identical with — , .
Writing (i.) in the form 9P
a==B + 12a~> (»■>
where B is therefore a known quantity, we see that to a first
approximation a2 = B; and, therefore, more closely
12 VB ^ ;
which determines a2 in terms of known quantities.
It is convenient here briefly to point out the chief advan
tages of the method. In the first place, it is independent of
any knowledge of the contactangle between the liquid and
the tube; Secondly, it is on two counts more sensitive than
the capillarytube method. Neglecting minor corrections,
and assuming h' to be zero, equation (i.) becomes V ia&jzau
2a2=rhu (iv.)
* Winkelmann, Handbuch d. P/iysik, ed. 2, Bd. i. p. 1173
t Phil. Mag. July 1914. *
Liquids in contact with Different Gases. 405
if we assume that the liquid in the manometer is identical
with that under investigation. It follows, therefore, that hi
is the height to which the liquid would rise in a capillary
tube of radius r if its contactangle happened to be zero.
But if the liquid has a contactangle 0, the height measured
in the Jaeger experiment is greater than the height measured
in the corresponding capillarytube experiment in the pro
portion 1/cos 6. (This fact might be made the basis of a
method for the measurement of contactangles.)
Moreover, we see from equation (i.), still assuming h' to
be zero, that very approximately
2a?p/=rp1h1 ;
so that, if h be the height to which the liquid would rise in a
capillary tube of the same radius, we have, assuming the
liquid to have a zero contactangle,
Pi
Thus, if a liquid of very low density be used in the mano
meter, it is easy to arrange matters so that hl, the maximum
height observed in the Jaeger experiment, shall be from 10
to 20 per cent, higher than the value of h observed in the
corresponding capillaryrise experiment, and corresponding
small variations can therefore be more easily traced.
In the experiments detailed below a watermanometer was
used, as the variations observed were well within its range,
and the change of density of water with temperature is very
exactly known.
Carbon dioxide and air were the oases chosen for in
o
vestigation, and experiments have been made on benzene,
turpentine, water, methyl and ethyl alcohols, chloroform,
and ether.
The experimental arrangements were quite simple. A
Ytube of glass was titled with two stopcocks on the bent
limbs of the Y« One of the limbs was connected through a
<=02499, p'=796, fc*=9566.
Gas.
9V 0. //,. a2.
T (air).
T(COa).
T„ (air).
T17 (GOJ.
Air ...
18°2 17% 2671 02969
2318
2322
C02...
182 176 2647 02931
2289
2294
Air ...
184 172 2689 02997
2341
2343
002...
184 174 2 649 02934
...
2292
2296
Air...
185 173 2682 02986
2332
2335
CO....
186 174 2*652 02939
2295
2299
Air...
186 172 2681 '02985
2331
2333
C02...
186 175 '• 2650 02936
1 !
2293
2297
Mean T17 (methylalcoholair) =2333 dynes
cm.
Mean T17 (methylalcoliolC02) = 22'97 <1ynes.
^=•0154,
410 Mr. A. Ferguson on the Surfacetensions of
V. Ethylalcohol.
^' = •794, 7i' = 9566, r = «02499.
Gas.
18°0
e.
hv
«2.
T (air).
T(C02).
TM (air).
T18 (C02).
C02...
o
178
2652
02947
2296
2294
Air . . .
180
176
2663
•02966
2310
...
2306
CO,...
180
175
2646
•02939
2289
2283
Air...
183
175
2669
02975
2317
2312
co2...
180
179
2627
•02909
2266
2265
Air ...
178
177
2664
•02969
2343
2310
CO,...
184
176
2648
•02942
...
2292
22*88
Air ...
185
176
2672
•02984
2324
2320
CO.,...
188
178
2647
•02941
2291
2289
Air ...
188
176
2668
•02974
2316
2312
...
MeanT18 (ethylalcoholair) = 23*12 ^'.
Mean T18 (ethylalcoholC02) =22*84
ST
T
cm,
dynes
•0121.
VI. Ether.
£' = •720, 7i' = 9566, r=*02499.
Gas.
0r
9.  hx.
a.
T (air).
1777
T(C02).
Tu (air).
T15 (C02).
Air ...
o
187
16°8; 2139
•02487
1799
CO,...
188
156 ! 2122
•02458
1736
1743
Air...
190
143 2477
•02554
1803
1795
C02...
190
147 i 2148
•02503
17 68
1764
Air ...
189 150! 2146
•02500
1766
1766
CO,...
189 152 2146
•02500
1766
1768
Air ...
189
147 2161
•02525
1784
1780
C02...
189
149 2162
•02527
17*85
1784
Air ..,
190 145 2177
02554
1803
1797
C02...
190J147 2150
•02507
1770
1766
Air ...
190 142 2176
•02552
1802
1792
...
Mean T15 (etherair) = 17*88 ^\
Mean T15 (etherC02) = 17*65 ^es,
ST
T
= •0129.
Liquids in contact with Different Gases.
VII. Turpentine.
^' = •865, r= 02499, 7i' = 9566.
411
Gas.
k
9.
hx. 1 a\
T (air).
T(C02).
T18 (air).
TM (C02).
Air .
.1186
o
184
3130 \ 03296
27'97
2805
CO,.
.186
186
310803263
2770
2782
Air .
.186
185
3120 03281
2784
2794
CO..
.1186
187
3109 03265
2771
2785
Air.
.J186
186
3122 03284
2787
2799
CO..
.1186
188
3108 03263
2770
2786
Air.
.186
187 j 311603275
2779
2793
CO..
.1186
1891 3106 03261
2768
2786
Air..
. 186
188 3117 03276
2781
2797
IMea
n T18 (turpe
ntineai
r) =2798 d?nes.
' cm.
Mean T18 (turpcntmeC02) = 27"85*nes.
v A ' cm.
ST
T
= •0040.
The table below summarizes the results obtained
Substance.
Temp.
T (air).
T (CO,).
ST
T
Benzene
o
17
17
15
17
18
15
18
2937
2779
7388
2333
2312
1788
27:98
2883
2722
7304
2297
2284
1765
2785
•0184
0205
•0114
•0154
•0121
•0129
•0046
Chloroform
Water
Methylalcohol...
Ethylalcohol ...
Ether
Turpentine
Although it is of the nature of a sideissue, an interesting
result may be deduced from the figure given above for the
surfacetension between turpentine and air. The usual value
given for this constant as deduced from capillaryrise ex
periments is about 2()*7 dynes per centimetre at 18° C. If
it be assumed that this value is really T cos 0, where 6 is the
angle of contact of turpentine with glass we have at once,
a 2G*7
•9541,
whence #=170,5 approximately, in very close agreement
with the value (17°), given by Magie * as the result of
measurements of the total depth of a large bubble of air
imprisoned in the liquid under a convex lens.
* Phil. Mag. August 1688, p. 102 .
412 Dr. T. M. Lowry on an
It is as well, perhaps, again to emphasise the fact that the
values for Fjr given above depend on two factors which are
usually assumed to be inseparable — the solubilityfactor and
the "natureofthegasfactor," — although there are a priori,
grounds for presuming that the solubilityfactor contributes
but little to the total effect. Whilst writing the above,
however^ it appeared to the writer that the two factors might
be separated by observations of the surfacetension made at
temperatures at or near the boilingpoints of the liquids in
question, when the effect of the solubilityfactor would pre
sumably be inappreciable. It is hoped shortly to communicate
results dealing with this point.
University College of North Wales, Bangor.
July 1914.
XLVI1I. An Oxidizable Variety of .Nitrogen.
By T. Martin Lowry, D'.Sc*
[Plates VIII.XI.]
ri'HE photographs which are reproduced in the present
JL paper are of interest both from the spectroscopic and
from the chemical point of view. On the spectroscopic side
they represent an application of this method of investigation
to gases which were so dilute that a column of gas 64 feet
in length was required to produce an adequate absorption.
