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\o THE fae

LONDON, EDINBURGH, anp DUBLIN

PHILOSOPHICAL MAGAZINE

AND 4

JOURNAL OF SCIENCE.

CONDUCTED BY

SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R.8.

SIR JOSEPH JOHN THOMSON, 0.M., M.A., Sc.D., LL.D., F.R.S.
JOHN JOLY, M.A., D.8c., F.B.S., F.G.S.

GEORGE CAREY FOSTER, B.A., LL.D., F.R.S.

AND

WILLIAM FRANCOIS, F.I.8.

“ Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster
vilior quia ex alienis libamus ut apes.” Just. Lrps. Polit. lib. i. cap. 1. Not.

VOL. XXXV.—SIXTH SERIES.
JANUARY—JUNE 1918.

LONDON:
TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET.

SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD.
SMITH AND €0N, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ;—
AND VEUVE J. BOYVEAU, PARIS,

“Meditationis est perscrutari occulta; contemplationis est admuirari
perspicua .... Admiratio generat questionem, queestio investiga
investigatio inventionem.”—Hugo de S. Victore.

“Cur spirent venti, cur terra dehiscat,

Cur mare turgescat, pelago cur tantus amaror,

Cur caput obscura Phoebus ferrugine condat,

Quid toties diros cogat flagrare cometas,

Quid pariat nubes, veniant cur fulmina ceelo,

Quo micet igne Iris, superos quis conciat orbes

Tam vario motu,”

J. B. Pinelli ad Mazonium.

CONTENTS OF VOL. XXXV.

(SIXTH SERIES),

NUMBER CCV.—JANUARY 1918.

Lord Rayleigh on the Theory of Lubrication..............
Prof. H. Nagaoka on the Calculation of the Maximum Force
between Two Coaxial Circular Currents.................
Mr. J. A. Tomkins on the Nodal-Slide Method of Foco-
RN eh xo oer eile ON at osc) nih ol AIS oy ao eile al re a | eG a/et em Sieh ate
Mr. T. Carlton Sutton on the Value of the Mechanical Equi-
ME MBE eMC saree tee cca Cure Siete visi VR ola was ee
Dr. L. Silberstein on Light Distribution round the Focus of a
Mma tG VALIOUS NperbUres 2. see y eels s Lb eel eles Ds
Mr. F. Twyman on Interferometers for the Experimental
Study of Optical Systems from the point of view of the
RSP MEME EY: - 2 Mere Mie reas Lew pa uke eared. Oba A Wranaare
Dr. M. Wolfke on a New Secondary Radiation of Positive
2 AEA SS Pree es e- Bhs or alate d aca osaenas ea ese a, SYR ent
Prof. Barton and Miss Browning on Variably-Coupled Vibra-
tions: IJ. Unequal Masses or Periods. (Plates I. & II.)..
Mr. Nalinimohan Basu on the Diffraction of Light by
Cylinders of Large Radius. (Plate III.) ..............
Prof. Sudhansukumar Banerji on Aerial Waves generated by
Beebe bart Ls Cr ape yee aise eel che dil ai ohate sta
Mr. Sisir Kuma Mitra on the Asymmetry of the Ilumination-
Curves in Oblique Diffraction. (Plate V.) ............
Dr. J. G. Leathem on the Two-Dimensional Motion of
Infinite Liquid produced by the Translation or Rotation of
SRMERABMEMBUINOREA To SUA Bel ss ioashd aon a aiain alee. alae:

27
30

49
59
62
79
97
112

2.

iV CONTENTS OF VOL. \XXV.—SIXTH SERIES.

Prof. G. W. O. Howe on the Relation of the Audibility Factor

of a Shunted Telephone to the Antenna Current as used

in the Reception of Wireless Signals. With Note by

Nir. wan. der Pol, ities Sy eee ah ea ede eee eee ae 131

Proceedings of the Geological Society :—

Mr. A. Holmes on the Pre-Cambrian and Associated

Rocks of the District of Mozambique..............

Dr. Felix Oswaid on the Nimrud Crater in Turkish

ATNODIA GG 6 /F Sue 2(4 aie diate) wate: aya is' 9 a4 0. = rr
Intelligence and Miscellaneous Articles :—

On Coupled Circuits and Mechanical Analogies, by

Sir G. Greenhill

ene © 0.0 wie we 8 v6 88 vy 8 8 Ue & Ce ee eee eee

NUMBER COCVI.—FEBRUARY.

Sir Oliver Lodge: Continued Discussion of the Astronomical
and Gravitational Bearings of the Electrical Theory of
Mather (ooo) 8.0 2 ce is oes one atds ok ern

Lord Rayleigh on the Lubricating and other Properties of
Thin Oily; Films | 2... ..« cote eweeies 2s...

Prof. Q. Majorana on the Second Postulate of the Theory of
Relativity: Experimental Demonstration of the Constancy
of Velocity of the Light reflected from a Moving Mirror. ,

Mr. Prentice Reeves on the Visibility of Radiation ........

Prof. Andrew Gray on the Hodographic Treatment and the
Energetics of undisturbed Planetary Motion .........-

Mr. F. E. Wood : A Criticism of Wien’s Distribution Dew

Prof. Barton and Miss Browning on Coupled Circuits and

Mechanical. Analogies, .....-..2.--....... 0

Dr. W. F. Smeeth and Dr. H. E. Watson on the Radioactivity
of Archean Rocks from the Mysore State, South Jndia. .

Dr. 8. R. Milner on the Effect of Interionic Force in
Blectrolytes:, | ae cdiee es aiireeteile V2 ti ale d <2) eer

Notices respecting New Books :—

Researches of the Department of Terrestrial Magnetism.
Vol. III. Ocean Magnetic Observations 1905-1916,
and Reports on Special Researches

Page

, 134

138

138

140

141
157
163
174

181
190

203

v,

.- 206

Bal

CONTENTS OF VOL. XXXV.—-SIXTH SERIES. Vv

NUMBER CCVIIL—MARCH.

Page

Mr. Rk. M. Deeley on Rain, Wind, and Cyclones .......... 22
Prot. R. W. Wood on Resonance Spectra of Iodine. (Plates

Oo VUE RICE CUE ca ROT A Rcd A eh a ea 236
Prof. R. W. Wood and Prof. M. Kimura on the Series Law

PemeorianCe SPCCUMM Mee LN Ue Lee ee Ws we ciel es 252
Dr. A. M. Tyndall and Miss N. S. Searle on the Pressure

Hacer in Oorona Diseharve’. 2... Se earl Gk

Dr. Harold Jeffreys on some Problems of Evaporation ..... 270

Dr. G. W. Todd on General Curves for the Velocity of
Complete Homogeneous Reactions between Two Substances

pieensiant. Volumes (Plate LX. )i.. 0. 5 oe ese 281
Prof. A. Anderson on the Coefficients of Potential of Two
PepeatciaTi SMMELERS fis Chee do ssh a, Bae ows ehieraoe wks 286

Notices respecting New Books :—
Centennial Celebration of the United States Coast and

Geodetic Survey, April 5and 6,1916 ............ 298
Proceedings of the Geological Society :—
Mr. J. Morrison on the Shap Minor Intrusions ...... 292
NUMBER CCVIIL—APRIL.
Dr. L. Vegard on the X-Ray Spectra and the Constitution of
Se 2S hs ee eee Seen (may a alaetery al ian oh meh oncce anes 293
Mr. G. W. Walker on Relativity and Electrodynamics.
Or OGRE SUS ean ak? SiGe OMIM acta He Na RE 327
Dr. H. Stanley Allen on Molecular Frequency and Molecular
2 LOE, DSL EARR ORAS A SRN LSAT SAN TnL SUES ee Ul Gree nO 338

Sir Joseph Larmor on Transpiration through Leaf-Stomata.. 350
Dr. 8. R. Milner on the Effect of Interionic Force in

PEO BES | ie ee Ns Lie Wim Sher Mises occa le Wedel) ova oitelavealy ake 302
Prof. G. N. Watson on Bessel Functions of Equal Order and
CE TSC US RII 2 ye ORS Ae EA igi ee ey eo A 364

Notices respecting New Books :—

Prof. R. A. Millikan’s ‘The Electron: Its isolation and
measurement and the determination of some of its
PEGPOULICR Meee Ws grtdln wi csnpe Neo Apia) ol obid\ x eh-ossisieps ah wah ee 370

Napier Tercentenary Memorial Volume.............. 371

Modern Instruments and Methods of Calculation: A
Handbook of the Napier Tercentenary Exhibition .. 372

Prof. Harris Hancock’s Elliptic Integrals (Mathematical
LSE Ge OE sis ot a ) Se ae eer a 372

vi CONTENTS OF VOL. XXXV.—-SIXTH SERIES.

NUMBER CCIX.—MAY.

E
Lord Rayleigh on the Scattering of Light by a Cloud of
similar small Particles of any Shape and oriented at random. 373
Mr. G. A. Hemsalech on Fox Talbot’s Method of obtaining
Coloured Flames of Great Intensity .................. 382
Prof. A. Anderson on the Problem of Two and that of Three
Electrified Spherical Conductors. .,..2 -:.»<:e meee 388
Mr. W. G. Bickley on some Two-Dimensional Potential
Problems connected with the Circular Arc ............ 396
Dr. H. Stanley Allen on Molecular Frequency and Molecular
Number.—Part II. The Frequency of the Longer Residual

age

TRAYS. ose oa ge cehsin's 4 'e eue' poaietene syn aliens milo (a, fete tr 404
Dr. Harold Jeffreys on Wood’s Criticism of Wien’s Distri-
rti.on, Loa cian ote tet ooh pha: aveties ans 0G 410
Dr. H. E. Ives on the Resolution of Mixed Colours by
Differential Visual Diftusivity,..... i... /2-/dale~ <5 wee ee 413
Prof. F. Y. Edgeworth: An Astronomer on the Law of
ROT OP o.oo Seis se ses, Oh i RI als oe oe 422
Dr. H. Jeffreys on Transpiration from Leaf-Stomata; with
Note by Bird « Garimor ss oeege. 3) a <2). 0. Soke eee 431, 433

Dr. G. W. Todd.on a Method of obtaining General Reaction-
Velocity Curves for complete Homogeneous Gas Reactions
at Constant Pressure: 3.7 ars. 66 ce et. ss 435
Notices respecting New Books :—
Annuaire du Bureau des Longitudes pour l’année 1918. 444

NUMBER CCX.—JUNE.

Dr. H. Stanley Allen on Molecular Frequency and Molecular
Number.—Part III. Inorganic Compounds. Lindemann’s

Wormilla ik Ll Cee Cele lee + err 4A5
Dr. F. L. Hitchcock on the Operator V in Combination with

Homogeneous Functions. Second Paper .............. 461
Mr. A. O. Allen on Graphical Methods of correcting Tele-

scopic Objectives .... 6... ee cece eee eee eee eet e ee 471

Prof. A. S. Eddington on Electrical Theories of Matter and
their Astronomical Consequences with special reference to
the Principle of Relativity .................5.5- wees 481

Prof. G. W. O. Howe on the Relation of the Audibility
Factor of a Shunted Telephone to the Antenna Current
as used in the Reception of Wireless Signals............ 487

Mr. R. F. Gwyther: A Doctrine on Material Stresses...... 490

Prof. C. V. Raman on the Wolf-note in Bowed Stringed

Mie EEUUTMPGTVUS) i005, cigs gee ee 0 SAVE > 0 MRM n ee Se et ee 493

CONTENTS OF VOL. XXXV.—SIXTH SERIES. vil
Page
Mr, Nalinimohan Basu on a New Type of Rough Surface the
Motion of a Heavy Particle on which is determinable by
PRAIRIES) GOVE uC E ye ake sui. stat ateauebeaaie ete afaiies «eee hce me 496
Mr. W. G. Bickley on Two-Dimensional Motion of an Infinite
LoL DTT gh SISTA on a eI A ren ere A 500
Proceedings of the Geological Society :-—
Mr. L. Dudley Stamp on the Highest Silurian Rocks of
the Clun Forest District (Shropshire).............. 502
Prof. W. Johnson Sollas on a Flaked Flint from the Red
REE) i ee eee cae tc are area tue sla ei toalang ) oluideale 503
Mr. R. Dixon Oldham: Some Considerations arising
from the Frequency of Harthquakes .............. 504
Intelligence and Miscellaneous Articles :—
On Relativity and Electrodynamics, by G. W. Walker... 508

(oe Ses hae ROUGE OO Ts or CRO Ale Ae ar es alee pr Rs eee 509

PLATES.

I. & IL. Illustrative of Prof. Barton and Miss Browning’s Paper on
Variably-Coupled Vibrations : II. Unequal Masses or Periods.

IIT, Illustrative of Mr. N. Basu’s Paper on the Diffraction of Light by
Cylinders of Large Radius.

IV. pao of Prof. S. Banerji’s Paper on Aerial Waves generated

y Impact.
V2 Tilasiatiee of Mr. S. K. Mitra’s Paper on the Asymmetry of the
Tllumination-Curves in Oblique Diffraction.
VI.-VIII. Illustrative of Prof. R. W. Wood’s Paper on Resonance
Spectra of Iodine.

IX. Illustrative of Dr. G. W. Todd’s Paper on General Curves for
the Velocity of Complete Homogeneous Reactions between Two
Substances at Constant Volume.

X. Illustrative of Mr. G. W. Walker’s Paper on Relativity and
Electrodynamics.

/ THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SERIES.) <—~ ~ vray ete :

JANUARY 1918. JAN 281918

ago

I. Notes on the Theory of Lubrication. ~~~
By Lord Rayuzies, 0.1, F.RS.*

ODERN views respecting mechanical lubrication are
founded mainly on the experiments of B. Towert,
condueted upon journal bearings. He insisted upon the
importance of a complete film of oil between the opposed
solid surfaces, and he showed how in this case the main-
tenance of the film may be attained by the dragging action of
the surfaces themselves, playing the part of a pump. To
this end it is “‘ necessary that the layer should be thicker.on_
the ingoing than on the outgoing side”, which involves a
slight displacement of the centre of the journal from that of
the bearing. The theory was afterwards developed by
Q. Reynolds, whose important memoir § includes most of
what is now known upon the subject. In a later paper
Sommerfeld has improved considerably upon the mathe-
matics, especially in the case where the bearing completely
envelops the journal, and his exposition || is much to be
recommended to those who wish to follow the details of the
investigation. Reference may also be made to Harrison {,
who includes the consideration of compressible lubricants
(air).
* Communicated by the Author.
+ Proc. Inst. Mech. Eng, 1883, 1884.
¢ British Association Address at Montreal, 1884; Rayleigh’s Scientific
Papers, vol. ii. p. 344.
§ Phil. Trans, vol. 177. p. 157 (1886).

|| Zettschr. f. Math. t. 50. p. 97 (1904).
{| Camb. Trans. vol. xxii. p. 39 (1918).

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan, 1918. B

/

2 Lord Rayleigh on the

In all these investigations the question is treated as two-
dimensional. For instance, in the case of the journal the
width—axial dimension—of the bearing must be large in
comparison with the arc of contact, a condition not usually
fulfilled in practice. But Michell* “has succeeded in solving
the problem for a plane rectangular block, moving at a
slight inclination over another plane surface, "free from this
limitation, and he has developed a system of pivoted bearings
with valuable practical results.

It is of interest to consider more generally than hitherto
the case of two dimensions. In the present paper attention
is given more especially to the case where one of the opposed

surfaces is plane, but the second not necessarily so. /As an

) alternative to an inclined plane surface, consideration is given
to a broken surface consisting of two parts, each of which is
parallel to the first plane surface but at a different distance
from it. /It appears that this is the form which must be
approached if we wish the total pressure supported to bea
maximum, when the length of the bearing and the closest
approach are prescribed. In these questions we may anti-
cipate that our calculations correspond pretty closely with
what actually happens,—more than can be said of some
branches of hydrodynamics.

In forming the necessary equation it is best, following
Sommerfeld, to begin with the simplest possible case. The
layer of fluid is contained between two parallel planes at
y=0Oand at y=h. The motion is everywhere parallel to a,
so that the velocity-component u alone occurs, v and w being
everywhere zero. Moreover wu is a function of y only. The
tangential traction acting across an element of area repre-
sented by dz is u(du/dy)dx, where pu is the viscosity, so
that the element of volume (da dy) is subject to the force
p(d?u/dy*) dx dy. Since there is no acceleration, this force is
balanced by that due to the pressure, viz. —(dp/da) dz dy, and
thus

dp as
ohae tr

In this equation p is independent of y, since there is in this
direction neither motion nor components of traction, and (1),
which may also be derived directly from the general hydro-
dynamical equations, is immediately integrable. We have
_ 1 dy» !

Vii Oa da! +A + Bye. ee

where A and B are constants of integration. We now
* Zeitschr. f. Math. t. 52. p. 123 (1905).

Theory of Lubrication. 3
suppose that when y=0, u=—U, and that when y=A,

u=(. Thus
_ Yahy dp _ y ie
ua Ue (1— z) Olle SA
The whole flow of liquid, regarded as incompressible, between
0 and his
Oe eR: apr. ORO
(ea=- 5 -F=-0

where Q is a constant, so that

7720) 2Q
de ae (A (4)

If we suppose the passage to be absolutely blocked ata place
where @ is negatively great, we are to make Q=0 and (4)
gives the rise of pressure as # decreases algebraically. But
for the present purpose Q is to be taken finite. Denoting
2Q/U by H, we write (4)

d 6uU
= = Ca ERE Aad eat ee (3

When y=0, we get from (3) and (5)

du ,4h—3H
dy = re ones Satie si). tele ys (6)

which represents the tangential traction exercised by the
liquid upon the moving plane.

It may be remarked that in the case of a simple shearing
motion Q=4AU, making H=A, and accordingly

dp/da=9, du/dy=U/h.

Our equations allow for a different value of Q and a pressure
variable with «.

So far we have regarded h as absolutely constant. But it
is evident that Reynolds’ equation (5) remains approximately
applicable to the lubrication problem in two dimensions even
when h is variable, though always very small, provided that
the changes are not too sudden, # being measured circum-
ferentially and y normally to the opposed surfaces. If the
whole changes of direction are large, as in the journal-bearing
witha large are of contact, complication arises in the
reckoning of the resultant ferces operative upon the solid
parts concerned ; but this does not interfere with the appli-
eability of (5) when h is suitably expressed as a function
of z. In the present paper we confine ourselves to the case

By

be

a

4. Lord Rayleigh on the

where one surface (at y=0) may be treated as absolutely
plane. The second surface is supposed to be limited at e=a
and at e=b, where h is equal to h, and hy respectively, and
the pressure at both these places is taken to be zero.

For the total pressure, or load, (P) we have

b > dp
Pa) pdv=—( wads,

on integration by parts with regard to the evanescence of
p at both limits. Hence by (5) |

PL ot eae °xvda
ay ee a

Again, by direct integration of (5),

odax odx
o= (5-H, _ . .

by which His determined. It is the thickness of the layer
at the place, or places, where pis a maximum ora minimum.
A change in the sign of U reverses also that of P.

Again, if be the value of « which gives the point of
application of the resultant force,

b b
a. p={ pede=3{ a dx,

z.P_ (?xdx o eda
ee = Wes nn
By (7), (8), (9) # is determined.
As regards the total friction (IF), we have by (6)
EF kee odx
“0 =4(°—3H( aa es eee (10)
Comparing (7) and (10), we see that the ratio of the tota
friction to the total load is independent of » and of U. And,
since the right-hand members of (7) and (10) are dimen-
sionless, the ratio is also independent of the linear scale.

But if the scale of h only be altered, F/P varies as h.

so that

We may now consider particular cases, of which the
simplest and the most important is when the second surface

Theory of Lubrication, 5

also is flat, but inclined at a very small angle to the first
surface. We take
Sa] ESE ON a, SOE a aman Oa

and we write \for convenience

a ded bib a= hey cy SEED)
so that
(ees. tre Sch a ae es)

We find in terms of ¢,.k, and h,
_ 2khy

a acai sar ie tea e,
(eg NOE ty
gat Gant eet ee f+ 8).
x k?—1—2klogk
Pea ilon kOe lon eva (16)
Seen mee Se) (17)
P~ ¢ 3(&+1) logk—6(k—]) °

U being positive, the sign of P is that of
2(k—1)
ona em

If k>1, that is when h,>/,, this quantity is positive.
For its derivative is positive, as is also the initial value when
k exceeds unity but slightly. In order that a load may be
sustained, the layer must be thicker where the liquid enters.

In the above formulz we have taken as data the length
of the bearing ¢ and the minimum distance h, between the
surfaces. So far &, giving the maximum distance, is open.
It may be determined by various considerations. Reynolds
examines for what value P, as expressed in (15), is a maxi-
mum, and he gives (in a different notation) k=2°2. For
values of k equal to 2°0, 2°J, 2:2, 2°3I find for the coefficient
of c*/h;? on the right of (14) respectively

"02648, -02665, -02670, -02663.

In agreement with Reynolds the maximum occurs when
k=2-2 nearly, and the maximum value is
pUc?

A

It should be observed—and it is true whatever value be

taken for kK—that P varies as the square of e/hy.

log k —

P=0-1602 arent ae.)

6 Lord Rayleigh on the

With the above value of k, viz. 2°2,
HS UOT hig.) ee

fixing the place of maximum pressure.
Again, from (16) with the same value of f,

#@—a=0-4231¢, . 9. . 2 ae
which gives the distance of the centre of pressure from the
trailing edge.

And, again with the same value of k, by (17)

BP=470h fe. ..

Since hy may be very small, it would seem that F may be
reduced to insignificance.

In (18) .... (21) the choice of & has been such as to make
P a maximum. An alternative would be to make F/P a
minimum. But it does not appear that this would make
much practical difference. In Michell’s bearings it is the
position of the centre of pressure which determines the vaiue
of k by (16). If we use (20), & will be-2°2, or thereabouts,
as above.

When in (16) £is very large, the right-hand member tends
to zero, as also does a/c, so that x—a tends to vanish, ¢ being
given. As might be expected, the centre of pressure is then
close to the trailing edge. On the other hand, when & exceeds
unity but little, the right-hand member of (16) assumes an
indeterminate form. When we evaluate it, we find .

X—-A= Ce.
For all values of k(>1) the centre of pressure lies nearer
the narrower end of the layer of fluid.

The above calculations suppose that the second surface is
plane. The question suggests itself whether any advantage
would arise from another choice of form. The integrations
are scarcely more complicated if we take

Line Uo)
We denote, as before, the ratio of the extreme thicknesses
(Ao/h,) by k, and ¢ still denotes b—a. For the total pressure
we get from (15)

Poa ft) tae ee
6uU is mee jes 1) (8n—2) postin
‘ i fi fp-2+2/n— 1
ea | Sia

from which we may fall back on (15) by making n=1.

Theory of Lubrication. 7

For example, if n=2, so that the curve of the second sur-
face is part of a common parabola, P is a maximum at

2

P=0163 ee, Dee ERGs eM)
1

when k=2°3. The departure from (18) with k=2:2 is

but small. In order to estimate the curvature involved we
may compare 4$(fh,;+A,) with the middle ordinate of the
curve, viz.

dm(a + b)?=44 Why + V(2°3h)) PP =1'58 Ay,
which is but little less than
3(hy + he) = ghy(1+2°3)=1 65h.

It appears that curvature following the parabolic law is of
small advantage.

I have also examined the case of n=~x. It is perhaps
simpler and comes to the same to assume

BPE IE RIN LS Oe (25)

The integrals required in (7), (8) are easily evaluated.
Thus
dz e 784—e~ Bb —
{G- 2B ~ 2BKh,?’
“dx e7?8a—e-38 8]
a) aes 38 BBR h,?’
the _ 3kh,(k?—1)
making ‘El = SCPE y ° : 5 3 - ‘ - (26)

In like manner

\: de k(1+2@Ga)—1—28d

| ae AB*k*h? ;
adz_ k(1+38a)—1—3Bb
Bayi 9h?

Using these in (7), we get on reduction

pa seu awe , B@-#)O—a)
~ PRREU 6 pay it

or, since Bc=log k,

DL ops Oo Cee ae eT) lop RD)?
= Fab | A aa ane i. “tia (27)

8 Lord Rayleigh on the

If we introduce the value of 8, the equation of the curve

may be written
R= ke).

When we determine & so as to make P a maximum, we get
b= 23 and

2
P0654", .,
1

again with an advantage which is but small.

In all the cases so far considered the thickness h increases
all the way along the length, and the resultant pressure is
proportional to the square of this length (c). In view of
some suggestions which have been made, it is of interest to
inquire what is the effect of (say) r repetitions of the same
curve, as, for instance, a succession ef inclined lines

ABCDEF (fig. 1). It appears from (8) that H has the

Hige 1.
ae a) ear
A Cc E
x
@) U

same value for the aggregate as for each member singly,
and from (5) that the increment of p in passing along the
series is 7 times the increment due to one member. Since
the former increment is zero, it follows. that the pressure is
zero at the beginning and end of each member. The
circumstances are thus precisely the same for each member,
and the total pressure is r times that due to the first, sup-
posed to be isolated. But if we imagine the curve spread
once over the entire length by merely increasing the scale
of w, we see that the resultant pressure would be increased
r* times, instead of merely 7 times. Accordingly a repeti-
tion of a curve is very unfavourable. But at this point it is
well to recall that we are limiting ourselves to the case of
two dimensions. An extension in the third dimension, which
would suffice for a particular length, might be inadequate
when this length is multiplied ~ times.

The forms of curve hitherto examined have heen chosen
with regard to practical or mathematical convenience, and
it remains open to find the ferm which according to (5)
makes Pa maximum, subject to the conditions of a given
length and a given minimum thickness (f,) of the layer of

Theory of Lubrication. 9
liquid. If we suppose that 2 becomes h+6h, where 6 is the
symbol of the calculus of variations, (8) gives,

oh dh dx ;
2 ede 8H | ede + 3H Fi =O: ctu)

and from (7)
OF i) Oh(—2h+3H eda _ sy (22
6nU a Hs Dh el ca

the integrations being always over the length. Hlimi-
nating oH, we get

are Sh-Seda> E 3G)
waa as bh eB foe a

The evanescence of SP for all possible variations 6h would
demand that over the whole range either

Jh-sada
L4= =
\h-3dax ?

(31)

Ort an) ORs 90.3) 638)

But this is not the requirement postulated. It suffices
that the coefficient of dh on the right of (32) vanish over
that part of the range where h>h,, and that it be negative
when h=/y, so that a positive dh in this region involves a
decrease in P,a negative 6h here being excluded a priori.
These conditions may be satisfied if we make h=h, from
2=0 at the edge where the layer is thin to =c,, where ¢, is
finite, and h=3H over the remainder of the range from ¢ to
(j+¢:, where c¢,+c,=c, the whole length concerned (fig. 2).
For the moment we regard c, and c, as prescribed.

Fig. 2,

For the first condition we have by (8)

Ey Ae ait €,/hy? + ¢2/he?
3h.=H= €,/fy? + Co] hg?”

so that
Glo (2E— SY ww FEU C84)

determining /, where as before k=h,/hy. The fulfilment of (34)

10 Lord Rayleigh on the

secures that h=3H over that part of the range where h=hy.
When h=h,, h—§H is negative ; and the second condition
requires that over the range from 0 to ¢

Sh-sade
Sh “8d

be positive, or since c, is the greatest value of 2 involved,
that

1

fh-8ada—e\h-da= +. Ji(cete (35)

The integrals can be written down at once, and the con-
dition becomes

B< C97 /¢;", ° i wee (36)
whence on substitution of the value of c,/c, from (34),
kK2b—3)?>1. . .

If & be such as to satisfy (37) and ¢/c, be then chosen in
accordance with (34) and regarded as fixed, every admissible
variation of h diminishes P. But the ratio c,/c, is still at
disposal within certain limits, while c, +c, (=c)lis prescribed.

In terms of £ and c by (34)

d ¢ _ o(2k®—27)
AS 1+2e23e, 3 14-2 ae ae
and by (7)

(38)

ey tsi : en Oe 2k—-3 se?
pee). Ry? {4 (3-28) + a 7 ie ee oes
(39)
The maximum of /(k) is 0°20626, and it occurs when
k=1'87. The following shows also the neighbouring values:

he, H(k). k(2k—8)?.
1°86 0°20624 0:964
1:87 0°20626 1°024
1°88 0:20617 1:086

It will seen that while £=1°86 is inadmissible as not
satisfying (37), k==1°'87 is admissible and makes

2
P—-20gpeeee 0

9 b)
hy

no great increase on (18). It may be repeated that & is

Theory of Lubrication. 11

the ratio of the two thicknesses of the layer (h2/h,), and that
by (34) |
CeO ees Taal Nia ow CeL

This defines the form of the upper surface which gives the
maximum total pressure when the minimum thickness and
the total length are given, and itis the solution of the problem
as proposed. But it must not be overlooked that it violates
the supposition upon which the original equation (5) was
founded. The solution of an accurate equation would pro-
bably involve some rounding off of the sharp corners, not
greatly affecting the numerical results.
The distance w of the centre of pressure from the narrow
end is given by
BV AZO Ce cM ip fete in NN CED)

differing very little from the value found in (20). From
(10) with use of (38) we get

F 4c(k—1)? 4¢

FFG GWE] eT) anny 02) at Ee i ids ae es)
and
F 4h (k—1)
P ¢(2k—3) ° >
It k=1°87, ,
P/PSs09ra te, Oy sok) ala)

a little less than was found in (21). The maximum total
pressure and the corresponding ratio F/P are both rather
more advantageous in the arrangement now under discussion
than for the simply inclined line. But the choice would
doubtless depend upon other considerations.

The particular case treated above is that which makes P a
maximum. We might inquire as to the form of the curve
for which F/P isa minimum, for a given length and closest
approach to the axis of « Inthe expression corresponding
with (32), instead of a product of two linear factors, the
coefficient of 6/ will involve a quadratic factor of the form

Bah+Ch?+De+Eh+F,. ... . (46)

so that the curve is again hyperbolic in the general sense.
But its precise determination would be troublesome and
probably only to be effected by trial anderror. Itis unlikely
that any great reduction in the value of F/P would ensue.

12 On the Theory of Lubrication.

Fig. 3 is a sketch of a suggested arrangement for a foot-
step. The white parts are portions of an original plane

surface. The 4 black radii represent grooves for the easy
passage of lubricant. The shaded parts are slight depressions
of uniform depth, such as might be obtained by etching with
acid. It is understood that the opposed surface is plane
throughout.

P.S. Dec. 13.—In a small model the opposed pieces were
two pennies ground with carborundum to a fit. One of
them—the stationary one—was afterwards grooved by the
file and etched with dilute nitric acid according to fig. 3,
sealing-wax, applied to the hot metal, being used as a
_ “resist.” They were mounted ina small cell of tin plate,
the upper one carrying an inertia bar. With oil as a lubricant
the contrast between the two directions of rotation was very
marked.

Opportunity has not yet been found for trying polished
glass plates, such as are used in optical observations on
‘“interference.” In this case the etching would be by hydro-
fluoric acid *, and air should suffice as a lubricant.

* Compare‘ Nature,’ vol, Ixiv. p- 885 (1901) ; Scientific Papers, vol. iv.
p. 546, '

[ 1B

II. On the Calculation of the Maximum Force between Two
Coaxial Circular Currents. By H. Nacaoxa, Professor
of Physics, Imperial University, Tokyo *.

< thors problem of calculating the maximum force between

two coaxial circular currents originated in the absolute
measurement of electric current by means of a balance. The
formula for calculating the force was developed by Lord
Rayleigh + in his investigation on the electro-chemical equiva-
lent of silver. Recently a similar method was used by
Rosa, Dorsey, and Miller{, in the determination of the
international ampere. The interesting question as to the
position of the coils and the maximum force acting
between them was taken up by F. W. Grover §, who
expressed the said quantities by means of Jacobi’s q-series.
In a note on the potential and the lines of force of a circular
eurrent ||, | have shown how the expansion in g-series of
S-functions converges very rapidly in calculations of like
nature. The expression for the mutual inductance between
two coaxial coils and the force between the currents passing
through them can be conveniently expressed in terms of q.
Grover extended the expression for the force to terms in-
volving g'* and g,° in the power series, thus increasing the
accuracy of the expression to decimal places scarcely needed
in practical measurements. From the integral expression
for the force, we can by differentiation arrive at an expres-
sion giving the condition of maximum force. This method
was followed by Grover, who obtained an expression for
calculating the maximum force that can be applied for given
coils in finding the distance between them. The expression
in its final form is sufficiently convergent to be of practical
value, but the approximation leading to the value of g which
corresponds to the required maximum seems to offer another
solution. Obviously the reduction of the integral involves
the use of elliptic functions, which can be expressed in terms
of S-functions ; there is, in addition, a factor containing the
distance between the coils. This factor gives rise to a very
convenient formula for finding the required distance when
once the value of gis known. Thus the first step is essen-
tially the evaluation of g. According to the method followed

* Communicated by the Author.

+ Rayleigh, B. A. R. p, 445 (1882); Phil. Trans. clxxv. pp. 411-460
(1884) ; Scientific Papers, ii. p. 278.

t Rosa, Dorsey, & Miller, Bull. Bureau Stand. viii. pp. 269-393 (1911).

§ Grover, Bull. Bureau Stand. xii. pp. 317-374 (1916).

|| Nagaoka, Journ. Coll. Sci., Tokyo, xvi. Art. 15 (1903); Phil. Mag.
vi. p. 19 (1903) ; Proc. Math. Phys. Soc. vi. p. 156 (1911).

¥

14 Prof. H. Nagaoka on the Calculation of Maximum

by Grover, this difficulty is overcome by finding the approxi-
mate value of the distance by means of Rayleigh’s formula,
and by using a relation which is rather empirical, and the
ultimate result is arrived at after a number of successive
approximations.

The process which lam now going to develop is similar
to that already used in my former papers and is characterized
by giving the value of g corresponding to the maximum force
by a simple relation ; it does not necessitate the knowledge
of the approximate value of the required distance, but by
two or three processes of approximation in finding the value
of q, it leads to results which can be used in measurements of
great accuracy even in the most unfavourable case.

Denote the radii of the coils by a and A, the distance
between them by z, then the mutual inductance is given by

ane = cos 0dé ;
AGiay af VA? +a?+22—2Aacos@’ ~ ~ Be
whence the force between the unit currents passing through
them is given by
oM : cos 0d0
=— =4r7 Aaz +
mae » (A? +a?4+22—2Aa cos 6)? (2)

For evaluating (1) and (2), we have to put, as usual,

4(s—e,)(s—e,)(s—e3) = 48° —gos —93=8,

where
9 ers
Qy= a eh ei) eee
cL 7 y
oe A? +a? + 2? re \3
o GAg 0? Soe”
and = = or) S= Ge (uw).

We easily find that (2) is given by

ea 24 hepa CS)
Oz oe Jas 1924) ~28
dey
2(e, — @2)(41— 3)

Tz 2 1 1 1 I
fe ara | 3,7(0) + on (sa) alg 5H) (0)32(0) ;
ae a ae

= —mAay'z} )— (m+ exor) }

Force between Two Coaxial Circular Currents. 15

Expressing the quantity under the parenthesis in terms
of g, we obtain

OM _ 19272 ,
ag ae
+ 680529! + 3374659"? + 15137409"!

= EL Ser aS Ge ae a ee cera gi

=

{1+ 209? + 2259*+ 184096 + 1212098

The expression in terms of g, is

uy. Te
mao Vv Aag;
+ 216489,’ —73600qi1° + 2269449,’ — 6481899,2+ ....)

} (1+ 129,—1929,? + 12329,3— 56349,4

—12q, logn 2 . (1—109, + 6091? — 3009,? + 13009,4
1
— 48849, + 16320;5— 499209," + 1425009,8+....) I.
(I’.)

These formulz include many terms extended by Grover.
The condition required for the maximum force is simply
given by
oly hates 07M

on 2

This evidently leads to the following relation between two
integrals

a:

act" | cos 6 dé
‘ : (A? + 9?+22—2Aacos 6)?

S cos 6 dé
- ch Sais a 2! 2
at (A24 a? 4+2°—2Aa cos 6)? Sh

This equation was utilized by Grover for finding the
distance between the coils when the said condition is
satisfied. The evaluation of the integrals is not an easy
process, and for finding g from (4'), we have to assume an
approximate value of < given by Rayleigh’s formula, and
arrive at the final result by successive steps. The solution
of the problem can, however, be obtained in another form
by finding the value of g directly from the known value of
a/A.

16 Prof. H. Nagaoka on the Calculation of Maximum
Reverting to the expression (I.) for F', we see that (4) is
equivalent tc )
ale:
oy
provided z can be expressed by means of 3-functions. For
this purpose we take advantage of the relation’

0,

éy—e, — (A—a)?+2?

€y— 3 4Aa i
éy— 63 ies (A +a)?+2?
€g— e3 ny 4Aa

Expressed by means of 3-functions

2, S04(0) _ (Aa)?

Aa ~3,4(0) Aa
and 2 33°(0)

Adding them, we obtain, by utilizing the relation
33°(0) —S0*(0) =52'(0),

J -1Q2NB)-(E49). - @

which enables us to expand < in terms of q.
For expansion, it is convenient to use the product series
of 3’s: thus

$,(0)=HA-g@)it+ge ty,
$(0)=M(1—g™)A—g""",

by which (1—g*) being common to the numerator and
denominator is eliminated. On evaluation we obtain

2?
Aa
_1+ 20g! 624" + 216g°— 641g! + 1686937789" + 82489"
Ag >
f ° +) eee
a.
where ety

Force between Two Coaxial Circular Currents. 17

For expressing z in terms of g,, we remark that

1
Pe) cia Cais, (Gee) A oy
Aa 3,4 (0, 25 Aa

—— === ummm

whence we obtain

2 ee (0, ei 7) + 3H(0; — ) -(< a (5')
ee. 6)
ACSC aaa
A similar remark as for (6) applies to the calculation of (7)
in terms of g;, resulting in the expression
32
7 =2+ 649;(14 89, + 4497 + 1929)? + 71894 + 24009,°
SM SOG AUT OAG ys nate I. Cowie cs A)
It is generally sufficient to retain g,°, the remaining terms
being negligible even in the most accurate work that can at
present be attempted.
a?
Squaring (I.) and substituting for Agia (6) and (6’), we
find the required condition to be equivalent to
on
0g

All the rest is a simple mechanical operation; the final
equations for obtaining g or qg; from the known ratio of the

=).

dimensions of the coils r ee (=a++ according. to

Grover’s notation) may be ae under the following
form

ahs - +6°8 g—-51°6 q+ 6144 g?—7934°6 g?
+ 103683-6 g°— 1353676"4 91-4 176493006 9!

—230°1 x 10% g%+.. MEN sooihatyy (AE)
Phil. Mag. 8. 6. Vol. 35. No. 205. Ten. 1918. C

18 Prof. H. Nagaoka on the Calculation of Maximum

The coefficients are exact up to that of g!*; the rest of
higher terms are insignificant and may be neglected even
in the most unfavourable cases. For practical calculation,
the above equation is applicable to values of » ranging from
r=a0 to r=2°'5, the latter corresponding to the case of
a/A=0°5. From this value down to r= 2, which is the smallest
occurring in practice for the case of coils of equal radii, we
have to use an expression in terms of q;, which can be ex-

2
pressed by substituting (5) in (I.) and making =(0: The

formula is as follows :— fe)
Lr {1-— 16q; + 3769,°>—46729;3 + 38948q,*— 2521929;°
+ 1365888 9,°—64633609,7 + 27500946q5—....

1209, logn = (1169, + 1549,2— 1120938
1

+ 66809,t—342729)5 + 156268g;°—. ...}
=2{1—2649,? + 409693 —36828y,* + 2457609)
— 860712q,° + 64143369,’ —273772629)° +..-.

4 249 loon = (3—64qi-+ 102g, - 53769,8
1
+ 327129,4—1693749,° + 775908q,°— ....)}. CII’.)

It is worthy of remark that (II’.) is to be used from
a/A=0°5 to 1, the value of g,; ranging from g,=0:0113 to
g:=0. The series in the numerator and denominator con-
verge very rapidly, and we can sometimes utilize the formula
for a somewhat larger value of g,; the only tedious process

of calculation is finding logn =
1

When once the value of either qg or g; is found, we can
calculate z by (6) or (6’) ; and then IF by (1.) or (I’.).

As an example of practical calculation, let us take the case
aj/A=0°5; i.e. r=2°5. From the first three terms in (I1.),
we find by inspection that g is nearly 0°11; putting this
value in (II.) and calculating to 97, we find that the right-
hand side is about 2°5059, giving Ar=—0-°0059, and hence
Ag=0:00054 ; next putting g=0°11054, we find

Ar= —0:00003476

by taking all the terms into account; thus the final value
of qg corresponding to the required maximum is

q=0°110543224, log g=1:04353213.

Force between Two Coaxial Circular Currents. 19

The number given by Grover is log g=1°4035322; the
value for 2/A is 0°38353439, while Grover gives for it
0°3835341, agreeing to within a ten-millionth part.

For the convenience of practical calculation, the following
tables of Ar and r are given for different values of g and qy
respectively, calculated for me by Mr. Shobei Shimizu;
Ar in the first table are residuals

614°4 g? —7934°6 97+ ....—230'1 x 10%?
in (II.), while rin the second table refers to those calculated
according to (II’.).

| |
| q: Ar. ne: | q- | ir. | A -
Lite SRA ETA | A Ieowes: (ere (ie
;
| 0-00 | o-co00000 _ | 0000 — 20000000
| aa 339002
0:01 | 0-0000001 | | 0-001 | 2-0389002
19 | | 366859
0:02 | 0-0000020 0:002 | 2:0705861
| 128 | | 389351
| 0:03 | 0-0000148 0-003 21095212
| 469 | 409255
— 0:04 | 0:0000616 0004 | 2:1504467
| 1244 | | 427502
| 9:05 | 0-0001860 | | 0-005 | 21931969
| 2705 | | 444584
| 0-06 | 0:0004565 | | 0-006 | 2-2376553
| 5147 | 460811
| 0:07 | 00009712 0:007. | 2-2837364
| | 8885 | | 476382
| 0-08 | 0:0018597 | | | 0008 | 23313746
| 14251 | | 491447
0:09 | 0:0032848 | | 0-009 | 2:3805193
| 21574 | | 506120
| C10 | 0-0054422 | | | 0-010, | 2-4311313 |
| | 31175. | | |

0-11 | 0:0085597

From the table, it will be noticed that the residuals ap-
parently cover a wide range in the calculation with g; this
is at once evident from the fact that the range covered by
the g-series is very great compared with that in which the
gi-series are applicable. This is in no way an impediment
in practical calculations, as the formula in terms of g is
simpler than that in g,; especially in numerical evaluation
of g from known values of 7, the formula (II.) is characterized
by the great facility with which all the terms can be calcu-
lated and the required approximations brought to test. As
all the rest of the calculation depends on the value of g,

C2

20 Maximum Force between Two Coaxial Circular Currents.

when once this is accurately known, the distance between
the coils and the force exerted at that distance can be found
by the formulze (1) and (6).

As regards the utility of the solution of the present pro-
blem, it would be unnecessary to spend words on its bearing
in the construction of the current-balance for the absolute
measurement of electric current. The numerical data caleu-
lated by Grover are of great value in researches of this kind.
The solution which I have here given may be of service in
the direct calculation when the dimensions of the coils are
given. How far the accuracy of the instrument can be
relied upon is of great interest to me, as I believe that the
instrument can be used for a purpose totally different from
the usual measurement of the electric current, and which
seems not yet to have been well noticed.

The most exact method of measuring relative values of
gravity is that of comparing the periods of invariable pen-
dulums at the place of observation with those at the standard.
station. The great inconvenience and difficulty accom-
panying the method of observation lie in the extremely
accurate measurement of time; the rate of the clock must
be known to 1/60th part of a second per day, if the period |
is to be exact to one part in five million. For this we have to
take a transit instrument of fairly large aperture, and when
obstructed by bad weather we have to wait for days. In
addition to this, the occasional change of the clock-rate
necessitates the unintermittent continuation of observation
which imposes a great burden on the observer. This
tedious and unwelcome obstruction to the usual method of
gravity determination may, to a great extent, be overcome by
using a current-balance instead of invariable pendulums.
The strength of the current is to be evaluated by means of
known resistance of the circuit and the terminal potential
difference, for which the electromotive force of the cadmium
cell must be relied upon. Itis a question if we can bring the
constancy of the cadmium cells and of the coils to the same
order as that of the pendulum and clock. The weight
counterbalancing the attraction of the coils is an immediate
measure of the force of gravity at the place of observation.
For this purpose it is perhaps necessary to design an instru-
ment anew in a transportable form, and construct the coils
such that the attraction is of sufficient amount to give the
desired accuracy. It must however be well noticed that
the method of current-balance is not free from objections, as
the current is liable to fluctuations and the coils are heated
in course of measurement and give rise to convection current;
moreover, the correction to be applied to such disturbances

eon,

On the Nodal-Slide Method of Focometry. 21

is difficult to calculate and almost impossible to estimate
exactly. ‘There may be other methods of dispensing with
astronomical observations in gravity measurements, but I
believe that the method of current-balance is one of the
most accurate that can be easily brought into practice with-
out sacrificing to any great extent the degree of precision
usually attainable with invariable pendulums.

In the theory of atomic constitution, it is generally as-
sumed that there are rings of electrons in rapid rotation :
these are no doubt equivalent to currents and must exert
mutual influence upon each other. When such atoms com-
bine into a molecule and are in such a position that the-
planes of the rings are parallel to each other, then the posi-
tion of maximum force between two circular rings here
discussed will be of some significance in the atomic
configuration of molecules.

III. On the Nodal-Slide Method of Focometry. By J. A.
Tomxnins, A.R.C.S. (Lond.), Lecturer in Physics, Technical
College, Bradford *.

ia A. ANDERSON has recently described (Phil.

Mag. Jan. 1917, p. 157) an elegant method of deter-
mining the focal length, and other constants, of a lens system
based upon a general theorem of which the ordinary nodal-
slide method is but a particular application. This theorem
is that for any lens system there is always one, and only one,
point on the optic axis such that a small rotation of the system
about a perpendicular axis through it will cause no lateral
displacement of the image of an object in a given position.
This point was shown by Prof. Anderson to divide the
distance between the object and image, andalso that between
the nodal points, externally in a ratio equal to the value of
the magnification. .

A numerical example of the determination of the focal
length of a diverging combination by this method was given,
butit was pointed out by Mr. R. EH. Baynes (Phil. Mag.
April 1917, p. 357) that these data yielded very different
results for the focal length and the positions of the nodal
points when calculated in different ways. These discrepancies
were explained by Prof. Anderson (Phil. Mag. July 1917,
p- 76), who discussed the effect of various errors and showed
that, while the method gave quite satisfactory values for
the focal length, it failed to do so for the distance between

* Communicated by the Author.

22 Mr. J, A. Tomkins on the

the nodal points, because small errors are multiplied by the
numerical value of d, the distance through which the com-
bination is moved, which may be large. He also described
another method which gave satisfactory values for this
distance. In a third paper (Phil. Mag. Sept. 1917, p. 174)
he gave some further properties of this point, which he terms
the nul point. There are, however, two possible sources of
error mentioned by Prof. Anderson, viz. (1) want of pre-
cision in determining whether there is any displacement of
the image, and (2) error in determining its position, which
seem to call for further consideration.

With reference to the first it is to be noted that in the
ordinary nodal-slide method there is one, and only one,
possible axis of rotation of the lens system, viz. that passing
through the second nodal point, whereas in the general
method described by Prof. Anderson there is an infinite, or
doubly infinite, number of possible axes. The object of this
communication is to investigate the best position, if any, for
the nul point, and to compare the results with those obtained
by the ordinary nodal-slide method.

For the purpose of observing the displacement of the image
the best position will be that for which a given small dis-
placement of the axis from the nul position will, for a given
small rotation of the lens system, produce the greatest
displacement of the image.

To determine this it is necessary first to find an expression
for the displacement of the image due to a small rotation
about any axis.

Pie. 1.

Fig. 1 shows the displacement produced by a convergent
combination in the general case in which the first and last
media are different, and in which, therefore, the principal
and nodal points are not coincident. P,Q, and P,Q, are the
object and image respectively, H, and H, the principal

Nodal-Shide Method of Focometry. 23

points, N, and N, the nodal points, Ff and F, the principal
foci. Suppose the system to be rotated about O through a
small angle @ so that the principal axis moves into the
position indicated by the dotted line. Then, to a first approxi-
mation, the nodal points N,; and N, will move into the positions
N,’ and N,’ and the image P,Q, will move in the same plane
into the position P,'Q,' obtained by drawing N,'P,' and Ny'Q,,
parallel to PN,’ and Q,N,’ respectively.

Let N,O = l, N.N, = a, NP, =U; Noes =v.
Bien!) NN 36 and NON = (a2 ye:

Hence the displacement of the image is given by
PN,

s=Q.Q.’=N.N,'+ NP, -N,NY
=(a+l)0—~.16
u
See WC Lm me) EO e. if, (LE)

where m= = , the magnification.
In order that the displacement of the image may be zero
for a given value of 8 we must have

s=at+l(1—m)=0,

or Fe TRS EN ANE

There is thus one, and only one, position of the axis of rotation
for which there wil! be no displacement of the image, viz.
that which divides the distance between the nodal points
externally in a ratio equal to the value of the magnification—
a result obtained in another way by Prof. Anderson. The
best position for the axis of rotation will, as already pointed

out, be that for which . is a maximum subject to the con-
dition given by equation (2).
Differentiating (1) and substituting from (2), we get

d g
qa Gd—mea- Fee 8)

The rate of change of the displacement thus varies directly
as a, the distance between the nodal points and inversely as /,
the distance of the nul point from the first nodal point.

It is greatest when /=0, i.e. when m=, and the nul
point coincides with N,. It thus appears that the best

24 Mr. J. A. Tomkins on the

position is attained when the axis of rotation passes through
the first nodal point, in which case the object, real or virtual,
will be situated at the first principal focus.

The light will then emerge as a parallel pencil, and the
image can be viewed through a telescope focussed for parallel
rays, as in one of the well-known methods of determining
the focal length of a thin lens. A further advantage of this
position is that the nodal points are determined directly as
in the ordinary nodal-slide method. Cares

We will now apply these formule to the example given in
Prof. Anderson’s second paper (Phil. Mag. July 1917, p. 76),

where

"4
OP, =2,=142 em.; OP, =yj4=9'4 cm.; and m= = 0°0662.
OP,'=2,=29'1em.; OP,’/=y,=8'3em.; and m= at =0'285.

#=113°3 cm.; HoH,=a@=2°43 cm.

The distance H,O in Prof. Anderson’s figure (Phil. Mag.
Jan. 1917, p. 158), in which the principal and nodal points
are coincident, is given by

OP, a OS yc. a

FD OF, OP, ne? ae ° ° (4)
which is but a particular case of the general expression
obtained by Prof. Anderson.

a
l—m

a

and in the second position H,O=— ae —3°40 cm.
sma?)

Hence in the first position H,O=— = — 2°60 cm.
1

These positions in relation to the nodal points are shown to

scale in fig. 2.
Hip, 2.

JT LES a een
(0) 0: lpg yy

2 \

Suppose now that 0==5°=0-0873 radian and that the axis
of rotation is moved 1 mm. from the nul position towards the
object.

Then, in the first case, l,==— 2-604+0-1=— 2°5 ecm.;
m,=0°0662; and the displacement of the image calculated
by equation (1) is 0:0083 cm. In the second case we have
l,= —3'40+0:1=—3:3 cm.; m,=0°285 and the dis-
placement is 0:0062 cm. If, however, the ordinary nodal-
slide method had been employed, in which case O would

Nodal-Slide Method of Focometry. 25

coincide with H., then J=—2°33, m=0, and the dis-
placement under the same conditions would have been
0°0087 cm.

These small displacements may also be calculated more
simply by using the approximate formula

ds aé
sl él. be a Cea tata C75

In the first case 1,:= —2°6 and 6s=0°0082, in the second
case /,=—3°4 and 6s=0:0062; while in the ordinary
nodal-slide method J=— 2°43 and 6s=0-0087. These
results agree closely with those obtained by the more
exact formula. Or, to put the matter rather differently,
if we suppose the smallest observable displacement to
be 0°01 cm., then the distances through which the axis
would have to be moved from the nul position to produce
this displacement would be 1°20, 1°61, and 1:15 mm. in the
three cases respectively. These examples show that in this
particular case the nul point can be determined with greater
accuracy by the ordinary nodal-slide method than by the
general method.

If, however, a convergent combination forming a real
image, as in fig. 1, had been employed, the nul point
would lie between N, and N,. and its position could be
determined with greater accuracy by the general method
than by the ordinary nodal-slide method. In this case thie
nul point is indicated by Oo, which is the point of inter-
section of Q,Q, with the optic axis.

Again, if, with the same convergent combination, a virtual
image were formed, a case not likely to arise in the deter-
mination of focal length, the magnification would be positive
and greater than unity. The nul point would therefore lie
on the side of N, remote from No», and its position could be
determined by the general method with greater or less
accuracy than by the ordinary nodal-slide end) according
as it is at a distance from N, less or greater than a
respectively.

If the nodal points are determined by either of the two
direct nodal-slide methods, which will here be distinguished
as the first and second nul methods respectively, the principal
foci and focal lengths may be most readily and directly
found by measuring the distances from the nul point of the
object in the first method and of the image in the second.

Now in obtaining the position of the image there will be
a certain range along the axis within which the object may

26 On the Nodal-Slide Method of Focometry.

lie and yet produce a distinct image ona fixed screen. And,
conversely, there will be a certain range within which the
screen may be moved and yet give a distinct image of an
object in a given position.

The former range is called the depth of focus of the in-
strument, and may be shown (Heath’s ‘ Optics,’ pp. 269, 270)
to be given by the expression

Nee

— m tan a’

where & is the distance of the object from the entrance pupil,
e the maximum value of the circle of indistinctness, m the
magnifying power, and « the angular aperture of the
instrument.

In the second case the range through which the screen may
be moved may be similarly shown to be given by the equation

ME

hia
APS,

where &' is the distance of the image from the exit pupil.

In order to determine with precision the positions of the
object and image the property required will be the inverse of
these. In the first case, it will vary directly as the mag-
nifying power and the angular aperture, and, in the second
case, it will vary inversely as the magnifying power and
directly as the angular aperture.

By comparing these magnitudes in the two cases just.
referred to it will be seen that the principal foci can be
determined with greater accuracy by the first than by the
second nul method.

Finally, the similarity of object and image will depend
on the resolving power of the system, which increases with
the angular aperture, and this again will be greater in the
first nul method than in the second.

The conclusions arrived at in the foregoing were tested by
the following experiments :—

Two thin plano-convex lenses, each having a focal length
of 25°8 cm., were mounted with their convex surfaces inwards
and ata distance of 12°9 cm. apart. The calcalated focal
length of the combination was 17:2 cm., and the distances of
the first and second nodal points were 8°6 cm. measured
inwards from the first and second lenses respectively, so that
the distance between the nodal points was 4:3 cm.

The positions of the nodal points were then determined by
the first and second nul methods.

In the first method it was found that the lens system could
be moved through a total distance 1:7 mm. before any
sensible disp!acement of the image was caused by a small

Value of the Mechanical Equivalent of Heat. 27

rotation, whereas in the second method a motion through
5 mm. was required to produce the same result.

The principal foci of the combination were also found,a
needle being used as object in the first method and to locate
the image in the second method. The distances through
which the needle could be moved before any sensible parallax
was observed were 2 mm. and 4 mm. respectively.

This particular combination was chosen in order to get a
fairly large distance between the nodal points, but as the
lenses were uncorrected quite a small rotation caused tie
image to become confused owing to oblique aberration.
A further test was therefore made with a 15 in. (38:1 em.)
Ross photographic lens in which the distance between the
nodal points was only 16 cm. The following results were
obtained for the ranges of adjustment in the two cases:—

First Method. For nodal points 0°2 mm.; for principal
foci 3°5 mm.

Second Method. For nodal points 1°5 mm.; for principal
foci 11 mm.

These figures must be taken as indicating the relative
rather than the absolute accuracies of the two methods,
since the nodal-slide employed, though not of the roughest,
had no fine adjustments. They appear, however, to show
that the first nul method is in every respect superior to the
second.

In conclusion, I wish to thank my colleague, Mr. J. E.
Rycroft, for his kindness in making the diagrams.

IV. On the Value of the Mechanical Equivalent of Heat.
By T. Caruiton Sutton, B.Sc.*
ier following values of the Mechanical Equivalent of
Heat (for references see Kaye and Laby’s ‘Tables’
and Guiiths ‘Thermal Measurement of Energy’) have
been reduced to joules per mean calorie :—

TSAS samen ers a sine ene 4173
ERPS) Enea oe oS ls oS lactis 4:184
DOS REA 8 Gh scl fa aie v's, «os 4:188
1894 Schuster and Gannon ...... 4:185
1897 Reynolds and Moorby ...... 4184
1899 Callendar and Barnes ...... 4184
1900 Griffiths (deduced) ........ 4184
1906 Jager and Steinwehr ...... 4:188
1908 Crémieu and Rispail........ 4:189
1909 Barnes (deduced) .......... 4:185
WORD). SECU se chk ke wal 2 caters, 3s 4179

1914-5 Sutton-Henning (see below).. 4185
* Communicated by Principal E. H. Griffiths.

eS

ae

—————

SS ———— cme
Sa

— SS

———

aS ee eS
= SSS

—
ee

———————————————————

28 Mr. T. Carlton Sutton on the Value

The results obtained since the year 1905 are not as concor- —
dant as might have been expected. It may be of interest
therefore to compare the value of the heat of vaporization of
water at 100° C. obtained electrically at the Reichsanstalt
by Henning (Ann. d. Phys. 1906-9) with that obtained by
the author in terms of the mean calorie directly (Proc. Roy.
Soc. April 1917). :

Henning’s results are shown in the following graph, which
indicates that the value at 100° C. is 538°5+0°3. Probably
the constant errors are even less than 0°3.

Henning’s work may also be checked by comparing it
with that of Griffiths’ at 30° C. The values for the heat of
vaporization at that temperature are in good agreement,
579°9 and 579°3 mean calories respectively.

Fig. 1.

4-188 joules)

-
_

1
73)
102)
1S)

HENNING

Latent Heat(J

oS V.&S. cals.

}
Ww
00

@ 100°C 100:5°C
Temperature

The corresponding values obtained by the author (Proce.
Roy. Soc. April 1917) give 538°8+0°1 mean calories at
OOS 0:

If, then, Henning’s mean value is correct to 1 part in 2000,
the deduced value of the Mechanical Equivalent should be
correct to within two units in the third decimal figure.

Henning uses the Jager and Steinwehr valne of the
15° calorie to convert joules to calories, and takes the Reicls-
anstalt value of the e.m.f. of the Clark Cell. What he
actually determines as the Energy of Vaporization at 100° C.
is 538°5 x 4°188 Reichsanstalt joules.

of the Mechanical Equivalent of Heat. 29

Fig. 2.
e
2
539-0K= Sate
> 4
ine
oes
i &
i
1a
w
SI
25/2 pa) Se — piaemree E : ae =
99:5..C ioar°c 100-5 C
Temperature
The theoretical curve is obtained from Clapeyron’s relation
=(V—v)T
L=(V—»v)T ST

to show the change of latent heat with temperature. (Since the values of
V and a are not known with the same accuracy as L, the slope of the

curve ouly has been taken from this relation. This is sufficient for the
purpose of reducing the results.)

It will be noticed that the more accurate methods of measurement
give values which show clearly the agreement with the thermo-

dynamical result.

On comparing this with the author’s results, which are

measured directly in mean calories, it follows that
538°5 x 4:188=538°88 J,
Bis J =4:185+0-002 joules per mean calorie.

It will be noticed that this is in exact agreement with the
values obtained by the earlier experimenters—Schuster and
Gannon, Reynolds and Moorby, Callendar and Barnes,—and
with the values deduced on separate occasions (from all data
then available) by Griffiths and by Barnes; on the other
hand, this value and those obtained by the more recent experi-
menters are certainly too discordant to be reconciled.

This agreement between the values given by the classical
experiments and the value given by a method so different as
the present substantially increases the probability that the
value of the Mechanical Equivalent lies between 4:184 and
4-185 joules per mean calorie.

Lesa

VV. Light Distribution round the Focus of a Lens, at

various Apertures. By L. SILBERSTEIN, Ph.D., Lecturer
in Natural Philosophy at the University of Rome”.

Bibliographic and*Introductory.

YINHE distribution of the intensity of light in the
neighbourhood of a caustic has been studied by Sir G.
Airy as early as in 1838 (Camb, Phil. Trans. vol. vi.), for
the case, however, of an unlimited beam only. Some of
the effects of spherical aberration of limited beams upon the
central intensity and the definition of the image have been
investigated by Lord Rayleigh in 1879 (Phil. Mag. vol. viii.
pp. 403-411). The chief problem considered by him relates
to a beam of cylindrical waves of rectangular section, their
aberration being assumed proportional to the cube of the
lateral coordinate «. The solution is reduced to the evalua-
tion of an integral of the form J cos (av+be*)dx. Availing
himself of the numerical results of the mechanical quadratures
recorded in Airy’s paper, Lord Rayleigh calculates and draws
three intensity curves for the focal plane (loc. cit. p. 406),
corresponding to the case of no aberration (b=0), and to
those in which the marginal aberrations amount to + and
4 period. The practically important consequence drawn
from the aspect of these curves is that ‘‘ aberration begins
to be distinctly mischievous when it amounts to about a
quarter-period.”” In the next case studied, that of sym-
metrical aberration proportional to #*, Lord Rayleigh
calculates, by the aid of a series, the intensity at the central
point only. Passing, finally, to beams of circular section,
he limits himself again to the calculation of the central
intensity, viz. in the case of axially symmetric aberration
proportional to the fourth power of the distance, and finds
that, as in the preceding cases, aberration begins to be
prejudicial when it mounts up to a quarter of a period.
This result has since become widely known, having been
incorporated into the Enc. Brit.t and several English
text-books. In 1884 Lommel investigated the distribution
of illumination in the diffraction image of a point given bya
circular aperture t. The problem in this case reduces to the

evaluation of integrals of the form

J Jo(aa) cos (b2*)ada, J Jo(az) sin (ba?) adx
* Communicated by the Author,

+ Cf. Lord Rayleigh’s article on “ Diffraction of Light” in the

11th ed. of Enc. Brit. vol. viii. pp. 2388-255.
t E. Lommel, Bayer. Akad. d. Wiss, vol. xv.

Light Distribution round the Focus of a Lens. 31

which are developed into power series of the upper limit or of
its reciprocal. Lommel’s results have been adapted, in 1891,
to the problem of pin-hole photography by Lord Rayleigh*,
whose paper, besides a theoretical and experimental discussion
of the subject, gives also five curves exhibiting the distri-
bution of light round the centre of the image corresponding
to different apertures. As will be seen later on, the typical
phase aberrations of wave-surfaces emerging from lenses
differ in kind from those involved in the pin-hole problem.
Investigationsiaiming directly at a diffractional treatment of
the images produced by lenses were undertaken, in 1893, by
R. Straubel, whose papers are quoted in Winkelmann’s
* Handbuch’ (1906, vol. vi. p. 106), but unfortunately are
not accessible to the writer, and a little later by K. Strehl in
a very attractive book entitled ‘Theorie des Fernrohrs auf
Grund der Beugung des Lichts’ (Barth: Leipzig, 1894)f.
The earlier part of the work being dedicated to preparatory
matter, Strehl investigates in Chap. V. and VI. the intensity
along the optical axis and in the focal plane of an aplanatic
object-glass, and since this is materially the same problem as
that of a circular aperture treated by Lommel (loc. cit.),
Strehl bases himself upon Lommel’s results, as had already
been done by Lord Rayleigh, and enunciates a number of
theorems on the general features of the light distribution for
the case in question. The effects of “ spherical aberration”
are treated in Chap. VII., where the intensity formula is
developed for the case in which the emergent wave is an
ellipsoid of revolution ; the series developments (pp. 62-63)
are very complicated, and it does not appear that they could
conveniently be applied to concrete numerical calculations.
They enable Strehl, however, to enunciate some general
theorems about the symmetry relations of the diffraction
effects associated with spherical aberration of the said kind,
and an important conclusion on the true measure of the mis-
chievous effect of spherical aberration (p. 65). In a sense,
Strehl is right in declaring that by his investigation “the
problem of spherical aberration is completely solved.” So

* Phil. Mag. vol. xxxi. (1891) pp. 87-99.

+ After that Strehl has published several papers in Zettschr. f.
Instrumentenkunde for 1895-98 which I have not been able to consult.
However, to judge from Winkelmann’s quotation, the ground covered
by these papers is essentially that of Strehl’s book. Winkelmann (Joc.
cu. p. 403) quotes also, in connexion with the diffractional theory of
the telescope, Ch. André’s “ Etude de la diffraction dans les instruments
d’optique ” (Paris, 1876), without, however, describing the contents of
a paper, which was published in Ann. se. de l’école norm. supérieure,
vol. v.

32 Dr. L. Silberstein on Liyht Distribution

fact it is, inits essence. None the less it seems desirable to
treat problems relating to concrete lenses with all numerical
(or graphical) details, and to express the results in terms of
the attributes of the given lens or lens system.

Ii is precisely the object of the present paper to give ~
a fully worked out example of this kind, as a part of investi-
gations undertaken at the instance of Messrs. Adam Hilger
in connexion with their Lens Interferometer which exhibits
ad oculos, through its ‘‘ contour map,” the phase retardation
of al! the elements of an originally plane wave produced
by the passage through a given lens. The example selected
for the present purpose relates to the simplest possible lens,
viz. the plano-convex lens, traversed by a beam of finite
circular section along the optical axis. It has seemed that,
owing to its extreme simplicity, it may be the best to show
the reader a practicable and easy way of dealing with more
complicated telescopic objectives.

To complete the above bibliographic sketch we have
still to mention that the remaining chapters of Strehl’s
work are dedicated to the diffractional aspect of astigmatism
and coma which are treated on similar lines as spherical
aberration, to cylindrical waves, etc. These subjects, how-
ever, are beyond the scope of the present communication.
Lastly, we have to mention a more recent paper by James
Walker (Proc. Phys. Soc. London, vol. xxiv. 1912, pp. 160-
164) in which the subject of Strehl’s Chapter VIL., viz. the
intensity due to a rotationally ellipsoidal wave, is again
taken up. Here the expression for the intensity is deve-
loped into a complicated double series (cf. last line of the
paper quoted) which, although mathematically unobjection-
able, does not seem convenient for actual calculation *.

It must be kept in mind that for physical applications
hardly more than two significant figures in the final light
intensity are required. Under these circumstances the
method of mechanical quadraiures or a graphic methed,
analogous to that of the Cornu spiral, seems by far the
most convenient. Although laborious for very accurate
work, it certainly becomes very handy when only the said
degree of precision is aimed at. It wiil be explained and
applied in what follows.

* Tt has occurred to me that some of Walker’s intermediate formule,
as for instance that on top of p. 163, would easily yield a more
“ eommodious” expression than is the final one.

round the Focus of a Lens, at Various Apertures. 33

General Formule.

Let the portion s of the surface of a fixed sphere, of
centre O and radius R, which we will take as our reference
sphere, be the seat of monochromatic luminous oscillations
of constant amplitude a, but of different phases 7,

a COs mtn)
(F oy

Given the distribution of 7 over s, find the intensity of light
at the centre O and at points P near O. Let ds be an element
of the reference surface, r its distance from the point P in
question, c the light velocity and X\=c7Z the wave-length, in
vacuo. Then the usual way of applying Huyghens’ principle
gives, for the luminous vibration at P,

avi kv. 20r \
—§ {Zain (nt <7» =) ds, shoge Ts (1)

where n=27/T. Let N (fig. 1) be the pole of angular
coordinates, 7. e. ON the axis, @ and @ the pole distance
and the longitude of anelement ds. Let $, be the longitude

Fig. 1.

tee,
%.

of the point P, further p its distance from the axis and
o its axial distance from O, away from N. Then, neglecting
the squares of p/R, o/R,

r= |1-§ sin 8. cos (¢-$2) +5 cos 6].
Phil. Mag. 8. 6. Vol. 35. No. 205. Jan, 1918. D

34 Dr. L. Silberstein on Light Distribution

Introduce this into (1), writing in the denominator, r—A,
and develop, remembering that n i" is constant all
over s. Then the integral will split into two others
multiplied by sin, and by cos n(t—2) Thus, taking the
intensity at the sphere s as our unit, i. e, putting a?=1, the

intensity of light at P will be

i= C48), 1.00

wa eRe |
where

Cos = { cos, sin | »— 228 sin 0. cos (6—¢p)
+o cos a]. ds.

These formule are valid for any distribution of phase,
n=n(0, ¢), and for any form of the edge of s (diaphragm).
If, as corresponds to the subject of the present paper, 7 is
a function of @ alone * , and if the edge is itself a circle of
latitude, 0=0,= const., then we have axial symmetry round
ON, so ‘that Oh become independent of the longitude of
the point Ps and we can put ¢,=0. The most convenient
integration vatinble: being now @ and ¢ themselves, take

ds = R? sin 0 .dé . dd

and integrate over 6=0 to 27 and over 0=0 to 0,. Develop
(3) and introduce the abbreviations

a= =P, B= 7 a

Then, after some easy transformations, the expression for the
light intensity at the point ee o) will be

where || is the absolute value of the complex integral
at
w ={ ertntB eos) Jo(asin@).sin@d@. . . (6)
0

For the focal plane, as we shall henceforth call the plane

* That is, if the “contour lines” exhibited by the Lens Interfero-
meter are circles of latitude,

round the Focus of a Lens, at Various Apertures. 35

o=(0, or B=0, for cophasal vibrations (n=O), and for
small 6,, the integral (6) reduces to the familiar form

w = | 129) 006 = 7 5, (abs),
giving for the intensity the well-known expression
i a) 2(27p )
L=(—) 4; (=? 41).
Formule (5) and (6) are valid for any angular aperture

0, < =, and for any given axially symmetrical phase dis-

tribution 7=7(8). The axial displacement @ (of P) from the
centre enters only through the factor e®°°8; the transversal
displacement a enters through the zeroth Bessel function,
and the phase heterogeneity through e”. Notice in passing
that, by (6), a phase distribution of the type »=g.cos@*
is equivalent to a rigid shift of the whole image (luminous
region) along the axis by 38 and is, therefore, unessential.
In the case of a wave issuing from a lens, with the centre O
placed in its focus, and NO along its optical axis, the series
development of » does not at all contain such a term, 2. ¢.,
practically, no term in 6”. In fact, it will be seen that, with
the above choice of the reference sphere, the series for 7
starts, for any “‘uncorrected”’ lens (such as the simple plano-
convex lens), with 6‘, the next term being in 6°.

In all practical cases, connected with lenses, the angular
semiaperture 6, hardly exceeds 4° or 5°. Under these
circumstances we can write in (6), both in the factor
of Jy and in Jy itself, sin@=@, and in the exponential,
Bcos@=8—t80?. The first term, @, giving only the
factor e** outside the integral, dees not influence the value
of |w|? and can, therefore, be rejected.

Thus, introducing the new variable

al
te

the formula for small 0, will be
2w =| elln—Bu) J (a /u).du,- . « . « (7)
0

where 7 is a given function of u. The corresponding
intensity at P will be determined by (5), a and @ being

* Which in the case of a small 9, becomes y= —5g6”, the additive
constant term g being irrelevant.

36 Dr. L. Silberstein on Light Distribution

the coordinates of P (with \/27 as unit length). The upper
limit of the integral stands for 9@,°, the suffix having been
dropped.

For certain forms of 7 the integral (7) can be developed
into more or less complicated series, as has been done by
various authors, especially for the focal plane (@=0). In
general, however, whether the phase distribution » be given
graphically or analytically, the gaol will be reached much
more easily and quickly by a number of comparatively small
steps Aw or “du,” starting from 0 and leading to the
required u=6,’, either by mechanical quadratures or by
a graphical construction analogous to that of the famous
Cornu spiral. The latter method can now and then be
checked by the former, which is particularly advisable
for the first stages of the procedure. A curve drawn in
this manner (for any fixed a, 8) has also the advantage
of exhibiting the local intensity as a function of the
aperture; the process corresponds, in fact, to a gradual
opening of (the pupil of) a lens, from no to the full
required aperture.

Consider the plane of the complex variable

2w = a+ = 2,
so that, L being the distance of the point z from the origin,
the corresponding intensity will be
I = (@RL/d)?.
To every fixed point P(a, @) of the luminous region belongs,
in the z-plane, a curve whose element is fully given by
dz = et PYJ)(a,/u) . du.
Thus the sloping angle ¢ at any point of the P-curve will be

eianig— Bu; .). ) 4) i) i

the length of an arc element
dl =|Jo(ar/u)|.du,. . . . .

and, therefore, the curvature

k= =a (e-P) a

By means of these formule any P-curve can easily be
drawn step by step, much in the same way as the Cornu
spiral. For any point P of the focal plane (@=0) the
angle e¢ is simply equal to the phase excess 7, and for
points outside the focal plane it is smaller by Bu. The

round the Focus of a Lens, at Various Apertures. 37

arc elements d/ corresponding to equal steps du, for axial
points P(«=0), are all equal, as in the case of the Cornu
spiral. Outside the optical axis, however, the steps d/
become smaller and smaller as we approach the first zero
of J), which happens the sooner the larger a. At the

same time the radius of curvature : dwindles to nothing
(unless an 2); at the apertures corresponding to the zeros

—

of Jo(#V u) the P-curves have cusps, from which they emerge
with increasing steps to be lessened again when the next
zero is approached, andsoon. Foruw=0,7=0 and, therefore,
e=0 ; thus, all P-curves start from the origin tangentially
to the z-axis. Again, since, in all cases of actual interest,
“1-0 for u=0, the initial curvature of all curves belonging

to the focal plane is nil ; and the initial curvature for any other

point P is k=—6= i But for any not highly corrected
lens the term = in (10) soon becomes the more important

one, unless o mounts to many wave-lengtls. Thus, with
the exception of the first steps, the sloping angle is given
primarily by 7, and the curvature by = :|Jo|, the modifi-
cations due to an axial displacement being comparatively
small.

In most cases, therefore, it will be found that it is sufficient
to draw in detail the P-curves for the focal plane only, when

- the construction data become

AL
Th dinar

If dn/du preserves its sign, the sense of the windings of a
P-curve remains throughout the same (say, anticlockwise),
even in passing through a cusp. If 7 passes through a
maximum or minimum (and J)¥#0), the curve becomes
flat and inflected. If a P-curve, no matter after how
many windings, happens to pass again through the origin,
the light at P * is extinguished, and while the curve passes
on (increasing aperture), the light will reappear there, and
soon. A good check at any stage of the curve construction
may be to measure the whole length / of the path already
covered and to compare it with its correct length, which can

ES dl — |g da.

* i,e., along the circle through P, centred upon and normal to the
optical axis.

38 Dr. L. Silberstein on Light Distribution

at once be derived from (9). Up to the first cusp, 7. ¢., as
long as the first root of J is not exceeded, we have, in re
of that formula,

a ze

Jy(ar/u), ar/u2-4048.. . (10’)

If a/u is contained between the first two roots 2, x, of

Jo(z)=0, then

aie PAP ei 4 2a
ie = ) Ju(e)ede— | Jo(a)rdz = —7 Ji(a1) —=5 Ji(2),

2°4969 wa

i ae ee ae VY Si(a/u), 2,Sa/u<a, (10a)

and so on.

The are length (10’) has, as well as the chord ZL, a note-
worthy physical meaning. In fact, it represents the value
of 2w, for any point of the focal plane, for a perfect wave
(q= 0), so that the normal intensity, due to such a wave, can

be written, by (5),
Ne =) B on/uX 24048. .

On the other hand, the intensity due to the defective wave
has been

5 ‘
More generally, for a ring-shaped aperture contained
between 0, and @, the chord Z in (12) is to be replaced by
the chord ad joining the corresponding pair of points wa, us of ©
the curve. Thus, up to the jirst cusp, the ratio of the two
intensities, at the same point of the focal plane, is
ieN S23: aoe ee
In words, the defective intensity is to the normal intensity
as the squared chord to the squared arc of the P-curve, from
the origin to the point win question. The latter will obviously
be the longer of the two, the more so the greater the value
attained by 7 at the aperture wu. More generally, for a ring-
shaped aperture 0,, 63, the ratio of the two intensities will be
equal to the square of the ratio of the chord Lig, to the are ly
of the curve. The simple relation (13) will enable us to see
at a glance on the P-curves the relative value of the intensity
due to the defective wave, for various apertures. The
“definition” of the image will be exhibited by the mutual
position of curves corresponding to various points of the

field.

r=(™) ii. . a

round the Focus of a Lens, at Various Apertures. 39

When the first cusp is exceeded, the are length J loses
its simple physical meaning *. But this will oceur the
farther away from the focus, the smaller the angular semi-
aperture \/u. |

If the wave-surface which we are projecting, so to speak,
on the sphere s, is produced by a centred lens or lens system
from, say, a plane, axially incident, perfect wave, and if the
centre O is the focus of that lens, then, as has already been
mentioned, 7 will be of the form 664+0/6°+..., or in
terms of wu,

=a tO Ue es PSB CA)

The constant coefficients 6, 6’, etc. will depend on the
individual properties of the lens.

The above generalities will be made plain presently by
working out in detail the case of the most simple lens.
Other cases can be treated on similar lines.

The Plano-Convex Lens.

Let a beam of parallel, cophasal, rays of monochromatic
light of wave-length impinge upon the flat face of a plano-
convex lens of refractive index n. Let 7 be the radius of
curvature of the convex (spherical) face. Let our centre O
be the focus, and R=ON its shortest distance from the

Fig. 2.

~

convex face of the lens. Asour reference surface s (hitherto
so called) let us take the sphere of radius R and centre O
(fig. 2). A ray incident at height h will be refracted at A,

* Tf 2, <a,/u<vx,, and if J, be the arc from the origin to the first
cusp, then /, in (11) and (13), is to be replaced by /, manus the arc length
from the first cusp to the point ~; and so on.

40 Dr. L. Silberstein on Light Distribution

and will continue its path in air along BO’, piercing the
sphere s at B. Write AB=m. Then, taking »=0 at N,
the phase at B, corresponding to the angle NOB=@,

will be
Qa h2
==T{ m—nr (Goa oe 4)

The required length m will be determined, for every h *, by
the following trigonometric set :
nh

e 0 e A ‘7; .
sing =-, sin? =—; wo =1'—1i,
7 Lis

]

/sin 2’
Peek. 'N A x ' / ! I
esate = (a ee

. mb 2 1 ;
? = w—in, where sin2, = R | As'| sin,

and, finally,

rsini—Rsin 6
m= a
sin @

By successive series developments I find, up to (FF
inclusively, the spherical aberration t ‘

RON ie Gy
As’ = 2n—-1\e 1+ 4 ?

peine- 041 ae

and o=s, = = of course ; next, for the angle 0,
eae 2
sind = (n—1)"f1-™@e¥) (‘)
? 2 r
wk teas 4
_ n(n a n?) (*) in (17)

and ultimately, for the phase, as defined by (15),
dar n?(n—1) ay TT ah eae ae
nA ee () 5}. as)

a series starting with the fou: th power of the relative aperture,
as has been expected [the coefficient of the second power in
the series of m is 4nr, and this is exactly cancelled by the

reflexion.
+ Which, although not needed ultimately, may be of some interest.

* Subject to the obvious condition h< S beyond which there is total

round the Focus of a Lens, at Various A pertures. 41

corresponding coefficient in the second term of (15)]. If, for
instance, n=1°5, then

Qarr 9 (h\4 =)
n= — Se eee) (1 +5 a)

and h (1 3h? 39 )

nn? Fae (igor las =
Passing to the application of formula (18) to our diffrac-
: Senos ire h
tional problem, we can readily limit ourselves to values of ‘

less than or at any rate not considerably exceeding 1).
Under such circumstances, (17) can be written, with sufficient
accuracy,

sin G== 9 = (wea,

and, therefore, :
See ES ig
oh ise 4(n—1)?
The negative sign can be dropped, which makes no difference
for points of the focal plane, and, for all other points, requires

only a sign reversal of their abscissee 8. Thus, in our previous
notation,

LG a eS NITTE ieh eh eG
ae 4An(n—1)® - Go)

Tt will be convenient to introduce as the integration
variable, instead of wu,

ie 2B / rh 20
pa un/ amy CE eR aT Male
so that

ee A oe land 'ona/an = alsa:
Aa):

where aa We Dero (F)
J pees = OL = ° ° e . AND
nee (55) » \20 et)

Thus the sloping angle, the arc element, and the curvature
of any P-curve will be, by (8), (9), (10), remembering the
sign reversal of @,

TV" 1, ch or \ 1? ae
=" +8(z) , a= (5) \Jo(a! Vv) | de, an
geen Ob 558s 8),

[Jol

42 Dr. L. Silberstein on Light Distribution

It will be enough to consider in detail the intensity * at
points P of the focal plane only. Then the required data
are

vw Me - V Qarbv
os al = (F) Tela) (dv, k= , (22)

where now «’ defines the distance of P from the optical axis,
and v the aperture at any stage. If Z be the chord from the
origin to the point v of the P-curve, then the intensity at P
is, as in (12),

Lear tee ith Sees :
I =(*) Basalt... + 8)
This is the graphical equivalent of the explicit expression

_ (7k Fue 2 2
1=(%) ap (At+B), . . . (23a)

v 2
A,B =| cos, sin (=) .Ju(a'V a) de, . (24)
;

where

12 °€- AE times the previous 2 and y, respectively.

The angle ¢ is independent of a’, i.e. of p. Thus the tangents
to the various curves at corresponding points (same 2, i. e.
same aperture) are parallel to one another, while their
curvature is inversely proportional to |Jo|. For v=1,
/ 2, 1/3, etc., ¢ becomes equal to one, two, three, etc., right
angles. For the central point or focus, A, B become
identical with the usual Fresnelian integrals,

v 2 v 2
Ay = C(v) ={ Cos a a, Bo = Se) =| sin ee ;
0 0

the corresponding curve is the Cornu spiral {, well-known
in connexion with the straight-edge problem. With in-
creasing distance p from the focus, 7. e. with increasing 2’,
the curves change rapidly, and soon lose their simple spiral
character, owing to the Bessel factor of the integrand.
Starting from v=0 and ascending by steps “dv” =0'1, the
desired P-curves, or «’-curves, are at first more conveniently
drawn by quadratures, the rectangular coordinates A, B
being calculated by (24) as far as v=1'5 or 2:0; after that

* In applying the diffraction formula to the present case we disregard,
of course, the inequalities of amplitude at points of the sphere s due to
the slightly different incident angles 7 and glass thicknesses traversed by
the rays.

+ In the present case, only one of its two branches comes into play.

round the Focus of a Lens, at Various Apertures. 43

the curves are more conveniently drawn by the help of (22 a).
Formule (10’), or (10 a), ete. may, now and then, be used for

a control of the length / of arc attained.

In this manner four curves have been drawn on milli-

metrated paper and are reproduced in the annexed fig. 3.

Fig. 3.

B-0.5

(0

C!0]

“0.5

The curves marked [1], [10], [20],and [30], correspond to

/
Ms =h 10, 20, and 30.

If, for example, n=1°'5, and r= 510% (say, X= micron,

and r=5°55...em.), then these curves will correspond, by
(21) and (19), to points of the focal plane whose distances
from the focus are p=X, 10A, 20A, and 30X, respectively.
Similarly for any other plano-convex lens. The circlets on

Q 1.2

Te "EE: 0.2 A=0.5 c 0.75

44 Dr. L. Silberstein on Light Distribution

the curves mark, by their centres, the apertures corre-
sponding to v=0'l, 0:2, and so on, up to v=3°0. The
solitary circlets near v=1:0, 2:0, 3:0 belong to p=0, 2.e. to
the central or focal curve which is the usual Cornu spiral.
It will be seen that the curve [1] deviates but little from the
central one. The intensity, proportional to the square of
distance from the origin, increases with v, or with the square
of the aperture, first rapidly and then more and more slowly,
and reaches its maawnum a little beyond v=1:2. Then it
falls to a minimum which is considerably smaller than that
maximum, and so on. The spiral character of the curve [1]
would go on for over 3000 windings (v over 140) which,
however, for any reasonable lens, would lie much beyond
the limit-aperture (h=7/n) ; the spiral would reach its cusp,
or first zero of Jo, much beyond that limit. The next curve,
[10], is very soon deprived of its simple spiral character ; the
Bessel function vanishes already at v=1'465, where k=,
and the curve has a cusp in the neighbourhood of which the
stations v=1:1, 1:2, etc. are more and more crowded. Then
the curve emerges from the singular point in an elegant
fashion and follows on a nearly circular path ; it is drawn
up to v=3'0 only, but even beyond that it would never go
far away from the spot. The intensity, for [10], reaches its
maximum a little beyond v=2°7; the value of this maximum
is only about 4 of that of the curve[1]. The curve [20] has
one cusp at, v=0°3662, and another at v=1'929. It also is
drawn up to v=3'0. Finally, the curve [30] has in the
interval studied as many as three cusps, one between v=0°1
and 0-2, another between 0°8 and 0°9, and yet another between
2-1 and 2:2. Neither the [30]-, nor the [20]-curve goes far
away from the origin. The illumination is here very scanty,
especially at v=1:2 which is the most favourable aperture for
the centre, and also for [1]. Other details can be seen trom
the drawing itself. Here but two more remarks: First, that
it seems very doubtful whether there is at all a curve re-
passing exactly through the origin, 2. e. whether there is
at all a rigorously dark ring round the focus of the lens
(physically it is enough that the light becomes very weak
already for [20]). And, second, that the best illumination at
the focus and, at the same time, the best definition is reached
a little beyond v=1°2. Opening the plano-convex lens so far
as v= 2°0, for instance, would not only darken the focus, but
make the intensity at [10] nearly as great as at the focus.
In the process of constructing the curves of fig. 3 many
of the values of A, B needed have been calculated by
quadratures ; to make the set complete all others have also

round the Focus of a Lens, at Various Apertures. 45

been calculated. These values of A, B, as defined by (24),
are not by themselves interesting, and will not be given
here; the square sums A?+ B’, however, proportional to the
intensity, (23.a),may be useful. They are collected, together
with those for a’ =0, in the following Table, which may be a
good supplement to the P-curves themselves. The second
column contains the sum of the squares of the Fresnelian
integrals, the third etc. the values of A?+ B?, for v=0'1, 0°2
up to v=3:0, corresponding to a’= 27 .10-4? etc., that is, to
the above curves [1], and so on.

| (0? av? a \2
A? + B?= \ eb 2 Jo(a! Vv) dv
0 |
v. EPO) yee te bi 10, 20. 30.
27
0:0 0°0000 0:0000 0:0000 0-:0000 0:0000
0-1 ‘0099, _ 099, "0096 ‘0068 0042
0-2 “0399 0399 "0327 ‘0174 0050
03 0898 "0895 ‘0663 "0243 ‘0025
0°4 *1591 °1583 "1058 "0253 ‘0003
05 ‘2466 *2450 "1474 0218 ‘0010
0-6 "3499 3471 "1879 . "0162 0042
07 “4648 ‘4604 "2243 "0202 ‘0078
08 "5849 ‘5781 "2542 “0099 0101
0:9 ‘7004 6913 ‘2758 0131 ‘0108
10 “8004 “7886 "2886 0213 0101
i tet | 0°8728 0°8571 0:2932 0:03387 0°0103
2 -9004 "8847 "2918 "0481 ‘0117
1:3 8788 ‘8668 "2875 70618 0183
1-4 *8040 "7952 ‘2838 0722 "0318
15 6848 *6792 "2833 ‘0774 “0494
16 5410 -5408 -2868 OF77 "0643
i Se 4065 ‘4106 2923 0747 "0715
1°8 *3164 3228 "2957 ‘0711 0712
1°9 *3029 3044 "2923 0691 0673
2:0 3564 *3630 "2795 0693 ‘0636
21 0°4781 0°4786 02599 0:0701 0:0611
22 6123 “6049 ‘2411 0687 ‘0614
2°3 6981 "6852 2332 0634 "0617
2°4 “6921 ‘6786 "2434 "0554 "0588
2°5 "5926 "5860 2693 0494 *0521
26 "4537 4578 2973 ‘0504 "0449
271 *3593 6712 *3094 0595 “0480
2°8 3718 -3820 "2957 ‘0714 ‘0486
29 “4846 "4832 *2636 ‘0771 ‘0589

3°0 0°6132 0°5987 02368 0:0715 0:0659

46 Dr. L. Silberstein ow Light Distribution

The curves in fig. 4 (which could also be taken from
those of fig. 3) are drawn * directly according to this table,
with v as abscisse and A?+B? as ordinates, and thus
represent the intensity at selected places of the focal plane

Fig. 4,

as a function of the aperture of the lens. It will be
remembered that v is proportional to the square of the
aperture, viz., by (20),

rn? hy? r
e/a oman (fh) Begs os (25)
Thus, for example, if n=1:5 and r= : 10°, then

9
2
(o= 1000 x) . The uppermost curve in fig. 4 corresponds

* My thanks are due to Mr. W. Widigjor for the execution of both
sets of curves.

round the Focus of a Lens, at Various Apertures. 47

to the centre or the focus, and to the usual Cornu spiral ;
that curve is well-known from the distribution of light
outside the shadow of a semiplane; here its physical
meaning is different: it represents the central intensity
as function of (the area of) the aperture. It has its first,
and highest, maximum at about v=1-23, as already men-
tioned. A number of isolated points below and near this

; 2

curve represent the intensity at the circle a! = (1. e.
p=xA, for the said example). These points are too near the
first curve to be joined into a continuous line. The next

BU, (i.e. p=10A for the
/ 1000

above example) ; its first, as well as its second, maximum is
very flat. Its interesting feature is that it nearly reaches
the focal curve at about v=1°9. The moral is obvious: at
v=1:23 the definition as well as the light intensity are
excellent, at v=1°9 very bad ; at about v=2-3 much better
407

Rae SIT:

curve, | 10], corresponds to «=

again. The next curve, [20], corresponds

607
a i
nd the lowest, | 30], to vat
To form an opinion about the intensity as function of
distance from the focus, at any fixed aperture v=3-0, it is
enough to read the above numerical table in horizontal rows
(instead of columns). The corresponding curves would again
corroborate the above remark, viz. that the greatest central
intensity and the best definition of the image produced by a
plano-convex lens, of refractive index n and curvature
radius 7, are obtained at an aperture corresponding to about
v= 1-23, and to be determined for any concrete case by (25).
If, for instance, n=1°5 and A= micron, then the best
relative aperture is, in round figures,

for * = 5°56.em. 273'em. \ 6-9 mm... 1mm:
hfe) 070352" O20 057 090
The last of these would be a very minute lens indeed.
Practically, net going much below r=3 cm., the best
relative aperture will not exceed 55. Opening the lens u
to v=1-'9 would spoil the central intensity and the definition
considerably. The next best aperture, after v=1°23, would
correspond to about v=2°3 ; the next favourable opportunity
will lie a little beyond v=3. The marginal phase retardation

is, for the best aperture v=1°23, n=1515 or a little over

3/8 of a period,

48 Light Distribution round the Focus of a Lens.

Returning once more to the table on p. 45, notice that the
v-values arrayed in the first column have themselves a simple
meaning, their squares being proportional to the “‘ normal ”
central intensity No which would correspond to a perfect
wave. Thus the relative central intensity, which is the
squared ratio of chord and arc of the Cornu spiral, is
given by

Ty: No = (Ag? + Bo?) : 2,

i.e. simply by the figures of the second divided by the squares
of those of the first column. Thus, for v=0°5, I): No= Sey
and for v=1, or y equal to a quarter-period, I,: No =*8004,
agreeing with Lord Rayleigh’s result of 1879 *, and so on.
The relative intensity at the focus decreases steadily, of
course; the absolute intensity, due to the defective wave,
increases only up to a certain maximum and oscillates in
decreasing amplitudes round a limit corresponding to the
asymptotic value A,’+B)"=3. The central intensity by
itself cannot, of course, inform us about the “ definition.”
It so happens however that, in the case of the plano-convex
lens at least, the best aperture, v==1°23, for the focal absolute
intensity is also sensibly the best for the definition of the
image.

So much about the distribution of intensity in the focal
plane of the lens. The intensity at points outside the focal
plane will require but a few remarks, owing to the relative
smallness of the axial intensity gradient (for small @) already
hinted at. Consider, for instance, the points of the optical
axis itself, that is, a’=a«=0. Then, according to (22), the

1/2
arc element is dl= 5 dv, precisely as for the foeal or

Cornu-spiral itself, and, the sloping angle and the curvature
of a 8-curve,

2 1/2 nei lt’
=" +8(5) vb = vv 2b 48,

where @=27rc/d measures the axial coordinate of the point
in question. Take, for instance, our previous example

Se Ne a Then, by (19),
a hae ob
e= 3 +7000” k=100070+ 8.

* Phil. Mag. viii. p. 409. Rayleigh’s figure, 0:8003, differs but
insignificantly from the above one.

Interferometers for Study of Optical Systems. 49

The B-curve even for c=102 will, therefore, deviate but
very little from the central or Cornu-spiral. In other words,
the axial intensity-gradient, for appreciable v, will. be very
small (as compared with the transversal one, dI/dp). Both
eurves will start from the origin horizontally. The Cornu
spiral is initially flat while the @-curve has the initial
curvature @=270/d, and, if «<0, an inflexion point for
Pick.
Be eaSOOX? t €
With the exception of such details the @-curves deviate
but insignificantly from the focal Cornu spiral, and it
would therefore be hardly worth the trouble te draw them
accurately.

London, October 1917,
Research Dept., Adam Hilger, Ltd.

.. even for |o| =50A, very near the origin.

VI. Interferometers for the Experimental Study of Optical
Systems from the point of view of the Wave Theory. By

F. TwyMan*.
CoNTENTS.

1. Description of the Interferometers.
2. Various Uses.

1. Description of the Interferometers.
rFX\HESE instruments in their simplest form resemble the
well-known Michelson interferometer, the main essential
optical difference being that the two interfering beams of
light are brought to a focus at the eye of the observer.
Optical elements or combinations suitable for examination
by means of these instruments may almost all be classed in
two categories. Into the one category fall those combinations
which are required to receive a beam of light which has a
plane wave-front and deliver it again after transmission with
a plane wave-front ; and into the other fall those the object
of which is to impart spherical wave-fronts to beams which
are incident on them with plane wave-fronts. The two corre-
sponding arrangements will be referred toas the prism inter-
ferometer and the lens interferometer respectively.

The Prism Interferometer.
The prism interferometer is shown in diagram (fig. 1) as
arranged for the correction of a 60° prism, such asis used for

spectroscopy.
The light used must consist of monochromatic rays. Such

* Communicated by the Author.
Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918. 1D)

50 Mr. F. Twyman on Interferometers jor

a light may be obtained from a Cooper-Hewett Mercury-
Vapour Lamp, combined with a suitable filter.

The light from the source is reflected by the adjustable
mirror A through the condensing-lens B, by means of which
itis condensed on the aperture of the diaphragm C.

The diverging beam of light is collimated by a lens D, and
falls as a parallel beam ona plane parallel plate K, the second
surface of which is silvered lightly so that a part of the

Fig. 1.

Diagram of Prism Interferometer.

light is transmitted and part reflected. The major part
should be reflected. One part passes through the prism L
in the same way as in actual use, and being reflected by the
mirror F passes back through the prism to the plate K.
The other part of the light is reflected to the mirror G and
back again to the plate K. Here the separated beams re-
combine, and passing through the lens E each forms on the
eye placed somewhat beyond the aperture in the diaphragm P
an image of the hole in the diaphragm C. One of the re-
flecting mirrors in its mount is shown in fig. 2. :

Experimental Study of Optical Systems. oI

When the mirrors are adjusted, interference-bands are
seen which form a contour map toa scale of wave-lengths of
the extent to which passage of the beam twice through the
prism has distorted the wave-front. This distortion can be

Fig. 2.

Mirror in Mount.

corrected by removing from each point of one prism face, by

local polishing,an amountof glass proportional tothe distortion

of the wave-front at that point; hence it follows that the bands

also form a contour map of the glass requiring to be removed

in order to make the optical performance of the prism perfect.
Fig. 3.

Fig. 3 represents in diagram a typical map, where Q
represents the highest point of a “ hill.” The procedure in
such a case is to mark out the contour lines on the surface
of the prism with a a hen ane dipped in rouge, and then to

2

o2 Mr. F. Twyman on Interferometers for

polish first on the region Q, subsequently extending the area
of polishing first partly, then wholly, to the next contour line;
and soon. The marking out of the prism surface can be
done while observing the bands.