On the chemical side they produce evidence, which appears
to be unique, of the existence of an oxidizable modification
of nitrogen, an allotropic form of the element which is
perhaps the first essential product in its fixation. A brief
summary of the conclusions arrived at has been published in
the i Transactions' of the Chemical Society f, but the photo
graphic evidence on which those conclusions were based is
now reproduced for the first time as as a contribution to a
subject which has figured frequently in preceding volumes
of the ' Transactions ' of the Faraday Society.
The gases under investigation were obtained by the action
of the electric discharge on air. Two forms of discharge
were used. First, the " silent " discharge in a large
Andreoli ozonizer containing sheets and grids of aluminium,
thirteen in number and 30 in. x 30 in. in area, separated by
sheets of micanite. Second, a sparking discharge between a
series of iron studs separated to a distance of fy incn Three
groups of 6 sparkgaps and two of the Andreoli ozonizers
were used at various times.
* Communicated by the Author. From the ' Transactions ' of the
Faraday Society, July" 1913.
t Trans, ci. pp. 115258 (1912).
Odidizable Variety of JSitrogen. 413
The chief interest of the work consisted in a comparison
of the gases obtained from the ozonizer and sparkgaps
separately with the " combined " gas obtained by using both
forms of discharge, in accordance with a process devised by
Mr. Sydney Leetham, of York, in 1903. At the time when
the experiments were made the bleaching gases manufactured,
by this process were usually obtained by blowing air at the
rate of about 70 c.f.m. through one of the ozonizers and then
through a box in which 3 sparkgaps were energized in series
with the ozonizer by an alternating current of O'l ampere
at 9000 volts. Other arrangements were, however, described,
and some of these were tested by the spectroscopic method.
The spectroscopic apparatus consisted of three parts :
(1) A wooden trunk 64 feet in length and 3 in. x 3 in.
in section, the volume being thus about 4 cubic feet. The
trunk was provided with glass windows at each end, and
inlet and outlet pipes were provided for the gases under
examination. The effective length of the trunk could also
be reduced to 16 feet by inserting a glass slide and using an
additional " blowoff" hole so that the gas was confined to
the earlier portion of the trunk.
(2) A Nernst lamp with filament vertical served as a
source of light. It was placed in the focus of a condensing
lens of 3^ in. focal length, fixed immediately in front of one
of the glass windows. At the other end of the trunk the
light was picked up again by a condensing lens of 11 in. focus,
but on account of the great length of the trunk the image
formed by the second lens was circular in form, all the light
from the upper and lower portions of the filament being lost
against the sides of the trunk, so that the effective illumi
nation was derived only from a minute portion in the centre
of the filament.
(3) The spectroscope was provided with achromatic quartz
calcite lenses of 13 in. focus, but when it was desired to photo
graph the spectrum a camera with a lens of 22 in. focus was
used. W ratten and Wainwright's "panchromatic plates"
were used with six minutes' exposure and two minutes'
development in the dark.
The exposures shown in Plate VIII. at once revealed the fact
that, whilst the ozonizer and sparkgaps separately produced
no visible absorption, the flutings characteristic of nitrogen
peroxide could be recognized clearly in the spectrum which
had been photographed through the " combined " gas. The
copper spectrum shown at the bottom of the plate served for
calibration, the two yellow and three green lines being
clearly seen at one end and two violet lines at the other end
of the spectrum.
414 Dr. T. M. Lowry on an
In order to estimate the amount of nitrogen peroxide in
the u combined " gas it was necessary to prepare a series of
•standard photographs. Some of these are shown in Plate IX.,
.and indicate that the proportion of nitrogen peroxide in the
u combined gas" is not far from 40*00 by volume. The
standard photographs were taken with mixtures of nitric
oxide and air, prepared with the help of the apparatus
shown in fig. 1. The nitric oxide was made from mercury
and nitric acid ; its purity was tested by absorption with
ferrous sulphate. It was collected in a cylinder funnel,
provided with a T piece and a capillary funnel through
which mercury could be poured ; the volume of mercury
used in a given time afforded a measure of the volume of
nitric oxide transferred to the " mixingjar." In this jar
nitric oxide was diluted with known volumes of air run in
from a standard meter. The mixture of nitric oxide and air
was taken from the bottom of the mixingjar and passed on
to the trunk for observation.
Plate X. reveals the remarkable fact that the proportion
of nitrogen peroxide in the " combined " gas is but little
affected by passing the air through the ozonizer and spark
gaps in the reverse order. Under these conditions the
ozonizer alters its function completely and becomes a very
efficient generator of nitrogen peroxide, producing this gas
in larger quantities even than the series of 17 sparkgaps
used for the last exposure on the plate.
Up to this point the efficiency of the " combined " ar
rangement appeared to be due to some alteration in the
function of the sparkdischarge when supplied with ozonized
air, and of the silent discbarge when supplied with sparked air.
Plate XI. shows, however, that the problem is essentially
chemical rather than electrical or electrochemical in cha
racter. The three exposures at the top of the plate show
that almost equal yields of nitrogen peroxide are produced
when part of the air is passed through the ozonizer only,
and part through the sparkgaps only, the two products
being afterwards mixed on the way to the trunk. In this
arrangement a greatly increased yield of nitrogen peroxide
is seen to result from the chemical interaction of two parallel
currents of air after both have passed away from the electric
discharge.
This result might be interpreted as meaning that the
ozone produced by the silent discharge oxidizes to nitrogen
peroxide some lower oxide of nitrogen produced by the spark
discharge. But this oxide could only be nitric oxide, as
nitrous oxide is too unstable to be produced by sparking.
The explanation would therefore amount to a suggestion
Oxidizahle Variety of Nitrogen.
415
that an insufficient time had been allowed for nitric oxide to
be oxidized to nitrogen peroxide by atmospheric oxygen
before the gas escaped from the trunk, unless the oxidation
were accelerated by ozonizing the air. That such an effect
may exist is shown by the fact that, whilst the proportion of
nitrogen peroxide present in the gas obtained from a series
•of 17 sparkgaps amounted to about y0Vo wnen the gas was
passed rapidly through the trunk (Plate X.e), it was in
creased to about 3 o^o (Plate XI. d) when the velocity was
reduced to \ c.f.m., by blowing off most of the gas, in order
to allow ample time for the gas in the trunk to mature.
But this explanation is not sufficient to account for the whole
•of the observations, since even under the very favourable
conditions of the lowvelocity experiments 17 sparkgaps
were required to produce a yield of nitrogen peroxide only
onethird greater than that obtained from '6 sparkgaps with
the help of ozone.
Fie. 1.
Apparatus for preparing Standard Mixtures of Nitric Oxide and Air.
The efficiency of the ozonizer must evidently be attributed
to its ability to oxidize to nitrogen peroxide some substance
which cannot be so oxidized by air. This oxidizable sub
stance can scarcely be other than a variety of nitrogen itself,
produced by the sparkdischarge in much the same way that
ozone is produced by the silent discharge. The most efficient
way of oxidizing it is to produce it in an atmosphere already
charged with ozone (Plate XI. c). A rather less efficient
method is to pass the air in which it is produced directly
416 Notices respecting Neiu Books,
into an ozonizer (Plate XT. b). Less efficient still is the
arrangement whereby the oxidizable nitrogen is mixed with
ozonized air in a mixing chamber a few feet away from the
discharge apparatus (Plate XI. a). The slight, but clearly
marked gradation in the first three exposures of Plate XL
affords unmistakable evidence of the instability of the oxidi
zable variety of nitrogen, which appears to revert in the
course of a few seconds to a form in which it can no longer
be oxidized either by oxygen or by ozone.
The observations now described are probably of importance
in connexion with the technical fixation of nitrogen. It is
not unreasonable to suggest that, since the whole of the
oxides of nitrogen must be produced by the union of dis
rupted molecules of nitrogen with disrupted molecules of
oxygen, the most important feature of the process may be to
generate the oxidizable variety of nitrogen and then to
provide a supply of atomic or of ozonized oxygen to oxidize
it before it reverts to the ordinary inactive form.
As a point of contrast it should be noticed tha t Strutt's
chemically active nitrogen is not oxidizable by oz me under
the conditions of his experiments.
The author is indebted to Mr. Henry Simon, of Manchester,
for permission to publish an account of these experiments.
XLIX. Notices respecting New Books.
Photoelectricity. By A. Ll. Hughes, D.Sc, B.A.
Pp. viii +144. Cambridge University Press. 1914.
rPHIS book deals principally with the emission of electrons from
■ solids, liquids, and gases under the influence of light, and is
a valuable addition to the Cambridge Physical Series. The most
interesting questions at issue relate to the velocities and the
numbers of the emitted electrons, and the way these quantities
depend on the frequency and the intensity of the exciting light,
and on the nature of the emitting substances. The treatment is
ftot limited to these questions, but includes a number of related
phenomena. The part dealing with the relation between fluores
cence, phosphorescence, and photoelectric emission is particularly
satisfactory.
The subject is of especial interest at present on account of the
light it throws on the socalled quantum theories. Professor
Hughes has made many valuable contributions towards its recent
development, and his opinions on the more controversial points
will be read with interest. The book bears evidence that it is
written by one who is familiar with the experimental difficulties
W'hich beset this line of investigation. It will be welcomed by all
who are interested in modern experimental and theoretical physics.
McLennan.
Phil. Mag. Ser. 6, Vol. 28. PL VI.
o
to
I
to
a
01
Fig. 1.
ISSEELIS.
Phil. Mag. Ser. 6, Yol. 28, PL VII.
LOWRY.
Phil. Mag. Ser. 6, Vol. 28, PI. VIII.
(a) Blank.
(l>) Ozonizer only.
(r) 3 Sparkgaps only
(
ozonizer)
('/) " Coaibined " gas (or
dinary arrangement)
(ozonizer ■ — ■ — > j
Bparkg ips).
( ozonizer).
" Combined " gas
(§ c.f.m.) ordinary ar
rangement (ozonizer
> 3 sparkgaps).
( + 1_ Va* Wr, 9 g8loga68loga
4D + 4=D'log&loga 10° r+4D' log 6 log a
riff. i.
Inasmuch as the concentration is zero at r = a and r=b, it
is obvious that there must be a position rx where the concen
tration is a maximum.
By differentiating (3) with respect to r we find
r{
21og
In the steady state the deposit particles may be regarded
as streaming towards the central rod or the case according as
they are situated within or without the cylinder r = rx (fig. 1).
The relative distribution on central rod and case will then
be proportional to the volumes cut off by the cylinder r = rij
activity on central rod tt^2—  a2)
total activity
tt(62
1
2 log
a2)
h b2a2
(4>
the Active Deposit of Radium. 421
The first term, which in most cases is the more important,
is the same as the electrical capacity per unit length of the
condenser.
By integrating equation (3) the total number of deposit
particles in the gas which are present in the steady state in
the volume corresponding to a height I of the condenser is
seen to be
When a = 0, i. e. if the central rod be absent, the number
present becomes — j^rrpj this case is of special interest in con
nexion with the determination or D (v. Section 6).
Let us now proceed to apply equation (4) in order to esti
mate the fraction of the total activity which should appear on
the central rod in a diffusion experiment, i. e. when the central
rod and the case are metallically connected. It is of interest
here to observe that the expression obtained in (4) for this
fraction is determined wholly by geometrical conditions,
being independent of the nature of the gas, pressure, amount
of emanation, etc.
For the vessel employed, a = *09l5 cm., 6 = 2*9 cm.,
= •145; «* =001;
2 loo 
fraction = '144.
This is greater than the fraction *105 which was obtained
for the activity distribution as a result of a series of diffusion
experiments under varied conditions; however, it should be
remembered that expression (4) refers to an indefinitely long
cylinder, and there is no doubt that the corrections for the
diffusion to the top and bottom would bring the calculated
and observed values into closer agreement.
Another diffusion experiment was performed with a central
rod of thin steel, the radius being '015 cm. The value
obtained for the fraction of the deposit on this rod was '066;
the value calculated from (4) is '095.
To revert to the experiment when the emanation was
mixed with ethyl ether wre see, therefore, that the values
given in Table I. indicate that very approximately all the
deposit particles are uncharged and reach the electrodes as a
result of diffusion. The limiting value of 10 per cent, which
422 Prof. E. M. "Wellisch : Experiments on
was assigned in the previous paper is incorrect on account o£
the erroneous assumption which was there made with regard
to the diffusion distribution.
It may not be out o£ place here to draw attention to the
fact that nothing has been said with regard to the charge
carried by the deposit particle at the moment of formation ;
what is here asserted is that the particles are uncharged in
ether at the end of their recoil path, even in the presence of
electric fields which are easily able to prevent recombination
of the gaseous ions.
It is probable that if the concentration of the emanation
were sufficiently great, the deposit particles in ether would
form positively and negatively charged aggregates, as has
been shown later (Section 5) to occur for the deposit particles
in air.
Distribution of Active Deposit in Air, Hydrogen, and
Carbon Dioxide.
4. A slight correction has to be made for the limiting
values of the cathode deposit in air, hydrogen, and carbon
dioxide, owing to the fact that in the previous paper the
diffusion correction was made on the assumption that the
diffusion distribution was proportional to the areas of
the exposed surfaces. The experimental values (p), un
corrected for the diffusion of neutral particles, which were
obtained for the percentage cathode deposit were as follows :
air 894, H2 89*4, C02 81*1.
The corrected values which give the number of positive
particles expressed as a percentage of the total number of
particles are obtained by evaluating the expression
200p2,100
179
These values are : air S8'2, H2 88% C02 78'9.
The broken curves given in fig. 2 of the previous paper,
which exhibit the dependence of the cathode deposit on the
amount of emanation when small potentials are applied to the
case, wTould take slightly different positions if the proper
diffusion correction were made. The effect of this correction
would be to make each activity curve lie still further below
the corresponding ionization curve than it actually does in
the diagram ; this would bring out with greater emphasis
than before the fact that the rate of recombination between
the positively charged deposit particles and negative ions is
greater than between the positive and negative ions among
themselves. Inasmuch as this point is brought out clearly by
the Active Deposit of Radium. 423
the curves there given, and as in any case the actual shape of
any one of the curves depends upon the particular vessel em
ployed, it was not thought desirable to reproduce the corrected
curves.
Experiments ivith large amounts of Emanation.
5. It was shown in Section 3 that when the emanation
mixed with any gas is introduced into a cylindrical vessel,
the amount of active deposit in the gas (i. e. gas activity)
when the steady state is established is inversely proportional
to the coefficient of diffusion (D) of the active particles in
the gas. If, therefore, we could determine the amount of
active deposit in the gas at any time, we would be able to
obtain the value of D. The determination of this coefficient
is of importance in the theory of diffusion because the
particles are of relatively large mass and also because they
are present in extremely small number.
Before describing the method employed for determining
the amount of active deposit in the gas, it is of interest
to record that when relatively small amounts of emanation
were employed the gas activity was found to be a definite
fraction of the total activity. With larger amounts of
emanation, however, variations of such an exceptionally
large order were found to occur in the values of the gas
activity, that it became of interest to make a systematic
investigation. The results obtained in this connexion are
given in the present section ; the determination of D by the
use of relatively small amounts of emanation is described in
the next section.
In the experiments now to be described it was not found
necessary to determine the total amount of the active deposit
present in the gas. The experimental procedure may be
briefly described as follows : approximately 2\ millicuries of
radium emanation * mixed with air was introduced through
dryingtubes and cottonwool into the cylindrical testing
vessel, which was closed at the top with a rubber stopper, no
central rod being present. Several hours and occasionally a
few days were allowed to elapse in order that any dust
present in the gas might have a chance to subside. The
outer case was connected to any desired potential, but as
there was no inner electrode the electrical conditions inside
the cylinder were not disturbed. When conditions were
steady an electrode permanently connected to earth was
introduced for 1 minute and then withdrawn, the cylinder
* Kindly supplied by Professor Boltwood.