Fig. 4.

Fig. 5.

et)

It should be noted that variations in the contour lines are
obtained by a tilt of the plane of reference. Thus a slight
adjustment of mirror F (fig. 1) might change a contour map

Experimental Study of Optical Systems. 53

from that shown in fig. 4 to that shown in fig. 5. The form
of surface is in each case the same (see the sectional diagrams
at the top of the figures), but correction can be carried out
according to whichever plane of reference is the most favour-
able from the point of view of the operator. In order to find
whether Q (fig. 3) is a hill or a valley, the cast-iron table M
(fig. 1) can be bent with the fingers so as to tilt the mirror F
in such a way as to lengthen the ray-path. If the contour line
at () expands to enclose a larger area, a hill is indicated, and
vice versa. Although the words “hill” and “valley” are
convenient to use, it must not be supposed that the imper-
fections necessarily result from want of flatness either of one
or of both surfaces of the prism. The contour map gives the
total effect on the wave-front produced by double passage
through the prism, and shows in wave-lengths the departure
from planeness of the resulting wave-surface.

The Lens Interferometer.
(See figs. 6, 7, and 8.)
Fig. 6.

fart

rs >
\

Diagram of Lens Interferometer.

In the lens interferometer all parts are left as in the prism
arrangement except that the mirror F is removed and replaced

54 Mr, F. Twyman on Inierferometers for
Fig. 7.

SSeS

Lens Interferometer.

Lens Interferometer.

Experimental Study of Optical Systems. 55

by the lens and mirror mount shown in fig. 9. T represents
the lens under test, U a convex mirror in such a position
that it reflects back along their own paths the rays received
from T. The mirror U can be moved by a screw motion

Fig. 9.

Lens and Mirror Mount.

actuated by the handle V, so that its distance from T can be
varied at will. It will be seen that when the adjustment of
this part of the apparatus is correct, the whole lens addition
will, if the lens T be perfect, receive the beam of plane wave-
front and deliver it back again with a plane wave-front.
If it does not do so the departures from planeness of the so
delivered wave-front will form a contour map of the cor-
rections which have to be applied to the lens in order to
make its performance, when in actual use, perfect.

The general procedure is the same as in the case of prisms,
namely, to choose such an adjustment as presents a favourable
aspect for working and to polish off those portions of the
surface corresponding with the parts of the contour map
which represent hills.

2. Various Uses of the Interferometers.
(a) Control of “ retouching.”

The instruments were primarily designed for the correction
by retouching of object-glasses and prisms.

The process of retouching (viz. local polishing away of the
glass) appears to have beenadopted byall the great constructors

56 Mr. F. Twyman on Interferometers for

of large astronomical lenses*. Although the necessity for
this device has not generally been referred by them to the want
of homogeneity of the glass, yet the almost invariable pre-
sence of such heterogeneity would render retouching essential
even were there no other causes of optical imperfection.
The method of carrying out the process with the aid of this
apparatus has been sufficiently described above. It has been
found both speedy and effectivein the manufacture of prisms
and lenses where the highest accuracy is desired.

Many other methods of testing telescope or camera objec-
tives have been devised with a view to the control of retouching.
With each of the well-known methods results have been
attained of an excellence commensurate with the reputation
for optical work of high quality which has distinguished the
individual exponents; but none of the methods appear to give
indications upon which the optician could take action without
a more or less complex process of reasoning.

The apparatus here described, on the other hand, produces,
as has been shown above, a system of interference-bands
which may be regarded as a “ contour map” of the imper-
fections. This contour map can for practical purposes be
considered as located at any of the optical surfaces involved ;
and in the case of the control of retouching, the observer
may, if he likes, draw this map upon the surface under
treatmeat. He is then in a position without further preli-
minary to remove the superfluous material from the pro-
minences by polishing with pads of suitable size and shape,
the “contour map” giving all that is necessary for him to
know both as to the location and the magnitude of the
sources of the imperfections.

(b) The testing of lens systems.

The apparatus obviously affords, when applied to any
optical system, all the data necessary for a complete and
precise statement of the degree of optical perfection of that
system. As regards small aberrations it is sufficient to quote
the well-known and valuable generalization :—

‘‘An obvious inference from the necessary imperfection of
optical images is the uselessness of attempting anything like

* The opinions of Schroeder, Grubb, Czapski, and Alvan Clark on
this subject are cited in a summary by H. Fassbender of the then known

methods of testing object-glasses: “ Altere und neuere Methoden zur
Priifung von Objektiven,” Deutsche Mechaniker Zeitung, July 1918,
pp. 1383-188 & 149-155. This report should be read by all interested
in the subject. It omits, however, an ingenious method due to
Dr. Chalmers, see Proc. of the Optical Convention, vol. ii. p. 56 (1912).

Experimental Study of Optical Systems. 57

an absolute destruction of aberration. In an instrument
free from aberration the waves arrive at the focal point in
the same phase. It will suffice for practical purposes if the
error of phase nowhere exceeds}. This corresponds to an
error of 3A in a reflecting and $A in a (glass) refracting
surface, the incidence in both cases being perpendicular ”’*.

In the case of larger aberrations (implying by that word
the effect on the “ image ” of the deviation from sphericity of
the wave-front) the writer is firmly of the opinion that the
departure (expressed in wave-lengths) of the wave surface
as it leaves the dioptric element from a spherical wave
surface, should form the basis of all statements of imper-
fections of definition, and believes that such a procedure,
besides being most rational according to our present know-
ledge, is at the same time very convenient in application by
the manufacturer.

The co-ordination of the phenomena observed on this
interferometer with the image-forming properties of the
optical system under test is one of the objects of Dr. L.
Silberstein’s investigations, part of which appear in another
paper in this Magazine.

(c) The use of aspherical surfaces.

In spite of the great advances in the technique of optical
manufacture during the past thirty years, the definition of
actual optical systems still leaves ample room for improvement.
Hiven if we confine our remarks to the best makers, it is only
in certain of the smaller optical instruments that the imper-
fections of definition are inconsiderable, and even then only on
the best part of the field of view. This isin great part due
to the difficulty with a limited choice of suitable dioptric
materials (for, although extensive, the choice is not as varied
as the lens designer could wish) of obtaining the results
desired by calculations based on the utilization of spherical
surfaces alone.

If we look for the reason why opticians have so far almost
entirely limited themselves to the production of dioptric
elements bounded by spherical surfaces, we find it at once in
the comparative ease with which such surfaces can be generated
with precision. But, given a sufficient incentive, it cannot
be doubted that surfaces other than spherical could be
produced. Indeed this has already been done with an

* Scientific Papers of John William Strutt, Baron Rayleigh, vol. ii.
Article on Optics, Encyclopedia Britannica, xvii. 1884.

58 Interferometers for Study of Optical Systems.

accuracy suitable for the purpose in the case of a few ex-
ceptional types of lenses, for instance the special spectacle
lenses designed by Von Rohr for those whose eyes have
been operated on for cataract*; and with considerable
accuracy (though with a smaller departure from true sphe-
ricity) in the “ figured ” lenses or mirrors for large telescopes
and for other special purposes.

Now, experience has shown that with the aid of this
apparatus, and even with the ordinary means of retouching,
the optician can face without dismay the task of making with
precision quite considerable departures from the sphericity
of his surfaces. It appears to the writer certain that by
modifying from true sphericity the surfaces of systems which
have been suitably computed for the purpose, results will be
attained more perfect than has hitherto been possible in, for
instance, camera lenses. It is even likely that valuable
results may be attained at some future time by using
apparatus of this kind to assist the correction to a pitch of
high accuracy of lens systems wherein definitely aspherical
surfaces have been generated by processes essentially different
from those ordinarily used by opticians.

(d) The experimental study of lens systems.

There are probably few interested in optical systems who
have not felt disappointment at the comparative lack of
success which has attended attempts to deal with the passage
of light through such systems by means of the Wave Theory.
The difficulties are mathematical, and not of course to be
directly relieved by such instrumental aids us those now
described.

But itis perhaps not too much to hope that the existence
of means of direct demonstration of the effect on wave
surfaces of passage through optical systems may attract the
attention of mathematicians to this aspect of the subject of
dioptrics, with good result.

Research Department,
Adam Hilger Ltd.

* See Proc. of the Optical Convention, 1912, p. 118.

[ 59 ]

VII. On a New Secondary Radiation of Positive Rays. By
M. WotrkE, Dr. phil., Lecturer at the Federal Technical
High School ‘and at the Universit, y of Zurich, Switzerland”.

P to now, two secondary radiations of positive rays

have been known: the slow electron rays { and the

very soft X-rays recently discovered by J. J. Thomson f,

which are probably caused by a retardation of the ions when
encountering a rigid body.

Some years ago Chadwick § and Russell || showed that
e-rays are capable of exciting the y-radiation of heavy
elements.

So far, however, it has not been known whether positive
rays were capable of producing a similar effect. Now this
question is of very vital interest, for if it can be proved that
positive rays are able to excite characteristic rays, it then
becomes possible to obtain a better insight into the connexion
between the process of excitation and the chemical nature and
charge of the exciting particles. And such an investigation
might throw new light on the questions relating to the
mechanism of excitation and emission of X-spectra.

These considerations have prompted me to investigate the
question as to whether positive rays are capable of exciting
the characteristic X-radiation.

The experimental method was based upon similar principles
to that employed by Chadwick{ in his investigation of the
excitation of the characteristic y-radiation of gold by a-rays.

Througha channel of circular section 10 mm. widea pencil
of positive rays was let fall upon a circular opening provided
in a brass box. This opening is divided into two halves, and
each of these is covered up by a double foil made up of one
foil of a heavy metal, say tin or lead, and of a second foil of a
light metal, say aluminium, laid over the first one. These
foils are so placed that over one half of the opening the
heavy-metal foil is on the outside with the aluminium foil
turned towards the inside of the box, while the second
similar double foil is so arranged over the other half of the

* Communicated by the Author.

+ J. J. Thomson, Proc. of Cambr. Phil. Soc. xiii. p. 212 (1905) ;
Ch. Fiichtbauer, Phys. Z. S. vii. p. 153 (1906) ; L. W. Mer Phys. Rev,
Xxil. p. 312 (1906).

t J. J. Thomson, Phil. Mag. [6] xxviii. p. 620 (1914).

§ J. Chadwick, Phil. Mag. [6] xxiv. p. 504 (1912) ; xxv. p. 193 (1913).

|| J. Chadwick & A. S. “Russell, Proc. Roy. Soc. A. lxxxviii. p. 217
(1913); A. S. Russell & J. Chadwick, Phil. Mag. 6] xxvii, p. 112 (1914).

q J. Chadwick, Phil. Mag. [6] xxv. p. 193 (1913).

60 Dr. M. Wolfke on a New

opening as to have its aluminium side outwards. Behind
these foils the photographic plate is placed.

Thus on one half of the opening the positive rays fall
upon a heavy-metal surface, and on the other half upon an
aluminium surface. The characteristic radiation of the
heavy metal being more intense and harder than that of
aluminium, it reaches the photographic plate with an in-
tensity not perceptibly diminished. On the other hand, the
characteristic radiation of aluminium is weak and soft and is
absorbed to a large extent by the layer of heavy metal through
which it has to pass. Therefore, if it is true that the cha-
racteristic radiation of the heavy metal is excited by positive
rays, then the impression produced on the photographic plate
must be stronger beneath that half of the opening where the
positive rays fall upon heavy metal, and less strong beneath
the other half, where they encounter an aluminium foil.
The higher the intensity of excitation of the characteristic
rays, the more pronounced will be this difference in the
strength of the impression obtained. ;

Such secondary cathode rays as might have been produced
by the positive ‘rays and the X-rays in the channel itself were
deflected behind the channel, and prevented from entering
the opening in the box by means of a field of sufficient power
created between channel and box.

In order to eliminate the effect that might have been pro-
duced by any irregularity of thickness in the two foils, every
test made was checked by a second exposure with the same
foils as in the first case, but turned about so as to have their
relative position reversed.

In order to avoid too strong heating of the foils by these
powerful positive rays, the exposures were intermittent so
that short exposures alternated with longer breaks.

Two heavy metals, tin and lead, were treated. The foils
used were ‘016 mm. thick in the case of tin and ‘028 mm. in
the case of lead; the thickness of the aluminium foil was
‘007 mm. ;

The experiments have shown that when acted upon by positive
rays, either of these metals emitted a penetrating radiation of
fair intensity which is probably tts characteristic radiation.

When tin was tested, all photographs without exception
showed a very marked contrast—that is, where the positive
rays fell upon the tin surface, the darkening of the plate was
strongly pronounced, while the other half of the circular
imprint showed but a faint darkening. The photograph
obtained is seen in the annexed figure, and this result.was in
accordance with anticipation.

Secondary Radiation of Positive Rays. 61

Several similar exposures were made with an induction-coil
and others with an influence-machine. The duration of ex-
posure was varied between 2°5 and 22 minutes, the potential
between 25 and 40 mm. spark-gap, and the pressure between
"0007 and :0037 mm. mercury.

The annexed photograph was obtained with an influence-
machine and an exposure of 22 minutes. The spark-gap was
25-30 mm. and the pressure ‘(0037 mm. mercury.

t
/ $ oy eee em
: : » fioge EAD een ek ted ee
‘ i Re oseec rs
a oat Ui a Ses
“ Ges y * Sak . -
; toe Bc ID oe tee
ys eae pee
| aera:
' -
t

Mie
<e

Cremer aN

When photographs were taken with lead foil, with
potentials ranging between 25 mm. and 40 mm. spark-gap,
the darkening of the plate was only faint, and no difference
was visible in intensity of the impression on either half of the
image. As soon, however, as the spark-gap was increased to
45 mm.a very distinct contrast became visible in the darkening
of the two halves.

The intensity of the photograph was greatest on that half
where the positive rays fell upon the lead surface, and there
only on that part where the intensity of the positive rays
would be a maximum.

From this it would seem that the energy required for the
excitation of characteristic X-rays has a lower limit, just as
Duane, Hunt, Hull, Webster and others have observed in the
case of cathode rays in a Coolidge tube.

For cathode rays the relation e>hv has been established,
where ¢ is the minimum energy of the electron necessary for
exciting the K-series, and v tlhe maximum frequency of the
line Kf, that corresponds to this series. Supposing this
relation to hold also for the excitation of characteristic rays
by positive particles, then to excite the KA, line of tin
(A='432.10-® cm.) the voltage required would have to
exceed 57 KV. In the described experiments, however, the

62 Prof. Barton and Miss Browning on

maximum width of spark-gap used was 45 mm., the spheres
being 31 mm. in diameter; so that according to tests made
by C. Miiller* the voltage applied could not have exceeded
50 KV. The K-series could not then have been excited, and
it is probable that the wave-length excited belonged to the
L-series of either metal.

I propose to investigate further the wave-lengths and other
properties of these radiations. A report of such studies will
be published shortly.

Summary of Results.

1. For the first time the excitation of a penetrating radiation
by positive rays was observed. This effect was retained
on photographic plates in the case of tin and lead, and
it is surmised that it is the characteristic X-radiation
of these elements.

2. A lower limit was found to exist for the voltage necessary
for excitation.

3. Hinstein’s quantum condition leads to the supposition
that the new effect that has been observed is excitation
of the L-characteristic rays of either element.

The Physical Laboratory,

Technical High School of Zurich.
August 1917.

VIII. Variably-Coupled Vibrations: Il. Unequal Masses or
Periods. By Evwin H. Barton, D.Sc., F.R.S., Professor
of Physics, and H. Mary BROWNING, B. Se., Lecturer and
Demonstrator in Physics, University College, ’ Nottinghamt.

[Plates I. & II.]

CONTENTS.
Page
L INTRODUCTION. Ree 2s sos 2c cS sclera done sleoe ere 63
Tl. THrory FoR UNEQUAL MASSES................5. 63

Equations of Motion and Coupling.
Solution and Frequencies.
Initial Conditions.
Dampings, Separate and Coupled.
Ill. THrory FoR UNEQUAL PERIODS ..........e0e0es 69
Equations of Motion and Coupling.
Solution and Frequencies.
Initial Conditions.

LV. RELATIONS AMONG VARIABLES ........000ceeeces i2

* ©. Miller, Ann. d. Phys. [4] xxviii. p. 585 (1910).
+ Communicated by the Authors,

Coupled Vibrations: Unequal Masses or Periods. 63

Page
Wes tee Bich MEWEARECESUELTN. . ... <<< 566-4 ie secs es 76
Masses 20: 1.
Logarithmic Decrements.
Masses 5: 1.
Length 3: 4
SLO ae ee eee be ee we 78

I. INTRODUCTION.

| fa a recent paper * two types of coupled pendulums were

experimented with, their lengths and the masses of
their bobs being in each case equal. The present paper, the
second of the series, deals with the double-cord pendulum
only, but in cases where either the masses of the bobs are
unequal or else the lengths of their suspensions are unequal.

These mechanical cases may he regarded as somewhat
analogous to the electrical cases of inductively-coupled circuits
with unequal inductances or unequal periods respectively.

With unequal masses and equal lengths it is noticeable
that with small couplings a great increase in the amplitude
of vibration of the small bob entailed very little loss in that
of the large bob. Indeed, for masses as 20:1 we almost
realised the case of forced vibrations.

The funnel of the light bob was here of cardboard and so
had an appreciable damping. This rendered it necessary to
make corresponding modifications in the theory.

With unequal lengths and equal masses the response
showed a great diminution for small couplings, whereas for
larger couplings the mistuning seemed without appreciable
effect.

The paper includes twenty-seven photographic repro-
ductions of double sand traces obtained simultaneously one
from each bob of the coupled pendulum.

Il. THrory ror UnEquat Massss.

Equations of Motion and Coupling.—Throughout the work
described in the present paper the double-cord pendulum
was used. This was shown in figs. 1, 2, and 4 of the first
paper. The equations of motion and coupling were given
as (27)-(29) and may now be rewritten here as follows :—

.

pty, P+Q+AQ pe get PQ 9g,

dt? © (1+8)(P+Q) 1 ere Ee Oe (1)

ee ee ees a |

dt * (1+8)(P+Q) *l~” (148) ° (P4+Q 77 J

sn 2 De) ae 9
Y— (POQ1AQ(P4EP1Q) ° «= ®)

* Phil. Mag. (6) vol. xxxiy. no. 202, pp. 246-270 (Oct. 1917).

64 Prof. Barton and Miss Browning on
Let us now write in the above
2 =p. ee i.

Also divide the two equations (1) by P and Q respectively,
and insert the frictional term 2k dy/dt in the first of them.
We then obtain

dy dy 1+p+ fp goal ys e 2 4
etn > Weep)! SR) nee
aie 0 a A
aft Uted+p)”” Ase) Cee)
These may be written

LOUD ae
dé + 2ka + ay = pbz, - + + + (6)
and a ;
aa ee by;, |. (7)
iui 00 a
: Al Bi ea \ (8)
es eS caw daa Ane
(L+)(1+p) J
Solution and Frequencies.—To solve (6) and (7) let us write

Boe ei, Q

and, on inserting in (7), we have

#" +c at |
pales |e.
Then (9) substituted in (6) gives

a2
(* 7) (2? + 2ka+a)=pb,

re w+ 2kar+(ce+a)a’+2ker+ca—pb?=0, . (10)
which is the auxiliary biquadraticin z. Though this equation
has the form of the general biquadratic, an approximate
solution, presenting all the accuracy needed for our purpose,
may be easily obtained by noting that & is small compared
with the other constants. For, as appears from the experi-
ments, & is of the order one-thousandth of the coefficient of
x? and of the constant term.

Then we may write for the roots of x in the biquadratic
(10) the values

(9)

OI OPTI, pre EO, 1 tice! a

Coupled Vibrations: Unequal Masses or Periods. 65

where 7 denotes »/(—1), and 7 and s (being comparable to £)
are to be treated as small quantities whose squares or products
are negligible in comparison with p and g which Hones
upon the larger constants of the equation.

Thus, with the roots from (11) we may write instead of

(10) the equivalent equation
(2+r—ip)(a+r+ip)(e@+s—ig)(a@+s+ig)=0,
OF 2442 (r+s)a®+(p?+qtrts? +4rs)a?
+2(p*st+q?¢r+r7?s+rs’) 2+ (p?+77)(q?+s7)=0. (12)
This, on omittin @ the negligible quantities, becomes the
approximate equation ce ote! accurate for our purpose,

a+ 2 (r+ 8)a8+ (pi + q?)a? + 2(pist gr)x+ peg’ =0. (13)
The epuparisen of coefficients in ae and (13) yields

res=k, Li Mpat ni isat oad neat)

PET G6 Pay hey a) aie) Oo) CED)

Pe ge Clay le ie hee aban maa (CLOW

VEG CUO a Win, Vinriis Shea CLe)

From (15) and (17) we may eliminate 7 and oe a
quadratic in p? whose roots may be called p? and g?. We
thus find

2p? =c+a+ VW{(a—c)?+4pb7}, > |
and ONS Ei eND 2 : (18)
27 =e+a— V{(a—c)* +4 pb}.
Again, from (14) and (16) we obtain

a —
r= gai)? and s= ag

ae
And by use of (18) these become
_ a—c+ VW {(a—c)*? + 4b}
T= te : : ° (19)
and ke —¢)?
c--a+t /{(a—c)?+4 pb? tas S808 BGs

nl 2V{(a—c)?+4pb"}
Then, inserting the values of a, b, and c from (8) in
ee (19), and (20), we obtain

m
t= me.) t= WA+8)’ ut iAan Mw)
BATE BY) oa POL 699)
Wa Ha |
Eaten, BO NLHY . TRATES Leet ees)
k
bg er ° - is : 3 : ° (24)

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918. F

66 Prof. Barton and Miss Browning on

Thus, using (11) in (9) and introducing the usual constants,
the general solution may be written in the form
ese" Ae + Be) +e "(Ce + De“™), « (25)
and, omitting 7? and s?,
Ne) ; oe a
y= ( a "(Ae + Be?) fy ( a 2" (ee De“) |

Or, by transformation of (25) and (26) and use of (21)-
(24), we may write the general solution in the form

fi) (— Ac? + Bett) eee (eee (26)

mt

z= He" sin (mt +e) + Fe-“sin( Frm +4), (27)
and
! —pst f | —st_s mt f
y= —pl'e sin (mt+e') + He ain ae +4"), (28)

where

(Bye we AU + 8)?k? |

B?m?
and eNO SB) 6 ae
tan(e Reena. ig
wi (F!)?= F? Bm? +4(1 + 8) k? 4
and ae i : (30)
tan(¢’—¢)= vt 4,

the exponential coefficient s being given by (24), and H, e,
F, and @ being the arbitrary constants dependent on the
initial conditions. In many of the experimental cases EH’
may be assimilated to H and F’ to F without appreciable
error. The changes (e’—e) and (¢'—@) of the phase angles
may be distinctly appreciable for very small values of B.
But in these cases the vibrations show a slow waxing and
waning of amplitude and the phase is of very little importance.
On the other hand, for 8 equal to unity, we have

tan(e'—e)=4k/m and tan(¢'—¢)=—2%/2k/m.
And the numerical values of these are of the order 0-020
and 0:014, hence e’—e=1° 10' and ¢'—g¢=0° 48’ nearly.
Hence for all our present experimental cases, we may drop
the four accents in equation (28).

Coupled Vibrations: Unequal Masses or Periods. 67

Initial Conditions. Case I—Suppose the heavy bob of
mass Q (which =pP) is pulled aside and that the light one
of mass P is allowed to hang at rest in its more or less
displaced position according to the coupling in use. Then
we may write:

For ¢=0 let z=/, ]

- then it follows statically that y= eae 5 |
Lt p+ Bp op setat 0H)

also put Pe dy

ae 0 and oh =(). J

Differentiating with respect to time (27) and (28) without
its accents, and writing in the latter n for m/(1+ 8), we
find
= = He? [m cos (mt + €)—ps sin (mt +e)]

+Fe~"[neos(nt+)—ssin(nt+¢)], . (32)
oy = —pHe~ "| m cos (mt +e) —ps sin (mt +e) ]

dt
+ Fe~“[n cos (nt +$)—ssin (nt+¢)]. . (33)

The conditions (31) introduced in equations (27), (28),
(32), and (33) give
j—lisne+ sind, . '. .) (34)

Bef co ote ie 3
itp fees pli sine--F sind). 2 <)> (393)

0=H(m.cose—ps sine)+F(ncos¢—ssing), . (36)
0=—pEH(m cos e—pssine) + F(ncos¢—ssin d). (37)

But, by reason of the smallness of ps in comparison with
m (of the order 0°01) and of sin comparison with n (still
less), we may write instead of (36) and (37) the following :

O=Hmecose+Fneos¢d, . . . . (38)
and o= —pHmcose+Fneosg. . . ~. (39)
These are satisfied by

e=F and gar. Ue Dek eee Pita ())
These values inserted in (34) and (35) give
r f=E+F,
an
Bef _
; 1+p+B8p ‘caeiok
whence
i iA d om (1+p)Bf
aap and F ore ye . (Al)

| ee

68 Prof. Barton and Miss Browning on

Hence, for the special solution with these initial conditions,
we have

eta a —pst CEPA) a mt
Biter ie! aces Wisi Wi
SE ES —pst (1+ p)PF —a
a T+p+Bp° eae <B5° te OEE (43)
where s= :
a: ea

Thus the ratios of the amplitudes of the quick and slow
components in the y and z vibrations are respectively given by

a a ee: | ep tae |
(T+ py B° and + p)8° a (43 a)

Case II.—Suppose now that the heavy bob (of mass
Q=pP) is pulled aside while the light one (of mass P) is held
undisplaced. Then we have:

For bem ay i), ;
d Ny dz ; at (44)

Putting (44) in (27), (28) without accents, (32) and (33),
and omitting small quantities as before, we find

peal ieee } (45)
O=—pHsine+F sing;

0=Em ce e+ Fncos ¢, (46)
O=—plim cose+Fncos ¢.

Then (46) is satisfied by

7 My
e=5 and $= 95>

and putting these values in (45), we obtain
di nd ih
ei F=
l+p tie l+p'

Hence, for the special solution with these initial conditions,
we have
dll pe ef oa mt
ee: J U+B)’
oon eo ?* cos mt + cp en

“a mt :
aire ip Sh aay - (48)

cos mt + e cos

p
l+p (47)

Coupled Vibrations: Unequal Masses or Periods. 69

Accordingly the ratios of the amplitudes of the quick and
slow vibrations in the y and < traces are respectively
—(p—1)st

p

Relation of Dampings in the Vibrations separate and
coupled.—The vibrations of a separate damped pendulum
of length / are derived from the equation of motion

d*y d
de + 2h +my=0, Ay Sure Ah (50)

ee and one (49)

where m?=q/l.
The solution of this involves simple harmonic vibrations
of approximate period
T= 27|m,
and of damping factor
7 ata

Thus the ratio of successive amplitudes is
pee eas eas /m
nearly.

But the logarithmic decrement » (per half wave) is
a as the logarithm to base e of this ratio.
ence

_ kr

Or ig ae
m

Soke (or riihe) OD

which gives the relation between damping coefficient and
logarithmic decrement for a separate pendulum.

We have now to express in terms of \ the two damping
coefficients r and s which apply to the superposed vibrations
when the pendulums are coupled. Thus, combining (23)
and (24) with (51), we find

ee ease D)
Aa Pa gk 7? e ° e e . ° e (52)
and
Emax 5
Se Geaanan (53)

Ill. THrory For UNEQUAL PERIODs.

Equations of Motion and Coupling.—Still using the
double-cord pendulum, as shown in figs. 1, 2, and 4 of the
first paper, we now make the masses of the bobs equal, but
the lengths of the suspensions unequal. (The droops of
the two bridles always remain equal.) In other words,
Q=P or p=1, while the lengths of the suspensions for the

70 Prof. Barton and Miss Browning on

y and z vibrations are now denoted by 7/ and / respectively,
the droop of each bridle being @/ as before.

Then the equations of motion of the pendulums may be
written at first in the form :

d?
oo =0, “i alta nn

QS + Qv=o, oh a

where @ and y are the inclinations of the suspensions to the
vertical.
But we have also

g—¥—Rlo . ee)
nl

nd y= —— (56)

where @ is the inclination to the vertical of the planes of
the bridles.

Neglecting masses of bridles, connector, and suspensions,
® must satisfy

Qo(y—o)=Po(w—8) =P9(p—o). . . (57)
Then (56) in (57) gives

ljo=
And (58) in (56) yields

_ (2+8)y—Bez _ (8+2n)z—By
~ UB+ Bn+2n) mee (B+Bn+2) ° ©
Then by (59), equations (54) and (55) become

YNZ
5 8n = oa ey

d*y 2+ Bet Bm?
dt? * eameyeea, B+ Bn+2n” sia
d*z B+2n m2 Bm?
— + 5 mz = = y, - -. (6
d® * B+Bnt+2y"*~ B+Bn+2n” ist)
where m? is written for g/l.
So, for the Te ry, we have
‘Si
SUNN e ECR Jbl
~ @4BBt Ry oe
Hence, for n=1, we recover the original relation
rare aileron un) we

which agrees with (32) of the first paper.

Coupled Vibrations: Unequal Masses or Periods. 71
Solution and Frequencies.—In equation (61) try
ee" 7
then we have 64
(B+ Bn+2n)+(B+2n)m?,,¢ > + (4)
Uke a ee ae a
Bm
And, by (64) in (60), we obtain
{2° (8+ 8n+2n) + (B+2n)m?} $2?(8B + Bn + 2)
+ (2+ 8)m?}=f?m*,
This reduces to the auxiliary biquadratic in z,
x(B+ Bn+2n)+2(14+84+7)m?x?+2m*=0. . (65)
Solving this as a quadratic in 2, we have

2 2 ee ae eee
z= —mM B4+Bn+2 eli ee COG)
Or, let us write

ea Ob. Gia sh eon a COG)

Then, for the sake of brevity putting A? for (l—n)?+ 6’,
we have

1+8+7+A
25 2
P= B+Bn+2n "’

== ne 2 . e e e
TBE Baton oe

ees Sos ea

q Les ase

sad

Thus, using (67) in (64) and introducing the usual con-
stants, we obtain

2=E sin (ptt+e)+Fsin(gtt+¢), . . . (69)
and

1—7+A,,. —(1-— :
y= — FE sin (pete) + 9 EY P sin (Ge + 9), (70)
p and q being defined by (68).

Initial Conditions.—Consider the case of pulling aside the
bob Q of the pendulum of length / whose vibrations are
denoted by z, the other bob hanging at rest in a more or
less displaced position according to the magnitude of the
coupling.

72 Prof. Barton and Miss Browning on

Thus, we may write :

For b=; z=f, \

when it follows statically that, y= sf | oe ae
and we have also a =) a =/0 |
NAS ag dt ine

Differentiating (69) and (70) with respect to the time,
and introducing (71) gives equations which are satishied by

7 T

e=5 and $= 5. wis a

Then, introducing (71) and (72) in (69) and (70) we find
pce a ld

if 2(2 + BYA ,

pa 2tB\=n+ayre, |"
ie 2(2+8)A 1A
Finally, (72) and (73) in (69) and (70) give as the re-
quired special solution
_ (2+8)(—1+7+4)—£?
a Rae LIGA qty f cos pet

2 1— A 2
Se a feosqt, . (TA)

(73)

and
14+8+7—A
yc mpperayary Coe

1+B+n+A

a a Bfecosgt. . . . . (7)

IV. RELATIONS AMONG VARIABLES.

It is instructive to plot graphs with the values of the
coupling y as ordinates, the abscissee being the corresponding
values of 8 (ratio of droop of bridle to pendulum length).
A different graph is needed for each value of » and p (which
are respectively the ratios of pendulum lengths and masses
of bobs).

The data for these graphs are derived from the equations
and are given in Tables I., II., and III. (and Table I. p. 265
of October paper).

Coupled Vibrations: Unequal Masses or Periods. 73

TaBLE I.— Masses 20: 1 and lengths equal

: Bridle Droo Actual Droo Frequenc
Soplng gc sea for total ar | Baio ‘
Lf Pendulum length. 229 em. em | BE :q= NV(1 a= N(1+B).
Per cent. cm.
0:0 0-0 1:00
0:05 10:9 1:025
0:10 20°8 1:05
0-151 29:9 | 1:07
0207 39°7 1:10
0°265 48°1 | 1-12)
0:60 85°9 1:27
1:53 138°6 | 1:59
3°00 171°8 | 2:00
|

TaBLE I].—Masses 5:1 and lengths equal.

Bridle Droop Actual Droop Frequency

Coupling ——_—— = 6} for total length Ratio
a Pendulum length. 229 em. p:g= N(1+ 68).
Per cent. em.
0 0 0 | 1:00
] 0°027 6:0 1-014
2 0:058 13°3 1-029
5:2 07141 28°2 1-07
94 0259 ail 1-12
25°5 1 114°5 1-414
39°5 2 ‘152°7 1°732
48°8 3 1718 2

TaBLE JI].—Masses equal and lengths 3

: Bridle Droo Actual Droo Frequenc
Coupling Ee sg for total on Ratio :
th Pendulum length. 229 cm. Pig:
Per cent. ‘ em. |
0 1:154
55 O01 20°8 1:16
10°3 02 38'2 1175
22°4 05 — 763 1:29
36°5 1 114°5 1:48
59°9 2°6 165°4 2-00
63°2 3 171°8 rg

74 Prof. Barton and Miss Browning on

The graphs referred to are given in fig. 1.

e may now, from the data in the same tables, plot
graphs with the values of the frequency ratios p:g as ordi-
nates, the abscissee being the corresponding values of the
coupling y.

These are shown in fig. 2, separate graphs being plotted
for mass ratios 1, 5, and 20 and lengths equal, and also for
lengths 3:4 and masses equal.

S Ay
JOT.

5S
oO

Coupling.

@®
°o

Droep + Length.

Fig. 1.—Couplings and Droop.

With the separate frequencies equal and a given coupling,
it may be noted that the greater the inequality of the masses
the greater is the inequality of the frequencies of the result-
ing superposed vibrations of the coupled system.

Coupled Vibrations: Unequal Masses or Periods. 75

he

pq p=e0 5 pz

ik Tie ae
ee.

Ratio of Frequencies.
}

Coupling. ¥

Fig. 2.—Frequency Ratios and Coupling.

76 Prof. Barton and Miss Browning on

When the coupling vanishes the frequencies of the separate
vibrations are of course undisturbed. Thus for equal lengths,
but any ratio of masses, we have for y=0, p:g equals unity.
But for different separate frequencies (2. e., 7 not equal to
unity) we have for y=0, p:q greater than unity. But with
large couplings the effect of unequal separate frequencies
gradually disappears.

V. EXPERIMENTAL RESULTS.

Masses 20:1.—The bobs used in these experiments were
of the order 1000 gms. and 50 gms. respectively. Figs. 1-11
of Plate I. give photographic reproductions of the double
sand traces simultaneously obtained when the masses of the
bobs Q and P were as 20:1, 2. e., p= 20.

The couplings vary from 1 per cent. in the first to a little
over 30 per cent. in the last, and are shown as percentages
on every figure.

Figs. 1-8 were obtained by drawing the heavy bob aside
horizontally, the light bob being allowed to hang at rest in
its more or less displaced position according as the coupling
was tight or loose. In figs. 9-11, while the heavy bob was
pulled aside, the light one was held in its undisplaced posi-
tion. Figs. 1-6 show a very marked effect due to the
inequality of the masses. Tor, as the resultant vibrations of
the light bob wax and wane in amplitude, those of the heavy
bob scarcely change. Thus showing that with masses 20: 1
we have in this respect almost reached the limiting case of
forced vibrations in which the reaction of the driven on the
driver is negligible. The frequencies, however, are still
appreciably affected. The contrast with the case of equal
masses may be seen by referring to figs. 1-5 in Plate V. of
the October paper, where the waxings and wanings occur
equally and alternately in both traces. Figs. 1-8 show that
as the coupling increases the inequality of the frequencies
of the superposed vibrations increases also. Hence there are
fewer vibrations in the beat cycle and this fulfils the theory.

In fig. 9 the initial displacement of the heavy bob was so
great that a collision occurred between the two as indicated.
But its effect passed away after a few vibrations. This may
be seen by fig. 10,in which with a slightly smaller displace-
ment the collision was avoided.

Fig. 11 shows appreciable damping of the vibration of
the light bob which was held undisplaced while the heavy
one was drawn aside, whereas that of the heavy bob is not
appreciably damped. Thisis exactly what might be expected

Coupled Vibrations: Unequal Masses or Periods. 77

from general considerations. But it seems at first sight in
direct contradiction to the theory which shows that the y and
z vibrations for the light and heavy bobs respectively involve
the selfsame damping factors. But by equations (23) and
(24) we see that one damping coefficient is p-times the other.
Again, by equation (48) the amplitude of the slow vibrations
of the heavy bob is p-times that of its quick ones. In the
present experimental case p equals 20, hence almost all the
vibration visible is the slow one with the negligibly small
damping coefficient. On the other hand, by equation (47)
we see that the amplitudes of the slow and quick vibrations
of the light bob are numerically equal. Consequently the
large damping coefficient, which is 20 times the small one,
affects at least half of the amplitude visible.

Logarithmic decrements.—The lower trace on fig. 11 just
dealt with, led to the theoretical introduction of the damping
of the light bob as expressed by the constant & in equation
(4). It also became necessary to estimate the experimental
value of k. To do this one pendulum with a light bob was
allowed to oscillate alone, the other being meanwhile dis-
connected. The traces for the lighter bobs P were taken
when their masses were respectively as used in the experi-
ments, so as to be one-twentieth and one-fifth of those of the
corresponding heavy ones. The results are given in fig. 12.
From the upper trace with the very light bob consisting
simply of a cardboard funnel, a few weights and sand (total
mass about 50 gms.), we find that the logarithmic decrement
is of the order \=0°017.

Then by (57) we have

b= =(0:005)m. . . . . . (76)

The lower trace with bob about 120 gms. shows consider-
ably less damping and the decrement need not be evaluated.

Masses 5:1—The masses of the bobs used in these
experiments were of the order 600 gms. and 120 gms.
respectively.

Figs. 13-19 in Plate II. show double traces obtained with
this arrangement. In figs. 13-16 we see very plainly the
beat effects on the lower trace which is left by the lighter
bob. The traces of the heavier bob also show distinct but
much slighter fluctuations of amplitude. In this respect
they are seen to present an intermediate state between the
cases of equal masses and masses as 20:1. And this is just
what we should naturally expect. Further, the beat cycles
contain fewer and fewer vibrations as the coupling increases.

78 Coupled Vibrations: Unequal Masses or Periods.

This again is in accord with theory, for the frequencies of
the superposed vibrations are then more unequal and there-
fore gain more quickly on each other.

Lengths 3:4.—Figs. 20-28 show double traces simul-
taneously obtained with the masses of the bobs equal, but
the lengths of the suspensions as 3:4. The lower trace on
each figure is that made by the shorter pendulum. In the
case of fig. 20, the short pendulum was pulled aside, the
long one hanging still in its slightly displaced position. In
the cases of figs. 21-25, the long one was drawn aside while
the short one hung at rest in its more or less displaced
position. In figs. 26-28 the long pendulum was pulled
aside while the short one was held in its zero position, as
this favoured the exhibition of the compound harmonic trace
which it was then sought to obtain.

The couplings in this set vary from about 5 per cent. to
over 60 per cent. In the 5 and 10 per cent. couplings the
response of the second pendulum is feeble and the beat
cycles contain very few vibrations. These are the effects of
the inequality of lengths. But as the coupling is further
increased these effects of the inequality of the separate
frequencies are seen to be overpowered. ‘This is exactly
in accord with the theory as exhibited in the graphs on
fig. 2.

"Hig. 26 shows an accidental collision of the lighter bob
with the releasing apparatus. But the effect of the blow is
seen to pass away after a few vibrations, as shown by com-
parison with fig. 27, which is a repetition of the conditions
first intended. Figs. 26 and 27 are seen to present almost
the appearance of the compound harmonic motion of a tone
and its octave, the latter being too sharp. Fig. 28 shows
the coupling reduced to 60 per cent., and this gives the
relation of frequencies almost exactly 2:1.

The pair of simultaneous traces in fig. 28 is almost
identical in type with those in fig. 11 of Plate V. in the
October paper, in which latter case the lengths were equal.
It may well seem surprising that the effect of the present
mistuning (in which the frequency ratio exceeds 8 : 7) should
be so completely obliterated by this coupling. But experi-
ment and theory agree that it should be so.

VI. SumMMARY.

1. This second paper describes further experiments with
the double-cord pendulum, but with the masses unequal as
20:1 and as 5:1, or the lengths unequal as 3:4. These

Diffraction of Light by Cylinders of Large Radius. 79

are somewhat analogous to coupled electrical circuits with
different inductances or different periods.

2. The case of masses 20:1 is seen to be very nearly that
of forced vibrations in which the light bob is driven by
receiving energy from the heavy bob or driver, while the
latter’s loss, though equal in energy, entails only a very
small decrease of amplitude. The case of masses as 5:1 is
about midway in character between that of 20:1 and equal
masses. Highteen photographic reproductions of double
traces are given for unequal masses.

3. It was noticeable on one of the traces that the light
bob showed diminution of amplitude as the trace proceeded.
This led to taking resistance into account in the equation of
motion. It was also necessary to determine experimentally
the actual damping of the light bob when vibrating sepa-
rately. The theory thus developed and numerically applied
fitted the observed facts.

4. In the case of unequal lengths but equal masses, a
feebler response and a shorter beat cycle may naturally be
expected than if mistuning were absent. Both these effects
are quite striking with loose couplings. But with the tighter
couplings the effect of mistuning is practically unnoticeable.
The theory agrees with this experimental result. Nine sets
of double traces are given for the unequal periods.

5. It is hoped that these methods may be shortly applied
to the illustration of important phenomena in other branches
of Physics.

Nottingham,

Nov. 19, 1917.

TX. On the Diffraction of Light by Cylinders of Large Radius.
By Nawinimowan Basu, M.S8e., Sir Rashbehari Ghosh
Research Scholar in the University of Calcutta *.

[Plate IIL]
Introduction.

1. C. F. Brusu has recently published a paper containing
some interesting observations on the diffraction of light by
the edge of a cylindrical obstacle t. Brush worked with

* Communicated by Prof. C. V. Raman.

+ “Some Diffraction Phenomena: Superposed Fringes,” by C. F. Brush,
Proceedings of the American Philosophical Society, 1913, pp. 276-282.
See also ‘Science Abstracts,’ No. 1810 (1913).

80 Mr. Nalinimohan Basu on the Diffraction of

cylinders of various radii (the finer ones being screened on
one side so as to confine diffraction to the other side only),
and observing the fringes formed within a few millimetres of
the diffracting edge through a microscope, found that they
appeared brighter and sharper with every increase in the
radius of the cylinder. The fringes obtained with a smooth
rod of one or two centimetres radius differed very markedly
from those formed by a sharp edge or by a cylinder of small
radius. They were brighter, more numerous, showed greater
contrast between the maxima and minima of illumination,
and their spacing was different from that given by the usual
Fresnel formule. Brush also observed that when the radius
of the cylinder was a millimetre or more, the fringes did not
vanish when the focal plane of the microscope was put
forward so as to coincide with the edge of the cylinder.
Sharp narrow fringes were observed with the focal plane in
this position, becoming broader and more numerous as the
radius of the cylinder was increased.

2. To account for these phenomena Brush has suggested
an explanation, the nature of which is indicated by the title
of his paper. The diffraction-pattern formed by the cylinder
is, according to Brush, the result of the superposition of a
number of diffraction-patterns which are almost, but not
quite, in register. He regards the cylindrical diffracting
surface as consisting of a great many parallel elements, each
of which acts as a diffracting edge and produces its own
fringe-pattern, which is superposed on those of the other
elements. Brush has made no attempt to arrive, mathe-
matically or empirically, at any quantitative laws of the
phenomena described in his paper. A careful examination
of the subject shows that the view put forward by him
presents serious difficulties, and is open to objection. One
of the defects of the treatment suggested by Brush is that
it entirely ignores the part played by the light regularly
reflected from the surface of the obstacle at oblique or nearly
grazing incidences. I propose in the present paper (a) to
describe the observed effects in some detail, drawing attention
to some interesting features overlooked by Brush ; (b) to show
that they can be “interpreted in a manner entirely different
from that suggested by him; and (c) to give a mathematical
theory together with the results of a quantitative expe-
rimental test.

3. Reference should be made here to the problem of the
diffraction of plane electromagnetic waves by a cylinder
with its axis parallel to the incident waves. The solution of
this problem for a perfectly conducting cylinder has been

Laght by Cylinders of Large Radius. 81

given by J.J. Thomson%, and for a dielectric cylinder by
Lord Rayleigh t. These solutions are, however, suitable for
numerical computation only when the radius of the cylinder
is comparable with the wave-length. A treatment of the
problem in the case of a cylinder of any radius has been
recently given by Debyet. He considers the electromagnetic
field round a perfectly reflecting cylinder, whose axis is taken
for axis of z, with polar co-ordinates r, ¢, and waves in the
plane zy polarized in the direction of z, the electric component
in z being e**, Expressing the disturbance-field in the form

eS in= J, (Ka)
an (ea)

(in which J,, is the usual Bessel function, H, is Hankel’s
second cylindrical function, and «= 277/X), Debye transforms
the solution into the simple form

a eon”

2r

. H,(«r) cos nd

: e7tk(r—2a oe)

ZL=—

Debye’s work is of considerable significance, but his final
solution is valid only for points at a great distance from the
surface of the cylinder, whereas the phenomena considered
in the present paper are those observed in its immediate
neighbourhood. No complete mathematical treatment of
the subject now dealt with appears to have been given
so far.

General Description of the Phenomena.
Fig. 1.

4, The experimental arrangements are those shown in the

diagram (fig. 1). Light from a slit 8 falls on a polished

* ‘Recent Researches in Electricity and Magnetism,’ p. 428.

+ Phil. Mag. 188). ‘Scientific Works,’ vol. i. p. 534.

t P. Debye, “On the Electromagnetic Field surrounding a Cylinder
and the Theory of the Rainbow,” Phys. Zeitschr. ix. pp. 775-778, Nov.
1908. Also Deutsch. Phys. Gesell. Verh. 10, 20, pp. 741-749, Oct. 1908 ;
and ‘ Science Abstracts,’ No. 258 (1909).

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918. G

82 Mr. Nalinimohan Basu on the Diffraction of

cylinder of metal or glass and passes it tangentially at C*.
The axis of the cylinder is parallel to the slit. A collimating
lens may, if necessary, be interposed between the slit and
the eylinder. The fringes bordering the shadow of the edge
C are observed through the microscope-objective M and the
micrometer eyepiece EH. The latter may be placed at any
convenient distance from the objective so as to give the
necessary magnification. The effects are best seen with
monochromatic light obtained by focussing the spectrum of
the electric arc on the slit with a small direct-vision prism.
For photographic work, the eyepiece E is removed and
replaced by a long light-tight box in front of which the
objective M is fixed, and at the other end of which the
photographic plate is exposed. Sufficient illumination for
photographing the fringes may be secured by using the are
and illuminating the slit by the greenish-yellow light trans-
mitted by a mixture of solutions of copper sulphate and
potassium bichromate.

5. The phenomena observed depend on the position of the
focal plane of the objective with reference to the diffracting
edge of the cylinder, and an interesting sequence of changes
is observed as the focal plane of the objective is gradually
moved, towards the light, up toand beyond the edge OC (fig. 1)
at which the incident light grazes the cylinder. Some idea
of these changes will be obtained on a reference to Plate III.,
figs. I. to VIII., in which the fringes photographed with a
cylinder of radius 1°54 cm. are reproduced. (A Zeiss
objective of focal length 1:7 cm. was used, and the magni-
fication on the original negative was 135 diameters.)

6. To interpret the phenomena it is convenient to compare
them with those obtained by a sharp diffracting edge in the
same position. Using the cylinder, it is found that when
the focal plane is between the objective and the cylinder, but
several centimetres distant from the latter, the fringes are
practically of the same type as those due toa sharp diffracting
edge. They are diffuse, few in number (not more than seven
or eight being visible even in monochromatic light), and the
first bright band is considerably broader and more luminous
than the rest. The fringes become narrower (retaining their
characteristics) as the focal plane is brought nearer the
cylinder till the distance between the two is about two
centimetres. At this stage some new features appear; the

* A glass cylinder may be used without inconvenience as the light
transmitted through the cylinder is refracted out to one side, and does

not enter into the field under observation. Very little light is, in fact,
transmitted through the cylinder at oblique incidences.