424 Prof. E. M. Wellisch : Experiments on
being again closed by the rubber stopper. It was demon
strated repeatedly during the course of the investigation that
less than 1 per cent, of the amount of emanation escaped
during the process of introducing and withdrawing the
central electrode ; the decay of the emanation was examined
from day to day, and in this way the loss could be estimated.
It was found that the most convenient method of comparing
the activities of rods which were introduced under different
conditions was to measure the activity at any convenient
period after removal, and then to calculate the activity
corresponding to the lapse of a definite period from the time
when the rod was originally withdrawn. The definite period
chosen was 4 hours 40 minutes. The measurements of the
activity were made at a sufficiently long time after removal,
so that thenceforth the activity decayed with the period of
KaB.
These experiments were designed to furnish information
with regard to the activity accumulating in the gas ; by
repeating them from day to day we could ascertain how this
gas activity varied as the amount of emanation decreased
through disintegration. In addition to these experiments
electrodes were often introduced with various potentials
applied to the case and allowed to remain for several hours
before removal ; their activity was estimated in the same
manner as before. Moreover, advantage was taken of these
long exposure experiments to determine the relative amount
of emanation in the vessel. This was accomplished by
measuring the ionization current by means of a sensitive
Siemens Halske galvanometer (4*5 x 10~9 ampere per cm.
when unshunted) ; in measuring the current a potential of
1000 volts was applied, because with potentials of only a few
hundred volts there was evidence of abundant recombination,
so that with these small potentials the currents would not be
proportional to the amount of emanation.
Two distinct series of experiments were made in this con
nexion ; in each instance about 2\ millicuries of emanation
were introduced, and each series occupied a little over one
month. Several hundred measurements of the activity were
made under various conditions, but for the sake of brevity
only a few results which are most typical are here given
(Table II.). These, however, are sufficient to illustrate the
interpretation which is given later.
The column headed u accumulation " gives the time during
which the deposit particles were allowed to accumulate in
the gas before the insertion of a central electrode. This
time is reckoned from the time of withdrawal of the electrode
the Active Deposit of Radium.
425
which had been previously inserted. Exposures referred to
as " long " were for periods of over 12 hours.
Table II.
Emanation introduced 29 April, 1914.
1
No.
Date
1914.
Electrode
removed.
Accumu
lation.
Duration
of
exposure.
Potential
on case.
volts.
Activity
(after
4h.40m.).
Saturation
current
xio8
h.
m.
b. m.
ampere.
1
1 May
12.45 p. m.
10
2 5
990
272
528
2
3.30
2
44
1
990
595
3
4.6
20
1
0
•03
4
4.30
23
1
220*
•09
5
2 May
11.00 a. m.
long
400
558
6
12.25 p. in.
1
24
I
400
521
7
12.39
13
i
400
324
8
4 May
9.28 a. m.
long
+ 160
544
9
10.55
1
26
1
+ 160
142
10
1.45 p. m.
2
49
1
+ 990
572
11
5.23
i
3 38
+ 990
1140
259
12
5.24*
%_
1
+990
•30
13
5 May
3.15
ll
4 29
990
•51
223
14
3.16

1
990
•06
15
8 May
10.23 a. in.
long
+990
528
121
16
9 May
12.15 p. in.
25
1
990
118
17
2.50
2
34
1
+ 990
149
834
18
7.55
5
1
0
•01
19
11 May
11.15 a. m.
1
45
1
990
•67
581
20
15 May
10.20
96
1
 80
•06
21
11.30
1
9
1
+ 80
•10
269
22
16 May
10.55
20
1
 4
•004
23
2.15 p. m.
3
19
1
+ 4
•006
(after 25 in.)
24
20 May
4.21 p. m.
96
1
160
877
•91
25
5.0
38
1
+ 160
943
26
21 May
7.15
7
30
1
160
667
27
8.10
54
1
+ 160
877
28
22 May
3.35
18
1
+ 160
617
29
4.45
1
9
1
160
4+2
30
23 May
75
25
1
160
115
31
8.00
54
1
+ 160
220
32
25 May
7.14
26
1
160
•19
33
7.48
33
1
+ 160
•52
* Alternating 60 cycle.
The point of chief interest in connexion with the results
of this table, is the large amount of deposit which goes to
the central electrode with negative potentials applied to
the case.
The results are readily explained on the view that there
are present in the gas charged aggregates of active deposit.
426 Prof. E. M. Wellisch : Experiments on
These aggregates diffuse very slowly 5 and therefore accumu
late in the gas, so that in the steady state there is a large
amount of gas activity. Experiments 15, 16, and 17 in
conjunction show that the amount of charged activity in the
gas constitutes alone approximately half the total amount of
active deposit in radioactive equilibrium with the emanation.
The slow rate with which the particles diffuse is well shown
hy the smallness of the activity in the absence of any applied
potential (Sand 18). With a large positive potential applied
to the case the inner electrode if exposed for 1 minute
receives practically all the positively charged aggregates,
plus the positively charged deposit particles formed in
1 minute ; the amount formed is, however, in the circum
stances to which Table II. refers comparatively small (12).
A large applied negative potential brings over similarly the
negatively charged aggregates.
An essential condition for the presence in the gas of a
large amount of active deposit is that sufficient time should
be allowed for the particles to accumulate. Experiments 10
and 12 show well the difference which results from varying
the time allowed for accumulation. From a knowledge of
the activity resulting from a long exposure (cf. 11) it is
easy to calculate by the aid of standard transformation
formulae the activity which should result from an exposure
of 1 minute in the absence of any accumulation. The theory
gives that the former activity is 40 times the latter, which
shows that the activity in experiment 12 is due practically
entirely to the formation during 1 minute ; experiment 10
shows in consequence the preponderating influence of the
deposit which has been allowed to accumulate in the gas.
There is abundant evidence that the aggregates owe their
charge to the ions present in the gas. Experiments 13, 14,
and 16 bring out this point most clearly ; in experiment 13,
where the duration of exposure is long, the ions are removed
from the gas by the applied field, and the activity measured
is due practically entirely to the diffusion of the neutral
deposit particles ; in experiment 14 there has not been
sufficient time for accumulation of deposit in the gas and the
consequent formation of aggregates ; in experiment 16 the
central electrode receives the negatively charged aggregates.
Moreover, for long exposures there is more deposit on the
central rod with an applied potential of —400 volts than
with —990 (cf 1 and 5). With the former potential it was
easy to show by readings of the ionization current that there
was considerable recombination of the ions ; this implies the
presence of ions in the gas, and hence an increased number
tlie Active Deposit of Radium.
4,27
of charged aggregates of active deposit *. Experiments 16
and 17 or 20 and 21 show that the amount of positively
charged gas activity is slightly greater than the amount
which is negatively charged ; the ratio is roughly the same
as that o£ the mobilities o£ the negative and positive gas
ions ; in other words, the amounts of positive and negative
activity are roughly in the same proportion as the numbers
of positive and negative ions present in the gas.
More direct evidence that the aggregates acquire their
charge from the ions present in the gas is furnished by
investigating the effect produced by causing Rontgen rays
to pass through the gas. A beam of fairly intense Rontgen
rays passed through the aluminium bottom of the vessel for
about 15 minutes before the introduction of the electrode ;
the rays continued to pass during the exposure (1 minute)
of the electrode and were switched off when the electrode
was removed. These experiments were performed only when
the original batch of emanation had decayed considerably ;
on this account the extra ionization produced by the rays
was relatively all the stronger. It will be seen from the
table given below that both the positive and negative gas
activity were considerably increased as a result of the extra
onization.
Date
1914.
Duration of
Exposure.
Potential
on case.
Activity.
2 June...
1 in in.
160
•15
1 min.
lfiO
180 Xrays
1 min.
+160
192
1 min.
+ 160
305 X rays
In these measurements the activities were measured
25 minutes after removal of the electrodes ; the capacity of
the electrometer system was only 1/21 of its previous value.
The second part of Table II. is a continuation of the results
obtained as the emanation decayed still further. The activity
of the central electrode being relatively small is now deter
mined by measurements made 25 minutes after its withdrawal
from the emanation, the capacity of the electrometer system,
however, remaining the same as before. The chief point of
interest in connexion with this table is the rapidity with
which the charged gas activity falls off when the amount of
* Of course the charged aggregates also suffer recombination, but they
acquire a fresh charge from the ions present in the gas.
428 Prof. E. M. Wellisch : Experiments on
emanation decays beyond a certain value. In Table I. the
amount of charged gas activity is approximately proportional
to the quantity of emanation in the vessel as measured by
the saturation current (cf. 2, 16, and 19). If in Table II.
we calculate the ratio of the gas activity obtained with
1 minute exposure and —160 volts to the saturation current,
we obtain for the five successive days the following numbers:
965, 974, 790, 2'62, 50.
A careful examination of all the results obtained indicates
that the proportion of charged deposit present in the gas
contained in the cylindrical vessel without a central electrode
remains approximately constant over a wide range ; but
when the emanation has decayed until the saturation current
obtained when a central electrode is introduced is about
5*0 X 10~9 ampere, this proportion suffers a sudden diminu
tion in magnitude. A current of 5*0 x 10"9 ampere implies
the production of 3'23 X 1010 pairs of ions per second in the
vessel, and as the volume of the vessel was 326 c.c. the
number (q) of ions produced per c.c. per second is approxi
mately 108. If we assume that the value of the coefficient
of recombination (a) is unaltered by the presence of the
emanation, and take for a the value assigned by Townsend,
viz., 3400 Xe, then the number of ions of each sign present
per c.c. in the vessel is s/q/a =7*95 x 106.
When the density of ionization in the gas falls below this
amount, there seems to be a rapid diminution in the amount
of charged activity in the gas. When the emanation decays
still further, the active deposit particles in the gas appear to
be uncharged and in amount to be a constant fraction of the
total amount of deposit in equilibrium with the emanation.
It appears from the results that aggregation of the deposit
particles ceases fairly abruptly at the critical stage, but there
is also evidence that a considerable fraction of the deposit
particles in the gas can still be charged even after they have
ceased to aggregate. Aggregation implies a slow rate of
diffusion, which is made manifest by the very small amount
of deposit collected on a central rod in the absence of an
electric field. It is easy to tell, by measuring the propor
tionate activity collected on such a rod, the stage when
aggregation practically ceases. As stated before, aggrega
tion ceased fairly abruptly when the saturation current was
5*0xl0~9 ampere, but evidence at any rate of negatively
charged deposit particles in the gas was found even with the
current as low as 10"9 ampere.
The general effect seems to be that the active deposit
the Active Deposit of Radium,
429
atoms situated in the gas charge up both positively and
negatively in appreciable numbers when the density of
ionization exceeds a certain amount; the atoms thus charged
act then as nuclei of condensation and build up aggregates
of active particles when these are present in sufficient
quantity.
There is of course a possibility that both the aggregation
and the chargingup may be conditioned by the presence in
the gas of foreign nuclei such as traces of dust particles, etc.
Against this view, however, is the fact that practically
identical results were obtained in different series of experi
ments, when in every instance great care was taken to
exclude foreign matter. Such matter might, however, have
been introduced on the insertion of the electrode. It should
in this connexion be mentioned that when the emanation had
decayed so that the saturation current was in value between
5'0 x 10"9 and 10~9 ampere, the number of charged particles
for any given amount of emanation, although small, was
subject to irregular variations.
Coefficient of Diffusion of the Active Deposit of Radium,
6. An expression was given in Section 3 for the amount
of active deposit present in the gas contained in a cylindrical
vessel when a steady state is established such that the
production of fresh deposit from the emanation is balanced
by diffusion to the surface. This expression is, however,
only approximate as it neglects
the diffusion to the top and bottom
of the cylinder ; inasmuch as the
exact expression comes out with
surprising simplicity, it seems
desirable to present here the exact
theory.
Let ABCD represent a longitu
dinal section of the closed cylinder
(without any central electrode).
Let q be the number of active
deposit particles produced per c.c.
per second from the emanation.
Let the axis of the cylinder be
taken as the axis of z, and let the
origin be midway between top and
bottom.
Let 21 be the height of the
cylinder and b its radius.
Using cylindrical coordinates we have as the equation of
Fif
430 Prof. E. M. Wellisch : Experiments on
the steady state
fB2^ l"dn o2n~\
where n is the number of particles per c.c. at (V, z).
The boundary conditions are
w ==0 when r = b for all values of z ;
91 = 0 when z= ztl for all values of r.
r = 01
At the point 0, i. e. at ^ > we must also have
~bn ~ftn
^— =0 and ^ =0.
or dz
Let N = ^+^(r262), (2)
(1) becomes
^ + i^ + P=0, (3)
Or2 r or o^
with the conditions
N = 0 when r = b for all values of s . . (i.)
N= jyt {r2 — b'2) when £ = ±Z for all values of r . (ii.)
a. r = (Tl BN n , BN .....
At n ^ ^r— =0, and ^— =0 (m.)
The function ~N = AJ0{oir) {ea" \e~a ), using the usual
notation of the Bessel functions, is a solution of (3) which
satisfies the condition (iii.). We have the constants A and «
at our disposal. (i.) will be satisfied if a be a root of
J0(Xb)~0. This equation has an infinite number of real
positive roots, X^ X2, Xn,
Now if we can^expand — L (r2 — b2) as a series
SA.(/*,+«w)J0&,r),
it is easy to see that
N=a,(^+rx*)j,M .... (4)
n = l
will be the solution of (3) which will satisfy all the condi
tions of the problem.
the Active Deposit of Radium. 431
Carslaw * gives the following expression for the coefficient
Bw in the expansion
/(r)=lB,J,(V)
where X„ is as above :
2fjbjf(r)J0(\nr)dr
B"=¥ ' {J,/ pa)}2
So that in the case under consideration we have
A _ w JO *U ...
n b* (eW + eMjiJo'Cknb)}'*
We have next to determine the integral
(5)
I
t
(r*b*r)J0(Kr)dr (6)
)
By making use of the integral
^n.1(x)da:=xn3nQc\
r
together with the known relations subsisting among the
Bessel functions of adjacent order f, the following indefinite
integrals were obtained (Tor the sake of brevity the analysis
is omitted) :
]xJ0(tv)dx = as Ji(x),
$jjfa)da = 2.i'2JoGr) + (.y34li')J1(^).
Applying these to (6) and utilizing the fact that J0(\nb)= 0,
we obtain 4.7.
as'the exact expression of (6). From (5) we obtain
A  2 2L 1
A>~ b2 D\tt3 (eX»'+ e*J)Ji(kJ>y
and from (2) and (4) in conjunction
n=W^^T^~ bD^iXye^l+eKljJx^y ' ' [n
* § 126, * Fourier's Series and Integrals,' Macmillan, 1906.
f Cf. Gray and Mathews, ' Bessel Functions/ p. 13.
432 Prof. E. M. Wellisch : Experiments on
Now the total number of active particles in the gas is
given bv
S J £r=b£z=l
G = 47r I I nrdrdz.
o Jo
Using (7) and also the integral
Cb o
I rJ0(knr)dr= — Ji(\nb),
Jo An
we deduce as the expression for the total number of active
particles in the gas contained within a cylinder of height 21
and radius b when the steady state is established,
P _ ^qbH __ 87T<7 s 1 e^nl — e—^nl
ID W^XJeM + eW ' * W
where Ai, ^2> are the roots of J0(\6) = 0.
So far no approximation at all has been made, so that the
series (8) is the exact solution of the problem ; the first
term was obtained in Section 3 as the approximate solution
when the diffusion to the top and bottom was neglected.
The cylinder used in the present experiment had a height
of 14 cm. and a radius of 2*9 cm. So that we have Z = 7*0
and 6 = 29.