Light by Cylinders of Large Radius. 83

contrast between the minima and maxima of illumination
becomes greater than in the fringes of the usual Fresnel type,
and the number that can be seen and counted in mono-
chromatic light increases considerably. These features
become more and more marked as the focal plane approaches
the cylinder, and the dark bands then become almost per-
fectly black. The difference between the intensity of the
first maximum and of those following it also becomes less
conspicuous. Figs. I., I1., and III. in the Plate represent
these stages. A considerable brightening-up of the whole
field is also noticed as the focal plane approaches the cylinder,
but this is not shown in the photographs, as the exposures
obtained with the light of the are were very variable. When
the focal plane is within a millimetre or two of the edge at
which the incident light grazes the cylinder, a change in the
law of spacing of the fringes also becomes evident, the widths
of the successive bright bands decreasing less rapidly than in
the fringes of the Fresnel type. Fig. LV. in the plate illus-
trates this feature, which is most marked when the focal plane
coincides with the edge of the cylinder. At this stage, of
course, the fringes due to a sharp diffracting edge would
vanish altogether.

7. When the focal plane is gradually moved further in, so
that it lies between the cylinder and the source of light, some
very interesting effects are observed. The fringes contract
a little, and the first band, instead of remaining in the fixed
position defined by the geometrical edge, moves into the
region of the shadow, and is followed by a new system
of fringes, characterized by intensely dark minima, that
appears to emerge from the field occupied by the fringes
seen in the previous stages. (See figs. V.and VI.) The
first band of this new system is considerably more brilliant
than those that followit. It is evident on careful inspection
that the fringes that move into the shadow form an inde-
pendent system. For it is found that the part of the field
from which the new system has separated out appears greatly
reduced in intensity in comparison with the part on which it
is still superimposed. When the separation of the field into
two parts is complete, a few diffraction-fringes of the usual
Fresnel type are observed at the geometrical edge of the
shadow of the cylinder. (See figs. VII. and VIII. in the
Plate, in which this position is indicated by an arrow.)

8. A comparison of the effects described in the preceding
paragraph and of those obtained with a sharp diffracting
edge in the same position, furnishes the clue to the correct
explanation of the phenomena observed and dealt with in the

G 2

84 Mr. Nalinimohan Basu on the Diffraction of

present paper. Withasharp edge, the fringes of the Fresnel
type disappear when the focal plane coincides with it, and
reappear without alteration of type when the focal plane is
between the edge and the source of light. As mentioned
above and shown in figs. VII. and VIII. of the Plate, fringes
of this type may also be observed with the cylinder when the
focal plane is in this position, and in addition we have,
inside the shadow, an entirely separate system of fringes
characterized by perfectly black minima and a series of
maxima with intensities converging to zero. This latter
system has nothing in common with the diffraction phe-
nomena of the Fresnel class, and has obviously an entirely
different origin. That it is formed exclusively by the light
reflected from the surface of the cylinder is proved by the
fact that it may be cut off without affecting the rest of the
field by screening the surface. It is accordingly clear that
the light reflected from the surface of the cylinder plays a
most important part in the explanation of the phenomena,
and that the edge of the cylinder grazed by the incident rays
alone acts as a diffracting edge in the usual way, and not all
the elements of the surface as supposed by Brush. We shall
accordingly proceed on this basis to consider the theory of
the fringes observed in various positions of the focal plane
of the objective.

Theory of the Fringes at the edge of the cylinder.

9. When the focal plane coincides with the edge at which
the incident light grazes the cylinder, it is permissible to
regard the fringes seen as formed by simple interference
between the light that passes the cylinder unobstructed and
the light that suffers reflexion at the surface of the cylinder
at various incidences; for, if a sharp diffracting edge be put
in the focal plane in the same position, no diffraction-fringes
would be visible. The positions of the minima of illumination
in the field may be readily calculated.

In fig. 2, let O be the centre of the cross-section of
the cylinder in the plane of incidence, and let C be the
point at which the light grazes the cylinder. It is sufficient
for practical purposes to consider the incident beam as a
parallel pencil of rays. The ray meeting the cylinder at the
point Q is reflected in the direction QP. Let 2 QOA=8@, so

that 7 OQP= Boe and ZOPQ=5 —20. Let a be the

radius of the cylinder and CP=a. The difference of path, 8,
between the direct ray and the reflected ray reaching the

Light by Cylinders of Large Radius. 85

point P is evidently equal to QP—RP, which can be easily
shown to be given by

d=asin O(sec 20—1).

Fig, 2.

Similarly, we shall have
“=a sec 20(cos @—cos 20).

Therefore, neglecting 4th and higher powers of 0, we have
6=Fae and | #=3aG?/2.

2x \8
so that 52a (=).
3a

Since by reflexion the rays suffer a phase change of half a
wave-length, the edge © will form the centre of a dark band,
and the successive minima are therefore given by

nr\3 3a “ee 1 \2
2= oa) = 7 Cah. (nary,

where n=1, 2, 3, &c. The resuits calculated according to
the above theory and those found in experiment are given in

Table I.

86 Mr. Nalinimohan Basu on the Diffraction of
TABLE L.
a==1°>4 em.) A=6562 x 107" am

if Calculated width Observed width
; of Band. of Band.
i; "001775 cm. 00174 cm.
| 2, 001019 ,, 00102 ,,
| 3. 000875. ,, 00086 _,,
4, 000781 ,, 00076 ,,
Be OOOCLT "5, 00069 ,,
6.

| 000671 ,, -00068_,,

The discrepancies are within the limits of experimental
error. When making these measurements, the focal plane
was, in the first instance, set in approximate coincidence with
the edge of the cylinder by noting the stage at which a further
movement of the focal plane towards the light results ina
movement of the fringes into the region of the shadow. There
was, however, a slight uncertainty in regard to this adjust-
ment, and the best position of the focal plane was finally
ascertained by actual trial.

10. The ratio between the maxima and minima of illumi-
nation in the fringes at the edge may readily be calculated.
Dividing up the pencil of rays incident on the cylinder into
elements of width asin @d@ or a@d@ approximately, the
width of the corresponding elements of the reflected pencil
in the plane of the edge is dw, that is, 3a0 d@. The amplitude
of the disturbance at any point in this plane due to the
reflected rays is thus only 1/3 of that due to the direct
rays, multiplied by the reflecting power of the surface. If
the reflecting power be unity (as is practically the case at
such oblique incidences), the ratio of the intensities of the
maxima and minima is (1+1/73)?/(1—1/v3)’, that is
approximately 14:1. The dark bands are thus nearly, but
not quite, perfectly black.

Theory of the Fringes at the edge of the shadow.

11. If the fringes be observed in a plane (such as O’P’
in fig. 2) which is farther from the source of light than the
edge of the cylinder, the diffraction and mutual interference
of the direct and the reflected rays have both to be taken
into account. Since the reflected rays form a divergent

Light by Cylinders of Large Radius. 87

pencil while the incident rays are parallel, the effect of the
former at any point sufficiently removed from the cylinder
would be negligible in comparison with the effect of the
latter. If d, the distance of the plane of observation from
the edge of the cylinder, be sufficiently large, the problem
thus practically reduces to one of simple diffraction of the
incident waves by the straight edge C. The positions of
the minima of illumination with reference to the geometrical
edge of the shadow would then be given approximately by

the simple formula
a! =/Qndr=/4nv/dr/2,
where v7’ =C’P’ and d=CCO’,
or with great accuracy by Schuster’s formula,
av’ =y/ (8n—1)drf4=V (8n—1)/2V/ dd/2.
The two formule give results which do not differ materially

except in regard to the first two or three bands, as can be
seen from Table II.

TasueE II.
ace 2, a | 4. | ;
13 Proportionate | Proportionate
n, V 4n. WV (8n—1)/2. {widths of bands |widths of bands
| as per column 2./as per column 8.
a SE eee ee ee | eee ee eee eee
)
ie i) «2-000 1871 2-000 1871
2. 2°828 2°739 0-828 0868
3. ) 3°464 3391 0°636 0-652
Bcie| 4-000 3:937 | 0-536 0-546
5. ) 4-472 4-416 / 0-472 0:479
ent 4-899 4-848 | 0427 0432
7 | 5-292 5244 | = 0398 0:396

}

12. If d be not large, the intensity of the reflected rays is
not negligible. The following considerations enable us to
find a simple formula for the positions of the minima of
illumination which takes both diffraction and interference
into account. We may, to begin with, find the positions of
the minima assuming the case to be one of simple inter-
ference between the direct and the reflected rays. The
expression for the path difference, 8’, of the rays arriving
at the point P’ is readily seen from fig. 2 to be given by the
formula

6'=(d+asin @)(sec 20—1).

88 Mr. Nalinimohan Basu on the Diffraction of
Also, a' =d tan 20+ a(cos 0 sec 20—1).

These two relations may, to a close approximation, be written
in the form

6’ = 2d6? + 2a6*,
and a! = 2d0+ 3a0?/2.

Putting d=0, we get the formula already deduced (see
paragraph 9 above) for the fringes at the edge of the
cylinder. On the other hand, if d be greater than a, we
may, to a sufficient approximation, write

Oo =2d6",
and g' =2d0,

and the positions of the points at which the direct and the
reflected rays are in opposite phases are given by the formula

we! =+/ 2ndnr.

13. But, as remarked above, the simple formula z'= VW 2ndxr
also gives the approximate positions of the minima in the
diffraction-fringes at a considerable distance from the cylinder,
where the effect of the reflected rays is negligible. Itis thus
seen that the formule

nv = 2d6? + 2a6’, ; (A)
and a! =2d0 + 3067/2,

suffice to give the approximate positions of the minima of
illumination at the edge of the cylinder (at which point the
fringes are due to simple interference of the direct and the
reflected rays) and also ata considerable distance from it
(in which case they are due only to the diffraction of the
incident light). A priort, therefore, it would seem probable
that the formule would hold good also at intermediate
points, that is for all values of d. That this is the result
actually to be expected may be shown by considering the
effect due to the reflected rays at various points in the plane
of observation. The reflected wave-front is the involute of
the virtual caustic (see fig. 3 below). Atthe edge C, the radius
of curvature of the wave-front is zero, and increases rapidly
as we move outwards from the edge of the cylinder. The
reflected rays accordingly suffer the most rapid attenuation

Light by Cylinders 07 Large Radius. 89

due to divergence in the direction of the incident rays, and
less rapid attenuation in other directions. In any plane
C’P’, therefore, the effect of the reflected light is negligible
in the immediate neighbourhood of the point C’, and would
be most perceptible at points farthest removed from 0'*,
On the other hand, the fluctuations of intensity due to the
diffraction of the direct rays are most marked in the neigh-
bourhood of C', that is, for the smallest values of 6. We
should accordingly expect to find that when d is not zero,
the first few bands are practically identical in position with
those due to simple diffraction, and ihose following are due
to simple interference between the direct and the reflected
rays. The formule given above satisfy both of these re-
quirements. For it is obvious from the manner in which
they have been deduced that they satisfy the second
requirement. The first requirement is also satisfied, as, by
putting @ small, the formule reduce to nXN=2d6? and
a' =2d@; or, in other words, 2'=1/2ndn, for the minima of
illumination, which is also the usual approximate diffraction
formula. Accordingly, the complete formuls n\= 2d + 2a03
and «' =2d0+ 3a6?/2 would (on eliminating @) give the posi-
tions of the minima over the entire field with considerable
accuracy.

14. The statements made in the preceding paragraph are,
however, subject to animportant qualification. ‘The validity
of the formula obtained rests on the basis that, for large
values of d, the positions of the minima of illumination are
given by the simple relation 2/=/2ndr. This, however, is
only an approximation, as the accurate values are to be
found from Schuster’s formula (see Table II., above), when
the effect of the reflected light is negligible. When d is so
large that the formule n= 2d6? + 2a6* and w' =2d0 +-3a0?/2
give nearly the same positions for the minima as the simple
relation 2’ = V 2ndx, they should therefore cease to be strictly
valid. The actual positions of the minima for such values
of d should agree more closely with those given by Schuster’s
formula, and should, when d is very large, agree absolutely
with the same. This qualification is, however, of importance
only with reference to the first two or three bands obtained
for fairly large values of d. The differences in respect of the
other bands would be negligibly small.

15. To test the foregoing results, measurements were made
of the widths of the bright bands for a series of values of d

* Debye’s formula (Joc, cit.) shows that the intensity of the reflected
light becomes very small as ¢ approaches =.

90 Mr. Nalinimohan Basu on the Diffraction of

up to 2. cm. Table III. shows the observed values, the values
calculated from my formule, and the values according to.
Schuster’s formula (which would be valid for a sharp dif-
fracting edge in the same position). To calculate the posi-
tions of the minima given by the relations nX=2d6? + 2a6,
and #'=2d0+ 3a0?/2, the first equation was solved for 6 by
Horner’s method, and the resulting values substituted in the
second equation. The measurements of the width of the first
band were rather rough on account of the indefiniteness of
its outer edge. The agreement between the observed widths
and the widths calculated from my formule is seen to be
fairly satisfactory for values of dup to 3mm. For larger
values of d the observed widths agree more closely with those
calculated from Schuster’s formula, as explained in paragraphs
11 and 14 above.

Theory of the Fringes between the edge and the source of light.

16. As already remarked in paragraph 7, the direct and
the reflected pencils tend to separate into distinct parts of
the field when the focal plane of the observing microscope is

Fig. 3.

put forward so as to lie between the edge of the cylinder
and the source of light. Why this is so will be readily
understood on a reference to fig. 3. The rays reflected

OF

Light by Cylinders of Large Radius.

Tas.E I1T.—Widths of Bright Bands in em. x 10-5.
a=1'54 cm. = 6562 x 1078 cm.

: Oalculated | Calculated : Calculated | Calculated 4 Calculated | Calculated
Nd. pee nie [Formula [Schuster’s ere au _ | [Formula | [Schuster’s ties [Formula | [Schuster’s
(A)]. Formula]. pea (A)). Bormulejeslee es (A)). Formula].
d="d5 mm, d=1 mm. d=1'5 mm.
tee 277 294 239 397 385 339 471 458 415
Deets 133 140 111 173 170 157 198 199 193
Duacnes 107 111 83 134 134 118 154 155 145
cee 95 98 70 119 116 99 133 133 121
Disses? 86 89 61 100 103 87 114 119 106
Garis 80 81 55 93 97 78 106 110 96
d=2 mm. d=2'5 min. d=3 mm.
ive eae 522 479 p94 578 536 606 636 587
Pee 224: 225 222 256 247 249 274 273 272
Draens 173 175 167 194 197 187 211 201 202
eee 141 146 140 161 160 156 179) 178 174
Ouncas 131 133 128 138 145 137 153 154 150
Baer: 108 ? 121 115 129 129 124 141 141 136
d=5 mm. d=10 mm. d=20 mm.
Beane 349 340 301 486 477 497 699 672 703
Bese 267 263 264 375 366 373 514 516 528
Bais 223 223 221 315 308 | 313 442 435 442

92 Mr. Nalinimohan Basu on the Diffraction of

from the surface when produced backwards would touch the
enveloping surface which lies within the cylinder. This
surface, which is virtually the caustic of the reflected rays,
terminates at the edge C of the cylinder, and when the focal
plane of the observing microscope is moved forward from
CP to a position P’C’P, in front of the edge, the boundary
of the field on the right-hand side would shift into the region
of the shadow, and would, in fact, lie on the surface of the
caustic at the point P,;. Ifthe focal plane P’C'P; is consi-
derably forward of PC, the field is seen divided into two
parts. The first part P’C’ consists of the direct rays alone
(the reflected rays meeting P’C’ being too oblique to enter into
the field of the microscope), and should obviously be bounded
at C’ by a few diffraction-fringes of the ordinary Fresnel
type. The second part of the field P,C’ is due to the reflected
rays alone, and requires separate consideration.

17. In the case considered above, that is, when the focal
plane is considerably in advance of the edge, the fringe-
system within the shadow due to the reflected light is of the
same type as that found by Airy in his well-known investi-
gation on the intensity of light in the neighbourhood of a
caustic. For the elementary pencilsinto which the reflected
rays may be divided up diverge from points lying along the
caustic, and if the point P, at which the focal plane intersects
the caustic is sufficiently removed from the edge C at which
the latter terminates, Airy’s investigation becomes fully
applicable, but not otherwise. ‘The rays emerging from the
point P, after passage through the objective of the micro-
scope become a parallel pencil, while pencils emerging from
points on either side of P; become convergent and divergent
respectively. The reflected wave-front after passage through
the objective has thus a point of inflexion on either side of
which it may be taken to extend indefinitely, provided the
arc CP, be long enough. Assuming the focal length to be f
and the equation of the wave-front to be = Av’, the value of
A may be readily found. ‘I'he equation of the caustic is

(Ax? + 4y?—a?)?—27a42?=0.

From this, or directly by an approximate treatment, it may
readily be shown that the radius of curvature of the caustic
at the point © is # the radius of the cylinder. For our
present purpose, it is thus sufficient to treat the caustic as
equivalent to a cylinder of radius 3a/4 touching the reflecting

Light by Cylinders of Large Radius. 93

surface at C. We have

nae o14 (28)
~ 6dr? 6 adn\dn?2)’

areY . ; é
a Ai (53) is the measure of the convergence or divergence
n

of the normals to the wave-front in the neighbourhood of the

point of inflexion. Substituting the values obtained from
the formule of geometrical optics, it is found that

jf a Z

oO rig

The equation of the wave-front is accordingly
E=a7/8/".

The illumination in the fringe-system alongside the caustic is
then given by Airy’s formula

0 2
I=4 a cos (10? + mac) do | :
0

—1 S24
where W459 24 8X) 2.4),

x, being the distance of any point in the focal plane measured
from the point of intersection with the caustic. The integral
gives a series of maxima of which the first is the largest, and
the rest gradually converge to zero. The minima of illumi-
nation are zeroes”. As the focal plane is moved further and
further towards the source of light, the fringe-system moves
inwards along the caustic, but remains otherwise unaltered.
18. The foregoing treatment of the reflected fringe-system
in terms of Airy’s theory ceases te be valid when the focal
plane is not sufficiently in advance of the edge, and the are
CP, of the caustic is therefore not large enough. For the
reflected wave-front on one side of the point of inflexion then
becomes limited in extent, and its equation cannot with
sufficient accuracy be assumed to be of the simple form
&=An?’, extending to infinity in either direction. In fact,
when the focal plane is at the edge of the cylinder and CP,
is zero, the point of inflexion coincides with the extreme edge
of the reflected wave-front. At this stage, of course, the

* Graphs of Airy’s integral and references to the literature will be
found in an interesting paper by Aichi and Tanakadate (Journal of the
College of Science, Tokyo, vol. xxi. Art. 3).

94 Mr. Nalinimohan Basu on the Diffraction of

fringes seen in the field are due only to the interference
of the direct and reflected wave-trains. The phenomena
noticed as the focal plane is advanced towards the source of
light, represent a gradual transition from this stage to one
in which Airy’s theory becomes fully applicable. In the
transition-stages the field of illumination is a continuous
whole, of which, however, the different parts present distinct
characteristics. First, within the geometrical edge of the
shadow, we have a finite number of fringes (one, two, or
more according to the position of the plane of observation,
but not an indefinitely large number as contemplated by
Airy’s theory); these may be regarded as the interference-
fringes in the neighbourhood of the caustic due to the
reflected light alone. Following these we have a long train
of fringes due to the interference of the direct and the
reflected pencils. The first few of these should evidently be
modified by the diffraction which the direct rays suffer at the
edge C before they reach the observing microscope. Finally,
we may also have a part of the field in which the illumination
is due only to the direct pencil, the reflected rays not entering
the objective of the microscope owing to their obliquity.
This part of the field should appear less brightly iluminated
than the rest.

19. A complete theoretical treatment of the transition-
stages described in the preceding paragraph is somewhat
difficult, and has to be deferred to some future occasion.
There is no difficulty, however, in caleulating the positions
of the fringes due to the interference of the direct and the
reflected pencils when the focal plane is in advance of the
edge, provided the diffraction-effect due to the edge is
neglected. It is easily shown that the path-difference
between the direct and reflected rays ata point 2’ is given by

6' = 206? —2d6?,
xv! =3a0?/2—2d0, ; Ge

where x’ is measured from ©’ and d=CC’. By putting
§'=nd and eliminating 0, the positions of the minima of
illumination may be calculated. A complete agreement
of the results thus obtained with those found in experiment
cannot, however, be expected, as the fringes are narrow and
the modifications due to diffraction are not negligible. As
regards the fringes alongside the caustic due to the reflected
rays, we cannot expect to find a complete agreement between
their widths and those found from Airy’s theory, so long as
the latter is not fully applicable. The divergence, if any,

Light by Cylinders of Large Radius. 95

should be most marked when the region of the caustic under
observation is nearest the edge of the cylinder, and for the
fringes which are farthest from the caustic.

20. The foregoing conclusions have been tested by a series
of measurements made with the focal plane in various
positions in advance of the edge. To prove that the
boundary of the field within the shadow is the caustic and
not the surface of the cylinder, measurements were made of
the length C'P,, the rays incident on the cylinder being a
parallel pencil.

Observed value of C’P,. Calculated value.
@— 1 mm.).2.2 700454. em: "00433 em.
d=1°3 mm.... *00750 cm. ‘00733 cm.

The following shows the widths of the fringes observed in
the neighbourhood of the caustic when the focal plane was
1:6 mm. in advance of the edge, and the widths calculated
from Airy’s theory.

Width of fringes in em. x 1075.

Observed ... 159, 69, S18 yi A5, A3
Calculated... 155, 70, Os 50, 46, 43
(Airy’s theory.)

The agreement in beth cases is satisfactory.

21. Table IV. shows the results of measurements made of
the fringes in the transition-stages when the focal plane was
only a little in advance of the edge, and Airy’s theory is not
fully applicable. ‘The observed results are in general agree-
ment with the indications of theory set out in paragraph 19.
It will be seen that the fringes farthest within the region of
the shadow show a fair agreement with Airy’s theory, and the
others are more nearly in agreement with the widths calcu-
lated from formula (B).

Summary and Conclusion.

22. C. F. Brush has recently published some observations
of considerable interest on the diffraction of light by cylin-
drical edges. The views put forward by him to explain the
phenomena, however, present serious difficulties and are
open to objection. My attention was drawn to this subject
by Prof. C. V. Raman, at whose suggestion the present work
was undertaken by me in order to find the true explanation
of the effects, and to develop a mathematical theory which
would stand a quantitative test in experiment. This has now

96 Diffraction of Light by Cylinders of Large Radius.
TaBie LV.

Widths of Bright Bands in em. x10~°,
a=1°54em: X= 6562 x 1073 em.

: Calculated | Calculated Calculated | Calculated
Deas [Airy’s | [Formula See [Airy’s | [Formula

LS Formula]. (B)}. yee Formula]. (B)].
|
d='2 mm. d='4 mm.

1546 LopO ane, 1555 1550 | jee
852 696 894 777 696 | 754
722 570 777 628 570 674
672 504 705 590 504 636
622 461 637 566 461 594
590 429 602 500 429 543

d=‘6 mm. d=*8 mm,

1513 TBO al pon ans 1466 1550 | sida
700 OS) CARS Net MRP 706 G9G.. |) Sieeoees
626 570 586 563 570 || os ame
536 504 549 503 504 508
493 461 520 475 461 483
446 | 429 497 448 429 463 |

been done, and in the course of the investigation various
features of importance overlooked by Brush have come to
light. The following are the principal conclusions arrived at:
(a) The fringes seen in the plane at which the incident light
grazes the cylinder are due to the simple interference of
the direct and the reflected rays, the positions of the
dark bands being given by the formula #=3. (2a)3. (nd)? ;
(6) the fringes in a plane further removed from the source
of light than the cylinder are due to diffraction at the edge
grazed by the incident rays but modified by interference
with the light reflected from the surface of the cylinder.
The positions of the dark bands in these fringes are (to a close
approximation) given by the formule * #=2d0@ + 3a6?/2, and
nn = 2d6? + 2a6%, from which @ is to be eliminated; (c) when
the focal plane of the observing microscope is on the side of
the cylinder towards the light, the direct and reflected rays
do not both cover exactly the same part of the field, and by
putting the focal plane sufficiently forward towards the light,

* This formula is subject to a small correction which is of importance
only when d is large.

On Aerial Waves generated by Invpact. 97

they may be entirely separated. When this is the case, the
fringes of the ordinary Fresnel type due to the edge of the
cylinder may be observed, and inside the shadow we have also
an entirely separate system of fringes due to the reflected rays,
the first and principal maximum of which lies alongside the
virtual caustic formed by oblique reflexion; the distribution
of intensity in this system can be found from the well-known
integral due to Airy; (d) but when the focal plane is only a
little in advance of the edge, the caustic and the reflecting
surface are nearly in contact, and Airy’s investigation of
the intensity in the neighbourhood of a caustic requires
modification. Itis then found that only a finite number of
bands (one, two, three, or more according to the position
of the plane of observation) is formed within the limits of
the shadow, and not an indefinitely large number as con-
templated by Airy’s theory. The rest of the fringes seen in
the field are due to the interference of the direct and reflected
rays, but modified by diffraction at the edge of the cylinder.

The Indian Association
for the Cultivation of Science,
Calcutta, 8th May, 1917.

X. On Aerial Waves generated by Impact. Part II. By
SupHansukumMAR Banersi, M.Sc., Assistant Professor of
Applied Mathematics, University of Calcutta *.

[Plate IV. |

1. Introduction.

ee origin and characteristics of the sound produced by

the collision of two solid spheres were discussed by me
at some length in the first paper under the same title that
was published in the Philosophical Magazine for July, 1916.
It was shown in that paper that the sound is not due to the
vibrations set up in the spheres, which in any ordinary
material are both too high in pitch to be audible and too faint
in intensity, but to aerial waves set up by the reversal of the
motion of the spheres asa whole. The intensity of the sound
in different directions for the case in which the two spheres
were of the same material and diameter, was investigated by
the aid of a new instrument which will be referred to as

* Communicated by Prof. C. V. Raman.

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918. H

98 Prof. Sudhansukumar Banerji on Aerial
“the ballistic phonometer *.” The intensity was found to
be a maximum along the line of collision, falling off gradu-
ally in other directions to a value which is practically zero
on the surface of a cone of semi-vertical angle 67°, and
rising again to a second but feebler maximum in a plane at
right angles to the line of collision.

In view of the interesting results obtained for the case of
two equal spheres, it was arranged to continue the investiga-
tion and to measure the distribution of intensity when the
colliding spheres were not both of the same radius or material.
A mathematical investigation of the nature of the results to
be expected in these cases was also undertaken. In order
to exhibit the results of the measurements and of the
theoretical calculation, a plan has now been adopted which
is much more suitable than the one used in the first paper.
This will be best understood by reference to fig. 1 (PI. IV.),
which refers to the case of two spheres of the same material
and diameter. The figure has been drawn by taking the point
at which the spheres impinge as origin, and the line of
collision as the axis of w, and setting off the indications of
the ballistic phonometer as radii vectores at the respective
angles which the directions in which the sound is measured
make with the line of collision. The curve thus represents
the distribution of intensity round the colliding spheres in
polar coordinates, the points at which the intensity of the
sound is measured being assumed to be all at the same
distance from the spheres. The results are brought much
more vividly before the eye by a diagram of this kind than
by plotting the results on squared paper.

2. Case of two spheres of the same material but of
different diameters.

Fig. 2, which shows the observed distribution of intensity
when two spheres of wood of diameters 3 inches and 24
inches collide with each other, is typical of the results
obtained when the impinging spheres are nearly of the same
density and are of different diameters. There is a distinct
asymmetry about a plane perpendicular to the line of impact.
In addition to the maxima of intensity in the two direc-
tions of the line of collision, we have the maxima in lateral]
directions, which are not at right angles to this line. The

* This name was suggested by Prof. E. H. Barton, D.S8c., F.R.S.,
writing in the ‘Science Abstracts,’ p. 399, Sept. 1916.

Waves generated by Impact. 99

directions in which the intensity is a minimum are: also
asymmetrically situated.

For the explanation of these and other results, we have
naturally to turn to the mathematical theory which rests
upon the fact that the sound is due to the wave-motion set
up in the fluid by the sudden reversal of the motion of the
spheres. Let a and 0 be the radii of the two spheres and
p, and p, be their densities. Then the masses of the spheres

are ST pat and smp,b° respectively. Denoting the changes

in velocity which the balls undergo as a result of the impact
by U, and U, respectively, by the principle of constant

momentum we have U_,/U,=p,0?/p,a?. The ratio Ue ras

Us
depends only on the diameters and the densities of the
spheres, while, of course, the actual values of U, and U,
would depend on the relative velocity before impact and the
coefficient of restitution. It is obvious that if we leave out
of account the duration of impact, that is, regard the changes
in velocity of the spheres as taking place practically instan-
taneously, the character and the ratio of the intensities of
the sound produced in different directions would be com-
pletely determined by the sizes of the spheres and the ratio
of their changes of velocity, that is, by their diameters and
their masses ; when the spheres are of the same material,
the nature of the motion in the fluid set up by the impact
would depend only on the radii of the spheres.

The complete mathematical problem of finding the nature
of the fluid motion set up by the reversal of the motion of
the spheres, taking the finite duration of impact into account,
would appear to be of great difficulty. In my first paper, I
have shown that when a single sphere of radius a undergoes
an instantaneous change of velocity U, the wave-motion
produced is given by the expression

= 3 _(e¢é+a-—r
v= — Me < (s ee (A4e=" im) Joos GF (1)

which indicates that it is of the damped harmonic type,
confined to a small region near the front of the advancing
wave. The wave-motion set up in the case of two spheres
in contact assumed to undergo instantaneous changes of
velocity would be of a more complicated type. In order to
obtain a general idea of the results to be expected, particu-
larly as to the intensity and character of the sound in
H 2

100 Prof. Sudhansukumar Banerji on Aerial

different directions, we may consider the analogous acoustical
problem of two rigid spheres nearly in contact, which execute
small oscillations to and fro on the line of their centres.
This problem may be mathematically formulated and approx-
imately solved in the following manner :—

Given prescribed vibrations

U, cos 0, . e** and U, cos @,. e%#

on the surfaces of two spheres of radii a and 6 nearly in
contact, it is required to determine the velocity potential of
the wave-motion started and the distribution of intensities
round the spheres, where 6, and @, are the angles measured
at the centres A and B of the two spheres in opposite senses
from the line joining the centres.

Supposing, now, that an imaginary. sphere is constructed
which is of just sufficient radius to envelop the two actual
spheres (touching them externally), it is possible from a
consideration of the nature of the motion that takes place in
the immediate neighbourhood of the two spheres, to deter-
mine the aerial vibration on the surface of the imaginary
sphere which would produce on the external atmosphere the
same effect as the vibrations on the surfaces of the real
spheres A and B. When the equivalent vibration on the
surface of the enveloping sphere has been obtained, we can,
by the use of the well-known solution for a single sphere, at
once determine the wave-motion at any external point.

The radius of the enveloping sphere is evidently a+),
and its centre is at a point CO, such that BC=a and CA=b.

If the point C be taken as origin, and if the equivalent
vibration on the surface of the enveloping sphere be expressed
by the series

BAP. (cos 6), 1s). )

nn nN

where A,’s are known constants, the velocity potential of
the wave-motion is given by

mt asl) (cos 0)

Wr F Gk.arp oe” . (3)
where
PNMUE ON ros) 6 [hae (Seal Le aR Cer
Foti) 1 ot oraaaae i 2.4. (ikr)? OY
| pay Rg Sei

VY Ckr)=(1+tkr)f (thr) —tkr f,'(ikr). | a

Waves generated by Impact. 101

To obtain the equivalent vibrations on the surface of the
imaginary enveloping sphere, we shall regard the small
quantity of fluid enclosed by this sphere as ; practically i in-
compressible, and use the well-known solution by the method
“ successive images for two spheres in an incompressible

uid.

We know that the velocity potential due to such a system
of two spheres in an incompressible fluid can be expressed
in the form

Be Ui gry ai) heen iee ce 2) CO)
where ¢ and @¢’ are to be determined by the conditions

V7p=0, Vg" =0,
Sg =—cos@,, and ae =0, when 7,=a,
1
of =—cos@,, and oF =(0, when 7,=8,
2 2

11, 7, being radii vectores drawn from A and B.

When ¢ and @¢’ have been determined so as to satisfy
these conditions, the equivalent vibrations on the surface of
the imaginary sphere can be taken to be very approximately
given by the ase 7

—[U. Ce ool ae SM BOE ie

— ete.

The functions ¢ and ¢', as is well known, can be deter-
mined by the method of successive images, and if the
expressions for the velocity potential due to these images be
all transferred to the coordinates r, 0 referred to the centre
C of the enveloping sphere, we easily obtain

Bb b° b?
a AP Reta. De: Ak ON
26=4 E (a+b)° a5 (a+ 26)° (2a + 2b)°
Poy Rive Jae?
(a aby ”
F,_P@+ab—8) , B(20—a") _ V'Ga? + ab— 28")
+2a k ha Ba + (a +2)! Cs
4 O(Bb'—2a") _ 7] Ps (cos 8)

Besa en coe

102 Prof. Sudhansukumar Banerji on Aerial

+30 [pV +H) |, VON 0) Re aa
(a +b)’ (a+ 26)’ (2a+ 2b)
b*(3b?—2a")* _ ie (cos 0)
(2a 4 3by oa
stat [PLE LADVY | HEM —a! _ B80 tab 20)
(a+ 6) ~ (a+ 26)* (Qa + 2by8
‘ar
(2a +30) ,
+é&., . . . |
and
ola elie Oa Ra
hex ak @+aj'* O+2a) (2b+ 2a)
Mone 1 et
(2b + 3a)’ ae y?

4 OF [a— a’(b*+ab—a’) , a'(2a°—b*) _ a*(26°+ab—2a’)
— (6+a)* (6+ 2a)4 (26 + 2a)*
a’ (3a?— 267) P, (cos @)
(26+4+3a)? | Y
_ 33° [a a’(b’+ab—a’)? a'(2a?— 67)? a’(26?+ ab— 2a’)?
(6+a)’ (6+ 2a)’ (2b + 2a)’
a’(3a?— 267)? P; (cos 8)
(204-3) ]
4p [ ; e(?+ab—a*)® | aX(2a°—6*)*? a*(26?+ab—207)*
(6+a)® (6+ 2a)° (26 + 2a)°
a®(3a?— 2b7)8 ] P, (cos @)
(264-30) J ee

— &e., «< » « = - nnn

the law of formation of the series within the brackets being
obvious.
Coming now to the present problem of two unequal spheres
of the same material, let us take
a=2 inches and 6=1 inch.

Since the changes of velocities of the two spheres are in-
versely proportional to their masses, we must have

U, = 8U,,.

Waves generated by /mpact. 103
Substituting the values for a and 8, we easily find that

1 1
2p=28| (1+ 5 ae stip t-. )

Pew t P, (cos @)

DEY ts
2. | 1- (J+ eto + 55: +---)

ee = 9.0 P, (cos @)
= (53 so 7 isn To: = ih —

pa Gees 2S )
pt ptigt-

a we 8 P3 (cos 8)
(Stet gt --)) Oa

AS

3 3 A3
+4.0(1- (4 ot tit.)

Page 9°" 198
= fe BN P, (cos @)
ie (7 petit te aut
HG. ee tne ee)

29 = -2| (+5 53 a :)
1 7] Bi (cos @)
“Gobo 7?
42. 2| (Gi + a +a ee.)
4 1 NE: Age s
(4 tat art ait. JAS
Z
ge Te a £3?
ne [(s+ hepa ay - -)
i an s P; Ps (cos 8) 8)
(5 aang? )

. ae # a
+4. 2[ (5+ aa

Be ata lO. 1a. \ 12: (cos 8)
H(atat ge tipt eae.

pear e N22 TO Fe) AMT (EO)

104 Prof. Sudhansukumar Banerji on Aertal

Summing the series, we easily find that the vibration on the
surface of the enveloping sphere

a u,[ 22 rr

r=8 inches
4
ae U.| 496 P; (cos @) + 3°180 Pz (cos 0)

— 1-708 P; (cos 6) + 2°600 P, (cos 8) + By a: (11)

We have seen that when the vibration on the surface of
the enveloping sphere is

> A,P. (cos 0) . et,
the velocity potential of the wave-disturbance is

(a+b)? ix(ct-r+a+s) .AnP, (cos 8) |.
SS ee —__—__——— f,, kr),
¥ a F(ik.aton”?

Now when r is large, f,(ikr)=1, so that the factor on which
the relative intensities in various directions depend is
P,, (cos 0)

Ae epee
F,(tk a+b)

Thus if we put this quantity = F +7G, the intensity of the
vibrations in various directions is measured by F?+ G?.

The distribution of intensities in different directions round
the spheres will be influenced to a considerable extent by
the value of the wave-length chosen. If we take k(a+6)=2,
the wave-length is 3a inches, and if we take k(a+6)=3,
the wave-length is 27 inches. From the expression (1) for
the wave-motion produced by a single sphere undergoing an
instantaneous change of velocity, it is seen that the wave-
length to be chosen is of the same order as the circumference
of the sphere. From this, it appears that for a system of
two spheres whose radii are 1 inch and 2 inches respectively,
the wave-length to be chosen should be some value inter-
mediate between 27 and 47, probably nearer 2m than 47;
for, in the actual case of impact, the smaller ball which
would undergo by far the greater change in velocity would
probably influence the character of the motion to a greater
extent than the larger sphere. At the same time, it must
not be forgotten that the analogy between the cases of
impact and of periodic motion cannot be pushed very far,
inasmuch as the fluid motion due to impact is undoubtedly

Waves generated by Impact.

105

of different character in different directions, and not all

throughout the same as in the periodic case.

Now taking k(a+6)=2, we find (neglecting a constant

factor)

F=:0992 P, (cos @) +°2840 P, (cos €)—:0354 P3 (cos @)
—°0146 P, (cos 0) + &c.,
G='0496 P; (cos #)—°4040 P, (cos 8) —:0177 P (cos @)

+°0315 P, (cos 8) + &e.

, (12)

The values of F and G for different directions have been
calculated and are shown in the following table :—

ngl
aa F X const.
0 +3829000
10 +325908
20 +297435
30 +251537
40 +189095
50 +114215
60 + 33341
70 — 43647
80 —107073
90 — 147250
100 — 158815
110 — 140329
120 — 95149
130 — 34099
140 + 35677
150 + 102819
160 + 157869
170 +192648
180 +201000

TABLE I.

G X eonst.

— 338000
— 326525
—~273545
— 2143387
—114659
— 24167
+ 74407
4.155306
4.205212
4213625
+178920
+106238
+ 8675
— 99267
~ 202159
289237
— 353605
—392229
— 404000

(E?+G?) x const.

223144
212552
162738
109300
48946
13572
6565
25961
53574
67405
57322
30836
9106
10957
42100
94130
150280
190913
203617

Now taking k(a+6)=3, we find (neglecting a constant

factor)

F= +105 P; (cos @)+1°060 P, (cos @)+°016 P3 (cos 6)
— ‘281 P, (cos €) —&c.
= — 122 P, (cos #)+ +186 P; (cos 0) —-024 P, (cos @)

— &. .

AES)

106 Prof. Sudhansukumar Banerji on Aerial

The values in different directions have been calculated

from these expressions and are shown in the following
table :-—

TABLE IT.
Angles F by is
(in degrees). x const. G X const. (F?+G?) x const.

0 +900000 + 38000 811444
10 +890608 + 28804 794265
20 + 849305 — 23890 720805
30 +752167 — 45754 567620
40 +572469 — 90446 335284
50 +309902 — 1238998 111476
60 — 5783 — 1353846 18261
70 —313014 — 118446 111893
80 — 542728 — 73554 300178
90 — 635375 — 9000 403306
100 — 571364 + 60786 329762
110 — 371618 + 118638 154545
120 — 96799 +149218 31610
130 +184472 +144494 54592
140 +412409 +105758 180980
150 +559907 + 44650 315625
160 +630625 — 21410 398602
170 +654606 — 69748 433925
180 +658000 — 88000 440708

The values of F?+ G? shown in Tables I. and II., have
been plotted in polar coordinates in figs. 3 and 4 (Pl. IV.).
It is seen that in both cases the intensity in the direction of
the larger ball is greater than in the direction of the smaller
ball. The asymmetry is more marked when k(a+6) has
the larger value.

The intensity of the sound in different directions due to
the impact of two spheres of wood of diameters 3 inches and
14 inches respectively has been measured with the ballistic
phonometer and is shown in fig. 5. It is seen that this
curve is intermediate in form between those shown in fig. 3
and fig. 4, exactly as anticipated. The agreement between
theory and experiment is thus very striking in this case.

Waves generated by Impact. 107

3. Two spheres of the same drameter but of
diferent materials.

We have seen in the preceding section that in the expres-
sions for F and G for two spheres of the same material but
of unequal diameters, the terms containing the zonal harmonic
of the second order Ps (cos @) usually preponderate, and that
the intensity diagram is, accordingly, a curve which consists
of four loops. A different result is obtained in the case of
two spheres of the same diameter but of markedly unequal
densities. The zonal harmonic of the first order preponde-
rates in this case, and the intensity diagram is a curve
consisting of only two loops. To obtain this result theore-
tically, we have to proceed on exactly the same lines as in
the preceding pages.

Taking a=1 inch and 6=1 inch, we easily find from the
expressions (7) and (8) that

1 1coe UN SO P, (cos 8)

1 : 1 P, (cos @)
+2 aoe i4 avd aL ela
[1-5 94 Se 5! eee ]

Bia dk 1aee Bi! P3 (cos @
+3[1 - =| ee)

eta pts

7?

+4[l-st+e—ptac i

Data? FAO PD? y
EY UC irae we taiss Ue snes UA.)
and
is |) eee ae)
a. E pega 4st eo oh ange
1 ] 1 1 wets 0)
+ [1+ x gto |
es [1- £ i 1 1 3 (1 cae, 0)
Di =F Say AP 1s ee =?
Doe a Gana ices P, (cos @)
+[l- n+ ppt RR ei
— &e. . (15)

108 Prof. ‘Sudhansukumar Banerji on Aerial

Summing the series, we find that the vibrations on the
surface of the enveloping sphere, namely

| Ue eh +Us sa) oa

or r=2 inches

can be expressed in the form
al (U,—U,) x °2254 P,(cos 0) + (U_+ U,) x 3550 P, (cos 8)
+ (U,—U,) x 3645 P;(cos @) + (U,+ U,) x °3080 P,(cos 8)
+. (U,U,) x"2320 Pe(eos 0) + &e. le
If the ball of radius 6 is four times heavier than the one
of radius a, we have
U, = 4U,.

So that the vibration on the surface of the enveloping sphere
is proportional to the expression

°6762 P,(cos 6) + 1°7750 P.(cos 8) + 1:0935 P3(cos 8)
+ 1:5400 Py(cos @) +6975 P;(cos @) + &e.

Now taking k(a+6)=1, which will give a wave-length
equal to the circumference of the enveloping sphere, we get
(neglecting a constant factor)

F= 13524 P,(cos 6)—-04987 P.(cos 0) —-0074 P;(cos 8)
+:°0007 P, (cos 8)+ &e.

G=— °0676 P;(cos 0)—-:0798 P.(cos 6) +0047 P3(cos 6)
+°0012 P,(cos@)— &. . . (17)

The values of F and G, and of F?+ G? in different directions
obtained from the preceding expressions are shown in
Table ITI.

The values of (F?+G’*) shown in Table III. have been
plotted in polar coordinates and are shown in fig. 6. It is
seen that the maximum intensity in the direction of the
heavier ball is greater than that in the direction of the
lighter one.

The experimental curve of intensity of sound due to
impact of a sphere of wood, diameter 2+ inches, with a billiard
ball of nearly the same size is shown in fig. 7. Itis found

Waves generated by Impact. 109

that the directions of minimum intensity are not quite in the
plane perpendicular to the line of impact, being nearer the

side of the lighter ball.
TaseE III.

(a dees). F x const. G xX const. (F?+G?) x const.
0 + 786000 — 1415000 2620021
10 + 793732 — 1374897 2521061
20 + 813819 — 1256037 2240132
30 + 835068 — 1068615 1839986
40 + 845629 — 826059 1397992
50 + 828667 — 549652 989741
60 + 768690 — 262257 660005
70 + 654594 + 7901 427780
80 + 482433 + 237049 288493
90 + 252125 + 403500 225913

100 + 228907 + 495515 297466
110 — 331298 + 509107 368642
120 — 647986 + 454821 626929
130 — 954405 + 347884 1031220
140 — 1229335 + 211923 1555385
150 — 1458496 + 71667 2130948
160 — 1629521 — 47667 2655850
170 — 1734880 — 128811 3026866
180 — 1770000 — 157000 3157549

A result of some importance indicated by theory is that
when one of the spheres is much heavier than the other,
replacing the former by a still heavier sphere of the same
diameter should not result in any important alteration in
the distribution of the intensity of sound in different direc-
tions due to impact. This is clear from expression (16).
For when U, is much larger than U,, any diminution in the
value of U, should not appreciably affect the value of the
expression. ‘This indication of theory is in agreement with
experiment. ‘Several series of measurements have been
made with various pairs of balls of the same size but of
different densities, e. 9., wood and marble, wood and iron,
billiard ball and-iron ball, and so forth. Generally, similar
results are obtained in all cases. It was noticed also that
the form of the intensity distribution as shown by the ballistic
phonometer was not altogether independent of the thickness
of the mica disk used in the instrument. This is not sur-
prising, as the behaviour of the mica disk before the pointer

110 Prof. Sudhansukumar Banerji on Aerial

attached to it ceases to touch the mirror of the indicator
would no doubt depend, to some extent, on the relation
between its natural frequency and the frequency of the
sound-waves set up by the impact. The best results were
obtained with a disk neither so thick as to be relatively
insensitive nor so thin as to remain with its pointer in contact
with the indicator longer than absolutely necessary.

4, The general case of spheres of any diameter
and density.

When the impinging spheres are both of different diameter
and of different density, the result generally obtained is that
the sound is a maximum on the line of impact in either
direction, and a minimum which approaches zero in direc-
tions asymmetrically situated with reference thereto.
Generally speaking, no maxima in lateral directions are
noticed, that is, the curve consists of two nearly closed loops.
The difference of the intensity of the sound in the two
directions of the line of impact may sometimes he very con-
siderable. As a typical case, the results obtained by the
impact of a sphere of wood 3 inches in diameter with a brass
sphere only 12 inch in diameter are shown in fig. 8. It is
observed that the sound due to impact is actually of greater
intensity on the side of the small brass ball. As a matter
of fact, the result generally obtained is that the intensity is
greater on the side of the ball of the denser material even if
its diameter be the smaller.

The mathematical treatment of the general case is precisely
on the same lines as in the two preceding sections. It is
found in agreement with the experimental result that in
practically all cases in which both the densities and the
diameters are different, the zonal harmonic of the first
order is of importance and that the intensity curve consists
of two nearly closed loops, as in the case of two spheres of
the same diameter but of different density.

5. Summary and Conclusion.

The investigation of the origin and characteristics of the
sound due to the direct impact of two similar solid spheres
which was described in the Phil. Mag. for July, 1916, has
been extended in the present paper to the cases in which the
impinging spheres are not both of the same diameter or

Waves generated by Impact. see a

material. The relative intensities of the sound in different
directions have been measured by the aid of the ballistic
phonometer, and in order to exhibit the results in an effective
manner, they have been plotted in polar coordinates, the
point at which the spheres impinge being taken as the origin,
and the line of collision as the axis of 2 As might be ex-
pected, the curves thus drawn show marked asymmetry in
respect of the plane perpendicular to the line of impact.

A detailed mathematical discussion of the nature of the
results to be expected is possible by considering the analogous
case of two rigid spheres nearly in contact which vibrate
bodily along their line of centres. By choosing an appro-
priate wave-length for the resulting motion, intensity curves
similar to those found experimentally for the case of impact
are arrived at. A further confirmation is thus obtained of
the hypothesis regarding the origin of the sound suggested
by the work of Hertz and of Lord Rayleigh on the theory of
elastic impact.

When the impinging spheres, though not equal in size, are
of the same or nearly the same density, the intensity-curve
drawn for the plane of observation shows the sound to be a
maximum along the line of impact in either direction, and
also along two directions making equal acute angles with
this line. The sound is a minimum along four directions in
the plane. In practically all other cases, that is when the
spheres differ considerably either in density alone, or both
in diameter and density, the intensity is found to be a
maximum along the line of impact in either direction, and
to be a minimum along directions which are nearly but not
quite perpendicular to the line of impact. The form of the
intensity curve is practically determined by the diameters
and the masses of the spheres.

The investigation was carried out in the Physical Labora-
tory of the Indian Association for the Cultivation of Science.
It is hoped when a suitable opportunity arises to study also
the case of oblique impact. The writer has much pleasure
in acknowledging the helpful interest taken by Prof. C. V.
Raman in the progress of the work described in the present
paper.

Calcutta,
15th June, 1917.

Beagle baa

XI. On the Asymmetry of the Illumination - Curves in
Oblique Diffraction. By Sistz Kumar Mirra, M.Se.,
Sir Rashbehary Ghosh Research Scholar in the University
of Calcutta *.

[Plate V.|

Introduction.

N the Phil. Mag. for May 1911, C. V. Raman has given
At the results of a photometric study of the unsymmetrical
diffraction-bands due to an obliquely held rectangular re-
flecting surface previously observed by himt. The measure-
ments showed a very marked asymmetry in the distribution
of intensity in the diffraction pattern, the theoretical expla-
nation of which is discussed in the papers quoted. The
following were the principal conclusions arrived at by Raman
as the result of the quantitative experimental study of the
case :—

(a) The illumination at the points of minimum intensity
in the diffraction pattern is zero at all angles of incidence,
and the positions of the minima are accurately given by the
formula |

d=+7, +20, +37, &e., ;

where 6= a (sin i—sin @), a being the width of the aperture,

»X the wave-length, and 7, @ the angles of incidence and
diffraction respectively ; the fringes are wider on the side
on which @>1: and their number is limited on that side, as 0
cannot be greater than a

(b) The formula of the usual type (I=sin? 6/6?) for the
illumination in the pattern fails to represent the observed
intensity-curves at oblique incidences except in regard to
the position of the minima (6= +77, +27, &.). The inten-
sities at corresponding points on either side of the central
fringe for which the values of 6 are numerically the same
are not equal.

c) The observed distribution of intensity was found to fit
in with the theoretical formula, if the latter is multiplied by
a factor proportional to the square of the cosine of the

* Communicated by Prof. C. V. Raman.
+ C. V. Raman, M.A., “On the Unsymmetrical Diffraction Bands due
to a Rectangular Aperture,” Phil. Mag. Nov. 1906, See also Phil, Mag.

Jan. 1909.

Asymmetry of Illumination Curves in Oblique Refraction. 113

obliquity, which, of course, is not the same at all points in
the diffraction pattern. In other words, the ordinates of the
illumination curve were found to be proportional to the ex-
pression cos?@ sin? 6/6”.

The question arises whether these results, practically those
indicated in (6) and (c) above, are peculiar to the case of a
surface of rectangular form, or whether similar phenomena
might be expected with other forms of surface as well.
The cases which it seemed of particular interest to examine
are tiose in which the reflecting surface is not a single
individual area but consists of two, three, or more parallel
elements lying in the same plane. A satisfactory surface of
this kind which can be used at very oblique incidences may
be prepared by etching out deep grooves on the optically
plane surface of a thick plate of glass with hydrofluoric acid,
the edges of the reflecting strips left on the surface being
subsequently ground so as to be sharp, straight, and parallel.
I have prepared several such surfaces containing two and
three equidistant reflecting strips respectively. By placing
one of these on the table of a spectrometer, the diffraction
pattern produced by reflexion at very oblique incidences
may be readily observed through the telescope of the instru-
ment. The present paper describes the results of the
quantitative study of the phenomena thus obtained. Inci-
dentally the opportunity has also been taken of testing the
results obtained by Raman for the case of a single’ aperture
using improved optical and photographic appliances. The
experiments and determinations have throughout been made
using monochromatic light. This was secured by illuminat-
ing the slit of the spectrometer with light of a definite
wave-length isolated by a monochromator from sunlight or
are light.

Unsymmetrical, Interference-fringes due to two
parallel apertures.

Fig. I (Pl. V.) reproduces a photograph of the diffraction
pattern due to a surface containing two reflecting elements
each of width 0°48 cm., and 3°60 cm. apart. The direct
image of the slit of the spectrometer also appears in the
figures to the right of the diffraction-pattern. The photo-
graph is reproduced from a dense negative taken to show
the perfect blackness of the minima of illumination, and the
progressive increase (from left to right) in the width of the
interference-fringes of the light diffracted by the two re-
flecting elements. It was obtained by replacing the telescope

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918: I

Mr. S. K. Mitra on the Asymmetry of

of the spectrometer by a camera with a lens of long focus
(176cm.). Figs. I, I1],and IV reproduce three photographs
taken at three different angles of incidence, the reflecting
strips in this case being 0°754 cm. wide and 1:446 cm.
apart. In all the figures, the central fringe of the pattern
is indicated by a small cross x. The asymmetry of the
luminosity curve will be evident on comparing the brightness
of the corresponding bands on either side of the central
fringe ; for instance, the second band on the right and the
second band on the left in figs. IT and III, or the first band
on the right and the first band on the left in fig. IV. ,

The positions of the interference minima in the pattern
are given by the formula of the usual type

T 30 5
a =e + _— — kK
Orgs) Ey) Eig ae

where 6=(a+0) (sin z—sin @)/X, a being the width of
each of the apertures, 6 their distance apart, and 2, 0,2r
having their usual significance. To test whether the formula
holds good at the oblique incidences used, the negatives
were measured under a travelling microscope. In photo-
graph I, the distances between the successive interference
minima were determined to find whether the relations

sin 6,—sin 0. = sin 0.—sin 0; = sin 0;— sin 04, &e.,

indicated by the formula were valid. The results are shown
in Table I.
TABLE I,

a=048 cm. 6 = 3°60 cm.

Minima on the |Observed value of| Minima on the | Observed value of

; right of the sin 0,41—sin 8, left of the sin @,—sin On+41
centra x constant. central fringe. < constant.
sin 6,—sin 6, 0°712 sin 6, —sin @, 0°705

sin 0,—sin 0, 0:703 sin @,—sin 0, 0-710

sin 6,—sin 0, 0711 sin 6,—sin @, 0-711

sin 9,—sin 0, 0-711 | sin @,—sin 6, 0°710

histecarat: Weipa CM voMteesets sin 0,—sin 0, 0713

EE — — er

the Illumination Curves in Oblique Refraction. 115
For photographs II, III, and IV, the actual values of 0

for the interference minima were calculated from the known
constants a, b, 7, X and compared with the observed values.

These are shown in Tables II., III., and IV.

TABLE II.
a=0°754 em. b=1:446 cm. A="0000435 cm.
4=89° 15'-27.
Calculated diffraction | Observed diffraction
Interference Minima. angle, 90° —@ angle, 90°—@
(in minutes). (in minutes).
2nd on the right. 35/-98 | 3585
BR hiss oe Al! 97 | 41-83
|
Wat 0 3, lett 47'°-21 46':95
|
51'-92 | 52-05
64:02 | 63'°95
TasBueE III.
a=0°'754cem. b6=1:°446 cm. 2A="0000435 cm.
Oo 22-1.

Interference Minima. res oe ee the
2nd on the right. . 26'°15 26'-12
SR i ne 9s + 33'°88 33'°85
ist 3104).) Jere: 40'-25 40'09
A207, A a 45’-69
5th a9, iss ” 58’90

116 Mr. 8. K. Mitra on the Asymmetry of
TaBLeE LV.

a='754em. b=1:446 cm. r=-0000435 cm.
i= 89° 29'-90.

Interference Minima. | aaa | pea |
Ist on the right. 25/72 25'-62
1st on the left, | Ste Mar (lr) 33'°45 =
Briel | 40'-08 39/94 |
bk Aa | 54'-94 54/96

The Asymmetry of the Illumination-Curves.

As remarked above, there is a very marked difference in
the luminosity of the corresponding bands on either side of
the central fringe of the pattern due to the reflecting surface
ef two elements. Similar effects are also noticeable when
the reflecting surface consists of three elements. Figs. V
and VI in the Plate reproduce two photographs obtained
with a reflecting surface consisting of three elements. The
difference between the intensities of the 2nd principal
maximum on either side of the central one is very evident
in the reproductions and might be made out even in respect
of the secondary maxima on either side. This asymmetry
demands an explanation. As is shown by the measurements.
given in Tables I. to IV., the positions of the minima of
illumination are in good agreement with those calculated
from the formula of the usual type, which are obtained on
the assumption that each of the elements into which the
reflecting surface may be divided diffracts light strictly
in proportion to its area, and that the phase and intensity of
the disturbance incident on the surface are the same as when
the waves travel undisturbed. Further, the intensities at
the points ef minimum illumination are shown by observa-
tion and by the photographs to be zero, in agreement with
the results indicated by these formule. On the other hand,
the difference in the intensity at corresponding points of the
pattern on either side of the central fringe remains unex-
plained according to such formule unless regarded as an
obliquity effect.

the lllumination Curves in Oblique Refraction. 117

A series of cemparisons of the intensities of corresponding
bands on the two sides of the pattern has been made for the
cases in which the reflecting surface consists of one, two,
and three reflecting elements respectively, for various angles
of incidence. For this purpose, I have used a rotating-
sector photometer of the Abney type supplied by Messrs.
Adam Hilger, in which the free disk, which can be adjusted
by handle while in rotation, is smaller in radius than the
fixed disk. The sectors when in rotation thus present two
annuli of different intensities, the ratio of which can be
adjusted at pleasure by moving the handle of the instrument.
The disk of the photometer is placed at the focal plane of
the observing telescope, so that the diffraction pattern can
be seen through it with an eyepiece, the fringes on the
brighter side being observed through the inner annulus of
the disk, and those on the fainter side through the outer
annulus. To enable the intensities at corresponding points
on the two sides of the pattern to be compared, a screen
with two vertical slits is interposed immediately in front of
the photometric disk so as to cut off everything except the
regions under observation, which are then adjusted to equality
of brightness by moving the handle of the photometer.
Several readings can be taken in succession and their
average struck. The diffraction angles @ and 6’ of the two
bands under comparison may then be measured under a
micrometer eyepiece. Tables V., VI., and VII. show the
observed ratios of the illumination and those calculated on
the assumption that the formula for illumination includes a
factor proportional to the square of the cosine of the obli-
quity. It is seen that the agreement is good except when
the ratio is so large that it cannot be measured accurately,
owing to the near approach of the fainter band towards the
direct image of the slit.

- Taste V.
Single Reflecting Surface, width 0:90 cm.

Ratio of intensity of the first band on the right and
the first band on the left.

: Observed ratio Calculated ratio
Se of illumination. cos? 0/cos? 6’.
Ege 49 | 1:80 | 1-78
88° 53’ | 231 | 2°46
88° 56’ 2°81 | 2°89
|

88° 4’ ) 4:09 4°21

118 Asymmetry of Illumination Curves in Oblique Refraction.
Tasie VI.

Reflexion grating of two elements.
a=0°7384 em. b=1°446 cm.

Ratio of the 1st maxima|Ratio of the 2nd maxim
on the right and left. | on the right and left.

Angle of Incidence.
Observed. | Calculated.) Observed. | Calculated:

89° 6’ 1°43 1-40 2:09 2:01

89° 23 2:25 2-01 3:91 4:88

89° 28’ OL 2°44. 7-50 11°06
TABLE VII.

Reflexion grating of three elements.
a=0°'440 cm. 6=0°741 cm.

Ratio of the 1st two Ratio of the 2nd two
Principal Maxima on either | Principal Maxima on either
Angle of Incidence. side of the central one. side of the central one.

Observed. | Calculated. | Observed. | Calculated.

88° 49' 148 | 148 1:83 1TH
89° 5! 163 1-71 2-90 S15

! ‘ .Qn more than
89° 27 2°65 2-87 x 100 eae

Summary and Conclusion.

1. The unsymmetrical interference fringes of the light
obliquely diffracted by two parallel reflecting surfaces in
the same plane have been observed and photographed.

2. The illumination curve in the diffraction pattern (of
the Fraunhofer class) due to an obliquely-held reflecting
surface (which may consist of two or more separate parts in
the same plane), is found to be markedly asymmetrical,
corresponding points on either side of the central fringe
being of very different intensities. As the positions of the
points of minimum (i. e. zero) illumination are found to be
in close agreement with those given by the formula of the
usual type, the asymmetry of the illumination curve may be
explained as due to the varying obliquity at different points
in the diffraction pattern. Measurements of the ratio of the

On the Two-Dimensional Motion of Infinite Liquid. 119

intensities at corresponding points have been made with
reflecting surfaces of rectangular form or consisting of either
two or three elements in the same plane for various angles
of incidence; the results show that the expression for the
illumination at any point of the diffraction pattern contains
a factor proportional to the square of the cosine of the obli-
quity at such point.

The experiments and observations described in the note
were carried out in the Palit Laboratory of Physics. The
writer hopes to carry out further work on the subject of
oblique diffraction by various forms of aperture, and parti-
cularly in regard to the positions of the points of maximum
intensity in the pattern, which would no doubt differ from
those given by the usual formule owing to the asymmetry
of the illumination curves.

Calcutta,
8th June, 1917.

XII. On the Two-Dimensional Motion of Infinite Liquid
produced by the Translation or Rotation of a Contained
Solid. By J. G. Leatuem, V.A., D.Sc., Fellow of
St. John’s College, Cambridge *.

+. ERIODIC conformal transformations—that is, trans-

formations by which doubly connected regions in
the plane of a variable z=x+1y, externally unbounded and
bounded internally by a closed polygon or curve, may be
represented conformally and repeatedly upon successive
semi-infinite strips of width % in the half-plane 7>0 of a
variable €=£+7y—have been studied by the present writer
in a previous paper ft. It has there been shown how the
knowledge of such a transformation for any particular curve
makes possible the specification of a field of circulatory
liquid flow (with or without logarithmic singularities) round
a fixed solid body bounded by the curve.

It is now proposed to show that such knowledge makes
possible also the determination of the field of irrotational
motion due to any translation or rotation of the same solid
in surrounding infinite liquid.

* Communicated by the Author.

+ J. G. Leathem, “ On Periodic Conformal Curve-Factors and Corner-
Factors,” Proc. Royal Irish Academy, vol. xxxiii. Sec. A, August
1916.

120 Dr. J. G. Leathem on the

2. The geometrical, or (z, €), relation may be either in the

form
2= (0), -) » see ee

or in the differential form

d= 6d. Vi

where &({) is a periodic curve-factor of linear period \ and
angular period 27. With this angular period itis necessary *
that both @(€) and /(€) should, for » great and positive,
tend to infinity like exp (Zan/X). In fact, G(€) is expan-
sible in the form

G(C) = exp (—2mC/r) . Scsexp (2rsif/rA), ~. (3)

where s=0, 2, 3, 4..., and the coefficients may be complex.
The periodicity of < makes it necessary + that c;=0.
From integration of (3) it follows that

z=f(€)=c+(ir/277) exp(— 26/2) . S{c,/(1—s) } exp (2arstgA), (4)
where c is another complex constant f.

It is also to be noticed that, if | (€)| =h, and if
C,=K, exp (2y,),

h?=exp (=") E + exp (— =o) 2kKoks cos( “Z + Y2 —n)

6 6
+ exp (- =) 2reoecos( ——* +7—1)

8 8
+exp(— ze) { 2eqeacos( ==" +u-90) +e} ax - . (5)

the terms containing ascending integral powers of

exp (—2z7rn/2).

FUE ORS 08

t Lic. § 4.

t The problem of obtaining a transformation of the type of formula (4),
so that a given closed curve shall correspond to n=0, is the same as that
of the parametric representation of the given curve by a formula of the
type

a-+i=mexp (—7¢)+p + IS p eX (2s¢),

where ¢ is a real parameter, m and the p’s are complex constants, and s
takes positive integral values.

It is to be noted that a formula of this type need not represent a curve
free from nodes unless the constants are suitably restricted.

Two-Dimensional Motion of Infinite Liquid. 12]

It is understood that the boundary in the z plane is the
locus corresponding to 7=0.

3. Field of flow due to translation of the boundary.—lt
the boundary have a velocity V in a direction making an
angle w with the axis of 2, the superposition on the whole
system of such uniform velocity as brings the boundary to
rest gives an irrotational fluid motion which has zero normal
velocity at the boundary and tends, for z infinite, to flow V
in the direction w+. If this motion have velocity poten-
tial @ and stream-function w, so defined that the velocity is
the upward gradient of ¢, and if w=¢+iy, w must tend,
for z infinite, to the form

—Vzexp(—im)+const., . . . . (6)

or, in terms of €,

— Vxo(tr/27) expi(yo—w—27G/rA). . « (7)
Now if
w=—(V«A/7) sin (27g/A—yotpu), - ~ (8)

this tends, for n—>+, to the form (7) ; and as
= — (Vx A/7) cos (27E/A—y+ ») sinh (27/2),

it is clear that is zero along the boundary 7=0. The
corresponding form of @ shows that there is no circulation
round the boundary ; and w is free from infinities in the
relevant region.

Hence formula (8) specifies that irrotational motion
past the fixed boundary whose limit form, at indefinitely
great distance, is the assigned uniform flow.

4. The impulse of ¢he motion due to translation.--Though
modern speculation tends to regard wave-motion as the
prependerating factor in suction and other inertia pheno-
mena of floating bodies, it can hardly be doubted that, in
the case of a submarine at least, the ordinary inertia coeffi-
cients measure approximately the resistance to quick changes
of velocity. Thus the evaluation of the impulse (X, Y) of
the combined motion of solid and fluid, when the solid has
translatory motion, is of interest.

If an approximation to w for | z| great, closer than that

afforded by formula (6), be
w=—Vzexp(—i)+C+D/z,. . . (6a)

122 Dr. J. G. Leathem on the

where © and D are complex constants, it is known * that
X+1Y =—27pD,

where p is the density of the liquid, and it is supposed that
the mean density of the solid is also p.

If formula (6a) be expressed in terms of € by means of
formula (4), it yields the approximation

in ; 2
w= —~Veo(5~ Jexpi(qo—p— “a +C'

yy : 2 hae j
+ {Ves exp (yeu) + D(5™ ) exp (—ino) boxp(2™

where C’ is a constant. On comparison of this with
formula (8), it appears that the coefficient of exp (2a7¢/A)
must equal Vxo(tA/27) expz(u—yo). Hence

D = —(VA?/4r?) {x o" exp (14) —Koky EXP 1(Y¥o + Yo— pf) },
and therefore
X+tY =(pVA*/2a7){ | ¢ |? exp (tu) —Coceexp (—tw)}. . (9)
5. Field of flow due to rotation of the boundary.— When

the boundary has a motion of rotation, the specification of
the liquid motion presents greater difficulty ; the outline
of the procedure is as follows.

One motion is known which satisfies the proper condition
at the moving boundary—namely, a rotation of the whole
liquid, as if rigid, with the same angular velocity as the
boundary. ‘This may be called the first motion. It is not
the required motion because it is rotational, and because. it
has infinite velocity at infinity.

Another motion, which will be called the second motion,
can be specified. This also is rotational, having the same
vorticity at every point as the first motion; but at the
boundary its normal velocity is zero. It has infinite velocity
at infinity.

If the second motion be subtracted from the first motion,
the result is an irrotational motion whose normal velocity
at the boundary is the same as that of the boundary itself.
This may be called the difference motion. If the velocities
of the first and second motions tend to equality at infinity in
such manner that the difference motion tends to zero at
infinity and has no circulation round the solid, the difference

* J. G. Leathem, “Some Applications of Conformal Transformation
to Problems in Hydrodynamics,” Phil. Trans. Roy. Soc., A, vol. cexy.
1915, § 17.

Two-Dimensional Motion of Infinite Liquid. 123

motion satisfies all the requirements of the problem, and is
the motion due to the rotation of the solid.

6. The jirst motion.—The first motion will be taken to be
a rotation, as if rigid, with angular velocity w, about the
point z=« exp (iy). There is no loss of generality in this
choice, as the substitution of another centre of rotation can
always be effected by superposing a motion due to trans-
lation of the boundary as explained in article 3.

The first motion may be specified by its stream function Wy,

namely

Views ~KeRp (iy) | 2p. Wei atic) CLO}
which is also expressible as a function of and 7. Another
specification is by w,, v1, where

m=dW/07, wu=—dW/os ~. . (11)
these also being regarded as functions of and 7. It is to
be noted that wu, and v, are not velocity components, but that
the velocity components in the directions corresponding
respectively to € and y increasing are u,/h and v,/h.

From formula (4) it follows that

r (ant
&=KcCOSyt ee { Koe274/- sin (e-m)

Ks 4 -on(e-Inid gin ( 27ST DE )}
ee rok 4 sin ( 1 i Ve Hotes

— 7 r 27/A 2a )
y=e« sin yt oe} woe “ cos(=ZE—yp

2 ey, e cos a ae He Yel Cs
so that, when 7 is great,

—wh? oe
Wi= ya | Keele" — Quote, 008 (= E+)

— Koyk3e 270A Cos (es 13—0) + eve \ 5 (12)
and
ye 1 6 |
uj= = } ReneS Kykge7271* cos (= E+s—Y0) “Fw. Me
(13)
OX . {4
N= 5 { 2KoK, Sin (Ze+y-m)

+5 eege20 sin (7 é+1-n) Te i } , (14)

the terms being arranged in descending order of importance.

124 Dr. J. G. Leathem on the

7. The second motion.—The second motion, a rotational
motion having zero normal velocity at the boundary, is got
by an imaging of the vortex distribution ; and the utility of
the periodic conformal transformation consists in the fact
that it makes this imaging process possible.

The specification of the motion may be by a stream-
function y, or by functions (us, v) equal to Ow,/dy,
—Ov,/0&, the corresponding velocity components in the
z plane being uo/h, vo/h. 3

It is convenient, for a moment, to think of Wp as the
stream-function of a motion in the € plane, of which (ug, v)
would be the velocity. If such a motion, unhampered by
any rigid boundary, were due to a line-vortex representing
a circulation m, situated at €=¢', and periodically repeated
at C+rr, (r=1, 2, 3,...), it is known * that the corre-
sponding stream-function would be

m a aT ,
—3, log sin 5 (6-0) |.

When the line »=0 is a rigid boundary an image must be
introduced at the point €={'', where €” is the complex
conjugate to ¢', and the stream-function is

sin 7 (6-8) /sin 7 E—£")

Instead of single vortices a continuous periodic distri-
bution of vorticity may be postulated over the whole area
between »=0 and n=7, the former line being still a rigid
boundary; and if m be replaced by o(&')dS', where d§'’ is
an element of area and o a density of distribution, the
stream-function is

— x fot log

EEL a
2a °8

sin (6—£')/ sin =(¢—0") | dS’, (15)

the area integral being taken over a rectangle of length ¢
and breadth 2X.

If dS in the z plane and dW! in the ¢ plane be corre-
sponding elements of area,

dS = §h(2)}2d8' ;

so, if the circulation round the contour of any area in the
€ plane is to be equal to 2 times the corresponding area in
the z plane, it is necessary that

o(t') = 2wfh(E) 2... (16)

* Proc. Royal Irish Academy, J. c. § 13.

; Yo 2.00

Two-Dimensional Motion of Infinite Liquid. 125

The stream-function is then

Ee tr sin 5 (6—£') | |
v= | | {h(&') }? log A aaa dé'dn', (17)
$ f 1a)

Be sin 5 (S—¢ )
and this specifies a motion in the z plane which has vorticity o
at all points for which t>7>0, and has the curve corre-
sponding to 7=0 as a fixed boundary.

The corresponding u, » functions are

- 27
Nears sinh — (n—7')
1 = OH zs | Uys *

Ce Pa cosh = (nn!) — cos" (E—-#')
sinh = (9 +7)
anes NS I
: cosh —- (n + 1H) 608 (ons)
i Pr sinh = (£—£')
ot Ov: @ 1) V2 :
oa =| \ Se credence aos HoRaCT a
ey). cosh (n—7) —e0s-—- (eo)
sina (£—£)
St)

21 A ae I
cosh 2 (1+ 9!) cos (E- £')

It is to be noticed that if € is inside the area of integration
the subjects of integration in (17), (18), and (19) have
infinities at ¢’=€; these infinities, however, are not suffi-
ciently powerful to make the integrals divergent.

8. These formule may be checked by noting directly what
conditions uw, v; must satisfy if they are to represent the kind
of motion in the z plane which has been described in the
previous article.

& and 7 are curvilinear coordinates in the z plane, and the
corresponding velocity components are U=u,/h and V=w,/h.
The boundary condition V=0 or v,=0, when »=0, is clearly

126 Dr. J. G. Leathem on the

satisfied. The equations of continuity and of vorticity are
respectively

fo) U 0 Vh) = ) VA) = 2260 = 26
5E! ie oa hi a 3E! i 3a 0
or Out , O% Ove Os _ Y12
Ee ioe hii Of On mat

when @ is inside the area of integration.

The testing of these equalities involves the differentiation
of the integrals of formule (18) and (19), and this cannot
be done by the ordinary rule of differentiation under the
sign of integration, since that would yield semi-convergent
integrals. It is, however, easy to apply the method of
differentiation explained in the Cambridge Tract on ‘ Volume
and Surface Integrals used in Physies,’ articles 21 and 23,
and it is then readily verified that «% and x satisfy both
conditions.

A single compact formula giving both u,; and x% is

U+W= a | (e/)}2 | cot (6— ¢’)—cot - (f— | dé' dn’. (21)

The integral on the right-hand side has the appearance of
being a function of the complex variable €; but this appear-
ance is deceptive, for if v and u were conjugate functions
there would be no vorticity *.

9. The next step that suggests itself is a passage to limits
for ¢ infinitely great. This is feasible in the case of 2%, but
the integral representing u; proves to be divergent.

It will be shown that this can be remedied by adding to uw,
before passage to limit, a suitable function of ¢ which does
not involve & or 7. An addition to uw means simply the
superposition of an irrotational motion with circulation
round the fixed boundary. This will serve to cancel an
undesired circulation at infinity in the motion defined by
Uuzand v%.

No corresponding addition to y need be or could be
made.

It being necessary to consider not only the convergence
of the w and v integrals but also the forms to which these,
regarded as functions of € and , tend for » very great and
positive, it is important to notice two expansions of the
function which appears in square brackets in formula (21).

* On this point compare the writer’s note “On Functionality of a
Complex Variable” in the ‘Mathematical Gazette,’ early in 1918.

Two-Dimensional Motion of Infinite Liquid. 127
If y'>n>0

cots ($—£') — cob (6-8)
By; [s+> exp ( 2571) cosh 4 fn —i(E— ey}, (22)

and if 7>7'>0

cot (f—£') cots (f—8")

= — 415 sinh( 2") oxp 7 GigE) —n},- (23)

where s=1, 2, 3,.... The formule can be verified by
noticing that each side of each equality is equivalent to

1 i!
2i 2 mt 2 aed ©
aay rare eer)
For 7' great the most important terms of {A(f')}? are
4arn! Arré'
{h(f’) |}? = Noe exp (FS2) + 2a COs = +%—1)

; 9 / “6 /
+ 2Kok3 EXP (=== ) cos (“ee +9590) + Ue Pe).

the terms decreasing by successive negative powers of
exp (277’/A); it is to be noted that the functions of &
which multiply these exponentials are sums which may
contain constant as well as harmonic terms.

In studying the form of the subject of integration in
formula (21), for great values of 7’, with a view to
examining the divergence for t->«, the product of the
series (22) and (24) may be used. Tor this purpose a term
of the product may be ignored if its integral with respect
to &' through a range J is zero, or if it contains as factor the
exponential of a negative multiple of 7'/A. By one or other
of these tests every term is negligible except one, namely,

21K, exp (41rn'/2),

which contributes to the integral, at its upper limit (after
integration with respect to &’),

(2x 9°A?/22r) exp (47rt/2).

128 Dr. J. G. Leathem on the

Hence the subtraction of (ix ’@A/277) exp (47t/X) from
the right-hand side of formula (21) gives an expression
which has a definite limit for too. This justifies the
definition

an 2) 2
0+ iu, =< Lim f ves oo
AV t>o 2a

+( ("a0 ¥ { cot= (£—£')—cot < (= ¢") bag dr} |. (25)

10. Limiting form of the second motion at infinity——The
formula (25) defines a motion which has all the characteristics
required for the second motion, with the, as vet, possible
exception of tending to the proper form at infinity. It
is now necessary to inquire what are the limiting forms to
which wz and v2, functions of £ and 7, tend with indefinite
increase of 7.

For this purpose the series-expansions of formule (22)
and (23) may be used, each within the appropriate range
of 7’. If [22], [23] be used as abbreviations for the
expressions on the right-hand side of these formule,

Or. tKy Amt/A pe: 12 Lge
Vg tu, = 1 Lim me é =F {h(g )} [ 22] dé dn
re n/ 0

As the integral of formula (25) is absolutely convergent
in respect of the infinity of the subject of integration
at €'=6€, it is safe to use the series [22] and [23] right up
to the critical value n'=7 which separates the ranges within
which they are respectively valid. For {h(&’)}? the series
of formula (24) is again employed.

In taking the term-by-term products of the two series
which are multiplied under the sign of integration, any
resulting term may be passed over whose integral with
respect to £&’ over a range X is zero. Thus a term of the

the type
cos { (2arm/2)(E' + «)} cos {(2arnfr) ('+ 8)}

need not be considered unless m=n. Further, when only
an approximation for 7 great is desired, an estimate of the
importance of an exponential in 7’ and 7 is to be made
on the hypothesis that 7 is very great but that 7’ is of

Two-Dimensional Motion of Infinite Liquid. 129

a higher order of greatness at one of the limits of inte-
gration, while »'=7 at another of the limits. These con-
siderations reduce the important terms in the equivalent
of the square bracket in formula (26) to

Ug ant 4 43 (* aL 4mn!)®
a “4 49 Fe eon
ke ele
92 —4iry'/X Aor : F Ar / d Md !
+2e xox.cosh {9 —(E— E')} cos ae +%2—Y0 ) | dE dy

yee ee Arn! 4

—1i{ { 2K Kg sinh (=") cos(=" £ +7-%)
Arr ..

x exp — {i(E—£')—n} dé'dy/,

which reduces, after omission of some negligible elements,
to a form whose limit, for t->oo, on substitution in (26)
yields the formula

, Ve mre ss . [4
v2 tin Os | — iko’ EXP (=") + 2K ok, Sin (ZF +9) | .
(27)

11. The difference motion.—If formula (27) be compared
with formule (13) and (14) it is seen that

Vo+ig—(¥,+iu,) >0. © . . . (28)
Hence if (u, v) specify the difference motion, so that
OSes Uo, «U == Uy—- 095

u and v tend to zero at infinity.

Thus the difference motion is an irrotational motion,
vanishing at infinity and having at the boundary a normal
velocity corresponding to rotation about the point « exp (7y).
It is free from circulation, as a circulation would involve,
for 7 infinite, a definite limit value of w different from zero.
It therefore constitutes the solution of the problem of motion
due to the rotation of the boundary.

12. Forms of boundary to which the method applies——The
applicability of this method to solving the problems of
motion due to translation and rotation depends upon the
knowledge of a periodic conformal transformation which
will make any particular form of boundary correspond to
the real axis in the € plane. That a considerable variety of
such transformations and their corresponding boundaries is

Phil. Mag. 8. 6. Vol. 35. No. 205. Jan. 1918. K

eee ee Se

130 Two-Dimensional Motion of Infinite Liquid.

available is demonstrated in the writer’s paper on the
subject referred to above. In particular, mention may be
made of polygonal boundaries (/. c. § 8); and it may be
noticed that for a regular polygon of n sides the trans-

formation is
2
dz mamta AY 5 sn K sme |e
perth TOs”. ta) Onna ae
ae K | a sins (¢ ~)| at sin > , eee

so that 2
K? 2 NTN 2nié n :
2 — ‘ iS re .f SESE S Pete. $
(i ig | 2 { cosh Wc 4] > Ve ie

K being a constant ; the latter expression, with accented
letters, would be the first factor under the sign of inte-
gration in formula (25).

In all cases where the periodic transformation is known
the solution of the hydrodynamical problems is reduced to
quadratures.

In certain cases the integrations can be completed ; this
is noticeably the case when /(f), and therefore also h?, is the
sum of a finite number of terms harmonic in ¢. The inte-
gration may be accurately effected by the method used for
approximation in article 10 above. Of the terms arising
from the multiplication of h? into the series [22] and [23]
there are only a finite number which do not yield zero result
when integrated with respect to & through a range 2, and
each of these can be integrated separately with respect
to 7’.

The simplest example is the ellipse, for which the trans-
formation is

3) bo

z= ccosh {a—(Qri/r)t}, . . . . (BI)
so that |

2.2 / 9
Higa aut { cosh 2 a+— 1) — Cos (= e)t. ae

The working out of this case may be used to test the
method, as the results are otherwise known.

Another simple integrable case corresponds to a boundary
whose polar equation is

r=a@+2bc0s 20, (a>2b). . ..).) (ao
The transformation is
z = bexp (—27ig/A) +a exp (2726/X) + bexp (671f/r). (34)
12th November, 1917.

pai gahs 4

XIII. On the Relation of the Audibility Factor of a Shunted
Telephone to the Antenna Current as used in the Reception
of Wireless Signals. By Prot. G. W. O. Hows, D.Sce.,
MLE E.*

[See paper with the same title by M. van der Pol, vol. xxxiv. p. 184.]
ie audibility factor of a radio-telegraph signal is defined

as the ratio of the actual sound-producing current in
the telephone-receiver to the minimum value to which this
current could be reduced for the signals to remain just
readable. It is assumed that the wave-form of the telephone
current, and therefore also the character of the sound,
remain the same in the two cases. This ratio is usually
determined by shunting the telephone-receivers with a non-
inductive resistance until the signals are only just readable.
If there is any possibility of the total rectified current being
affected by the decreased resistance of the detector circuit
due to the addition of the shunt, a resistance should be
inserted in series with the shunted receiver to maintain the
total resistance of the detector circuit approximately constant.
From the value of the shunt it is then necessary to calculate
the ratio of the total or joint current to that through the
receiver.

It is not clear from Mr. van der Pol’s paper how he deter-
mined the resistance of the receiver which he gives as
1240 ohms; but since nothing is stated to the contrary, it
would appear that he has treated the receiver as a non-
inductive resistance equal in value to the actual resistance
of the receiver to continuous current. If so, the results
obtained will be in error for two reasons: firstly, because
the effective resistance of a telephone-receiver at the fre-
quency employed, viz. 467, is considerably greater than its
resistance to continuous current ; and secondly, because an
alternating current divides between two alternative paths in
a manner depending on the impedances and not on the
resistances.

As an example of the magnitude of the error thus intro-
duced, the following figures may be quoted: a 3200-ohm
receiver had an effective resistance Ry at 750 cycles per
second of 6200 ohms and an impedance Z of 9320 ohms,
whilst at a frequency of 1000 these values were increased to
7250 and 11,200 ohms respectively. Thus Z75>=2°9R, and

* Communicated by the Author,

132 Prof. G. W. O. Howe on the

LZio0=3'15 Ry. In the case of a 60-ohm receiver, it was
found that Ze37=4°27R, and Zi135=6'35R,. As a rule at
such frequencies the reactance is of the same order as the
effective resistance, so that the current lags about 45° behind
the terminal P.D.; this is, of course, merely a rough
approximation.

There may be some doubt as to the correctness of treating
the pulsating telephone current as a simple alternating
current; but in the opinion of the writer, the pulsating
current of audible frequency produced by the detector as the
result of the successive wave-trains may be regarded as a
steady current with a fundamental alternating current and a
number of harmonics superposed upon it, the fundamental
giving the pitch, and the harmonies the character of the sound
heard in the receiver. If the character of the note remain
constant, it would appear sufficient to consider the amplitude
of the fundamental, and to assume that this sinusoidal current
divides between the receiver and the shunt in accordance
with the ordinary laws of alternating-current circuits.

The writer is well aware that references can be given to
papers in which the ordinary continuous current-resistance
of the receiver was apparently used in calculating the audi-
bility factor, but in a recent paper Austin, who has done
much experimental work on this subject, is careful to point
out that the effective resistance of the receiver must be
determined for the given frequency and telephone pulse
form *,

Since Mr. van der Pol refers to papers by Hogan and
Love, both of whom refer to the impedance and not the
resistance of the telephone-receiver, it is possible that he has
also used the impedance, notwithstanding the statement in his
paper. If so, the paper would be of greater value and
interest were this definitely stated.

If Mr. van der Pol did not take the precaution to keep
the resistance of the detector circuit approximately con-
stant, as mentioned above, the correctness of his experimental
results is open to some doubt.

In order to see in what direction his results would be
modified by employing the impedance of the receiver
instead of the resistance, it has been assumed in the follow-
ing table that the impedance Z is equal to four times the
resistance Ro, and that the telephone current lags 45 degrees
behind the P.D.

The values obtained from the simple vector diagram are
as follows :—

* Proc. Inst. Radio-Engineers, 1917, v. p. 289.

Audibility Factor of a Shunted Telephone. 133

| | Ros Z+8 | I (total) R,+8| I
) Fal® S | S I (receiver) 108 S | 108 Ir
To ee el be Zeal) Les 0-097 | 0:267
0) ee 3 2°8 0176 | 0447
1 ia | 5 4°77 0301 0-679
3 4 vos ks 12-72 0602 | 1104
10 Ls aed Irs seed 40°71 1041 | 161
30 es | 121 120-7 1491 | 2-081

100 101 | 401 | 400-7 2-004 2°602 |

In view of the unavoidable lack of precision in all audi-
bility tests it is obviously sufficientiy accurate to neglect the

phase of the telephone current and take z z : as the
audibility factor. ~

The figure is a reproduction of fig. 2 in Mr. van der Pol’s
paper, with the addition of two dotted lines which represent
log M plotted against log + instead of log ae a For the
upper dotted line it has been assumed that Z= ks whilst for

x (total)

iL (receiver)

Sates 06 OS) VEO 2) 4 / 66) FE 20922 3 24 26

the lower Z=4R, as in the table above. Itisseen that if the
impedance of the receivers is three or four times their con-
tinuous current resistance, the resultant curve is considerably
modified, and with it the conclusions based thereon. If
further tests show that the results are not modified when the
resistance of the detector circuit is kept constant, it would
appear from the slope of the dotted lines that the audibility
factor varies as a higher power than the square of the radio-
frequency current. This appears improbable.

cease? |

Notre By Mr. vAN DER POL, JUNR.

The well-known facts respecting the division of current
between a telephone and its shunt circuit are correctly stated
by Prof. Howe, but he has rather lost sight of the motive of
my paper and of certain experimental difficulties. The
principal object in view was to test whether the audibility
factor could be considered to vary as suggested by Prof.
Love with the received antenna-current.

In discussing some experiments by Dr. Austin, Prof. Love
makes use of the audibility factor defined as R+8/S, where
S is the resistance of the shunt and R the telephone resistance*.
Jn order to test experimentally his suggestions as to the
proportionality of the so-defined audibility factor to the first
power or square of the antenna-current I had to use the
same constants. My experimental results appeared to be in
close agreement with Prof. Love’s suggestions.

The same definition (with the aid of the telephone resist-
ance) of the audibility factor is used by several other writers.
Prof. Howe refers in his paper to a very recent publication
by Austin which was published after my paper had been
sent to the Philosophical Magazine. Here Austin refers
to the telephone-impedance, but on the other hand, in a
former paper by the same experimentalist, he defines the
audibility factor using the telephone resistance instead of the
impedance f.

It is by no means clear whether Austin or Hogan em-
ployed the true impedance of their telephones in the
audibility factor, as in their papers cited no references at
all are given how they determined these impedances.

Farther, itis a matter of considerable difficulty to measure
the true impedance of a telephone when used as in Wireless
Telegraphy in series with a crystal detector, and therefore
traversed by an intermittent or pulsatory current, the wave-
form of which is not known. From the pronounced variation
in character of the tone in the telephone-receiver with dif-
ferent couplings it may further be concluded that, probably
as a consequence of the irregular shape of the characteristic
of mosi crystal detectors, the telephone current, while varying
in intensity also (opposite to the suggestion of Prof. Howe)
varies in wave-form, so that it is doubtful if the ordinary
well-known theory of sine-form currents may be applied to
the shunted telephone method.

Moreover, the current in the telephone circuit at the

* Phil. Trans. Roy. Soc. Lond. cexy. A. p. 128 (1915).

+ Bull. Bureau of Standards, vi. no. 4, p. 581 (1910). See also
J. ae Murray, ‘A Handbook of Wireless Telegraphy’ (1914),
p- 349,

Geological Society. 135

moment when the measurement is made is extremely small
and quite beyond reach of any thermo-electric ammeter.
The writer is therefore of opinion that an exact experimental
determination of the telephone impedance under actual
working conditions is a matter of higher order of difficulty
than the measurement of received antenna-current itself.

It must further be borne in mind that in any ease the
shunt value which quenches the telephone sound is difficult
to determine in practice with any but a rough approxima-
tion. Ina very quiet room it may perhaps be determined to
within 5 or 10 per cent., but in a wireless station or on board
ship perhaps not within 30 or 40 per cent.

No assumptions as that made by Prof. Howe that the true
impedance of the telephone under actual working conditions
is equal to four times the steady resistance has been justified
by any experiments. Hence, to avoid suppositions not based
on experiment, the value taken for the calculation of the
audibility factor in the case of my experiinents was the
steady resistance, although I was perfectly well aware that
this was not identical with the true impedance for the wave-
form and frequencies used.

Having regard to the uncertainty attending the constants
employed by Austin and Hogan, and the difficulty of deter-
mining exact values, it seemed better to base the reduction
of the observations on known measurements rather than on
assumptions as to the ratio of impedance to resistance.

XIV. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
{Continued from vol. xxxiv. p. 528.]

June 20th, 1917.—Dr. Alfred Harker, F.R.S., President,
in the Chair.

{i as following communications were read :—

1. ‘The Pre-Cambrian and Associated Rocks of the District
of Mozambique.’ By Arthur Holmes, A.R.C.S., D.I.C., B.Sc.,
F.G:S.

Beyond the coastal and voleanic beds of Mozambique (described
in a previous contribution—Abs. Proc. Geol. Soc. 1916, No. 994,
p- 72) the country assumes the form of a gently undulating plateau,
gradually rising towards the west and diversified by innumerable
inselberg peaks and abruptly-rising clusters of hills. The
dominant rock throughout is a grey biotite-gneiss. Interfoliated
with this are occasional lenticular masses of hornblende-gneiss and

136 Geological Society :—

amphibolite, and within these smaller bands of crystalline lime-
stone are sometimes preserved. In many places the gneisses
become garnetiferous, while eclogites and basic granulites also
occur. Schists—referable to arenaceous sediments—are found
only near the coast, where they are interbanded with gneisses;
and, as the latter are mainly of igneous origin, they are thought
to be intrusive into, and therefore younger than, the schists. As
a general rule, the foliation and the banding of the gneisses are
well defined in parallel uncontorted planes, the strike being com-
monly along, or somewhat north of, a north-east to south-west
direction. In certain inselberg peaks, the strike sweeps round
the contours, while the foliation-surfaces dip quaquaversally from
the summits. Into the gneisses later granites, belonging to at
least two different periods, have penetrated, riddling them with
enormous numbers of small intrusions, lit-par-lit injections,
tongues, and apophyses. Rocks of later age are rarely met with ;
but, in a few places, dykes of picrite and pyroxenite have been
found cutting the youngest pegmatites.

The succession of rocks in eight of the better-known districts is
described, and the following general classification is based on the
details thus provided :—

Ultrabasic Pyroxenites.
Dykes. Picrite. | Age unknown,

Intrusive Contact.

Granites and f Biotite-Granites. Pre-Cretaceous
Coarse Graphic Granite and other Pegmatites. and Post-Middle
Pegmatites. Quartz- Veins. Pre-Cambrian.

Intrusive Contact.

eee hoe if acne ae (including porphyritic
Granites: | varie ties).
Poemiaribos ie Pegmatites and Aplites. | Middle
A eat fi a | Pyroxene-Granite and Pyroxene Quartz- | Pre-Cambrian.
Aya | Diorite Series. ; Pb/U=0'14 to
he | Pyroxene-Granulites ? ) 0°17,
Intrusive Contact.
( Biotite-Gneisses and )
Gneisses | Gneissose Granites. |
and 4 Hornblende-Gneisses. Pb/U=0°21.
Associated | Amphibolites.
Rocks. ; Garnetiferous Gneisses,
| Granulites, and Eclogites.
Intrusive Contact. | yee

Pre-Cambrian.

Quartz-Mica-Schists.
artz-Magnetite-Schists.
ornblende-Garnet-Schists.

. Qu
Crystalline H
Heematite-Schists.
Fo
Cr

Schists and i
|

Limestones. rsterite-Marbles, and other

ystalline Limestones.

ee

Pre-Cambrian and Associated Rocks of Mozambique. 137

The above correlations of certain groups of rocks with the
Lower and Middle Pre-Cambrian of other regions are based on
the determination of lead-uranium ratios of zircons derived from
the gneisses and granulitic granites respectively, the zircons
having been obtained by crushing and panning the rocks in the
field. The gneisses give a ratio of 0:21, comparable with a ratio
of 0:24 obtained for Canadian zircons of Laurentian age. The
granulitic granites give ratios of 0-14 to 0:17, comparable to those
of radioactive minerals of late Archean: that is, late Middle Pre-
Cambrian, age in Scandinavia (Moss 0°12 to 0:15, Arendal 0°16 to
0-18, and Ytterby 0°15 to 0°17), Canada (Villeneuve, Quebec, 0°17),
and India (Singar 0°14).

The rocks are described in detail, with tables giving the quanti-
tative mineral composition and the specific gravities and radium
contents. Numerous examples of contact-phenomena between
crystalline limestones and various types of igneous rock are
recorded: pyroxene, amphibole, sphene, and soda-lime felspar
being the new minerals chiefly developed between granite and
limestone, with garnet and scapolite also in special cases.

With reference to the origin of the crystalline limestones and
gneisses, the following conclusions are arrived at :—

(a) The crystalline schists and limestones are interpreted as arenaceous
and caleareous facies respectively of an ancient sedimentary series,
their argillaceous complements being unrepresented unless they
enter into the composition of the biotite-gneisses.

(b) The limestones have controlled the formation of hornblende -
gneiss and amphibolite by their interaction with a granitic
magma that elsewhere is represented by biotite-gneisses. The
cores of the limestones have been enabled to resist further
silicification by being thus enclosed within a blanket of rocks
impoverished in silica.

(c) If the ancient sedimentary series included argillaceous formations,
it is thought probable that the gneisses are composite rocks
produced by the concordant injection of granitic magma into
such formations. This view, although not proved, is supported
by mineralogical and radioactive evidence, and by the fact that
in certain inselberg peaks the banding of the gneisses gradually
dies away as the slopes are ascended, the rocks passing into
granulitic granite nearly free from biotite and showing few
traces of foliation. These peaks are interpreted as the irruptive
foci of granulitic magmas which fed the lateral intrusions repre-
sented by the surrounding gneisses.

It is shown that there are at least three types of inselberg
peaks that owe their survival to peculiarities of structure and
composition. The first type is that just mentioned, in which
the foliation is less marked and the biotite-content appreciably
lower than in the surrounding gneisses. In the second, the
peaks are mainly composed of granulitic granite (again poor
in biotite compared with the gneisses), and in the third type
the peaks are riddled with tongues and apophyses of pegmatite

Phil. Mag. S. 6. Vol. 35, No. 205. Jan. 1918. L

138 Geological Socvety:—

and aplite. In each case the greater resistance offered to denu-
dation is related to the presence of less foliated and more felsic
rocks than are found in the adjacent plains. There remains a
fourth type—perhaps the most abundant—in which no differences
have been recognized. Many of these seem to be isolated relics
of gneissic escarpments; and it is suggested that desert erosion,
involving the attack of slopes at their base by arid weathering,
and the removai of disintegrated material by wind, is the most
favourable condition for the development and maintenance of
an inselberg landscape. Existing conditions of denudation are
considered to be unfavourable to inselberg survival; for the
peaks appear to be worn down by the removal of superficial
layers by exfoliation more rapidly than the surface of the plateau
is lowered.