Using Gray and Mathews' Tables the following are the
first five values of Xn :
A* = '829; X2 = l93; Xs = 2'98 ; X4 = 4«07 ; X5 = 5*15 :
— Xnl
A I a. A £*s Practit>ally indistinguishable from unity, and
S A = 2592.
From (8) we obtain for the number of active particles in
the gas in the steady state :
G = 1031 ^ (9)
Without the correction for vertical diffusion we obtain
123'8 ^y, L e. 20 per cent, in excess. Let Q be the number
of particles produced per minute in the gas contained in the
cylinder,
.. Q = UOirbHq,
G 1031 1 0146
Q~" 120 b2W~ D
9
G*
so that D=0146^ (10)
the Active Deposit of Radium. 433
It is apparent from the above that to obtain D we must
determine the ratio p ; in order to accomplish this it is
necessary to measure the ionization currents resulting from
Q and G under similar conditions. Put briefly, the method
employed was to deposit certain ascertainable fractions of Q
and of G on a central rod introduced into the cylinder in the
manner described in Section 5. These deposits could then
be compared by measuring the ionization currents to which
they gave rise in a second cylinder ; it was easy to verify
from the decay curves thus obtained that these fractions
were deposited on the rod chiefly in the form of RaA.
The experimental procedure was as follows : the emana
tion mixed with air carefully dried and freed from dust, was
passed into the cylinder and allowed to remain for several
days. A potential of — 160 volts was applied to the case,
and ah earthed rod was introduced for 1 minute and then
withdrawn, the cylinder being closed as before with a rubber
stopper. About half an hour later the experiment was
repeated with rod and case both earthed ; alter another
interval of about half an hour the experiment was repeated
with earthed rod and +160 volts applied to the case. The
activities of the rods were measured about 25 minutes after
withdrawal, when the decay curve is practically flat ; on this
account no correction had to be made for the decay of RaA
during the exposure of the electrode.
The activity deposited on the rod in the first experiment
( — 160 volts) will be due to two causes : in the first place,
the rod will receive a certain fraction (/) of the activity (Gr)
in the gas at the time of introduction ; it will also receive a
certain fraction (k) of the neutral deposit particles formed
during the exposure (1 minute). Suppose a fraction p of
tlio deposit particles are positively charged when a potential
of 160 volts is applied ; this fraction p of course takes into
account a small amount of columnar recombination which is
independent of the amount of emanation in the vessel.
In the second experiment (0 volts) the rod will receive /G
together with the fraction (IS) of all the deposit particles
formed in 1 minute. In the third experiment ( + 160) the
rod will receive the same as in the first experiment together
with pQ.
If a, />, and c represent on an arbitrary scale the measured
ionization currents due to the active deposit collected on
these rods, and if m denotes a factor of proportionality
between amount of deposit and ionization current, including
Phil. Mag. S. 6. Vol. 28. No. 166. Oct. 1914. 2 F
434 Prof. E. M. Wellisch : Experiments on
also the correction for decay from the maximum value3 we
will have the following relations :
fG + k(lp)Q = m.a (160 volts) ;
fG + kQ =m.b( 0 volts);
f&+pQ + k(l—p)Q = m.c ( + 160 volts).
We deduce
pQ = m. (c — a) ;
/,_ b — & .
c—a
G a(lp)b ^1±;
The assumptions made in obtaining the above relations
may at first sight appear doubtful in their applicability ;
still they appear to be entirely justified by experience. The
separation of Q into pQ positive and (1— •p)Q neutral has
been justified repeatedly by the author in numerous
experiments. The separation of G and Q as far as diffusion
effects are concerned is merely an application of the un
doubted principle that in diffusion processes we may treat
any large number of atoms chosen at random as a separate
gas to which the laws of diffusion may be applied. Finally,
there is the most important assumption that the activity in
the gas is entirely neutral. In order that this should be the
case it is necessary, as explained in the previous section,
that the experiments should be performed with small amounts
of emanation ; all the deposit particles in the gas appeared
to have lost their charge when the saturation current fell
below 10 ~9 ampere. The method of putting this point to
experimental test was to perform the three experiments with
various amounts of emanation in the testing vessel ; if a, £>,
and c remained in constant proportion over the series, this
was taken to imply the absence of any appreciable quantity
of charged gas activity, because it has already been shown
that the amount of charged activity decreased extremely
rapidly with decreasing amounts of emanation. The fact
that we are thus restricted to the use of small quantities of
emanation means of necessity an extremely small value for
the activity in the first experiment, in which the chief con
tribution comes from the gas activity ; on this account every
precaution was taken in measuring the corresponding ioni
zation current, the electrostatic shielding being made thorough
and the measuring vessel preserved carefully from radio
active contamination.
the Active Deposit of Radium. 435
A very large number of experiments were performed to
obtain the ratios a : b : c. There was little difficulty in getting
an accurate determination of: the value  ; this came out 7.
c y
for practically all the experiments. The most accurate
determination of the ratio  is 7^. ; this was obtained about
8 times under favourable measuring conditions. Still the
ratio > was obtained on a few occasions ; of course there
15
was always the possibility in these instances of the presence
in the gas of charged deposit, but the amount of emanation
was, as always, quite small. The ratio was never less than
— . The value of p was taken as *83 ; this value represents
the fraction of the active deposit that settles on the cathode
when 1G0 volts is applied to the case, the quantity of
emanation being supposed to be so small that volume recom
bination is absent (v. Section 4). The value of / was taken
as '10, this value representing the fraction of the deposit that
settles on the central rod in the absence of an electric field
(v. Section 3). It was ascertained experimentally that the
exposure of 1 minute afforded ample time for this fraction to
reach the central rod.
If we tike the ratios a : b : c as 9 : 20 : 180, then we have
from equations (10) and (11) D = *0445 ; £='064.
With a : 1> : c=12 : 20 : 180, we deduce D = '029.
With a:b:c= 9 : 24 : 210, we deduce D = '061.
The value D = *0445 cm.2 sec.1 represents the most probable
value for the coefficient of diffusion of uncharged atoms of
radium active deposit in dry air at a pressure of 1 atmosphere
and at room temperature. The other values for D are to be
regarded merely as extremes between which the true values
of D appear to he.
The value obtained for D is smaller than the values which
have been obtained experimentally for the diffusion of radium
emanation in dry air. In the case of the emanation the
values obtained are not very different from "10. The value
of D for the active deposit is approximately the same as that
found by Townsend for the diffusion of a gaseous ion in air.
The Experiments of Eelman n.
7. In the Jahrbuch der Radioaliintat (1012), Gerhard
Eckmann describes the results of a series of experiments
bearing on the diffusion and distribution of the active deposit
2 F 2
436 Prof. E. M. Wellisch : Experiments on
particles of radium. These experiments cover in many
respects the same ground as those here described ; this fact
was not recognized till the present series was completed.
Eckinann's work may be, for convenience, summarised under
the following heads : —
(1) Distribution of the deposit in electric fields ;
(2) Investigation with regard to aggregates ;
(3) Determination of the coefficient of diffusion (D).
With regard to (1) he comes to the conclusion that about
98 per cent, of the particles are positively charged, the
remainder being negatively charged. This is of course
opposed to the results of the present investigation. Eckmann
assumes that the total deposit is the sum of the deposits
obtained on a central electrode when it is first raised to a
high positive and then to a high negative potential. The
present experiments show that this assumption is incorrect,
the 2 per cent, which is obtained in the latter instance being
due to the diffusion to the central electrode of a small
fraction of the neutral particles. Eckmann ascribes the
deposit on the anode to negatively charged particles, and
accordingly neglects a considerable amount of deposit which
settles on the case of his vessel.
Eckmann's work with regard to aggregates anticipates to
a large extent that described in the present paper. He
shows that they are charged, and is undoubtedly correct in
his assumption that they owe their charge to the ions present
in the gas. He shows also that the anomalous results
obtained by Debierne * in a series of experiments with
regard to the diffusion of radium active deposit can be
explained by taking into consideration the formation of
aggregates. The present experiments bring out several new
points of interest, but on the whole serve merely to confirm
the results of Eckmann in this connexion.