2. ‘The Inferior Oolite and Contiguous Deposits of the Crew-
kerne District (Somerset).’ By Jinsdall Richardson, F.R.S.E.,
BG:S:

November 7th.—Dr. Alfred Harker, F.R.S., President,
in the Chair.

A Lecture on ‘The Nimrud Crater in Turkish Armenia’
was delivered by Fetrx Oswaxp, B.A., D.Sc., F.G.S.

The Nimrud voleano, one of the largest volcanic craters in the
world, is situated on the western shore of Lake Van, and was
surveyed and investigated geologically for the first time by the
speaker in 1898. The western half of the crater is occupied by a
deep lake of fresh water, while the eastern half is composed of
recent augite-rhyolites, partly cloaked in white volcanic ash. The
erater-wall is highest on the north (9903 feet), rising in abrupt
precipices over 2000 feet above the lake (7653 feet). The southern
wall is also precipitous, but only reaches the height of 9434 feet
(the south-eastern part). A large slice of the crater-wall has
slipped down on the south-west, so as to form a narrow shelf,
800 feet above the lake. The crater is nearly circular, 8405 yards —
from west-south-west to east-north-east, while the transverse axis
is 7905 yards. The lowest points lie on the long axis, reaching
only 8139 feet on the western, and 8148 feet on the eastern rim.

The ecrater-wall has an external slope of 338° on the south and
east, where it consists exclusively of overlapping lenticular flows of
augite-rhyolite and obsidian. On the south-west, west, north-west,
and north these are capped by thin sheets of cindery basalt which
must have possessed great fluidity, extending for many mules to
form wide plains of gentle slope and great fertility down to Lake
Van on the east and into the Plain of Mush on the west. These
basalt-flows dammed up the north-east to south-west valley between .

The Nimrud Crater in Turkish Armenia. 139

the Bendimahi and Bitlis rivers, and thus brought Lake Van into
being.

The history of the Nimrud voleano may be summarized.as follows
from the speaker’s observations :—

1. Its forerunner was the Kerkur Dagh on its southern eget
a denuded mass of grey augite-trachyte, rising to 9000 feet, and
crowned by many peaks. It was probably erupted i in the Pliocene
Period, subsequently to the folding of the Armenian area, in which
the latest folded rocks are of Miocene (Helvetian—Tortonian )
age, occurring north of the Nimrud Dagh and consisting of lime-
stones with corals (Cladocora articulata, Orbicella defrancez, &c.),
Tithothamnion, Foraminifera (Lepidocycline Orbitotdes, Amphi-
stegina, &c.), beds of Pecten (P. urmiensis, &c.) and of oysters
(Alectryonia virleti). Nimrud and the other numerous volcanoes
of Armenia came into existence at a period when the sedimentary
rocks could no longer be folded, but were fractured along definite
lines, and Nimrud is sce on the great fracture transverse to the
Armenian folds at the apex of their bending round from the Anti-
tauric (west-south-west to east-north-east) to the Persian (north-
west to south-east) direction, and it also marks the point of inter-
section of this fracture with a great north-east to north-west
fracture (Caucasian direction), which delimits on the south Lake
Van and the faulted depression of the Plain of Mush, abruptly
cutting off the Tauric horst of pre-Devonian marbles and mica-
schists.

2. Numerous flows of augite-rhyolite built up the vast cone of.
the Nimrud Dagh, and the 1 increasing pressure on the central vent
became relieved by extrusions of ‘augite- trachyte along radial
fissures, forming the present promontories of Kizvag, Zighag, and
Karmuch.

3. A presumably long period of inactivity was followed by violent
explosions destroying the summit of the cone, and from this crater
(smaller than the present one) vast lava-flows of a very fluid basalt
(crowded with phenocrysts of labradorite, pale-green augite, and
some olivine) flooded the country and filled up the Bitlis and
Akhlat valleys, which have since then been eroded a little below
their former depth. The Sheikh Ora crater of basic tuff (now
breached by Lake Van) probably belongs to this period.

4. Further explosions widened the erater in which a large lake
was formed, while the eastern half of the crater became filled by a
succession of outflows of augite-rhyolite, in which numerous blow-
holes were drilled, bringing to the surface large blocks of basaltic
agglomerate and also affording sections showing the transition
downwards from obsidian, spherulitic obsidian, and spherulitic
rhyolite to banded augite-rhyolite (with sanidine and green augite
in a micropecilitic ground-mass).

. The last eruption was. recorded in 1441 by a contemporary
thio chronicler, and resulted in the extrusion of a very viscous
augite-rhyolite along a north-to-south zone of weakness, both inside

140 Intelligence and Miscellaneous Articles.

the Nimrud crater where it separated off part of the large lake to
form the shallow, so-called ‘hot lake,’ and also to the north of
Nimrud, where it rose up fissures and in a small crater.

6. A violent earthquake in 1881 which destroyed the village of
Teghurt, at the eastern base of the crater-wall, was the last sign of
activity; but earthquakes are still frequent in the Plain of Mush at
the western foot of the Nimrud Dagh, and recent fault-scarps are
clearly visible along the borders of this faulted depression.

The speaker mentioned that he had presented his model of the
crater to the Museum of Practical Geology (Jermyn St.) and the
rocks and slides to the British Museum (Natural History), where
his fossils from Armenia are already preserved.

XV. Intelligence and Miscellaneous Articles.

COUPLED CIRCUITS AND MECHANICAL ANALOGIES,
Phil. Mag. Dec. 1917.

To the Editors of the Philosophical Magazine.

GENTLEMEN,—

(GENTEN ARIAN PERIGAL is clean forgotten today, and his

_ yaluable kinematic work on his lathe. His method should
be revived of drawing the ellipse or other Lissajous figures of
combined vibration, as on p. 515, fig. 2.

The enveloping rectangle is divided up into elementary rect-
angles by lines spaced, not equidistant, but in equal time of
simple vibration.

Perigal does this by describing a semicircle on each side of the
rectangle, and then produces the ordinates of points at equal
angular interval round the circumference.

Starting at any point of crossing and tacking across the
diagonal of an elementary rectangle, a succession of points is
made on an ellipse inscribed in the rectangle, and the points
are close enough to be joined up in a continuous curve, such
as Perigal could cut in his lathe.

If m and 7 steps are taken for a diagonal, the Lissajous curve
appears for a combination of two vibrations of m and n fold
frequency, and the phase difference of lead or lag is settled by

the position of the starting point.
Yours sincerely,

Dec. 18, 1917. G. GREENHILL.
1 Staple Inn, W.C. 1,

ash

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Phil. Mag. Ser, 6, Vol. 35, Pl. IT.

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Basv. Phil. Mag. Ser. 6, Vol. 35, Pl. TIT.

The iteenon of Light by a Cylinder of radius 1°54 em.

Phil. Mag, Ser. 6, Vol. 35, Pl, I.

ROWNING.

3

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Masses 20:f

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| Mag. Ser. 6, Vol. 35, Pl. IV,

sity Curve due to impact of Spheres
s but different materials.

ball; Diameter, 23 inches; Mass,
aterial, wood; Diameter, 23 inches;

Fie. 4

—s +o

(Material, 4 —.

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nsity Curve due to impact of two
ameters and densities.

iameter, 14 inches; Mass, 118 gms.
lameter, 3 inches; Mass, 158 gms.

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BANERSTI.

Phil, Mag. Ser, 6, Vol. 35, Pl. IV,

: - =
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Fie. 7.—Observed Foym of Titensity Curve due to impact of Spheres
Zone atciniine oneal Entenaiy avounaiea of nearly equal diameters but different materials,
Fic. equal colliding spheres. Sa Sphere on left: Material, billiard ball; Diameter, 22 inches; Mass,
SiR
!

ies, HO) eseeont Tn . : ss oar
Pia. 3.—Culeulated Form of Intensity Curve due to two Spheres Tre. 5. pe ef eRe) due to impact of _—
of diameters 2:1. [k(a+-b)=2.] (Material, wood; diameters 3 inches and 12 inches respectively, :

. ) ; p Y

—-

150 gms. Sphere on right: Material, Wood; mi

Mass, 66 ons Diameter, 2) inch oS;
ss, 66 ons,
2,— Observe! t) i i s im t of
Fie. 2. Ob ved distribution of intensity due to impac'
ix two unequal spheres of wood.

sl eS Iie. 8.—Observed Form of Tntehsity Curve due to impact of two
\

Spheres of different diameters and densities,

1c, 6.—Caleulated Form of Intensity Curye due to two equal Sphere on left: Material, brass ; Diameter, 1} inches; Mass, 118 gins.
nk Sphere on right; Vi, 4.—Same as fig. 3, but with k(+2)=3, Spheres of densities 4:1. Sphere on right: Material, wood ; Diameter, 3 inches; Mass, 168 gine.
Sphere on left ; 21 inches dinmeter.
24

3 inches diameter.

a) a ee ty
rs ies of iNet: a7 A
ee ED oe

ro ie oh
¢ = a

a ‘ ad

Mirra. Phil. Mag. Ser. 6, Vol. 35, Pl. V.

Illustrating the asymmetry of the fringes and of the illumination-curves in
oblique diffraction by a reflecting surface consisting of two parts (Figs. I
te IV) or three parts (Figs. V and VI) in the same plane.

eH
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SHRIMB.J AAD 5 .y
FEBRUARY 1918. eda ee

XVI. Continued Discussion of the Astronomical and Gravi-
tational Bearings of the Electrical Theory of Matter. By
Sir Ottver Lover *.

Part I.

M* short summary in the December number of the
Phil. Mag., page 519, put prominently forward the
idea that the expected effect required that the additional
inertia due to motion should be independent of gravitative
influence ; for the conclusion seemed obvious that if weight
and mass varied together there would be no change in accele-
ration, and that in that case it did not matter how much the
mass of a revolving body varied. But I soon perceived that
this was only attending to the transverse acceleration and
neglecting the longitudinal, which is taken into account in
Professor Eddington’s completer theory in the October
number of the Phil. Mag., page 322. He there re-deter-
mines the fundamental equation of particle dynamics, with
momentum a function of speed, and shows that not the ratio
F/m, but the product Fm, enters into the absolute term of

’ : u — Ef/m m
that equation, so that it becomes 1 + Y= Fae rh

I take up the thread again here, and point out that that
being so, the unexpected result follows, that if the additional
inertia is acted on by gravity, in accordance with the ordinary

* Communicated by the Author.
Phil. Mag. 8. 6. Vol. 35. No. 206. Feb. 1918. M

142 Sir Oliver Lodge on Astronomical and

Newtonian law F=ymm’/r?, the varying factor m will enter
twice into the equation of motion, and the whole perturbation
will be increased instead of being annihilated. In other words,
if the gravitative pull on the planet increases in the same
way as the inertia increases, the effect is not to cancel, but to
double the perturbing effect. On the other hand, if the extra
inertia is not affected by gravity the perturbing effect is as
already calculated. Consequently from this point of view
some perturbation seems inevitable ;—either the value reckoned
by me in August, with the extra inertia independent of gravity,
or else double that value, if the extra inertia is fully subject
to the Newtonian law of attraction. 7

The question arises therefore, rather pressingly, how much
dependence can be placed on the theory ? It willbe granted
I think that the fact that.a correct value for Mercurial apsidal
progress can be deduced from the electrical theory of matter
by a reasonable assumption of solar drift is not a negligible
fact. For if the theory were completely inapplicable the
value of drift required might have been of an altogether
unreasonable order of magnitude. The fact that the same
drift gave a Martial apsidal progress also of the right
magnitude (see August Phil. Mag. pp. 91 & 92) seemed to
me at the time very confirmatory. But I admit that the
changes in excentricity are not thus accounted for correctly,
and that the calculated perturbations for Earth and Venus
exceed any probable value for those planets.

I perceived in my August paper that a difficulty of this
kind would arise, but thought that it might be got over by
choosing a particular longitude for the projected component
of the solar drift which should almost nullify the result for
those two planets; and so I chose the longitude 294° as
being half-way between the perihelia of Mercury and Mars,
and inclined to their major axes at a reasonable angle, while
at the same time it happens to be practically half-way
between the perihelia of Harth and Venus, though inclined
to their major axes at a much smaller angle, and so being
less effective. I hoped therefore that it might be possible.to.
contrive to get rid of the calculated too great progress of
perihelion for Harth and Venus, especially as the roundness
of their orbits must make the exact position of perihelion
difficult to determine.

Professor Eddington, however, countered all this con-
trivance, in September, by working out the theoretical
changes of excentricity more thoroughly than I had done,
and showed that it was highly improbable that the calculated
perturbations could be admitted for some and evaded for

Gravitational Bearings of Electrical Theory of Matter. 143

others of the four inner planets, by any judiciously selected
direction for solar drift.

If therefore the theory fails to give all the known pertur-
bations correctly, something must be wrong ; and by finding
out what is wrong, we may perhaps discover something
instructive. At first I thought that it would suffice to say
that the extra inertia must after all be fully subject to
gravity, and that therefore the acceleration was unaltered.
But in the light of Eddingion’s improved theory, that
loop-hole is closed; for the ratio F/m turns out not to be
really involved. Sothe alternative that next occurred to me
was to suppose that the gravitative constant y may likewise
be a function of velocity, and that its changes act in a com-
pensating manner; that is to say, that the y for bodiés
in motion through the ether differs from the value appro-
priate for bodies at rest in one or other of the two

following ways :—
Hibs
r=w/ Js =)

if extra inertia is independent of gravity,
‘or uv
Y= Ve i 7)

if the extra inertia has its full complement of weight. But
then, so far as I know, there has never been any reason for
suspecting sucha variability in the Newtonian gravitative
constant.

If there should really turn out to be such a change in the
value of the gravitative attraction between bodies in rapid
motion, it would be a very remarkable and noteworthy fact,
and one that would serve to strengthen belief in some form
of the Principle of Relativity. Hitherto the experimental
foundation of that Principle has been the fact that whether
luminous or telegraphic signals were used, or whether the
interval of time between reception and transmission of
signals was made to depend on cohesion or some other
property of solid matter, there was always a disconcertingly
complete compensation, so that no motion of matter as a
whole through the ether could be demonstrated. I had
hoped that by aid of the astronomical perturbations calculable
from the electrical theory of matter this compensating
influence might be overcome, and that the force of gravity
would not, so to speak, join the conspiracy to defeat our

M 2

144 Sir Oliver Lodge on Astronomical and

object. We must, however, now face the possibility that
gravitation too obeys a compensating law, and declines to
enable us to receive information about absolute motion of
matter through ether.

We must face the possibility: I do not say we must
accept it; but it becomes necessary to consider what other
loop-holes there are out of the conclusion which has thus
suggested itself.

Is it possible that there is anything wrong about the
theory that the inertia of matter is increased as a certain
function of the speed? There can hardly be any doubt
about it for an isolated charge such as an electron, inasmuch
as the calculation based upon the behaviour of its lines of
force is very straightforward, and because in the case of
certain extra-high-speed ejections from radium the extra
inertia has been observed and measured and found to cor-
respond with theory.

But it may be argued that when electrons are packed
together into an atom,—positive and negative together, so as
to be on the whole neutral,—this effect, due to their individual
electrostatic lines of force, is masked, and that the whole
neutral atom ought not to show any but perhaps a residual
effect of that kind. |

Against this argument I urge several considerations :—

First, that if the inertia of the negative electron is wholly
electrical, it becomes exceedingly probable that all inertia is
of that kind; and if so, the abolition of electrical inertia
would mean the abolition of all inertia.

Secondly, although we may speak of electronsas “ packed
together ” into an atom, the packing is very far from being
close; for to all intents and purposes they are well sepa-
rated from each other and almost isolated, even though several
are contained in a sphere the ten-millionth of a millimetre
in diameter. It will be remembered that inertia is due to
the concentration of the electric charge, and this concentration
is only marked within a minute distance from a sphere so
small as an electron. Taking its linear dimensions as 107},
all but one-hundred-thousandth of the inertia lies within a
range smaller than 107%. Regarding the electron of a
hydrogen atom as a sphere an inch in radius, the adjacent
opposite charge is something like a mile away. Moreover,
the fields of opposite charges are subject to the law of simple
superposition. And again, the inertia in superposed fields of
force does not depend on the sign of each component, but is
proportional to the square of each component. Consequently
there seems no adequate reason against merely adding their

Gravitational Bearings 0) Electrical Theory of Matter. 145

inertias in simple arithmetical fashion, even for atoms much
more closely packed than hydrogen.

It may be said that little is known about the positive
nucleus, wherein most of the mass of an atom resides
according to the now prevalent view. That is true; but
then, according to analogy, it would appear likely that the
additional mass is due to still further concentration, and that
the size of the positive unit, at least in a hydrogen atom,
must be a thousand or say seventeen-hundred times smaller
lineally than even the electron ; in which case the argument
for its practical isolation, and for its effective inertia being
within an exceedingly small distance of its surface, is
intensified.

On the whole then, | think that the weight of argument is
strongly in favour of the full applicability of the theory of
electrical inertia to every kind of atom, and to all masses
of matter ; though it is just possible that for atoms of high
atomic weight some modification may have to be made
owing to their presumably more complicated structure,
especially the more complicated structure of their nuclei.

If this loop-hole is to be considered closed, what other is
there? Yetin view of the importance of the threatening
deduction that the gravitative constant is a function of
velocity, we must seek every way out of the negative con-
clusion that the perturbations predicted by the electrical
theory of matter do not in the case of the two planets Harth
and Venus really occur ; or do not occur to anything like
the extent required by the quantitative explanation of the
perturbations of Mercury by the same theory. If the theory
fails to account correctly for the outstanding perturbations
of the four inner planets—especially if it makes those per-
turbations too great,—it seems at present as if a variation of
the gravitative constant for bodies in motion is proved: a
result too important to be lightly regarded.

It seems therefore worth while to expend some labour in
calculating what those perturbations would be for the four
inner planets, given some favourable value of the solar

drift.

Part II.

It may be remembered that in the October Philosophical
Magazine Professor Eddington tentatively adopted but
partly modified my theory, and also introduced terms
depending on the excentricity, getting as his final result
two equations representing the main perturbations to be
expected from varying inertia, correct up to the first power

146 Sir Oliver Lodge on Astronomical and

of e. These equations, for convenience, I will here quote
from page 326, vol. xxxiv. as they stand :—

z 2 \

eda= — ease sin ao + Bae plc, cos 2, |
2c? 2c? 8c? 1)
upVO 2 c -

je

COs a+ e sin 2a,

J

where V is the component of true solar drift projected on
plane of orbit,
@ is longitude of planet’s perihelion reckoned from
the direction of V as zero,
uy is that constant component of the velocity of a
planet which is normal to its radius vector,

6 is the angle turned through by radius vector per
century,

and ¢ is the velocity of light.

Dor

I proceed to apply this improvement on what I published
in August, so as to ascertain whether or not the theory can
be made to work.

Let the solar drift be & times the planet’s velocity as
specified, say V=fuy; and introduce an aberration angle
a=u,/c ; then we can write the above equations thus :—

eda =%4e°0(—k sina + the cos2a +e), (2)
de=420(kcos@a+ ike sin 2a).

The common factor outside the brackets, 3«70, is inde-
pendent of solar drift and cannot be evaded. It varies as.
the reciprocal of the 5/3rd power of the periodic time for
different planets, or as the —2°5th power of their distances ;.
as can be seen thus :—

The large angle @ is 2an, so it is inversely as T; «@ is
proportional to uw), which is practically the same in magni-
tude as the average orbital velocity ; so # varies inversely

as ,/7 or as 1 by Kepler’s third law.

Hence 4a24 is proportional to T° or n or 7 It
becomes small, therefore, for the outer planets. It is also
plain that the values of $«°0xk for different planets vary
inversely as the square of their distances from the sun.

The value of 444 for the Earth is 1007 x 10-*=0'"648,
and from this it can be reckoned for the other planets by

5 —5
=

Gravitational Bearings of Electrical Theory of Matter. 147

dividing by r’?. The result is tabulated here, along with
other fixed planetary data for convenience of reference.

I. Fixed Planetary Data.

| Perturbation

Excentricity | Longitude of | Distance Bou Fe
of orbit. perihelion. | trom sun. 12°0 :
a Ti (064877 7”).

Mercury..........-. 2056 75 0-387 6-95
Wenust oi.te sais. "0068 129 0723 1-46
Barling. .02.2383.2 95: ‘0167 100 1-000 0-648
RESP Ae. 0933 333 1-524 0-227
dimptten 26.2 52005: "0483 12 5°203 0:0105
SRGMEI |) ce cans 0559 90 9-539 0:0023
MEANS | oe ic: 50-4 ‘0463 168 19°18 00004
Neptune............ 0090 / 47 30°04 0:00013

Now consider the bracketed factors of (2). We see that
the dominating part of both of these factors is &, and
that whatever longitude is chosen for @ neither factor can
exceed + to any considerable extent ; they will, in fact,
usually be both smaller than &. By suitable choice of w
either of the factors may be made small or zero; but if so,
the other will thereby usually tend to be big.

To make this more obtrusively clear we might write them
respectively

—ksin o(1+ $he sin ce We (3)
keosa(1+4kesin a). hes

If & is zero or small,7.e.if the solar system is nearly
at rest in the ether, the de perturbation vanishes, but not
the dw. It is rather remarkable that there should be any
residual perturbation due to fluctuating mass in a stationary
solar system. But of course the velocity in an orbit with
any excentricity is not quite constant, and the equations
show that when & is 0, whatever the value of e, there will
still be a cumulative da (progress of perihelion) equal to
300; that is, 4a? times the angle turned through by the

148 Sir Oliver Lodge on Astronomical and

radius vector in any given time. This is a curious and
interesting result, since it is independent of V and of e and
of w. The fact may require attention in another branch of
physics later on (see concluding remarks in Part III.).

Over the main factor, $270, we have no control; but we
can partly determine the bracketed factors of (2) by judi-
cious selection of # and k for any particular planet ; though,
having chosen for one, the others all follow.

The observational values to be accommodated by theory
are stated by Mr. Harold Jeffreys in the ‘ Monthly Notices ’
of the Royal Astronomical Society for December 1916,
whence I get the permissible range :——

Il. Unexplained Secular Variations per century, as observed.

ed. de.
eure Permissible de. Permissible
range. range.
Mercury ...... 3:48+0-43 |+8-91 to +8:05|—0:88+0:50 |—1:38 to —0:38
Venus oe... —0:05+0-25 |—0:30 to +0:20] 0-21+0°31 |4+0:52 to --0-10
Earth ......... 0'10+0-13 |-+0-23 to —0-03| 0:02-+0:10 |+0-12 to —0-08
Mars.......0..0. 0°75+0:34 |+1-09 to +0°41) 0-29+0:27 |+0:56 to +002

Hence to get results for Mercury and Mars from the
-above equations, such as will correspond in sign with the
entries in the above table, a solar drift must be chosen so
as to make sina negative for both planets, with cosa
negative for Mercury and positive for Mars. This suggests
an obtuse negative angle between Mercury’s perihelion and
the solar drift, and an acute negative angle for that of Mars;
but this can be seen to be impossible, though the converse
would be easy. A troublesome accommodation difficulty
lies in the fact that the major axes of these two planets
happen to cross nearly at right-angles, so that what suits
one is hardly likely thoroughly to suit the other.

The above table shows moreover that both the observed
perturbations for Earth and Venusare small, and might even
be zero. But looking at the equations (3), we see that in
general both the theoretical perturbations cannot vanish or
be small together for any reasonable value of k. (There is
no sense in a negative value for k such as —2/e.)

Gravitational Bearings of Electrical Theory of Matter. 149

Alternatives.

Now let us consider the numerical values of the theoretical
perturbations for different planets, with specially selected
values for the solar drift. Write them

eda =4070.k.(—sina+tke cos 2a+e/k),
de=4a°?.k.(cosa+ikesin 2a),

(4)

so as to show in each case

(1) A purely planetary factor 4276, concerning which
we have no choice ;

(2) A numerical factor k, depending on the magnitude
of assumed solar drift and varying directly as ,/r
for different planets ;

(3) A trigonometrical factor, depending mainly on the
direction of the assumed solar drift.

For moderate values of &, and for round orbits, the first
term of each direction factor (viz. —sin a for the one, cosa
for the other) is by far the most important.

Large values of & require delicate and practically im-
possible adjustments, so that if for any planet both pertur-
bations are wanted small (as they are), it is essential to keep
k down to the smallest value which will give anything like
the desired result for some one definite planet.

Let us then choose k or V/u) as small as possible. We
- might even try it no greater, for the Earth, than suits the

AGNES, fee
directed
see.

towards R.A. 18° 2™ or longitude 270° 46’, and declination
34° or latitude 57° 30’), of which the projected component
will be about half of two-thirds of the earth’s orbital speed,
or 4x10-‘*c, which makes & for the earth =}. Then for
the other planets we shall have & varying as 1/ug or ,/7, and
so get the column f, in Table III. below.

But the resulting value of & for Mercury, viz. °21, is far
too small to explain the progress of Mercury’s perihelion:
for that purpose Mercury’s £ must be comparable to unity,
in order to give a reasonable factor with which to multiply
32°@: but what its necessary value is will depend on a,
7. e. on the direction chosen for the solar drift with reference
to the planet’s major axis. Consider then what is the best
direction to choose.

The idea on which we started (see Phil. Mag. for last

known solar motion towards Vega (viz. 19°5
eo

150 Sir Oliver Lodge on Astronomical and

August) was to explain simply the outstanding perihelion
progression of Mercury, 43’’ per century or eda=8''"6; so
it is natural to select values of £ and @ which shall give an
approximation to this value in the case of Mercury and yet.
keep & small enough not to spoil agreement for the other —
planets. The most favourable possible direction is 90° beyond
the perihelion longitude of Mercury, which is 75°. For this.
drift longitude of 165° will make «= —90 and sina=—1.
The drift-factor for eda then attains its maximum, viz.

k+e(1—4),

while the factor for de is zero. In that case the value of &
for Mercury may be as low as ‘9 ; for, since e="2, this will
give a factor, 1:1, sufficient to convert the 7” belonging to
3a°0 for Mercury into very near the desired 8”. Corre-
sponding values of & for other planets are tabulated in the-
column k, below.

But then the perihelion longitude for Mars happens to be
333°, so that with drift longitude 165° the a for Mars would
be + 168° or 180—12° ; and the value of sin a will be small
and positive, and of cos@ big and negative—an arrange-
ment which does not suit Mars at all; hence a compromise
is necessary.

If we choose the compromise which I suggested in the
August Phil. Mag., page 91, viz. 294°, or rather that direc-
tion reversed, viz. 114°, we have got a direction equally
inclined to the major axes of Mercury and Mars, and also,
as it happens, practically equally inclined to the major axes
of Earth and Venus, which are another pair that may be
considered together. The angle a for Mercury in this case
is —39°, and for Mars is 180+39°; so in both cases sin @ is
negative, as wanted, though the less said about cos@ the
better. The corresponding values of & for all the planets,
including that needed to suit Mercury’s perihelion for this
drift direction 114°, namely 1°5, are tabulated below
as hae

The table shows some corresponding values of k and @ for
all the planets, with the corresponding assumed longitude
for solar drift. This is here called 7; and it must be
remembered that this solar drift direction, and not the first
point of Aries, is the artificial zero from which to reckon oa.
The @ are got by subtracting / from the perihelion longitude,
as cited. The & vary as “7.

Gravitational Bearings of Electrical Theory of Matter.

Ill. Alternative values of k and aw.

Longitude of | Distance
perihelion, | from sun.

a+. | r.
75 ) “4
129 | i
100 1-0
333 15
12 5:2
90 9°5
168 19
47 30

Mercury.
Venus.
Earth.
Mars.
Jupiter.

Saturn.

| Uranus.

| Neptune.

1,=271°. 1,=165°.

218
189

101
179
257
136

1:03

1-45} 3/61

1:82 | 242| 7:5 | 293 |13-1

1,=114°.

151

(ema Tae

54

10°5

il igs | 2a

4°

————l

262 |1-056
316 |1-44
287 |1°7

160 |2°1

199
277
305
234

The columns with suffix 1 represent the known solar motion
with reference to the stars; but for the present purpose
this has no necessary importance.

The columns with suffix 2 represent the most favourable.
direction for explaining Mercury’s perihelion-progress.

alone.

The columns with suffix 3 represent a compromise intended
to suit the perihelion-progress of the four inner planets.
so far as possible.

The columns with suffix 4 represent the values got by satis-
fying both the observed perturbations for Mercury, and
letting other planets look after themselves; i. e. by
solving the equations,

eda—to?8 = —ksina+thk’e cos 2a +e

= §"-38—6'"95= 1:206,

de—4e?8=kcosa+th'e sin 2a

= —0!"90-+-6"-95= —-13;

152 Sir Oliver Lodge on Astronomical and

or, since e=*206, or say ‘2 for correction terms,
ke?
=. cos 2a —k sina=1,

a i

k 2 ». e
50 Sn 2@a+keosa=—'13.

A solution of these equations * is very nearly
@ = 262°= —98°,
k =1-056.
And since the perihelion position for Mercury is
75S =at+l,

we get as the longitude of solar drift proper to account for
both perturbations of Mercury

l= 75°— 262°= —187°=173°.
We attend to this case further in Table V. below.

Numerical calculation for special cases.

In taking out the trigonometrical functions so as to get
the proper factors in each case we have to pay particular

* I solved these by successive approximation, the upper equation
mainly determining k, while the lower mainly determines w; general
consideration, about signs, etc., showing that w must be something big
in the third quadrant, 2. e. not far from 270°; but my brother has now
solved them in much neater fashion, as thus :—

x* cos 20—20x sind=20, |
2 sin 20-+202 cos 0= —2°6, {
x?(cos 20-+-7 sin 26) +20 tx(cos 0+7 sin 6) =20— 2°62,
(xe'®)” +202(xe"*) +102) = —80—2°6:,
(we'® +102) =(a+7b)’,
where a’—b’=—80 and ab=—1'3.
“. cceos@=a, xsiné=b—10, and 2?=a>+é?+100—208.
The value of a?-+-6? is 80°04224 nearly, and
a= +'1453252, b= +8'94545.
x cos = + °1453825 x cos 0= —'145325,
2 sin @= —18-94545 OF asin 0=—1:05455 ;
x=18°9460 or x«=1:0640,
6=270° 963), { 0 = 262° of

Gravitational Bearings of Electrical Theory of Matter. 153

attention to sign.

It may be worth while to give details

just for one case, and we may as well choose the one (No. 3)
in Table III. when solar drift has longitude 114°.
The perturbations are

eda =1270 kA
i ; - (6)
de=ta @ k o B
where A=-— sinw+ the cos 2a +eJk,

and

B= cosa+Hihe sin 2a.

They consist therefore of a purely planetary factor
depending only on wp and T; a solar drift magnitude factor
k=V/uo; anda relative direction factor A and B respectively.

We have now to reckon the direction factors A and B for
the particular case marked above with the suffix 3.

For the values of $70 and e see Table I.

IV. Calculated Perturbations for the Case of
[=114 tond, V-—-2-4 x 105%;

/ )
.|cos w.|sin2@.cos2a.} &. | A.

wz. sine B. | eda. | de.
—-—— —_- eee | eee ee [Seas
Mercury, —39 63 4777 | 98 | +21 15] -78 | -70l+81 [+72
Venus.... +15 |+-26 |+-97 ]+:50 | +87] 20) —-26] -97/-0-76 |+2:8
Earth... —14 |—-24 |4-97 ]—-47 | +8824) -24 | 96/4037 +15
Mars ...180+39—63 |—-777]+-98 | +21] 3:0) -675|—-71]+0-46 |~-048
Jupiter . 180-++78 —-978 |~-208] +41 |—-91] 55, 93 |—-18/+-054 —01
Saturn. —24 —-407/+°91 |—74 | +67] 74) -48 | -83]+-0083 +014
Uranus. +54 (+809 +°50 | +95 | "31 05-84 | -71)—-0035 |+-003
Neptune —67 —-92 |+-39 13-1) -92

—72 | —-70

-37 fon + ‘0007

The last two columns of this Table show that the superior

planets will give us no trouble, whatever their aspect, and in
spite of their comparatively large value of £; the smallness
of their own factor 4270 reduces all perturbations due to
varying inertia to practically nil. Even if V were as high
as 10 times the earth’s orbital velocity, or say 200 miles a
second, the superior planets would still give no trouble.

Of the four inner planets the entries for eda, compared
with those in Table II., are not bad; but the de are deplor-
able. And no choice of direction or of drift which satisfies
one set of perturbations seems likely to satisfy all.

|

|
;

154 Sir Oliver Lodge on Astronomical and

Perhaps the simplest plan of calculating perturbations for
-a given V, is to reckon $270k for one planet, say the Harth ;
‘divide this by relative r? for all the others; and then
multiply the numbers so obtained by the respective direc-
‘tion factors A and B; which consist of — sina and cosa
primarily, corrected by addition or subtraction of small
functions of k and e and sin2@ and cos 2a. Thus a first
approximation is |

648k,

v2

.sinaw, for eda,

oe 648 hy

yan | COS Gy LOR te,

7

where fp refers to the Harth, being V/10~4 c.

The next approximation can be taken from equations (3)
above, viz. the faetor (1+4hkesinaw) for both, with a
supplementary term for one.

The only other case which [ will work in detail is the case
when a direction is chosen to satisfy both edw and de for
Mercury. We can then sce what happens to the others.
This is the case which we previously tabulated as k, and
@, in Table I[LI., and which depends on the solution of
equations (5).

V. Calculated Perturbations for the Case of
1=173° and V=1:7x 10>‘ ce.
| l

w. sinw.|cosa.| &. | ey a eae cdw. | de.

Mereury...| 1804-82 |—-990 |—-139] 1-056] 1-14 |—-124/]+8:34 |—0-91

Venus...... —44 |—°695|+°719] 1-44 70 | °72 ||+1°46 |4+152
Earth...... —73 |—'956 |+--292] 1:7 "96 | 29 ||+1:04 |+0°32
Mars. :5.: 180—20 |+ 342 |— 940] 2:1 —'26 |—‘97 ||—0°12|—0°46

On the whole perhaps these perturbations agree rather
better with observation than do those in Table IV., though
now Mars is disappointing. If they are in any degree
tolerable, one may note that the drift, of magnitude 1:7 in
-direction 173°, thus postulated, can be considered as the

=

Gravitational Bearings of Electrical Theory of Matter. 155

resultant of a component of the known solar motion with

reference to the stars
ee x cos 075°='352, x 10~*¢, in direction 270%46

compounded with a true drift in plane of ecliptic
1°78 x 10~‘ ¢ in direction 162°.

I really do not know whether astronomers could pass, as
at all possible, outstanding perturbations such as those last
tabulated. Comparing Tables V. and II. they are clearly
not of orthodox size ; they are too big for Harth and Venus,
and too small for Mars. But I suppose that the recognized
values are in reality dissected out from a group system of
small discrepancies of which the total is more certain than
the precise distribution among individual members.

I submit also that even the forced agreement for Mercury
is not to be wholly set aside as mere algebra; for the
postulated solar drift is of a not unreasonable order of
magnitude, and the figures are got from it absolutely by a
theory which if not in the least degree valid might have
‘given them millions of times wrong. So the fact that
absolute values not quite hopelessly discordant with ob-
servation can be thus reckoned ought to be taken into
account.

Assuming that astronomers will not pass them, however,
we must face the question why not. Full gravitational
influence on the extra inertia might halve the necessary
values of k, but would not otherwise improve things. Total
absence of solar drift is unlikely ; so in order to explain the
hypothetical absence of perturbations which ought to occur
but do not, we may be driven to conclude that the gravitation-
constant itself is a function of the speed of the attracting
masses, in some such way as that suggested in Part I. above :
thus adding to the evidence for an uncompromising Principle
of Relativity.

Part Ill.—Suggested Possibilities.

In support of the idea that gravitative attraction may be a
fuuction of speed, I may point out that if the attraction were
of an electrical order, such dependence on speed would be
reasonable, and even the amount of the dependence would

156 The Electrical Theory of Matter.

be appropriate; for the attraction between two charges

Kr
one charge is revolving round another, the attraction between
them presumably has for its main term

Vu sin 6
F= ren £5(1- een 5

(See J. J. Thomson’s ‘‘ Report on Hlectrical Theories,”
British Association volume for Aberdeen, p. 110 (1885).)
Hence if
- yy We 2V :
m=m(1— “) = mo( 1— bi Bs Bh a

c?

2 2
moving together in parallel lines is ee (1- “5 while if

then, u being in this case far larger than any probable V,

; okie
Fin = Fyanal qi Vu He g in 2Vusin@ ui Ve +u ),

2¢? 2c?

and the terms involving sin @, which are responsible for the
cumulative terms in the solution of the differential equation
quoted in paragraph 1, cancel.

Were it not so, some curious consequences could be deduced
for an electron revolving at immense speed inside an atom
round a nucleus under the inverse-square law, especially
when such an atom is shot away at high speed ; for the
angle @ or 27n is enormous.

Assuming it possible, then, that a quantitatively similar
law holds in the case of gravity, the force of attraction
F=ymM/r* will diminish as m increases (M the central
body, moving steadily at speed V, will not change its value.
whatever it is), and accordingly the product Fm (involving
ym") will remain constant at whatever varying speed m
moves through the ether: the variation of the gravity-
constant y just compensating for the double variation of
mass m.

But it will be very remarkable if such compensation really
occurs ; and if such a fact is established it may begin to
throw some light on the family relationship of the force of
gravity.

tat

XVII. On the Lubricating and other Properties of Thin Oily
Films. By Lord Rayurien, O0.M., F.R.S.*

| Cir experiments about to be described were undertaken

to examine more particularly a fact well known in most
households. A cup of tea, standing in a dry saucer, is apt
to slip about in an awkward manner, for which a remedy is
found in the introduction of a few drops of water, or tea,
wetting the parts in contact. The explanation is not obvious,
and J remember discussing the question with Kelvin many
years ago, with but little progress.

Itis true that a drop of liquid between two curved sur-
faeces draws them together and so may increase the friction.
If d be the distance between the plates at the edge of the
film, T the capillary tension, and « the angle of contact, the
whole force is + -

2AT cose
d

A being the area of the film between the plates and B its
circumference. If the fluid wets the plate, e-=0 and we
have simply 2AT/d. For example, if d=6 x 10~’cm., equal
to a wave-length of ordinary light, and T (as for water) be
74 dynes per cm., the force per sq. cm. is 25 x 10° dynes, a
suction of 24 atmospheres. For the present purpose we
may express d in terms of the radius of curvature (p) of one
of the surfaces, the other being supposed flat, and the dis-
tance (#) from the centre to the edge of the film. In two
dimensions d=.2?/2p, and A (per unit of length in the third
dimension) =22, so that the force per unit of length is
8pT/z, inversely as x On the other hand, in the more
important case of symmetry round the common normal
A=72", and the whole force is 4apT, independent of x, but
inereasing with the radius of curvature. For example, if
T=74 dynes per cm., and p=100 cm., the force is 925
dynes, or the weight of about 1 gram. The radius of cur-
vature (p) might of course be much greater. There are
circumstances where this force is of importance ; but, as we
shall see presently, it does not avail to explain the effects
now under consideration.

My first experiments were very simple ones, with a slab of

+ BT sin «,

* Communicated by the Author.
+ See for example Maxwell on Capillarity. Collected Papers, vol. ii.
p- 571.

Phil. Mag. 8. 6. Vol. 35. No. 206. Feb. 1918. N

158 Lord Rayleigh on the Lubricating

thick plate glass and a small glass bottle weighing about
4 oz. The diameter of the bottle is 44 cm., and the bottom
is concave, bounded by a rim which is not ground but makes
a fairly good fit with the plate. The slab is placed upon a
slope, and the subject of observation is the slipping of the
bottle upon it. If we begin with surfaces washed and well
rubbed with an ordinary cloth, or gone over with a recently
wiped hand, we find that at a suitable inclination the con-
ditions are uniform, the bottle starting slowly and moving
freely from every position. If now we breathe upon the
slab, maintained in a fixed position, or upon the bottle, or
upon both, we find that the bottle sticks and requires very
sensible forces to make it move down. A like result ensues
when the contacts are thoroughly wetted with water instead
of being merely damped. When, after damping with the
breath, evaporation removes the moisture, almost complete
recovery of the original slipperiness recurs.

In the slippery condition the surfaces, though apparently
clean, are undoubtedly coated with an invisible greasy
layer. If, after a thorough washing and rubbing under the
tap, the surfaces are dried by evaporation after shaking off
as much of the water as possible, they are found to be sticky
as compared with the condition after wiping. A_ better
experiment was made with substitution of a strip of thinner
glass about 5 cm. wide for the thick slab. This was heated
strongly by an alcohol flame, preferably with use of a blow-
pipe. Ata certain angle of inclination the bottle was held
every where, but on going over the surface with the fingers,
not purposely greased, free movement ensued. As might
have been expected, the clean surface is sticky as compared
with one slightly greased; the difficulty so far is to explain
the effect of moisture upon a surface already slightly greased.
It was not surprising that the effect of alcohol was similar
to that of water.

At this stage it was important to make sure that the
stickiness due to water was not connected with the minute-
ness of the quantity in operation. Accordingly a glass plate
was mounted at a suitable angle in a dish filled with water.
Upon this fully drowned surface the bottle stuck, the inclina-
tion being such that on the slightest greasing the motion
became free. In another experiment the water in the dish
was replaced by paraffin oil. ‘There was decided stickiness
as compared with surfaces slightly greasy.

The better to guard against the ordinary operation of sur-
face tension, the weight of the bottle was increased by
inclusion of mercury until it reached 20 oz., but without

and other Properties of Thin Oily Films. 159

material modification of the effects observed. The moisture
of the breath, or drowning in water whether clean or soapy,
developed the same stickiness as before.

The next series of experiments was a little more elaborate.
In order to obtain measures more readily, and to facilitate
drowning of the contacts, the slab was used in the horizontal
position and the movable piece was pulled by a thread which
started horizontally, and passing over a pulley carried a small
pan into which weights could be placed. The pan itself
weighed 1 oz. (28 grams). Another change was the substi-
tution for the bottle of a small carriage standing on glass
legs terminating in three feet of hemispherical form and
2 mm.in diameter. The whole weight of the carriage, as
loaded, was 72 0z. The object of the substitution was to
eliminate any effects which might arise from the compara-
tively large area of approximate contact presented by the
rim of the bottle, although in that case also the actual con-
tacts would doubtless be only three in number and of very
small area.

With 4 oz. in pan and surfaces treated with the hand, the
carriage would move within a second or two after being
placed in position, but after four or five seconds’ contact
would stick. After a few minutes’ contact it may require
13 oz. in pan to start it. When the slab is breathed upon it
requires, even at first, 34 oz. in the pan to start the motion.
As soon as the breath has evaporated, $ oz. in pan again
suffices. When the weight of the pan is included, the forces
are seen to be as 1:3. When the feet stand in a pool of
water the stickiness is nearly the same as with the breath, and
the substitution of soapy for clean water makes little
difference.

In another day’s experiment paraffin (lamp) oil was used.
After handling, there was free motion with 1 oz. in pan.
When the feet stood in the oil, from 22 to 3 oz. were needed
in the pan. Most of the oil was next removed by rubbing
with blotting-paper until the slab looked clean. Atthis stage
2 oz. in pan sufficed to start the motion. On again wetting
with oil 2 oz. sufficed instead of the 22 0z. required before.
After another cleaning with blotting-paper 4 oz. in pan
sufficed. From these results it appears that the friction is
greater with a large dose than with a minute quantity of the
same oil, and this what is hard to explain. When olive oil
was substituted for the paraffin oil, the results were less
strongly marked.

Similar experiments with a carriage standing on brass

N 2

160 Lord Rayleigh on the Lubricating

feet of about the same size and shape as the glass ones
gave different results. It should, however, be noticed that
the brass feet, though fairly polished, could not have been
so smooth as the fire surfaces of the glass. The present
carriage weighed (with its load) 64 oz., and on the well-
handled glass slide moved with + oz. in pan. When the
slide was breathed upon, the motion was as free as, perhaps
more free than, before. And when the feet stood in a pool
of water, there was equal freedom. A repetition gave
confirmatory results. On another day paraffin oil was
tried. At the beginning + oz. in pan sufiiced on the
handled slab. With a pool of oil the carriage still moved
with + oz. in pan, but perhaps not quite so certainly. As
the oil was removed with blotting-paper the motion became
freer, and when the oil-film had visibly disappeared the
+ oz. in pan could about be dispensed with. Doubtless
a trace of oil remained. ‘The blotting-paper was of course
applied to the feet and legs of the carriage, as well as to
the slab.

In attempting to interpret these results, it is desirable to
know what sort of thickness to attribute to the greasy films
on handled surfaces. But this not so easy a matter as when
films are spread upon water. In an experiment made some
years ago * I found that the mean thickness of the layer on
a glass plate, heavily greased with fingers which had touched
the hair, was about 4 of the wave-length of visible light,
viz. about 10-* mm. The thickness of the layer necessary
to induce slipperiness must be a small fraction of this,
possibly ;1,, but perhaps much less. We may compare this
with the thickness of olive oil required to stop the camphor-
movements on water, which I foundt to be about 2 x 10-* mm.
It may well be that there is little difference in the quantities
required for the two effects.

In view of the above estimate and of the probability that
the point at which surface-tension begins to fall corresponds
to a thickness of a single layer of moleules{, we see that
the phenomena here in question probably lie outside the
field of the usual theory of lubrication, where the layer of
lubricant is assumed io be at least many molecules thick. We
are rather in the region of incipient seizing, as is perhaps not
surprising when we consider the smallness of the surfaces

* Phil. Mag. vol. xix. p. 96 (1910); Scientific Papers, vol. v. p. 538.
t+ Proc. Roy. Soe. vol. xlvii. p. 364 (1890) ; Scientific Papers, vol. iii.

p- 349. :
¢ Phil. Mag. vol. xlviii. p. 321 (1899); Scientific Papers, vol. iv.
p- 430.

and other Properties of Thin Oily Films. 161

actually in contact. And as regards seizing, there is
difficulty in understanding why, when it actually occurs,
rupture should ensue at another place rather than at the
recently engaged surfaces.

It may perhaps be doubted whether the time is yet ripe
for a full discussion of the behaviour of the thinnest films,
but I will take this opportunity to put forward a few remarks.
Two recent French writers, Devaux * and Marcelin t, who
have made interesting contributions to the subject, accept my
suggestion that the drop of tension in contaminated surfaces
commences when the jayer is one molecule thick ; but
Hardy tf points out a difficulty in the case of pure oleic acid,
where it appears that the drop commences at a thickness of
1-3 x 107° mm., while the thickness of a molecule should be
decidedly less. Many of Devaux’ observations relate to the
case where the quantity of oil exceeds that required for the
formation of the mono-molecular layer, and he formulates a
conclusion, not accepted by Marcelin, that the thickness of
the layer depends upon the existence and dimensions of the
globules into which most of the superfluous oil is collected,
inasmuch as experiment proves that when a layer with fine
globules exists beside a layer with large globules, the former
always contracts at the expense of the latter. As to this, it
may be worth notice that the tension T of the contaminated
surface could not be expressed as a function merely of the
volume of the drop and of the two other tensions, viz. T;
the tension of an air-oil surface and T, that of a water-oil
surface. It would benecessary to introduce other quantities,
such as gravity, or molecular dimensions. I am still of the
opinion formerly expressed that these complications are
the result of zmpurity in the oil. If the oil were really
homogeneous, Devaux’ views would lead one to regard the
continued existence of two sizes of globules on the same
surface as impossible. What would there be to hinder the
rapid growth of the smaller at the expense of the greater
until equality was established? On the other hand, an
impurity, present only in small proportion, would naturally
experience more difficulty in finding its way about.

The importance of impurities in influencing the transfor-
mations of oil-films was insisted on long ago by Tomlinson §;

* A summary of Devaux’ work, dating from 1903 onwards, will be
found in the Revue Gén. d. Sciences for Feb. 28, 1913.

+ Annales d. Physique, t.1. p. 19 (1914).

{ Proc. Roy. Soc. A, vol. lxxxviii. p. 319 (1913).

§ Phil. Mag. vol. xxvi. p. 187 (1863).

162 Lubricating Properties of Thin Oily Films.

and as regards olive oil, Miss Pockels showed that the
behaviour of purified oil is quite different from that of
the common oil. She quotes Richter (Nature, vol. xlix.
p. 488) as expressing the opinion that the tendency of oil
to spread itself on water is only due to the free oleic acid
contained in it, and that if it were possible to completely
purify the oil from oleic acid, it would not spread at all*.
Some confusion arises from the different meanings attached
to the word “spreading.” I suppose no one disputes the
rapid spreading upon a clean surface which results in the
formation of the invisible mono-molecular layer. Miss
Pockels calls this a solution current—a rather misleading
term, which has tended to obscure the meaning of her
really valuable work. It is the second kind of spreading in
a thicker layer, resulting in more or less rapid subsequent
transformations, which is attributed to the presence of oleic
acid. Miss Pockels says :—‘‘The Provence oil used in my
experiment was shaken up twice with pure alcohol, and the
rest (residue) of the latter being carefully removed, a drop
of the oil was placed upon the freshly formed water-surface
in a small dish by means of a brass wire previously cleaned
by ignition. The oil did not really spread, but after a
momentary centrifugal movement, during which several
small drops were separated from it, it contracted itself
in the middle of the surface, and a second drop deposited
on the same vessel remained absolutely motionless.” I have
repeated this experiment, using oil which is believed to
have come direct from Italy. A drop of this placed upon
a clean water-surface at once drives dust to the boundary in
forming the mono-molecular layer, and in addition flattens
itself out into a disk of considerable size, which rapidly
undergoes the transformations well described and figured by
Devaux. The same oil, purified by means of alcohol on
Miss Pockels’ plan, behaves quite differently. The first
spreading, driving dust to the boundary, takes place
entirely as before. But the drop remains upon the water as
a lens, and flattens itself out, if at all, only very slowly. Small
admixtures of the original oil with the purified oil behave in
an intermediate manner, flattening out slowly and allowing
the beautiful transformations which follow to be observed at
leisure.

Another point of importance does not appear to have been
noticed. Water-surfaces on which purified olive oil stands
in drops still allow the camphor movements. Very small
fragments spin merrily, while larger ones by their slower

* ‘Nature,’ vol. 1. p. 223 (1894).

Second Postulate of the Theory of Relativity. 163

movements testify to the presence of the oil. Perhaps this
was the reason why in my experiments of 1890 I found
the approximate, rather than the absolute, stoppage of the
movements to give the sharpest results. The absolute
stoppage, dependent upon the presence of impurity, might
well be less defined.

If, after the deposition of a drop of purified oil, the
surface be again dusted over with sulphur or tale and then
touched with a very small quantity of the original oil,
the dust is driven away a second time and camphor-
movements cease.

The manner in which impurity operates in these phenomena
merits close attention. It seems pretty clear that from pure
oil water will only take a layer one molecule thick. But
when oleic acid is available, a further drop of tension ensues.
The question arises how does this oleic acid distribute
itself? Is it in substitution for the molecules of oil, or
an addition to them constituting a second layer? The
latter seems the more probable. Again, how does the
impurity act when it leads the general mass into the un-
stable flattened-out form? In considering such questions
Laplace’s theory is of little service, its fundamental postulate
of forces operating over distances large in comparison with
molecular dimensions being plainly violated.

Terling Place, Witham, ;

Dec. 31, 1917.

XVIII. On the Second Postulate of the Theory of Relativity :
Experimental Demonstration of the Constancy of Velo-
city of the Light reflected from a Moving Mirror. By
Q. Masorana, Professor of Physics at the Polytechnic
School of Turin*.

HE Theory of Relativity is based upon two well-known
fundamental postulates. The first affirms the impos-
sibility of discovering the movement of a system without
referring this system to other systems ; that is to say, it
denies the physical reality of absolute motion. The second
postulate affirms that the velocity c of propagation of light
in vacuo is a universal constant. Both these postulates are
generalizations of facts or principles already admitted by
physicists.
In fact, we may regard the first as the extension to optical
or electrical phenomena of a classical principle of mechanics,

* Communicated by the Author.

164 Prof. Q. Majorana on the Second

an extension justified by the negative results of certain ex-
periments (Michelson and Morley, Trouton and Noble) by
which it was sought to discover the absolute motion of the
earth, or the ethereal wind which must traverse all terrestrial
objects. The second postulate is the generalization of a
fundamental principle in the theory of ethereal or electro-
magnetic undulations.

But if these two principles, derived from quite different
chapters of physics, have been fully accepted severally by
modern physicists, their origin has been almost forgotten ;
an ingenious structure arose upon their union: the theory
of relativity. This theory, while repudiating according to
Einstein and others a theoretical conception which had given
occasion for the formulation of the second postulate (the
ether), serves well to explain the insuecess of the above-
cited experiments.

Now our imagination, accustomed, as W. Ritz has said,
to “ substantialize ” physical phenomena, if it easily grasps
the essence of the first postulate, does not do so in the case
ot the second ; and the more so since, as has already been
said, some adherents to relativistic theories do not retain as
necessary the existence of a medium of transmission (the
zether) in order to explain the constancy of c. Moreover on
the second postulate, or, more precisely, on a certain portion
of this, depend the conclusions which appear artificial or
extraordinary in the whole relativistic theory *. The second
postulate must be understood in the sense that an observer
who measures the velocity of light finds always the same
value if both he and the source be at rest, relative or (if the
possibility be admitted) absolute, or if the source or the
observer, or both, have a uniform motion of translation. In
short, the second postulate affirms the absolute independence
of ¢ of any contingency whatever of uniform motion of
translation of the source or of the observer.

It is known that an !:ypothesis of a mechanical character
(emissive or ballistic), according to which to the ordinary
velocity of light must be added that of the source, can
explain, like the theory of relativity, the failure of the above-
quoted experiments. But this hypothesis is radically in
contrast with the electromagnetic theory, and consequently
is not much favoured f. But in any case laboratory experi-
‘ments can be conceived which should decide between the

* Carmichael, Phys. Rev. xxxv. p. 168 (1912).

+ In this connexion should be recalled the important critical work of

W. Ritz (Gwvres, p. 317) which perhaps has not been taken into sufii-
cient consideration by physicists.

Postulate of the Theory of Relativity. 165

above-mentioned hypothesis or mechanical theory and the
relativistic one. It is indeed possible to see that some
method, one moreover already in use, adopted for the verifi-
cation of Doppler’s principle may serve for the solution of
the above-quoted problem.

In order to see this, let us consider a luminous source 8
which emits waves of length A and of frequency x moving
towards the observer fixed at O (fig. 1). If we suppose

fig. 1.
SS ee ae oD Rs
RY, aS SS a RE, SRA YS

ێ-
eae

that the waves are transmitted through a stationary ether,
the n waves emitted in a second by 8 will be distributed
over the segment S’A=c—v. In the same time all the n
waves distributed in the segment OB=ec will have passed
through O ; we have therefore
c—v ¢ ; c

— =—, or n'=n—.

n n c—v
If we put v/c=8 and neglect terms of higher order than the
first in 8 we have

n'=n(1+ 8).
The new wave-length is obtained by the relations
fT == n'D'
A’=A(1— 8B).

If now instead of the hypothesis of a stationary medium we
adopt the ballistic or emissive hypothesis of which we have
spoken above, we shal! find that in one second the n waves
emitted by S will be distributed over the segment §’A’=c.
In the same time there will pass through O, n'’ waves which
will be distributed over the segment OB'=c+v. We have,
therefore,

Co G28 “an

Sr orn =n(1+).
And since c=nX and e+v=n'd' we see that, in this case,
=A.

As regards the frequency we arrive, therefore, at the same
conclusions (with the exception of the terms in 8”) whether
we adopt the ethereal or the ballistic hypothesis ; but for
the wave-length we obtain different values from the two

166 Prof. Q. Majorana on the Second

hypotheses, and these values differ by a term of the first
order in ®. If then the Doppler effect is measured, by
observation of the wave-length, different results should be
obtained according as the one or the other hypothesis is
adopted *. Now observations of the Doppler effect have
already been made by measuring the displacement of the
spectral lines, employing either prisms or diffraction-gratings.
In the case of prisms it may be observed that all the theories
of dispersion hitherto admitted lead to the supposition that
this phenomenon can only depend on the frequency of the
incident luminous vibrations. Consequently the displace-
ment of the spectral lines may be caused by the simple
variation in frequency due to the Doppler effect, and this
whether, for the light, the hypothesis of a stationary ether is
adopted, or a ballistic or emissive theory. From this point
of view, therefore, the question whether the velocity of
propagation of the light emitted by a source does or does
not change with the velocity of the latter remains unanswered.

But the Doppler effect has been established with diffraction-
gratings as well as with prisms, and for astronomical as well
as terrestrial sources tf. Now the function of a grating, from
the geometrical point of view, may be regarded as depending
exclusively on the values of the incident wave-lengths ; the
positions of the successive spectral lines remain exactly
determined by those values. But since, according to the
ballistic or emissive hypothesis, the value of X does not vary
with the velocity of the source, we see that the grating
should not give an appreciable result in the study of the
Doppler effect, and this, as is known, is not in agreement
with experience. We may then conclude from observations
of the Doppler phenomenon in the stars and the limb of the
sun with moving mirrors (Galitzin & Wilip), or again in
the canal rays (Stark, Paschen), that the velocity of light is
absolutely constant and independent of the movement of the
source ; this is equivalent to the rejection of the ballistic or
emissive theory. Thisis Tolman’s opinion f, in contradiction
to that of Stewart §. Indeed, it should be borne in mind
that the ordinary grating theory || may not apply exactly in
the case of a mechanical (ballistic or emissive) theory of

* These conclusions are identical with those already published by
other authors; see, e. g., Tolman, Phys. Rev. xxi. p. 26 (1910).

+ Galitzin & Wilip, Communications Acc. Russe, 1907, p. 2138: Stark,
Ann. d. Phys. xxviii. p. 974 (1909).

ft Phys. Rev. xxxv. p. 136 (1912).

§ Phys. Rev. xxxii. p. 418 (1911).

i| La Rosa, Nuovo Cimento, ili. p. 356 (1912).

Postulate of the Theory of Relativity. 167

light. In any case it should be remarked that astronomical
observations of the Doppler effect are not always made with
an @ priort knowledge of the relative velocities of source and
observer. In the case of the solar limb it is necessary,
moreover, to be cautious in establishing a relation between
the measure of displacement of the lines and the velocity of
the limb established by observation of the solar spots; in
fact, the light from the limb may be strongly refracted by
the perispherical incandescent gases, and consequently the
value of the Doppler effect may vary considerably *. So
far as terrestrial observations are concerned, and those on
the canal rays (Stark, Paschen), they give measures of the
phenomenon of only small precision, and it is impossible to
foresee by another method the exact velocity of the luminous
particles ; finally, observations made with moving mirrors
bear no relation to those with moving sources, and these may
produce different consequences f.

From all this we may conclude that up to the present
time we do not possess any quite certain evidence of the
immutability of ¢ with variable veloeity of the source if, be
it understood, we are not willing to admit as conclusive the
simple electromagnetic theory or that of bodies in motion
according to Lorentz or else Hinstein’s theory of relativity.
This conclusion is confirmed by the study of the works of
the chief supporters of the last theory, and, implicitly, of
the second postulate. In these works we frequently find
expressed the desire to discover further facts which will
definitely confirm the said theory: this desire corresponds
with the crisis of the latter years of the said theory.

But on the other hand, as Levi-Civita observes, after the
latest researches of Hinstein, which collect in an admirably
comprehensive synthesis all the physical pnenomena (gravi-
tation included), it is difficult to avoid the impression that
we are, as regards the theory of relativity, face to face with
some definite acquisition. But, while taking account of this,
it is not expedient to neglect any attempt ata definite con-
firmation, from an experimental point of view, of a theory
which has subverted to so large an extent our simplest
physical notions. This confirmation may follow from a
precise study of the velocity of propagation of light emitted
from a moving source, or, which is equivalent, of the wave-
length of this light.

To realize this study we must devise an arrangement

* Michelson, Astrophys. Journ. xiii. p. 192 (1901) ; Harnack, Ann. d..

Phys. xlvi. p. 558 (1915).
+ See the theory proposed by Ritz, Guwvres, pp. 821, 371, 444.

168 Prof. Q. Majorana on the Second

which will permit us to identify the structure of the luminous
wave, freed from all external action, in its free propagation
(or transmission) when the velocity of translation of the
source can be varied at will. But, apart from the fact that
we must inevitably experiment under the eventual action of
our earth *, two serious and almost insurmountable difficulties
oppose themselves to the realization of such a programme.
In the first place, it is not easy artificially to endow a
luminous source with rapid movement f, especially if this
source (as is necessary in some interference methods) has to
be very rigorously monochromatic ; moreover, I shall publish
an account in a forthcoming paper of a disposition of this
nature with which I am about to experiment. Secondly, in
order to be able to examine the structure of the light emitted
by a moving source, with whatever disposition, the light
itself has to be subjected tu reflexions, refractions, &c., some-
times fairly numerous ; that is to say, the luminous ray must
encounter ponderable matter after leaving the source. It
does not follow, therefore, that even if cin a vacuum varies
with the particular velocity of the source, this quantity does
not return to the same fixed value after the said phenomena
of reflexion, refraction, &c. It will be well, therefore, to
endeavour to eliminate as far as possible, in an experiment of
this nature, all causes tending to complicate the phenomenon,
and in every case to consider its results carefully.
Meanwhile, to begin with a relatively simple experiment,
we may undertake the study of the wave-length of a ray of
light reflected by a moving mirror{. This may correspond
with the experiment already realized, some years ago, by
Belopolski, and afterwards repeated by Galitzin & Wilip ;
but if the first of these authors employed prisms for the
observation of the Doppler effect (and consequently the
question of the eventual variation of \ remained unsolved),
the other two made use of a diffraction-grating, by which
the controversy spoken of above arises. It would be better

* T cannot succeed in imagining an interplanetary experiment of the
nature of that proposed (in jest) by Rose-Innes ; see Phil. Mag. xxvii.
p. 150 (1914). |

+ I understand by this a velocity higher thansome hundreds of metres
per second; this value may perhaps be attained, but it is difficult to
conceive a practical disposition for a higher velocity. Naturally I leave
out of account the employment of canal rays, which do not give simple
and well-known velocities.

+t While this article was in the press, M. Michelson has called my
attention to his paper on the same subject, published in the ‘Astro-
physical Journal,’ April 1913, the conclusions of which agree with those
that I am stating.

Postulate of the Theory of Relativity. 169

therefore to examine the ray reflected from a moving mirror
by an interference method simpler than that on which the.
action of the diffraction depends, as has been said above.

Before expounding this method it is well to recall that
many theoretical researches have been made on the influence
of the motion of the mirror upon the reflected luminous
wave, amongst them those of Abraham, Brown, Edser,
Harnack, Larmor, Planck. These researches make of the
problem either a simply geometrical investigation, or an
application of the electromagnetic theory of light. But
without discussing the result of these researches we may
accept the conclusion of Harnack * respecting the frequency
of the vibrations reflected by a mirror in uniform motion.
Let v be the velocity of the latter, normal to its plane,
reckoned as positive towards the source; c the velocity of
the luminous ray zn vacuo which makes the angle of inci-
dence @ with the mirror ; n, x’ the frequencies of the ray
before and after reflexion, the source and observer being at
rest. If we put 8=v/c we shall have

,__1+28 cos 0+?
Tk at 1-- .

which, neglecting the terms in 8”, reduces to
n'=n(1+28 cos 6).

This relation is the same as that of Ketteler Tf, which was
employed by Belopolski f in his investigation of the Doppler
effect, and follows simply from the consideration that the
image of the source moves with the velocity 2v in the direc-
tion of the normal to the mirror and, consequently, the
component of this velocity in the direction of the reflected
ray is 2v cos @.

If now we suppose that the ray is, by suitable arrange-
ments, reflected with the incidence 6, k times from several
mirrors in motion with the velocity v, we shall have

n'=n(1+ 2kB cos 8).

Therefore, according to the hypothesis of constant velocity
of light, neglecting the terms in 6? we shall have

r’=A(1—2kB cos A).
If, on the contrary, we suppose that the velocity of the

* Ann. d. Phys, xxxix. p. 1053 (1912) and xlv. p. 547 (1915).
+ Astronomische Undulationstheorie.
t Communications Acc. Russe, xiii. p. 451 (1900).

170 Prof. Q. Majorana on the Second

reflected light is variable (and equal to the sum of e=3.10!°cm.
and the component of the velocity of the image in the direc-
tion of the ray) we shall have

ce’ =c+ 2kv cos 6.

And since c'=n/v' and c=nav we have A’=nxr. (It remains

then to see by experiment whether or not we can observe,
in addition to the Doppler effect, a variation in the value
of X; from this we can ascertain whether c remains constant
or not on reflexion from a moving mirror. I have not pro-
ceeded to the observation of the Doppler effect in these
researches since there is no doubt about its existence, already
proved experimentally by the authors quoted ; I have rather
sought to find out whether and in what way A varies when
the velocity of the moving mirrors changes.

Belopolski’s arrangement for the study of the Doppler
effect was inconvenient on account of the excessive subtility
of the luminous ray necessary to obtain multiple reflexions
from the same mirrors ; for this reason the author mentioned
was unable to observe the displacement of the rays except
by photography. I prefer to adopt the arrangement shown
diagrammatically in fig. 2. A horizontal brass wheel R, 35 cm.

Fig. 2.

= ee ee oe

see ae kw es ow ew

Sen
¢
a
aot
/
4
4
]
bey

fs
Fiein
7
ji
|
ji
J
/
/

in diameter (6 mm. thick), which can be made to revolve
with a maximum velocity of 80 turns per second, bears on its
‘periphery ten mirrors similar to M, rectangular, plane,
vertical, of glass silvered at the back. The velocity of the
centres of the mirrors, corresponding to the greatest velocity
of rotation, amounts therefore to more than 100 metres per

Postulate of the Theory of Relativity. Bik

second. The number of revolutions of the wheel was deter-
mined acoustically in each experiment. The mirrors, at
equal intervals on the periphery of the wheel, are inclined
to the radius from R passing through the centre of each of
them at an angle 2=29°. They are fixed solidly to R by
screw movements capable of permitting a rigorous adjust-
ment. The support for the bearings of the axle of R carries
also fixed mirrors F, vertical like M, of which the number
in the figure is three ; ; but this number may, at will, be
reduced, or increased up to nine. The position of the F’s
and M’s is such that a parallel beam of light L, after a
certain number of reflexions from the F’s and M’s (seven in
the figure), may be received at L’ when R has determinate
angular positions. Naturally the intensity of L’ is much
weaker than that of L, and this enfeeblement is much more
marked if R is in rotation, because in this case the light
arrives at L’ only during certain very short instants (ten
times per revolution). I have observed in practice, however,
that the four moving and three fixed reflexions of the fioure
allow of experimenting with light sufficiently intense at L!
even if R is in motion: that is to say, that direct observation
(without photography) suffices to ‘establish the luminous
phenomenon of which we have spoken above.

To study the value of X the light L’ was examined with
the well-known interferometer of Michelson, shown diagram-
matically in the figure. It is known that ‘if the distances
8,8; and 8,8; are exactly equal fringes are observed with
the telescope C even if the light is not monochromatic ;
these fringes then have the coloration of Newton’s rings.
As soon as a difference of path occurs (even if only of a few
microns) observation with white light is no longer possible.
Monochromatic light must then be used, and the order of the
interference fringes increases with this difference. Their
visibility is greater, the simpler the luminous vibrations.
From the researches of Michelson * it is known that from
this point of view the line that gives the greatest visibility
of the fringes with the greatest difference in path is the
green one of mercury (A=546uy). In this case numberless
circular fringes are visible even for a difference of path
1=2(8,8;—S8,83) =40 cm. I have therefore employed as
source La mercury arc zn vacuo the light of which is con-
veniently filtered by solutions of chromate of potassium
and chloride of nickel to absorb the violet and yellow rays ;

ee, Travaux et Mémoires, Bur. Int. de poids et mésures, xi. p. 146
(

172 Prof. Q. Majorana on the Second

in this manner I have been able to observe with the tele-
scope CU, with sufficient clearness, countless circular fringes,
even forJ=32cm. But for these researches I have limited
the difference of path to =13 cm., or still less.

The disposition described above is particularly suitable
for detecting very small differences in the value of the
incident wave-length ; in fact, the value of / being large a
very great number of wave-lengths is comprised in this length
(e. g., 200,000 if A=0'5u, and J=10 em.), and correspond-
ingly for the same variations very sensible displacements
can be observed in the position of a fringe.

With the apparatus disposed as above, let us note with
the micrometer wire of the telescope the position of a fringe,
for instance the first central bright one, when R is in the
position shown in the figure, or, still better, when it revolves
with a negligible velocity (one turn per second). If, now,
this velocity be increased to sixty turns per second a displace-
ment of the fringe under observation is distinctly visible ; if
the mirrors are moving against the incident ray this dis-
placement indicates a diminution of A, and it changes sign
when the direction of rotation of the wheel is reversed, and
this indicates an increase of X. In order to define the sense
of the displacement, I will say that on examining the system of
circular fringes with the telescope focussed for infinite distance
the diameter of each of these increases when the mirrors
move against the incident ray, and the fringes themselves
crowd together as those of large diameter are very little
displaced; at the same time some new fringes come out
from the centre of the system. On the other hand, when
the mirrors are moving in the sense of propagation of the
incident light the diameter of each fringe diminishes ; they
become more widely separated, and some of them are as it
were swallowed up by the centre.

Before stating the measure of the displacement observed
we will see what it should amount to, making the hypothesis
that the velocity of the light reflected from a mirror is the
same as that of the incident light. Let g be the number of
revolutions of R per second and d its diameter, reckoned
between the centres of two opposite mirrors M, then wdg
will be the instantaneous velocity of translation of the latter.
Since the mirrors are inclined at an angle « to the radius of
the wheel passing through each of them, the component
of the given velocity in the direction normal to the plane of
each mirror will be

v = dg COs 2.

Postulate of the Theory of Relativity. 173
We have, therefore,

n'=n(1+

3

2kadg cos « cos |
C

and, by the hypothesis of the invariability of c,
N= fz 2krdg 08 a COS =

C
If 1 is the difference of path of two interfering rays in
Michelson’s apparatus, the number of fringes which are seen
to cross the micrometer thread of the telescope when »
becomes 2’ (that is to say when the velocity of rotation
varies between zero and g turns per second) is

__ & 2kmdg cos x cos 0

I Xr Cc

If the observation is made by noting the position of the
fringes when the wheel turns in one sense with the velocity
g, and that corresponding to an equal and contrary velocity,
the number of fringes crossing the micrometer thread will
be 2f.

Now, in my apparatus d=38 cm., a=29°, @=27°, k=4
(as in the figure); if X is put equal to 0°546u (green mercury
line), J=13 em., c=3.10'° cm., and g=60 (turns of R per
second in one sense and afterwards in the other), we may
expect, according to the preceding formula, a fringe dis-
placement 2/=0°71.

Experiment gives, for the case mentioned, a displacement
of between 0-7 and 0°8 fringes ; and it is not possible, for
reasons of visibility, to carry the precision of the observa-
tions further. But, as is seen, the agreement between the
predicted result and observation is sufficient ; this agreement
is confirmed by observations made by choosing other con-
venient values of J and g, of which for brevity’s sake I shall
not speak here.

Experiment, therefore, authorizes the conclusion that
reflexion of light by a moving metallic mirror does not alter
the velocity of propagation of the light itself, in air, and con-
sequently, with great probability, also in vacuo; at least, in
the conditions of the experiment above described. This
experimental result, as to which no doubt can be entertained,
is contrary to the hypothesis of some physicists who, like
Stewart *, basing themselves upon the electromagnetic
emission theory of Thomson, maintain the possibility that

* Phys. Rev. xxxii. p. 418 (1911).
Phil. Mag. 8. 6. Vol. 35. No. 206, Feb. 1918. ‘)

174 Mr. Prentice Reeves on the

light, after reflexion, is propagated with the velocity c+v,
where v is the component of the velocity of the image in the
direction of the reflected ray.

To complete these researches I intend, as I have said
above, to investigate further with the same interferential
arrangements, the velocity of propagation of light from a
source set in motion artificially ; but of these, and of the
general conclusions to be drawn from these investigations, I
reserve mention for a future occasion.

XIX. The Visibility of Radiation. :
By PRENTICE REEVES *.

fae theory of this subject has been given previously by
Nutting t and Ives {, and in those papers may be
found a thorough treatment of the early literature. In
this paper the writer wishes to present further data obtained
by a method similar to that employed by the above writers
but using a different apparatus. The writer has data from
thirteen subjects, five of whom were also used as observers
by Nutting in his list of twenty-one subjects. The values
for the spectral energy distribution of acetylene were those
offered by Nutting, and were obtained by weighting the data
accessible up to that time as well as his own results in this
laboratory. By using these values the writer was able to
directly compare results with those of the other writers, and
by using the values offered by Coblentz$ and revised by
Coblentz and Emerson ||, we can see the effect of various
values for the spectral energy distribution of acetylene.
The variations in the acetylene values are probably due to
the different kinds of burners used, as Coblentz has shown
that the spectral energy distribution in the longer wave-
lengths is affected by the thickness of the radiating layer of
incandescent particles in the flame.

The apparatus represented in fig. 1 is a modification of
the Nutting monochromatic colorimeter{] as manufactured

* Communicated by Dr. C. E. Kennett Mees, being communication
No. 55 from the Research Laboratory of the Eastman Kodak Company.

+ P. G. Nutting, Phil. Mag. xxix. p. 801 (1915); Trans. lum. Eng.
Soc. ix. p. 633 (1914).

+ H. E. Ives, Phil. Mag. xxiv. p. 149 (1912).

§ W. W. Coblentz, Bull. Bur. Stds. vii. p. 248 (1911); reprinted, ix.
p- 109 (1912).

|| W. W. Coblentz and W. B. Emerson, Bull. Bur. Stds. xiii. p. 1
1916).
q 3 G. Nutting, Bull. Bur. Stds. ix. p. 1 (1918); Zsch. f. Instrument-
enkund., xxxiii. p. 20 (1918),

Visibility of Radiation. 175

by Adam Hilger of London. This type of apparatus has
been described by Jones*, but the modifications made for
its use in this experiment warrant a separate description in
this paper. The light from an acetylene burner 8, is

Fig. 1.

Modification of Nutting Colorimeter.

focussed on the slit O, by the lens L. The pressure of the
gas was kept constant at 9 cm. as indicated on a water
manometer, and the width of O, was determined by a series
of preliminary experiments. A pair of nicol prisms N con-
trolled the intensity of light from S,, and by means of the
vernier V and a graduated quadrant attached to the movable
nicol we are able to determine the ratio of the incident to
the transmitted intensities. C, is a collimating lens, P a
constant deviation dispersing prism operated by a screw
carrying a wave-length drum D, which indicates directly
a 1b). A. Jones, Phys. Rey. iv. p. 454 (1914); Trans. I. E. S. ix. p. 687
7 L. A. Jones, Trans, I. E. 8. ey 716 (1914).
2

176 Mr. Prentice Reeves on the

the quality of the light through this part of the system. §,
isa gas-filled tungsten lamp the light from which passes
through a daylight filter, F, and by means of the collimating
lens C, strikes the matte surface on the Whitman disk W.
This disk, shown in fig. 1 a, is rotated by a motor belted to
M and turns so that at one instant a reflecting quadrant
sends a beam of standard white light from 8, to the eye,
and the next instant a blank quadrant allows the coloured
light through P to reach the eye. The eye sees the light
image at O,, which is screened down so as to restrict vision
to the fovea, and an artificial pupil A was used.

The white light source 8, was regulated so as to give an
illumination of 13 foot candles at W, and was kept constant
at this intensity by means of connexions through a Wolff
potentiometer and a sensitive galvanometer. With both
light sources constant and three independent series taken on
different days, we may safely assume the resultant average
curves to be representative of the observers.

A constant width of slit was used throughout. In order
to determine the necessary slit-width correction, the wave-
length interval corresponding to the width of the slit, as one
edge of the image is moved across the field, was determined
throughout the spectrum by sighting on certain lines of mer-
cury, hydrogen, and helium. The relative slit-width for any
wave-length thus determined multiplied by the corresponding
acetylene emission gave the value of the relative energy.

When taking a series of observations the necessary pre-
liminary adjustments were made, the movable nicol N (fig. 1)
was set for maximum intensity and the balance was made by
shifting the wave-length drum. For observations between
wave-lengths 500 and 680 the balance was made by setting
the drum and moving the nicol. An electric tachometer
was belted to the motor and the speed could be changed by
variable resistance. As all observers were familiar with
the theory of the so-called “critical frequency.” in flicker
balances, each observer regulated the speed, but no record
was kept of these values. The relative energy for a given
wave-length and the sine square of the angle read at V
gives us the relative energy for equivalent luminosity. The
values for the three independent series are averaged and
plotted for each individual’s curve. From this average curve
the visibility curve for that observer was obtained, and the
visibility curves were reduced to equal areas by weighting
the ordinates according to height in order to compare
separate curves and obtain the average curve of the group.
These results are shown in Table I., and fig. 2 shows the

177

Taste [.—Individual Visibility Results.

Subject. wA0 <0. “ONS “O28 Ot op OR "7 1B ee 50" 60> SO ee OL oe Sea rren.

JB sievsesvssrcees “LTS “258 512 “740 ‘876 1006 1'086 ~-986 -:804 «808 7666 “586 “406. “260 “1b6) “TtU) on0

2 FAVE, ...es..ss00.., 189 °882 °558 -712 -799 ‘880 ‘946 -955 -898 ‘799 ‘662 ‘558 ‘895° -287 °196 ‘116 “bdo
3 M.F\F, ............... ‘199 °803 -471 -706 ‘907 -996 1:018 1°005 -907 -778 ‘626 ‘502 ‘860 ‘261 ‘152 ‘101 ‘bb2
S MBH. ............ “151 ‘212 -881 674 ‘840 986 ‘990 1:008 “891 “786 “681 ‘577 464 “8b4 ‘225 “169 “bb6
6s K.H. .....ccccsc00002. 153 308 +466 °671 ‘808 ‘920 ‘990 ‘985 ‘941 ‘862 ‘722 ‘551 ‘395 284 ‘200 +185 ‘bb4
=, LAW. iscsssscseevese “206 ~*B80'- “401 - *749 864 065 1:001 982 882 -764. 422 -5I8 <80ll= “276 «168 "10n. -bb4
=> OHM. jisceeccee (145 “268me474--:778.— 913) 1-021 1088-962. “B48 728-607 “bOH “SB "Bee 9102" “ly S647
= P.G.N. .........0... ‘236 ‘318 467 -716 ‘861 ‘984 -996 ‘986 ‘891 ‘785 ‘654 ‘519 ‘888 282 -190 ‘120 ‘bob
“% PLR, wcesccssseeeseeeee “118 +209 392 +594 748 ‘867 986 ‘986 -890 ‘841 ‘775 ‘667 “494 ‘854 ‘242 ‘155 ‘55D
> BER. wc. ccc. (216 334 508-726 «862 «950 1-011 1-007 981 ‘884 -6h2 -496 861 ‘243 ‘141 ‘095 ‘553
BGS. ....c.c0000... 153 254 -406 616 ‘749 “875 “985 ‘958 ‘985 ‘869 ‘766 ‘626 °474 ‘345 239 -161 ‘560
ALBLW. weccesecceeeeee “187 °3821 “580 “750 ‘857 (970 ‘994 ‘970 ‘882 ‘788 650 ‘Ol ‘874 ‘271 ‘198 “186 bb2
R.B.W. .....cc000. (161 274 +485 “701 ‘865 "976 ‘980 ‘957 887 ‘818 -700 ‘671 ‘427 ‘806 ‘211 ‘138 ‘548

ee

Average ............ ‘175 ‘289 ‘475 ‘702 ‘842 ‘950 990 ‘977 ‘898 ‘807 ‘676 ‘548 409 ‘293

194 °127 ‘653

178 Mr. Prentice Reeves on the

mean curve as compared with the curves of other writers.
In Table II. are shown the results obtained by using the
different values for the acetylene emission and the mean

ign i { NY,

f fi
,

WAVE | LENGIY int |

AG 50 whl 52 59 54 5S 56 57 58 59 60 6) 62 63 .6F

—— Authors mean of [3 subjects.
a Nattings i ha as

SD -ofyes. ee ae oe
= == Curve calculated from formula ( by
Netting).

from Nutting’s and Ives’s results, as well as the results com-
puted from the formula offered by Nutting. This formula
was found to represent visibility very closely between the
wave-lengths of -48 and ‘67 and is of the form

V = Vm Re" -®)

where R=) max./A and a=181. When the writer used
Coblentz’s revised data for acetylene the results agreed re-
markably well with these computed results.

In Table III. the acetylene values used by the writer and

Visibility of Radiation. ig

Nutting are given, as well as the two sets of values published
by Coblentz. The greatest differences in these values occur
in the region of the longer wave-lengths and, as has been
said, are probably due to the different types of burners used.

Taste [l.—Comparative Visibility Results.

Mean V- Coblentz Coblentz SAEs : Computed
ash oh. from data publ. data publ. Se ; is ii

Table I. 1911. 1916. * formula.
“49 “175 172 172 227 °235 232
50 "289 °283 215 330 363 358
‘bl “475 ‘471 474 °477 596 514
52 “702 *705 686 ‘671 794 675
"53 "842 851 ‘841 "835 912 *824
D4 “950 "947 "935 "944 tere 933.
"5D "990 ‘988 “993 "995 1:000 "994
56 "O77 "982 "985 "993 ‘990 “993:
“5 "898 ‘926 935 ‘944 948 "939
58 ‘807 "825 "836 ‘851 875 839
‘59 ‘676 693 ‘710 "735 ‘763 af Ay
60 548 "DD2 “580 605 *635 “585
‘61 ‘409 ‘417 —°446 ‘468 ‘509 "456.
62 "293 "294 319 "342 387 343
63 194 "185 214 ‘247 272 “235.
64 "127 "125 "140 163 al I 65 "158

Taste II1.—Spectral Energy Distribution of
an Acetylene Flame.

2 i oblentz oblentz
oe 7 - ea - ere i oe.
“48 Me 16°5 170
“50 14-4 21-7 21°9
“52 184 27°6 219
54 23°2 34'8 39°0
was) 26°1 A 389
56 29°1 43°7 42:9
58 36°2 54°0 522
‘60 44:2 66°3 62:1
62 33°7 80°5 73°0
‘64 63°8 96°5 84:7
66 746 112°8 97°4
68 861 130°1 110-9
‘70 98:2 147°0 124°6

The spectral energy distribution of acetylene is probably
better known than that of any other light source, and the

VISIBILITY

180 On the Visibility of Radiation,

burner used in this experiment gives an extremely constant

quality of light *. |
In comparison with the other curves shown in fig. 2, the

writer’s visibility curve for the thirteen subjects is slightly

10

58

~

36 : J2 ry ; y
raf WAVE LENGTH IN l—Aythor’s mean of 5S subjects.

==<Nulftings: oe 4

more contracted. The maximum visibility occurs at wave-
length -553 in agreement with Ives’s curve, as against *555
for Nutting’s curve.

* Standardized burners may be obtained from the Research Labora-
tory, Eastman Kodak Company, Rochester, N.Y.

Hodographic Treatment of Planetary Motion. 181

Fig. 3 shows the mean of the writer’s resultsand Nutting’s
results on the five subjects who served as observers in both
experiments. The average maximum visibility found by the
writer is °555, and by Nutting -554.

This work was carried out at the suggestion of Dr. P. G.
Nutting, and the writer wishes to thank him and the other
members of the laboratory for their assistance.

Author's Note.—This paper had been completed before a
paper by Coblentz and Emerson* appeared, so that their
latest data have not been included.

XX. On the Hodographic Treatment and the Energetics of
undisturied Planetary Motion. By Professor ANDREW
Gray, #.B.S.t

J. * Ga following bit of Newtonian dynamics was sug-

gested by some passages in a recent discussion
in the Philosophical Magazine. It sets forth a mode of
dealing with the elementary theory of planetary motion
which may have some novelty, and states certain results,
which J think are interesting, with regard to the energetics
of such motion. Into problems of relativity I do not at
present enter.

As a matter of scientific history there would seem to be
no doubt that the first idea of the hodograph is due to
August Ferdinand Mobius, the inventor of the barycentric
calculus. In § 22 of his book, Die Elemente der Mechanik
des Himmels, which was published in 1843, Mobius specifies
a point H which moves so that its distance OH from a fixed
point O is continually equal in length to, and in the same
direction as, the line which represents the velocity of a
particle moving in a given path. He derives the result
that the acceleration of the particle in the path is repre-
sented in magnitude and direction by the velocity of the
point H. This of course is the whole idea of the hodo-
graph very clearly stated. So far as I have been able
to see, Mobius does not make any particular use of the
idea; and he does not seem to be aware that the curve
described by the point H is, for a planet moving in a conic
section, a circle, with O as an eccentric point within it.

* Coblentz, W. W., Emerson, W. B. ‘Relative Sensibility of the
Average Eye to Light of Different Colors and some Practical Appli-
cations to Radiation Problems.” Scientific Paper 303, Bureau of
Standards, issued Sept. 12, 1917.

+ Communicated by the Author.

182 Prof. A. Gray on Hodographic Treatment and

2. Apparently Hamilton was ignorant of the work of
Mobius, when he published his paper on “ The Law of the
Circular Hodograph” in the Proceedings of the Royai
Irish Academy in 1847. More than ten years after, in
a letter of date March 3, 1858, to De Morgan, he says :—
“Do you know much about Mobius and his works? . .
He wrote to me a couple of years ago... that he had been
lecturing on my theory of the circular hodograph, to which
he might, very plausibly, have put in a sort of claim, or
at least a claim to the general conception.” [Graves, ‘ Life
of Hamilton,’ vol. iii. p. 543. |

That the hodograph for the motion of an undisturbed
planet is a circle, was of course the great discovery, and
without doubt this discovery was Hamilton’s, and Hamilton’s
alone. That a name was given to the curve may have had
something to do with the progress of discovery. When the
curve had thus been endowed with a kind of personality,
such questions naturally presented themselves as: What
is the nature of the hodograph of a planet? What is the
hodograph of an unresisted projectile? and so on. The
names given to electric and magnetic quantities have
certainly helped to distinguish sharply between one idea
and another closely related to it, and led to the evaluation
of the individualities which the names emphasized.

3. I was surprised to see Professor Eddington’s statement
in the Philosophical Magazine for October that the theorem
of the resolution of the orbital velocity of a planet into two
components of constant amount, one of them also in a
constant direction, seemed to be overlooked in dynamical
textbooks. It was given in Frost’s ‘Newton’ in 1854, and
is stated on p. 147 of the fourth edition. It is to be found
in Routh’s ‘ Dynamics of a Particle,’ § 397, and in a book
on Dynamics by Dr. J. G. Gray and myself, § 134. In the
last mentioned work there is a rather full treatment of the
subject referred to recently in the Phil. Mag.—the motion
of bodies of varying mass.

The existence of these two components of velocity, one of
constant amount 2, say, and constant direction, the other
of constant amount ve, directed always forward at right
angles to the radius vector, affords the most elegant mode
of passing from the circular hodograph to the orbit. I shall
show first that the hodograph is a circle, and then pass from
the hodograph to the orbit.

Let the planet, denoted by P and supposed to be of unit
mass, be acted on by a force directed towards a fixed point 8,

and varying inversely as the square of the distance SP. If@

Energetics of undisturbed Planetary Motion. 183

be the angular speed with which SP is turning about S, and
r=SP, we have 7°6=h, where h isaconstant. Thus @=h/r’.
From a centre C let a circle be described with some radius 2a
in the plane of the path or in a parallel plane. Let a radius
CH of the circle be always parallel to SP. The veiocity of H

is 2a, and is therefore proportional to 1/r?._ If the force per
unit mass on the planet at distance r be u/r?, the acceleration

is 6/h, and so

velocity of H= = E a's} ty aise teen ae

7?

Thus, to the factor 2ah/p, the velocity of H represents the
magnitude of the acceleration of the planet. Its direction is
at right angles to SP, aud represents the direction that PS
would have if SP were turned 90° about P in the opposite
direction to that in which SP turns, as P moves in the
orbit.

Drawing the chord H,H, which on the scale adopted
represents the total change of velocity, between a previous.
point Hy, corresponding to a radius vector SPo, and H, we
see that, if we choose the proper point O in the plane of the
circle, OH, and OH will represent the velocities at the
beginning and end of the time ¢. It is clear that O must lie
within the circle, for the vector OH, which represents the
velocity, must turn through an angle 27 while the planet
traverses the orbit once, which would not be the case if O
were outside the circle.

The position of O must be independent of the value of ¢,
otherwise the speed for the radius vector SP would depend
on the choice of the initial radius vector SP). Thus O is a
definite point. This result, taken along with the repre-
sentation of the acceleration by the motion of a point in
the circle, shows that the hodograph of the planet is a circle.
The velocities are represented in magnitude and direction by
the radii vectores drawn to the circle from the eccentric
point O.

OH, is perpendicular to the direction of motion at Py, and
OH to the direction of motion at P. The members of the
family of lines drawn from O to the sequence of points H
are perpendiculars to the corresponding tangents to the
pat :

4. It is obvious that OH may be resolved into the two
components OC, CH, that is into two components of fixed
amounts, v, v2, at right angles respectively to the line OC

184 Prof. A. Gray on Hodographic Treatment and

and to the radius vector SP. We can now instantaneously
find the orbit.

Let 6 be the angle which SP makes with the direction OC.
The angular momentum of the planet about § is 7(v.+ v, cos @),
which has a constant value h. Thus if we write e for v/v,
we get

r(1+ecos 0) = fi .,)) Ce

Us

“

that is the path is a conic section of which § is a focus.

The length of the major axis is the sum of the lengths 7, 72
of r obtained by putting cos@=1, cos 0=—1, respectively.
If it is denoted by 2a we have

h i if Qh
2a = Vo (5+ a = v(1—e)' - : . (3)

Thus the perihelion and aphelion distances are a(1—e) and
a(l+e).

5. It is convenient to make the radius of the hodograph
equal to the 2a just found. The length of the line OH,
representing the velocity, is then equal to twice the length of
the perpendicular let fall from O on a line which is parallel
to the tangent to the orbit at the corresponding position of
the planet. In fact a circle of radius 2a described from 8 as
centre serves very conveniently as hodograph. The hodo-
graphic origin O is then coincident with the empty focus, as
will easily be seen from the fact that the perpendicular from
the empty focus to any tangent of the ellipse at a point P
intersects the radius vector SP on the circle.

6. As has been seen in (2) the two components 4, v2 of
velocity give an instantaneous integration of the differential
equations of the orbit. The radial differential equation

es 4 B& >
r— 76? =— a ©) eit) ye) oh ee (4)
shows this more clearly, and incidentally gives the value of h
in terms of v.. Since the component v, is at right angles to
the major axis, and is directed towards the side of that axis
to;which the planet passes at perihelion, we have

pS ysind, 6 = ON ee
where @ is the angle traversed by the radius vector from
perihelion to the position considered. Hence

T = v, 0080.0 = v1, cos 8(v2+ 2% Cos 8)/7,

Energetics of undisturbed Planetary Motion. 185.

and the equation of motion is
V_(Vg+v, cos 7) = =

or bb
os

— eeeos: - “ - s ° . (6)

the equation again of the path. But we have already
obtained by considerations of the angular momentum the
equation

He

U2

= I+ecos @ oh htoAn | hc tat Drernd., 6 (7)

ff

and therefore we obtain the very simple and remarkable
relations
Laven

rig Ra) aN (8)
that is the double rate of description of area by the radius
vector is equal to the ratio of the “intensity of the centre”
to the constant component of velocity at right angles to the
radius vector.

The time ¢ occupied in describing any part of an orbit is
thus equal to 2Av,/u, where A is the corresponding area
swept over by the radius vector. This, expressed in terms
of the focal radii and the chord of the are described and the
axes, is Lambert’s theorem.

A steamer rounding a buoy, in a uniform tidal stream,
with constant speed v, always directed at right angles to a
line joining it with the buoy, describes a conie section with
reference to the land. The curve is a hyperbola if vp is less
than the speed v, of the stream, and an ellipse in the contrary
case. The focal axis is at right angles to the stream, and
e€=v,/v.. [I believe that this illustration is originally due
to Greenhill. |

When 0=0, (7) becomes

a(1—e)(1+e) = h/v,
or (with v,/v,=e)

vaca antes} (8)
|

1

2

See ae
m=laq a |

186 Prof. A. Gray on Hodographic Treatment and

7. We deduce the energy equation as follows. The
resultant speed v is given in terms of v4, v2 by

Ut == 0)? + Ug" + 20102 COS.0,,) 2 |. 2
which by (6) and (9) can be written

? = 2(-—3-). oe V0 Sr

i
The term p/r represents the potential energy exhausted by
‘the passage of the planet under the central attraction from
infinity to the distance 7. Writing the equation in the
‘form
1
3 ae = Tae os! at Oe a
and taking —,/r as the potential energy, we see that the
kinetic and potential energies have a constant negative sum,
—p/2a. From this equation we shall draw some conclusions
which appear interesting. [It is to be understood that when
e>1, that is when the orbit is a hyperbola, the sum of the
energies 4v? and —y/ris + p/2a.}
The constant angular momentum is p/vy=eu/v,=)(p/a)?.
Hence the period of revolution is

24
T= 20 = an(“), ho A
Ps ro)

and is therefore independent of the eccentricity.

8. This gives an interesting instantaneous solution of the
elementary problem of polar dynamics. The orbital motion
of a planet is annulled when the distance from the sun is d;
Jind the teme which the planet will take to fall to the centre of
force. Since the period is independent of the eccentricity,
let e be less than but very nearly equal to 1. The foci are
practically at the ends of the major axis. When the planet
has just passed (not rounded) the aphelion end, the speed
along the major axis is zero. The time taken to reach the
other focus is half the period, and so the time taken by
the planet to fall into the sun is w(}d3/u)?, as may be
verified at once by direct integration.

The earth in the circumstances stated would fall into the
sun in 17/2472, that is in about 65 days.

[It may be noticed incidentally that if we use the value of
v? given on the right of (11), with the values of 2, v, given
in (9) and (10), and the values of the action and the period
just found, we obtain after a little reduction the integral

27 cos 0 dd e

DAD cs A On Pea a
> (1 +e cos 8)? i Tee

Energetites of undisturbed Planetary Motion. 187

Some years ago I came upon the examination question :
Prove that the action for a complete revolution of a planet in
its orbit 1s independent of the eccentricity of the orbit. The
following proof of this interesting proposition presented
itself: it is simple and I think elegant. The action, A, is
given by the equation

KS Vrdt — \ ods, SCs nN a

where the space integration is taken round the orbit and the
time integration for a complete period. Let p, p’ be the
lengths of the perpendiculars let fall from S and the empty
focus on the tangent at P to the orbit. We have

py=h=plv, or v= (u/v.b’)p' = (u/ab?)*p', by (9),
Hence

A = jvds = (4,) Sv'a Si Darl gud etn EEN)

since \ pds is 27ab, twice the area of the (elliptic) orbit.
The action is thus independent of the eccentricity, and
there is no variation of the total action from one orbit
to another, provided both possess the same major axis.
The action may also be written as 2abv.. [There is no
difficulty in writing down an expression for the action in
any finite part of the orbit.]
lf T be the period we get by (13)

Hy uae as Y (eal
Ai regen ats het ss oy) AB)

This is the time integral of v?. The time average of the
kinetic energy in the orbit is p/2a.

From this proof * it appeared that the action is pro-
portional to the area swept over in any time by the radius
vector from the empty focus to the planet. Thus while the
radius vector from the sun to the planet is the timekeeper,
measuring as it does time in the orbit by the area it sweeps
over, the other radius vector “keeps” the action. This

proposition I found had already been stated by Professor
Tait.

* Another proof naturally occurs in which the integration is effected
with the aid of the eccentric angle: but it is long and unsuggestive.
Since this paper was sent in I have found a memoir by Grinwis, Akad.
van Wetens., Amsterdam, ix. 1891-2, in which the proposition regarding
the action is given. Probably this was the origin of the examination
question.

188 Prof. A. Gray on Hodographic Treatment and

9. The term y/r in the energy equation is the potential energy
exhausted when the planet is brought by the sun’s attraction
from infinity to the distance r. ‘Thus we obtain by (11) the
curious theorem that the time-average value of this term for a
complete revolution of the planet is twice the time-average
of the kinetic energy in the orbit, that in fact the time-
average of this exhaustion of potential energy is equal to
the action. This is also, of course, independent of the
eccentricity.