With regard to the determination of D, Eckmann collected
the total amount of gas deposit on a piece of gauze ; he
found, for the particular vessel which he employed, that this
deposit was equivalent to a formation for "6 minute ; in
other words, S = If (with the notation of Section 6). He
then employed the formula Gr = "/fy (Section 3) in order to
estimate D, which came out as *06 ; this formula, however,
neglects the diffusion to top and bottom, which was impor
tant for the cylinder which he used. It is of interest to
* Debierne, Le Radium, vi. p. 27 (1909).
the Active Deposit of Radium. 437
apply formula (8) in order to evaluate D ; we find b given
as 4 cm. and I equal to 8 cm.
Equation (10) of Section 5 becomes now
D = 0266^;
Q
using Eckmann's experimental value for ^ given above we
W D = '0443.
a *'
Such an excellent agreement with the most probable value
obtained in the present investigation must be to some extent
fortuitous, but there can be little doubt as to the order of
magnitude of D.
Summary.
8. It is advisable to give here a brief synopsis of the
results obtained in the series of experiments dealing with the
active deposit of radium, including for the sake of con
venience several results previously published.
Suppose a quantity of radium emanation is mixed with a
dry dustfree gas; of the active deposit particles which are
produced a certain fraction are positively charged, the
remainder being neutral. This fraction depends upon the
nature of the gas, but not upon its pressure provided the
pressure be high enough to prevent any appreciable recoil on
to the walls of the vessel. The values of this fraction are
for air *, hydrogen, and carbon dioxide respectively, 8$"2,
88'2, and 78*0. When the emanation is mixed with ethyl
•ether, practically all the deposit particles are neutral.
In a previous paper the view was put forward that the
distribution of charge among the deposit particles had its
origin in the motion of recoil of these particles when expelled
from the atoms of emanation. The recoil atom during its
motion produces a large number of ions, and in all proba
bility is ionized itself so that it acquires a positive charge ;
however, it is always liable to lose its positive charge by
collision with an electron, and all the more readily in the
case of those gases or vapours such as carbon dioxide and
ether where the electrons do not quickly leave the columns f.
The general effect to be expected from such a process is that
when the deposit particles are brought to relative rest among
the gas molecules, there will be a definite fraction positively
charged, the remainder being neutral.
* 1 1 is of interest to recall here that Lucian has found the number
049 for the traction of the deposit particles of actinium which are formed
with a positive charge in air.
t Cf. Wellisch and Woodrow, Phil. Mag. [6] vol. xxvi. p. 511 (1913).
438 Experiments on the Active Deposit of Radium.
Let us now consider what happens after formation of the*
deposit atoms, and let us, for convenience in exposition,
assume that the emanation is situated in a cylindrical vessel
with a central electrode.
If we wish to bring over to the cathode all the positively
charged deposit particles, an electric field must be established
of sufficient strength to prevent both the columnar and
volume recombination with negative ions to which the
particles are subject. In this connexion it should be pointed
out that the recombination between these particles and the
negative ions is more intense than that between positive and
negative ions. As long as the electric field is large enough
to prevent volume recombination, the fraction of the total
deposit which settles on the cathode is independent of the
amount of emanation in the vessel, because if there is now
any loss after formation, it is due to columnar recombination,
which is conditioned by the value of the electric field.
With large values of the electric field the neutral deposit
particles reach the electrodes by the process of diffusion ; let
us now consider the process at work when there is no electric
field present or when the electric field is so small that there
is considerable volume recombination present in the gas.
What happens in these cases depends on the amount of
emanation in the vessel. With fairly large quantities of
emanation, the active particles which exist in the gas form
large aggregates which acquire positive and negative charges
from the ions present in the gas ; the aggregates become
then of course liable to lose their charge by recombination,,
but they can then regain it as before, so that on this account
a moderately small field is able to bring over to the central
electrode, either as anode or as cathode, a considerable amount
of active deposit. In fact, with negative potentials applied
to the vessel, the active deposit on the central electrode
(anode) increases with increasing potential, because over a
wide range the number of ions present in the gas remains
approximately constant ; but when the applied potential
becomes large enough to effect a diminution in the number
of ions present in the gas, the anode activity diminishes. In
the circumstances now under review the number of charged
deposit particles in the gas is in general greatly increased
by the production in the gas of extra ionization, e. g. by
means of Rontgen rays.
The formation of large aggregates appears to cease abruptly
when the concentration of the emanation and the density of
ionization sink below certain values ; the deposit particles
remaining in the gas may still carry electric charges, but as
the emanation decays still further the deposit atoms in the
Tlie Theory of Molecular Volumes.
439
gas are practically all neutral and reach the electrodes
through diffusion. On this account the general process
becomes again the relatively simple one o£ electric convection
and diffusion, to which reference has previously been made.
Under conditions which are such that the deposit atoms in
the gas no longer form aggregates nor are charged by means
of gas ions, their coefficient of diffusion through dry air at
one atmosphere pressure and at ordinary room temperature
(about 20° C.) is approximately *045 cm.2 sec.1.
Sloane Laboratory,
Yale University,
17th June, 1914.
LI. The Theory of Molecular Volumes.
By Gervaise Le Bas, B.Sc*
Part IV.
Completed Ring Systems and the Volxme Anomaly.
Systems of Completed Rings.
THE influence of Ring structure upon the magnitude of
most physical properties is usually not great — only a
small fraction of the whole. The more important properties
besides volume which are affected by this constitutive
feature are, Refractive Power and Magnetic Rotatory Power.
The extent of the influence exerted by Ring structure upon
these, is indicated by the following table. The columns
marked A represent the differences between the observed and
calculated values when all other constitutive influences have
been taken into consideration.
lief, active Power.
Magnetic Rotatory Power.
Compound.
Paraldehyde ....
Epichlorhydrin.
■ Pipiridene
Cyclopentanc...
Cyclohexane ....
; Cyclohexanone .
I Pinene ,
Camphene
 Tricyclodecane .
M"a(obs).
A.
324
4030
2047
+019 ;
2657
0
2309
+008
2762
+0 05
2782
+002
4372
4059
4309
4056
4165
4035
i
Compound.
Trimeth. CarboxylicAcid.
Tetraineth. Carb. Acid....
Ethyl Cvclobutvl Ketone.
M.
4141
5 048
6*911
A.
Cyclohexane 5'664
Camphor 9 265
Borneol 9807
0331
0465
0634
0982
2 x 0645
2 x 0636
Communicated by Prof. W. J. Pope, F.E.S.
440 Mr. Grervaise Le Bas on the
Refractive Power may be influenced to a slight extent by
Ring structure, but the evidence is inconclusive, and at any
rate no reliance can be placed on the conclusions arrived at.
Magnetic Rotatory Power shows a much more marked and
reliable influence, but this development of Perkin's theory
has not been much exploited.
The influence, such as it is, is positive for refractive power,
and negative for magnetic rotatory power.
Perkin in Trans. Chem. Soc. 1902, xci. pp. 2934, has, by
one method of calculation, found differences of varying mag
nitude for different classes of rings, but has discarded this for
an alternative method, which apparently shows that the cor
rection for ring structure is the same for simple rings of all
magnitudes. This is contrary to the observed effect on
molecular volume, for, as will be shown, the correction
depends on the size of the ring. Perkin did not realize the
significance of the result obtained by the first method of calcu
lation, and so discarded it for the second. The latter result
is only arrived at by utilizing the first and second members
of series, such as formic and acetic acids, acetone, &c, which
are anomalous in respect of their molecular magnetic
rotations. The first is thus the correct method of calculation,
and the results obtained are the true ones. If this be so, then
we must conclude that both magnetic rotatory power and
molecular volume show effects due to ring structure, which
are dependent on the size of the rings. The first physical
property is so much affected by constitutive influences that it
is doubtful if sufficiently accurate results have been, or even
can be obtained by means of this property. This is not the case
with molecular volume. This property is much less subject
to constitutive influences, and the effects due to ring structure
are relatively and actually large.
There does not thus appear to be any good method for
the elucidation of the Ring structure of a compound till we
turn to Molecular Volumes.
The history of the subject can be briefly stated. Kopp
certainly recognized that the densities of benzene and its
derivatives were abnormal, but notwithstanding this he
adhered to his atomic values C = 11*0 H = 5*5, which were
derived from comparisons made between the volumes of
paraffin and benzene derivatives. This is unjustifiable.
No reference was made to the possible effect of Ring
structure on volume, till Traube (BericJde, xxviii. p. 2926)
deduced the fact, that a contraction of —8'2 exists for the
hexamethylene ring, and one of —13*2 for the benzene ring
Theory of Molecular Volumes. 441
at 0°, with an additional contraction o£ 3 X 1*7 for three olefin
linkings. No reason exists for such a large difference
between rings of such similar natures as those of benzene and
hexamethylene,
A very full treatment of the effect of Ring structure on
molecular volumes was made by the author in the Phil. Mag.
s. 6, vol. xvi. no. 91 (1908), pp. 7786, and ' Chemical
News/ 99. p. 206 (1909). This work seems to have been
ignored in recent textbooks, although the evidence is very
complete and striking. Since then, in a paper read before the
British Association, Portsmouth 1911, and reprinted in the
* Chemical News/ 104. p. 151 et seg., an extension of the
work was indicated.
Homocyclic Rings.
Tlie Benzene Nucleus.
At the critical point we have the following results : —
V Benzene, C6H6 256'3 W 30. V/W 8'54
Hexamethylene, C6H]2 306'7 AV36. V/W 8'52
6H 504 = 6x84.
H in paraffin series has a volume of 9*67.
The contraction for benzene C6H6is found to be 32*4, and
for hexamethylene CGH12 40' 1. These values are proportional
to their complexity.
The diminished value of the H atoms in the nucleus of
benzene at the boilingpoint, vizv ~   = 3*20, is shown by the
following fact. If we gradual]} diminish the H atoms in the
nucleus and substitute methyl groups which are paraffin
residues, we diminish the contraction by 0*5 for each sub
stitution, that is, the difference between 3*70 and 3*20.
CH3=25'45.
Benzene.
CH M.V. Benzene, C6H6 96'0
nf \„ Dipropargyl,C6H6... 1110
 A for Ring LfrO
CH CH
^ / CH = CCH2CH2C = CH.
CH Dipropargyl.
442 Mr. Gervaise Le Bas on the
Toluene.
M.V. 118*25.
// \ VC6H6 96032 928
CH CH CH3 2545
'  2A.V 11825 Schid
CH C — CH3 ^=aa
^ / A —145
CH
pXjlene.
C — CH3 M.Y. 14040.
P1f \R VC6H4 9602x32 896
I ij1 2CH3 509
CH CH 2A.V 1405
C — CH3 a —141
Mesitylene.
CH3 M.V. 162.
I V C6H8 9603x32 864
C 3CH3 7635
CH CH ^5
CH3C C — CH3 A ~~U6
CH
The contraction of —15 in benzene is thus in part due to
the diminished volumes of the hydrogen atoms.
The almost strict accuracy of the above results is not
usually met with, because minor effects cause the values to
vary a little. The average contraction for benzene deriva
tives is about —14*7.
%
Theory of Molecular Volumes,
Benzene, CgH, M.V. 960
Ethyl Benzene, C0H5 . C2H3 1393
Propyl Benzene, CGH5.C3H7 162'1
Cinnamene, CCH5 . CH : CH., . 1319
Cymene, CGH4CH, . C3H. . 1849
Anisol, 06Hg.O.OH3 ... 1255
Pbenetol, CaH5 . O . C2H, . . . 1476
443
A
2A.Y.1110 150
1546 153
1764 143
1473 154
1996 147
1406 151
1628 152
Mean 150
Hexamethylene.
CH2
/ \
CH, CH2
CH2 CH,
\ /
CH„
The Hexamethylene Kino.
M.V. Hexamethylene, 66H12 116*3
V Hexylene, C6H12 132*4
A for Ring 1G1
CH2 : CH . CH2 . CH2 . 0H2 . CH3
Hexylene.
This is a somewhat larger contraction than is common
with benzene derivatives, and it shows tbat there is a ten
dency for the aromatic nucleus in
C0 Ho • H0 to extend
its influence to the extrahydrogens. If this were entirely
the case the volumes would be 115 2.
When, however, another substituent of a hydrocarbon
character is introduced, the extra hydrogens partake of the
nature of paraffin derivatives, and the volumes increase to
4'0 units.
Aromatic Derivative. Hydroaromatic Derivative.
M.V. 6xH M.V.
Toluene. CGH. . CHa 1183 23'8 1421 Hexahydrotoluene, C6HnCH3
Xylene, C6H4.(CHS)2... 1404 248 1652 Hexahydroxylene, 0GH10(CH3)2
Naphthalene, C,0HS 1472 240 171'2 Naphthyl Hydride, C10H1(
Mei
242=6x40
We find that pipiridene C5HnN stands in exactly the same
relation to pyridene C5H5N, as hexamethylene CGH]2 stand
to benzene.
Pyridene C5H5N. M.V. .
GH= 6x3*2 .
893
192
2A.V 1085
Pipiridene C5HnN ... 108*8 (obs.]
44:4 Mr. Gervaise Le Bas on the
The result of this is that when it is sought to discover the
contractions of substituted hexamethylene compounds, they
will be found to be apparently smaller than the contractions
for aromatic compounds.
y y
The Naphtenes. C ^<^ \.^
A~/\
W. Compound. M.V. 2A.V. A.
42 Heptanaphtene C7H14 142*1 155*4 13*3
\/ \/
XT' \
^ .^ XCH3
/\ /\
4* Octonaphtene C8H16 165*0 177*6 12*6
\/ \/
A A
CH3
54 Nononaphtene C9H18 [187*6]* 199*8 12*0
\/ \/
ch,"'x. / "
CH3
CO Dekanaphtene C10H20 [208*8] 220*0 13*2
\/ \/
CH.'X /^CH.(CH3)2
A A Mean... 12*8
* Calculated by means of formula given "below.
Theory of Molecular Volumes.
445
The contractions are all smaller than in the Aromatic Series,
the mean being —12*8 instead of — 15*0. This is due to
the increase volumes of the extra hydrogen. We thus have
in the following diagrams an attempt to indicate this
difference.
R
'' c \
/ / \ \
■ CH CHJ
i 1 II !
■CH CH
\ % / /
E. H
,' C \
/ / \ \
nfCH chvh
i I  ;
HVCH. CHfH
\ \ / /
^ CH /
H
Aromatic.
Hydroaromatic.
means of a formula
"b.p.
'+*(>&)
it is possible to calculate the volumes of compounds at the
B.P. to a close approximation, especially if we know the
volumes of related compounds. Thus Cymene C10HU gives
us the means of calculating the volumes of all the reduction
products thereof, including the single ringed terpenes.
Cymene, C10H14
d0 08718 dBV 07248 (Sclriff).
^ = 1'2028
<*B.P.
B.P. = 17G2 (4492 A).
27:>> 273
1_ —1— — ■'■W?
B.P. 4492 ~~
Ratio (const.) K= 0.^q9 =0*52.
446
Mr. Gervaise Le Bas on the
The values of M.V. enclosed in square brackets [ ] have
been calculated by formula.
Reduction Products of Cymene.
W.
Compound.
Cyiuene,
Carvene,
Menthene,
C10H14
C10H1C (Terpene)
C10H1S
Hexabydrocymene, C1 0H20
M.V.
£ A.V.
A.
1847
1998
151
192 8
2072
144
[2008]
214 6
138
[2083]
2200
137
The compounds just studied are : —
CH3 CH3
\ /
CH
I
CH CH
CH CH
% /
C
I
CH3
Cymene.
CH3