As a particular case of this the kinetic energy of a planet
in a circular orbit is half the potential energy exhausted in
the journey from infinity. Hence also, if the planet were
transferred from one circular orbit to one of (say) smaller
radius, the increase of kinetic energy would be only one
half of the additional potential energy exhausted in the
passage.

This result for the circular orbit is of course well known;
the corresponding relation which holds for the mean kinetic
energy in an elliptic orbit was, I believe, first stated by myself
in a letter in ‘ Nature,’ August 7, 1913. The fixed ratio (4)
of the mean. orbital kinetic energy to the mean potential
energy, exhausted to the different points of the orbit, is
curious ; but there is always a fixed ratio of the energy
dissipated in the interactions of bodies to the whole available
energy. For example, a body of mass m moving with
speed v collides inelastically with a body of mass m’ at rest;
and the kinetic energy dissipated bears to the original kinetic
energy the ratio m’/(m+m’), which is quite independent of
the details of the action between the bodies.

The theorem for passage from one elliptic orbit to another
is exactly parallel to that stated above for a circular orbit.

In the planetary case then, the bodies are only left moving
in elliptic orbits, when the proper adjustment has taken place; —
others if left moving too slowly will fall into the central
body, or if moving too quickly may recede from the central
body to undergo further energy modifications by collision or
otherwise.

Now let us comparea hyperbolic orbit with an elliptic orbit
as regards this affair of energy. We have the theorem that
the kinetic energy at distance r in a hyperbolic orbit exceeds,
and in an elliptic orbit falls short of, the potential energy
exhausted from infinity to this distance by the mean value of
the kinetic energy in the orbit. This result has not been
formally proved for the hyperbolic orbit, but has been inferred
by analogy from the result for the ellipse. There is no

Finergetics of undisturbed Planetary Motion. 189

difficulty in framing a formal proof, but the following may
be sufficient.

At a considerable distance trom the centre of force the
kinetic energy in the hyperbolic orbit will have become
practically constant and equal to w/2a. For r has become
very great and p/r very small, so that $v?=p/2a. The
planet therefore moves along the curve, ultimately along
the asymptote, with more and more exactly constant speed,
(w/a)*, and as it continues at this speed, in its coming and its
going, for an infinite time, the time-average of the kinetic
energy is /2a, as in the elliptic orbit.

It may seem a hard saying that the time-average of the
kinetic energy in a parabolic orbit is zero, but in this case
the energy equation is $v’=y/r, and so v is very small when
ris very large. Thus during an infinite time the value of
3v? is the evanescent quantity u/r, and thus the time-average
of the kinetic energy is evanescent.

This theory leads to the result that the exhaustion of
gravitational potential energy alone cannot have led to
motion of a planet or comet in a hyperbolic orbit. It would
seem that the necessary excess of mean kinetic energy must
have been produced by some cataclysm within a body, from
which the planet was thrown off after the sun had done the
work of bringing the body within a finite distance r of the
centre of force. Only by exhaustion of internal energy
does it seem possible to make up the necessary additional
energy.

The question of equipartition of energy between the
different stars has been a good deal discussed. If a system
of stars has its origin in the exhaustion of potential energy
by the attraction of some great central system, the energy
relations here discussed would appear to negative equipartition.
If there is an approach to equipartition of kinetic energy of
translational motion, and there is evidence apparently that
the more massive stars move the slower, such an origin
becomes on one more. ground improbable.

The University, Glasgow,
Dec. 31, 1917.

Phil. Mag. 8. 6. Vol. 35. No. 206. Feb. 1918. P

[ 190 ]

XXI. A Criticism of Wien’s Distribution Law.
By Frank Epwin Woop *.

1. Introduction.

ae purpose of this article is to criticise Wien’s distri-

bution law from a mathematical point of view :—To
show (1) that although the derivation of Wien’s distribution
law is generally made by steps which are not mathematically
justified, or for which no rigorous justification is given, still,
by using Wien’s assumptions}, a rigorous derivation is
possible; (2) that the law obtained by Wien is inconsistent
with other results which follow from the same assumptions ;
and (3) that this inconsistency is eliminated and a new law {
obtained if Wien’s implicit assumption be replaced by a
simpler and more probable one.

Incidentally several theorems, new so far as the author
knows, in the kinetic theory of gases will be obtained from
the Maxwell law for the distribution, with respect to their
velocities, of the molecules in a gas; a simple proof of the
Wien displacement law will be obtained from these theorems
and the Wien assumptions, and two interesting relations
regarding the dependence of the radiation of a molecule upon
its velocity will be given. Also there will be found some
criticisms of the treatment of the distribution !aw and of
the displacement law as given in the standard treatises.

Mendenhall and Saunders$, Waidner and Burgess |,
Rayleigh { and others have criticised from a physical point
of view the assumptions used by Wien to prove his distri-
bution law, while Lummer and Pringsheim **, Paschen ff,
and others have considered the agreement of this law with
experimental results. So far as the author knows, no
criticism from a mathematical point of view has been

published.

* Communicated by the Author.

+ It will be necessary to include under Wien’s assumptions one which
he has made implicitly, but not explicitly; or else to assume that he
has made a fundamental error.

+ [ am indebted to Professor Lunn, of the University of Chicago, for
this new formula, and for the observation that Wien either made a
mistuke or an unstated assumption.

§ Astrophysical Journal, xiii. p. 25 (1901).

|| Bull. Bur. of Standards, i. p. 189 (1904).

q Phil. Mag. xlix. p. 539 (1900).

*#* Verh, d. Deutsch. Phys. Ges. i. p. 1 (1900).
+t Astrophysical Journal, x. p. 40 (1899); xi. p, 288 (1900),

A Criticism of Wien’s Distribution Law. Toe

Wien’s distribution law is
eer?

f(A, 7) = aap. ? a) ee ise ne, | Stee ts (1)

where $(A, @)dX represents the intensity of radiation of a
black body at a temperature @ produced by waves whose
lengths lie between >X and X%+dX, aud where C and ¢ are
constants *. M. Planck { obtained this same formula (and
also another {t) but from considerations entirely different
from those used by Wien. The criticism of this article does
not apply to Planck’s work ; however, it has been suggested
that a similar criticism might apply to Planck’s derivation
of this same law. Other formulas for ¢(A, 0) have been
obtained by Callendar§ and Rayleigh ||; although these
formulas may be in closer accord with experimental results,
still the Wien formula has considerable importance due to
its use by Drude and many other investigators.

2. A derivation of the Distribution Law along the lines
proposed by Wien.
Wien takes a gas as the black body, and uses Maxwell’s
law that the number of molecules whose velocities lie between
v and v+dv is proportional to

Hagges da, inch ws (6:49

where «?= 2%", and v is the root-mean-square velocity ; a? is
proportional to @, the absolute temperature of the gas.

Wien makes the hypotheses :

(a) That the length of the wave sent out by a molecule
depends only upon the velocity of that molecule :—then v is
a function of 2X only.

(6) That the intensity of the radiation for wave-lengths
between X and X+dX is proportional to the number of
molecules, as given by Maxwell’s law (2), which send out
waves with lengths between A and A+ dn.

Wien states that it follows from these two hypotheses that

Fa)
Di eco mee 5s es (8)
where F(A) and f(A) are two unknown functions].

* Wied. Aun. lviii. p. 662 (1896).

+ Wied. Ann. i. pp. 69, 719 (1900).

t Verh. d. Deutsch. Phys. Ges. 11. p. 202 (1900).

g Phil. Mag. xxvi. pp. 787 (1918) ; xxvii. p. 870 (1914).

| Phil. Mag. xlix. p. 5389 (1900).

q In our development of Wien’s distribution law, it is assumed that
F(A) and f(A) are continuous functions. Just what physical significance
these assumptions have can be seen from § 6 of this article.

P2

192 Mr. F. E. Wood: A Criticism of

He then states: ‘‘ Now the variation of the radiation with
the temperature according to the law given by Boltzmann
and myself consists of an increase of the total energy in
proportion to the fourth power of the absolute temperature,
and a variation of the length of the waves associated with an
energy quantum and lying between X and A+4dA in such a
way that the corresponding wave-lengths are inversely pro-
portional to the temperature. So if one plots for any one
temperature the energy as a function of the wave-length,
then for any other temperature this curve will be the same
if the scale units of the graph are so varied that the ordinates
are made smaller in the ratio = and the abscissee are made
larger inthe ratio 8. This latter is possible for our value of
(A, 8) only when A and @ appear in the exponential as a
product X@. Then

ee)

fa = 0 ) ° . ° . ° ° (4)
where c denotes a constant.”

Since I have found no simple proof of (4) from the above
statements, I propose the following derivation of the form
of F(A) and /(A) based entirely upon Wien’s statements.
Let C, be the curve obtained by plotting X as abscissa and
y=@(A, 8) as ordinate for an arbitrary temperature 0, ; then
the equation of Q,; will be

I),
Bem (X)e "4.

Let C, be a corresponding curve for a temperature 0, ;

then the equation of C, will be i
(A)
Get n)e 9.

Now Wien’s statement, as corrected, is that the trans-

formation

ale an ! :
= a, a
2
will transform ©, into a curve congruent to ©,. This trans-
formation gives as the equation of the transform of C,

-p (2%
/@ ) Q he 65

— ge Fr ( Je 62 6).

i (3, 0, (6)

* This ratio should be = see Wien, Berlin Sitzungsberichte, vi. p. 55

18938) ; Lorentz, ‘ The Theory of Electrons,’ p. 74.

Wien’s Distribution’ Law. 193
Now if we make the transformation y'=y, =A, this
curve with equation (6) coincides with C,, and so
a

_ fA Pe rg) i \i ee oe
F 01 =(3 F “| JS a Gy
OD: z) (% ¢ (7)

This is an identity for all values of 6,, 6,, and X.
Now (7) can be written in the form

|

Fe *) 12 p(@A) Legg?
7 @5 02 6,
(3:) :

ee (8)

If @, and @, be replaced by £0, and k@,, where & is an
arbitrary constant different from zero, the value of the left-
hand member of (8) is unchanged, and therefore the value
of the right member is also unchanged. This gives
iif 32) - Foy} ita Va, — 5 ft

2 2
3

which is an identity for all values of k40. Therefore

é

xt (= — ZF 0) =0, AOS Saeioy
or
(lo) ») =e eee al)

Substituting (9) in (8) gives

0,
(ale) _, PPS WRU ats

In order to obtain the form of F(A) and f(A) we will
prove the Lemma: The most general solution of the functional

equation sb (ke)
“D( ek

eek iw ee (12

50 a

is b(x 2) — = , where k (\k\60), C, and a are constants. and >

1s a ee function of #.
Let ee lays, - 5/1 aieiuee ne cuss ile)
then A(z) is a continuous function of « From (12) and

(13) it follows that
ACT ACB): 9 shu inentavte sorters: wx: (hap

194 Mr. F. E. Wood: A Criticism of

Suppose first (4|<1,; then the A function has the same
value at the points z, tae seis k*x,... This set of points
converges to the point x=0, and therefore A(z) ak Algye
but since A(z) is continuous

lim A(w) = A(0), and A(2) AO)

This is true for every value of 2, and therefore A(z) is a
constant.
Now suppose |£|>1; make the transformation kv=y in

(14), then (14) becomes A(y)=A(2), and therefore the
function has the same value at the set of points y, = pe
oe ... which converges to the point =0, and therefore, by
the same reasoning as for |k| <1, A(z) is a constant.
Therefore A(v)=C in every case, and by (13) d(#) =

which proves the lemma.

Equations (10) and (11) are of the form (12), where

= Zi and therefore
0,

od ==;) 5)

F(A) = 55- a” Lorihgieiat eae aan (15’)

These values, substituted in (3), give Wien’s distribution
law.

3. A criticism of Drude’s proof of the Wien Displacement *
and Distribution Laws.

Equation (5) is the Wien displacement law and is
generally derived from the Stefan-Boltzmann law, which
states that

{Sp an=a const, . .

or that the total radiation varies directly as the fourth power

* The relation \6=a const. is sometimes known as the Wien dis-
placement law, and sometimes as a part of that law, but in this paper
will be regarded asa distinct law. The equation \@=a const. of itself
means nothing, since A and ¥) are independent. The equation is a short-
hand way of stating that the radiation for A\=A,, 6=6, will become the
radiation for A=A, when 6=62, where A,9,=A oe Wien obtains this
relation by actually determining the change of wave-length, the tem-
perature being increased by an adiabatic compression of the gas. No
criticism of the derivation of this relation is intended in this article.

Wien’s Distribution Law. 195

of the absolute temperature. Another form of the Stefan-
Boltzmann law is that >(6)=C6*, where W(@) is the total
radiation, and C is a constant. 7

Drude™* attempts to prove a statement that is equivalent
to (5), namely that

ced, oy was Oe a eae rg

where f is an unknown function of X@ alone. He states
that (17) follows from

wa) = {25 as), . 50 (05 fai6)

an equation which is an immediate consequence of the

definition of (@), viz.
¥@=( $0, Hdr.
0

e

Now (17) cannot follow from (18) unless ba =a const. ;

since this is true, by the Stefan-Boltzmann law, Drude’s
conclusion is not impossible. Suppose then that

{259 aney=s Coustl o . eo))
0

Now (17) does not follow from (19) alone, as the following
: EVEL E f(r, 7)
example will show. Let ¢(A, @)= ipe? then ra
is not a function of X@ alone ; however,

64
t+ x 6 1
{ 75 ans)={ Bp agi tO) =a T-

fF)
Even if we put d(,@)=F(Aj)e ®° , it is still possible to
choose F(A) and /(A) so that (19) is satisfied, while (17) is
not.

For if F(A) =A, f(A)=A, then _ @) is not a function
of X16 while

© Nee
0

* Lehrbuch der Optik, p. 480 (1900). This treatment is repeated in
the second edition.
T Pierce, ‘A Short Table of Integrals,’ formulas 480 and 493.

196 Mr. F. E. Wood: A Criticism of

Therefcre, Drude has not proved ihe displacement law ;
if, however, (17) be regarded as proved, then the remainder
of Drude’s argument for proving the distribution law can be
made rigorous *.

Let FA) -& |
F a ¢ =O(A0)) 4
where F(A), f(A), and @(A@) are unknown functions. Put
AA) , BA
Fa)= 8), py=— 2,
then (20) becomes
a 1
A(A)je °° =—D(rA0). « . . oe

~ 0
Now if X be replaced by kA and 6 by 7 the right-hand

member of (21) is unchanged, and therefore

Ba) BOA)
Aime” =ACkr)e ™. . 0)
Ba) Bla)
If the expressions e ** ,e¢ ** be expanded, (22) becomes
Bayo (BA)
A(n) ba +31 ey A
bi B(kA) 1 B2(KA) ]
= A(kA) [1+ a tee te |

or
[A (A) — ACAA) ] +5914 Q)BA)—A()BOA)] +...=0.,(23)
Since (23) is an identity in 0,

AW) = AKA) 300 a.
and since (24) is an identity in k, A(A)=C where Cisa
constant. Also from (23)

A(A) BA) — A(AA) BAA) =0,
from which B(X)=c, where c is a constant; so
C

C

* This derivation of the form of F(A) and f(A) is due to Professor
Moulton, of Northwestern University.

\

Wien’s Distribution Law. 197

4. A proof of some auxiliary theorems in the Kinetic theory

of Gases.

Before giving our derivation of Wien’s displacement law,
it will be necessary to obtain some laws in the kinetic theory
of gases. Let us consider the distribution of the molecules
of a given homogeneous gas at a temperature 0, with regard
to their velocities, v; if we plot v? as abscissa, and y, the
corresponding number of molecules, as ordinate *, then the

curve ©, which gives the distribution, will by Maxwell’s +
2 ly2

law have the equation y= pe’ 4, where k and J are inde-

1
pendent of v and 6. The equation of the corresponding
curve (', for the temperature 0,>6, (this restriction Is
convenient, but unnecessary) upon the (V?, Y) plane will be

Biph Ve
Y= ai Bact e a
2 . 2 0, 0, X °
The transformation V =o Ye a,Y takes Cy into
1 2

C,. Let v, and x/ 2 v, be called corresponding velocities for
1

the temperatures 6, and 0, respectively; and the intervals

Vj<v=v, and A ee Ay “o be called corresponding
2 1

velocity intervals for the temperatures 0, and 0, respectively.

OW
Me ky? = es dv?) 0, 0 Oa kV? 7 d(V?) (25)
v,2 6°? 6, bap 63”

1

and (25) holds for all finite values of v, and v, and also for
=O: Y= ,, 1. €.

odd PR Iv? A. 2 Zoe ONS
{ me edo) =a/ are # (VW), (26)
0 1 4/0 2

It iollows from (26) that the total area under the curve
C, does not equal the total area under the curve C,, as one
might expect from tie fact that in each case the area is
proportional to the total number of molecules in the gas, a
number which is unchanged by the rise in temperature.

* Strictly speaking, there will be in general no molecule of the gas
haying a given velocity » at a given instant; however, if N molecules
have their velocities within a very small interval dv about v, where the
width of 6v is preassigned, then we shall say for brevity that N molecules
have the velocity v.

t Scientific Papers of James Clerk Maxwell, i. p. 381.

198 Mr. F, E. Wood: A Criticism of

However, the areas associated with corresponding velocity

intervals have the same ratio, by (25), as the ratio of the
total areas, and so we have .

Theorem I. The number of molecules in a gas at temper-
ature 6, which have velocities in a given interval is equal to
the number of molecules of this gas at temperature 0, having
velocities in the corresponding velocity interval.

From this theorem we have the

Corollary. The number of molecules in a gas at temper-
ature 0, which have velocities less than or equal to an arbitrary
velocity V1 1s equal to the number of molecules in the same gas
at temperature 0, which have velocities less than or equal to

the corresponding velocity a / a “P
1

The velocity of any particular molecule of this gas at
temperature 0, is constantly changing, owing to impacts, etc. ;,
so it is difficult to consider the change of velocity due to a
rise of temperature alone. Let us consider an ideal gas in
which the velocities of the various molecules are such that
(1) the distribution of velocities obeys Maxwell’s law for
any temperature, and (2) the velocity of each molecule does
not change while the temperature remains constant. We
assume that the results obtained from a study of this ideal
gas will hold also for an actual gas. By this device, we can
consider a single molecule, or a group of molecules, at a
given velocity instead of the succession of molecules, or
groups, which have that velocity at various moments of time.
Moreover, if the temperature of this gas be raised to 05, each
molecule. in general, will have a new velocity, since the
mean-square velocity has been changed, and the new distri-
bution of velocities will also obey Maxwell’s law. The new
velocity of a molecule for given values of @; and @, will be a
continuous function of the former velocity ; also molecules
whose velocities for = 6, are zero, will again have zero (or
nearly zero) velocities for 06=@,. Then the molecules
having velocities in the interval 0<v<v, for 92=0@, must have
their velocities in some interval 0<v<v, for 62=60,. From

* If the velocity at which the maximum number of molecules move
be called the mode, then it follows from this theorem and corollary,
(1) that the mode varies directly as the square root of the absolute tem-
perature, (2) that when the temperature is increased, the number of
molecules whose velocity is at the new mode is less than the number
whose velocity was at the former mode, varying inversely as the tem-
perature, and (3) that the fraction of the total number of molecules
which have velocities less than or equal to the mode is independent of
the temperature.

Wien’s Distribution Law. £99
this and the Corollary to Theorem I. we have

Theorem II. The molecules having velocities in a given
interval for @=0, will have their new velocities in the corre-
sponding velocity interval for = 6.

If the given velocity interval be taken sufficiently small,
we have

Theorem III. The velocities vy and vz of a molecule of a
gas at temperatures 0, and 0, respectively, satisfy the relation
VAS

Theorem III. may be restated in this way :—If with
certain units v,7=«0,, where v, is the velocity of a molecule
of a gas at temperature 0; and « is independent of @, and y,
then the velocity of this molecule when the temperature of
the gas is 6, will be v, where v,?=«0,, and this relation
holds for any velocity v, and any two temperatures 0; and 63.
Consider two molecules m, and m, of a gas at temperature 0,
having velocities v; and v, respectively, and producing waves
of length A, and A, respectively ; if the temperature he
increased so that the velocity of m, is v2, the temperature
now being @,, then by Wien’s first hypothesis the length,
A,', of the waves produced by m, will be the same as the
length, X», of the waves produced by mz, when the tem-
perature was 0,. Now, from (27) and the law X@=a const.,

and since A, =A, ,
my _ Vm
ee ae
This proves
Theorem IV. The length of the waves produced by a

molecule in a gas at temperature 0 varies inversely as the
square of the velocity ; 1. e.

v= = , where bisa constant. . . . (28)

5. A derivation of the Displacement Law from Wien’s
hypotheses.

Let us consider the distribution of ¢, the intensity of
radiation from a gas at temperature 6, with regard to X, the
wave-length. Denote by K, the curve obtained by taking
> as the abscissa and ¢ as the ordinate. We will fix our
attention upon the radiation associated with values of X in
the interval, Ay<ASA, + Ay, where A, and 8A, are arbitrarily
chosen (except that 6A, is very small). Suppose that the

200 Mr. F. E. Wood: A Criticism of

temperature be raised to 0, and that the corresponding curve
be plotted. From the above theorems and the relation
’9=const., it follows that the molecules which produced

waves of length 2», will now produce waves of length oy

A.
and those which produeed waves of length \,+ 6A, will now

produce waves of length 4 (A; +6r,). Therefore the number
2

of molecules producing waves with lengths in an interval
with width 6A, about.d, in a gas at a temperature 0, is equal
to the number producing waves with lengths in an interval

with width A about a when the temperature is 09.

A, A,
Now (Aj, 0;)6A, is the intensity of radiation carried by
waves with lengths in the interval A;X<A<A,+ 6A, for 0=6),

and 8(F ms, 02) 5 dh is the intensity of radiation carried by
; z 6 7
the waves with lengths in the interval q, MEAS + ou) R
2 @ @ 2
When 6, and 0, are given, (Aj, 0;)6A; and $(3m, 05) SBA,
2 2
will be called corresponding radiation elements. In general,
two corresponding radiation elements are not equal, since
the total radiation increases wlien the temperature is increased
(Stefan-Boltzmann law).
The intensity of radiation produced by a molecule is a
function of the velocity of that molecule ; then

e=9(0"), <0 4) a rr
where ¢ is the intensity of radiation produced by a molecule
with velocity v and g is an unknown function of v? alone.
Therefore, using Wien’s second hypothesis,

OGre — _ Note ge (30)
. (0 0 Notas?) oe.) ae
(7m, 02) G5d 9(vs") —-g(v2")
2 2
where N is the number of molecules producing waves with
lengths in the interval AySAXA, + 6Aq, and v; and v, are the
velocities of these molecules when the temperature is 0, and
6, respectively.
It follows from (30) and Theorem III. that

b(dy 61)8a.__ _ glebh)
0 \ Oy a kana ere
(Bris Os) gdm ‘

(31)

Wien’s Distribution Law. 201

Therefore, the ratio of corresponding radiation elements is a
Junction of 0, and 6, respectively ; in particular this ratio
does not depend upon the value of X about which one element 1s
chosen. ? :
Since the area of each radiation element under K, is
equal to a constant times the area of the corresponding
radiation element under K,, then the total area under Kg is
equal to the total area under K, multiplied by this constant ;

4
but by the Stefan-Boltzmann law this constant. is (3) ,
Therefore 2

B(A,, 0,)6r Gas ;
Sa eet ee as Gem cao

1 1
p (Gide 02) g,o™

and therefore

b(Aa Gi) O( gv 0.)

This is another form of equation (5), and our derivation of
the displacement law is complete.

From (31) and (82) it follows that

g(KO;) _ 4
g(«0s) O32?
from which g(kO;) __ g(«O,) 2
6, . 2

where ¢ is a constant. Then
CASE | 5 aR ROD em (Fs

and from (29) it follows that e=c?v'; i. e. the intensity of
radiation given off by a molecule varies directly as the eighth
power of its velocity.

* In Wien’s first article upon the displacement law he assumed that
(32) followed immediately from the Stefan-Boltzmann law; in his
second article the error is corrected, but the derivation given is difficult,
even though rigorous, so most authors have followed Wien’s first
method and have not remedied the errors: ef. Wood, Physical Optics,
p- 620; Sachur, Thermochemie und Thermodynamk, p. 302; Winkel-
mann, Handbuch der Physik, 111. pp. 379-80.

202 A Criticism of Wien’s Distribution Law.

6. The inconsistency in the Wien Distribution Law, and
the revised form.

Let us consider the method by which Wien obtained (3)

from his two hypotheses. From the first hypothesis one
can write

p= (V), > oe oe
where m is an unknown function of X*.
Tf e=p(r) o:de (2) bye

denote the intensity of radiation produced by a molecule
giving off waves of length A, then by using (34), (35), and
Wien’s second hypothesis

Kym?(n)

b(r, O)=hyp(rA)m(rA)m'(AJe®

where k, and k, are constants and m'(A)= £ m(Q). So
Wien has written for brevity r

FO)=hpmaym'y; . . . (36)
JHNS=—hWA). ~.. . 2)
Now, from (37) and (15) ide
: BS. Te é
m(X) = Ex? (38)
and from (28), (29), (33), and (35)
p=, ees

where a is a constant. Substituting (38) and (39) in (36)
gives
A
F(A) = 13/2? e e . ° e e (40)

where A is a constant depending upon the constants a, c, hy,
and k, But (40) is inconsistent with (15'). Therefore,
the Wien formulats inconsistent with other results obtained
from the same hypotheses.

Now by the Maxwell law Tt, the number of molecules
having velocities between v and v+dv is

i
Vi |
* However, the form of m(A) has already been obtained in this article
from the relation \9=a const., and certain theorems in the kinetic theory
of gases; see Theorem IV. That this form agrees with the form obtained

by Wien’s method is an agreeable fact.
+ Scientific Papers of James Clerk Maxwell, i. p. 381.

es 2
N 5 ACCME, ek

Coupled Circuits and Mechanical Analogies. 203

where N is the total number of molecules in the gas, and «”
is proportional to the absolute temperature 0—see §2. If
one assumes that N does not change with 6 *, then (3) does
not follow from Wien’s assumptions, but instead

F Sia)

t Xr mat A. <
OL eee oh? ool bys 242)
If (3) be replaced by (42) and the argument given in §1
be repeated, it will be found that
A 6
FQ) = susp} SM= >;
where A and ¢ are constants.
Moreover (43) is consistent with (40) and (27), and so
Wien’s assumptions lead to the law

(43)

Oy )
d(A, 0) = i152 © Ag . ° e ° 9 (44)
Evanston, Illinois,
July 5, 1917.

XXII. Coupled Cirewts and Mechanical Analogies. By

HK. H. Barton, D.Sc., #.RS., and H. M. Brownine,
B.Se.F

if S the valuable ‘“‘ Note on the Action of Coupled

Circuits and their Mechanical Analogies” by

Prof. H. C. Plummert seems in places to imply a slight mis-

understanding of a previous one dealing with the same
subject, a brief reply appears desirable.

Prof. Plummer regards the matter chiefly from the
mathematical standpoint, whereas the authors of the October
paper§ were concerned chiefly with the physical phenomena
and their visible representation to average electrical students
(see pp. 246-7). And in some colleges not one per cent. of
these are masters of the problem mathematically.

2. Prof. Plummer expresses “a doubt whether a simple
electrical problem really is made easier for the average
student by a complicated mechanical analogy” (p.510). It
was never intended that the mathematics of the mechanical

* If N=o(6), where o is an undetermined function, and if Theorem
IV. §4 be regarded as valid, then it can be proved that ¢(6)=a const.
Wien’s implicit assumption is that N=c'6®”, where c’ is a constant.

+ Communicated by the Authors.

{ Phil. Mag. pp. 510-517, vol. xxxiv., Dec, 1917.

§ Phil. Mag. pp. 246-270, vol. xxxiv., Oct. 1917.

z04 Prof. Barton and Miss Browning on

analogy should explain or render easier that of the electrical
case. In its mathematical aspect the electrical case is prob-
ably simpler than that of any mechanical model yet put
forward.

The utility of the mechanical analogies given in the
October paper lies in their power to give with such simple
apparatus actual traces of the vibrations in question, and
that under the most various conditions as to masses of bobs,
lengths of the separate pendulums, and looseness or closeness
of their coupling. And these advantages can be reaped by
students who are too weak to assimilate the mathematics
either of the original problem or its analogies, though for
completeness’ sake the equations of both were naturally
included in the paper.

There is also another aspect of the matter. For it may
be hoped that the study of these mechanical models may
throw some light on the hidden mechanism of the electro-
magnetic phenomena.

3. Equations (27) and (28) of the October paper for the
double-cord pendulum were in the form

dy | P+Q+AQ P4y = BBQ ae
dt? ~ (1+8)(P+Q) 1+6 Pee

dz, P+®8P+Q og, B PQ g on
Qa t (+B P+Q) 21° 1+8°P ue

Prof. Plummer prefers the form

B

a? d?z )
(P + Q+,P) “ + (P+ Q)gly=—BQ7,, |
Cr
(P+ Q+8Q) 5 + (P+ Qygl-!e= PY, |
and adds “the analogy is now exact.” Is not this too much
to claim? As to the occurrence of the variables and their
derivatives the equations (2) do indeed present a formal
agreement with those for the electrical circuits, viz.,

dt?

Ben aaa}

dt +R= oo

Ge 2 = | (
oe 5 = Se

Hence in a certain restricted mathematical sense the analogy
is exact. But this exactness of analogy does not extend to

Coupled Circuits and Mechanical Analogies. 205

the coefficients and their physical significance. Thus in (3)
for the electrical case it will be seen that the same coefficient
occurs on the right side of each equation. This is M the
coefficient of mutual induction. Also the first coefficients at
the left of each equation respectively are Land N the separate
self-inductions. Now this correspondence of coefticients does
not hold between (3) and (2) but does hold between (3) and
(1). And for this reason the form (1) is probably preferable
to the experimental physicist, though the other form (2)
is distinctly illuminating and may be preferred by the
mathematician.

4. That the pendulums represented by equations (1) and
(2) are not in the complete sense an exact analogy to the
electrical case of (3) may also be seen from the relations of
the frequencies of the coupled vibrations in the two cases.
Suppose the two separate vibrations for pendulums or those
for the electrical circuits to be equal and denote them by
cos mt. Let the superposed coupled vibrations for each
system be denoted by cos pt and cos gt. Then, for the
electrical case, we have

prm> gq.
Whereas, for the mechanical analogy, we have
p=m, and m > q.

5. What would seem to the writers to be an exact
mechanical analogy to the electrical case would be one
capable of representation as follows:

d*y y d*z )

aiean ae | m
e 4

d*z ee

Qa — Qnte = 3 TY, |

In these P and Q denote the masses which vibrate in the
two systems. Their separate vibrations are to be obtained
by writing J=0. Thus giving as their separate vibrations
cos mt and cos nt respectively. Further, it should be noted
that in the above we are supposing that the introduction of
the cross-connexion terms on the right has not modified
the coefficients on the left. A model fulfilling these condi-
tions seems to be still a desideratum. In these equations it
may be seen by the theory of dimensions that J must be a
mass like P and Q.

Phil. Mag. 8. 6. Vol. 35. No. 206. Feb. 1918. Q

206 Drs. Smeeth and Watson on the Radioactivity of

6. Prof. Plummer appears to consider it a mistake to
regard asa mass the mutuai induction M of the electrical
case.

But does not the current view regard the mutual induc-
tion as an inertia factor of some sort? Thus in Sir J. J.
Thomson’s model referred to in the October paper (p. 251)
both self and mutual inductions are represented by masses.
In Prof. J. A. Fleming’s ‘ Alternate Current Transformer’
(vol. i. pp. 97-98, 1889), the electrical energy 3Li? is
likened to the mechanical energy of rotation 4lw?. Again,
in Sir Oliver Lodge’s ‘ Modern Views of Hlectricity ’ (p. 496,
1907), coefficient of induction (self or mutual) is given as
inertia per unit area.

It is true that the coupling in the electrical case is made
by a change of configuration which fixes the value of M.
But this does not prevent M from being a mass (7. e., an
inertia) like the inductances L and N which are also
dependent solely on configurations, provided no iron or
other magnetizable substances are present. As to whether
the coefficient M in the electrical equations is to be repre-
sented by a mass in any one mechanical model incompletely
analogous to it, is another matter.

Nottingham,
Dec. 17, 1917,

XXIII. The Radioactivity of Archean Rocks from the Mysore
State, South India. By W. F. Smuetu, D.Sc., ARS,
and H. EK. Watson, D.Sc., A.L.C.*

Preliminary Investigation.

HIS investigation was started some years ago on a

number of samples of the hornblendic schists of the

Kolar Gold Field, selected by Mr. H. M. A. Cooke, Super-
intendent of the Ooregum Gold Mining Company.

The samples were taken from the Kolar mines at different
depths, with a view to ascertaining whether the radium con-
tent varied with the depth from the surface in rock of fairlv
uniform character and composition. These hornblendic
schists and epidiorites are all ancient lava flows, or sills, of
fairly uniform composition, notwithstanding petrological dis-
tinctions in texture and structure.

* Communicated by the Authors,

Archean Rocks from the Mysore State. 207

An account of the method used and results obtained was
published in the Philosophical Magazine (6) xxviii. p. 44,
1914, and the results are repeated in Table I., Nos. 1 to 15.
It will be seen that the radium content is very low, remark-
ably constant, and that there is no variation in depth down
to a vertical depth of some 3500 feet from the surface.

Two of the samples—Nos. 12 and 15—gave results con-
siderably higher than the others, but microscopic examination
showed that these samples did not represent normal types of
the hornblendic schists or “ country” of the mines, but bad
undergone considerable mineral alteration, such as is common
in the immediate vicinity of the quartz veins or other acid
intrusives, and there is no doubt that the higher values are
due to the intrusion of acid material of higher radium
content.

Further Investigation.

It was then decided to obtain a number of representative
samples from the various components of the Archean com-
plex of Mysore, in order to ascertain how far the various
formations or groups might be distinguishable from one
another by their radioactivity, and what variations existed
amongst the members of each group as a result of magmatic
segregation. The experimental procedure was the same as
before, viz., 10 gms. of the finely powdered rock were fused
with potassium hydroxide, under reduced pressure, and the
resulting gases led to an electroscope after removal of the
hydrogen and drying. ‘Towards the middle of the experi-.
ments the leaf system in the electroscope broke down, and
was replaced by a smaller and more sensitive one, which was
subsequently carefully standardized. With this a leak of
1 scale division an hour corresponded to 1°67 x 10~"* gm. of
radium. Oontrol experiments showed no discontinuity be-
tween the two series of values obtained. All experimental
details have already been given (loc. cit.), and will not be
repeated. Altogether, fifty samples have been selected from
specimens in the Department of Mines and Geology of
Mysore and the radium determined, but each group is itself
so complex and variable that a much larger number would be
required before fair averages or estimates could be obtained.
In spite of this, certain interesting variations appear to be
indicated, and the results obtained have been grouped, in
Table I., under the various formations taken in order of aye
from the oldest to the youngest.

G2

208 Drs. Smeeth aud Watson on the Radioactivity of

The following classification is inserted for convenience of
reference, and shows the order of succession and relationship
of the various formations in Mysore as at present adopted by
the Mysore Geological Survey.

Classification of Mysore Rocks.

1. Recent soils and gravels.
Possibly Tertiary. 2. Laterite. Horizontal sheet capping Archezans,
Pre-Cambrian

(Animikean) . Basic dykes. Chiefly various dolerites.

Great Eparchean Interval.

. Felsite and porphyry dykes.

. Closepet granite and other massifs of corresponding
age. ;

; Ohaesockite, norite, and pyroxenite dykes.

. Charnockite massifs. (Complex.)

. Various hornblendic and pyroxene-granulite dykes,

. Peninsular gneiss. (Granite and gneissic complex.)

. Champion gneiss. (Granite porphyry, micaceous
gneisses, felsites and quartz porphyries ; usually con-
taining opalescent quartz and frequently associated
with autoclastic conglomerates.)

Ooonto Of

=

Eruptive Unconformity.

Including also :—
Amphibolites, peridotites, &c.,
mostly intrusive. RE
Conglomerates (autoclastic). Res «
and echloritic | Banded ferruginous quartzites,
een schists). origin doubtful, possibly igneous.
ei 412, Lower (horn- { Quartzites and quartz-schists,
wp ee ual blendic) di- mostly intrusive.
eewatin). vision (epi- | Limestones, probably secondary.
diorites and | Mica schists; metamorphic ig-
hornblendic neous.
L schists). Intrusive masses of dioritic} and
\ diabasic character.

11. Upper (chlori-
PP
tic) division
(greenstones

Archeean.

Se

Dharwar

\ Unknown.

We may now very briefly consider the groups of figures
presented in Table I., and the following summary of the
results will help to bring out such points of similarity or
distinction as exist amongst them, although the cbservations
are too few in number to permit of final conclusions being
drawn.

* ‘Outline of the Geological History of Mysore,’ by W. F. Smeeth,
Bulletin No. 6—Department of Mines and Geology, Mysore State.

Archean Rocks from the Mysore State. 209

(Numbers in brackets refer to the classification given above. )

Radium, gm. per

eee 10! gm. of rock.
Limits.
; Hornblends serie (Ia) eects. -nnen ss sc cssenees 0:14 to 0°25
Diarwar x 3) walbened( Gyjpes! os... 60.205: 0°34 ,, 0°96
Piloritic'series (I) oaie ee os kes cca ben 0:20 ,, 0°54
Basic intrusives (11) and (12) .................. 0:05 ,, 0-16
Obum pion gneiss 0). case ease. si dasies en ecens st 0°85 to 1°45
2 3 auriferous quartz °.....4..-... 1:28
Peninsular’ eneiss:(9) sores nese. vets boaeces cos 0:40 to 1:50
is sf POP MIA IbeSH S04 La.8. 6226500. 1°44 to 6°90
@hurnockites (A). eee alsa cSacetoee | 0-04 to 0°12
2 hypersthkenite (G)).... sb. .cc0c.cec00 0-06
Hf quartz-magnetite ore ............ 0:08
losepet: cranite tO) ic. peat teen Sedan te dees 2 = | 0°27 to 2:14
5 3 porphytiesH@):)...56. BS | 1°37 ,, 2°42
Dolerite dykes (Post-Archwan) (8) ............ 0°45

The hornblendie rocks of the Dharwar System (Nos. 1-18)
are low in radium and exhibit no great variation from the
mean, though many petrological types are included, such as
hornblende schists, hornblende diabases, amphibolite, and
hornblende granulite. When, however, these rocks are
altered in contact with intrusions of the Champion gneiss
and of the related quartz veins of the Kolar Field (Nos. 26-
30), all of which contain much more radium than the
normal schists, the radioactivity of the altered types is con-
siderably increased (Nos. 12, 15, and 16) to a point inter-
mediate between the radioactivities of the two reacting
masses.

The rocks of the Chloritic series (Nos. 19-21) do not appear
to differ much in radium from those of the hornblendic series.
The higher value in No. 21 may possibly be the result of
alteration due to a neighbouring granitic intrusion.

The basic intrusives of Dharwar age—that is to say, intru-
Sives into the general body of the Dharwar schists prior to
the period of the Peninsular gneiss—contain much less
radium than even theschiststhemselves. This is particularly
noticeable in the Bellara trap (No. 23) and the Grey trap of
Chitaldrug (No. 24), and these rocks afford an interesting
example of the possible use of such determinations in the
correlation of these very old and much altered Archean
types. Some years ago the Grey trap was considered to be
a modification of the less altered Bellara trap, but subsequent

210 Drs. Smeeth and Watson on the Radioactivity of

work showed that the Santaveri trap of the Kadur District
possessed a considerable resemblance to the Grey trap of
Chitaldrug, so much so that it had almost been decided to
class these two formations together and separate them from
the Bellara trap. The radium determinations, however,
show that the Grey trap and the Bellara trap have practically
the same radioactivity (wide Nos. 23 and 24), which is about
one third of that of any of the schists proper, while the
Santaveri trap (No. 19) contains about three times as much
radium and falls within the limits so far ascertained for the
schists. This appears to confirm the original classification,
which correlated the Grey trap with the Bellara trap, and
which, consequently, has been allowed to stand. The case
affords an illustration of the possible use of radium deter-
minations as an aid to correlation of the highly metamorphosed
members of the Archzean complex,amongst which it may often
happen that the chemical and mineral composition, and the
field relationships, do not afford sufficiently definite points of
similarity or distinction. The value of such determinations
will depend on the possibility of ascertaining fairly definite
limits for the radium contents of the various rock groups or
of the various members of such groups.

Amongst the various gneisses and granites, which have
|een divided into four great groups of distinctly different
ages, it will be noted that the Charnockites stand apart from
the others in virtue of their excessively low radioactivity,
which is much lower even than that of the Dharwar schists.

The Champion gneiss, Peninsular gneiss and the Closepet
granite—which last is also a variable complex—all contain
from 12 to 15 times as much radium as the Charnockites
and four to five times as much as the Dharwar schists.
The Charnockites have been shown by Holland to form a
distinct petrographical province amongst the gneisses of
Southern India and vary, as the result of magmatic segre-
gation, from highly acid granites to norites and hypersthenites
—all characterized by the presence of hypersthene and certain
physical features—and the radium determinations fully con-
firm the distinct individuality of the parent magma. The
varieties of Charnockite which have been examined show an
increase of radium with increasing basicity, but the hyper-
sthenite and quartz-magnetite ore—which are considered to
be end products of the segregative process—show a relapse
towards the mean value.

The other gneisses and granites are very complex, and the
determinations are not sufficiently numerous to permit of

Archean Rocks from the Mysore State. 211

very definite conclusions. It may be noted that the
members of the Champion gneiss series show least variation,
those of the Peninsular gneiss rather more, and those of the
Closepet granite the greatest variation. |

The auriferous quartz of the Kolar Field (No. 29) is inter-
esting as falling into line with the members of the Champion
gneiss series, with which it has been correlated on other
grounds, and, as already pointed out, the altered schists (lode
matter) in contact with, or in continuation of, the auriferous
quartz veins show values intermediate between those of the
normal schists and of the quartz.

It is interesting also to note that the matrix of the Con-
glomerate (No. 27) has the same radioactivity as the clearly
intrusive granite (No. 28), and this is in accordance with the
view that the conglomerate is autoclastic and due to crushing
of portions of the Champion gneiss series.

The pegmatite cross-course (No. 39) is remarkable as
yielding the highest result so far obtained. The pegmatite
contains a large amount of tourmaline, and it was thought
that this mineral might account for the high value. A small
quantity of the tourmaline was separated and, though a
definite determination was not made, the test was sufficient
to show that it was not abnormally high, and that the high
result of the rock as a whole was not due to this mineral.

A single determination (No. 50) has been made of one of
the very numerous dolerite dykes which are considered to be
of Pre-Cambrian age but subsequent to the formation and
folding of the Archean complex. This rock is very similar
in composition to the old hornblendic schists, which probably
were originally diabasic flows and sills of a much earlier
period. The resuit shows that the later rock contains more
than twice as much radium as the earlier type, but no further
inference can be drawn from a single observation.

| Summary.

1. These very ancient rocks, all of which are considered
to be of igneous origin, contain remarkably little radium.

2. Amongst the various groups which have been differ-
entiated on geological grounds, there are some striking
differences in the radium contents of some of them.

3. In the case of a fairly uniform group of rocks (viz., the
hornblendic schists of the Kolar Field) the radium content
does not appear to vary with the depth from the surface.

4, Different igneous magmas appear to contain very
different amounts of radium, and the latter, or the minerals

212 Drs. Smeeth and Watson on the Radioactimty of

which carry it, is subject to magmatic segregation. The
amount of radium in the segregated portions of a magma
sometimes increases and sometimes decreases with increase
of basicity.

5. Amongst magmas, the more basic appear to be lower in
radium than the more acid, and, in the products of granitic
magmas, the pegmatites appear to carry more radium than
the corresponding granites. The Charnockite magma, which
was probably of intermediate composition, forms a striking
exception, and is notable for its extremely low radioactivity.

6. In the case of rocks of somewhat similar character and
composition, and in which other means of distinction or
identification are lacking, a marked difference in the radium
contents may afford a means of correlating them with known
groups or formations for which the radium limits have been
sufficiently determined.

Bangalore, March 1917.

TaB.e I.
Radioactivity of Rocks from the Mysore State.

(The rock groups are arranged in order of age from the oldest

to the youngest. The radium is given in units of 107” gramme
per gramme of rock.)

ea ie Sa Description. Radium. Remarks.
Group 1. Dharwar System—Lower (hornblendic) series.
ft.
A Balaghat Mine — depth 1000) 0-21 | (The first 15 samples are
2. 9 a ,» 2000} O14 all hornblende schists
3. r » SUOG! * OE and epidiorites from the
4. Nundydroog Mine ,, 1000) 017 Kolar Gold Field. Tho
5. | » " » 9000) O14 depths are measured on
6. |Ooregum Mine ,, 1000; O18 the dip of the lode.)
(fe 3 i »y 2000) 20:17
8. | et ‘S », 4000) 0-21
9. |Champion Reef Mine,, 2000} 0:24
10. ” ” Dan) 99 3000 0-18
i) 2 a: AGED wOule
12. | Mysore Mine », 1000; 034 | Altered schist.
13. fs 53 js 2000) F021
14, i x », 3000} 0:23
1: 4 gt », 4000) 096 | Much altered schist, part

to quartz of lode.
16. | §2/877 | Amphibolite, footwall oflode,| 0°82 | Contact alteration,

Ooregum.
17. | Z3/510 | Fine hornblende schist, Baba-| 0°25

budan Hills, Kadur District ‘
18

.| 91/535 | Hornblende granulite; Kolar} 0°19

schists.

of the lode formation;
contact alteration close

Archean Rocks from the Mysore State.

213

Radium. Remarks,

ag a Deseription.
Group 2. Dharwar System—Upper (Chloritic) series.
19. | Z3/258 |Santaveri trap, Bababudan
| Hills.
20. | Z4/757 | Banded chlorite schist, Sacre-
bail, Shimoga,
21. | 24/703 = Cale- chlorite trap, Tarikere

series.

0:20 | Chloritic trap, with some
hornblende.

0:27

0:54 | Probably altered horn-

blendic trap.

Group 3. Intrusives of Dharwar age—subsequent to the schists.

22. | S2/943 |Hornblende diabase, Baba-!
budan Hills.

23. | W23/69 |Bellara trap, Tumkur Dis-
trict.

24. Z4/95 |Grey trap, Chitaldrug oa
trict.

25. | J4/117 | Titaniferous iron ore, Ubrani,|
Shimoga.

Group 4.

26. | J1/335 | Grey micro granite, Kolar.

27. | J1/156 | Fine micaceous granite-ma-
trix of crush conglomerate,
Kolar Gold Field.

28. | §3/313 | Fine mica granite, Mysore
Mine at depth of 3000 ft.

29. Auriferons quartz from the
Mysore Mine.

30. A/624 | Quartz-felspar porphyry.

31. $/402 | Massive phyrrhotite from

Champion Reef Mine

0°16

0:05 | Altered diabase intrusive
into echloritic series.

0:07 | Chloritie trap, probably
related to Bellara trap.

0°05 | Associated with ultra-

basic intrusives.

Champion Gneiss Series—intrusive into Dharwars.

0.85 |

1°45 |

1:34 |

1:28 | | These are considered to be
products of the Cham-

1:05 | pion gneiss magma.

007 | From Gifford’s shaft.

Group 5. Peninsular Gneiss Series—later Bs Champion Gneiss.

32. | J3/238 : pee porphyritic granite,
| Kolar.

33. | J3/278 | Dark grey granite, Kolar.

34. | J8/212 | Fine dark grey granite, Kolar.

35. | J3/276 (Fine light grey granite,
| Kolar.

36. J3/63 | Light crushed granite, east of
| Koiar Field.

37. | J3/122 | Uniform granite mass, Patna,
| Kolar,

38. $/326 | Pegmatite cross-course, Bala-

ghat Mine.
39. $/354 | Pegmatite cross-course, with

| tourmaline, Ooregum Mine.

0-91 )

| | These are in order of in-

0°99 }| trusion, beginning with
1:28 || the oldest.

0-47 J

0:40

1:50

1: 44 \ These probably belong to

the Peninsular gneiss.
In No. 39 the tourma-

Group 6. Charnockite Series—later than the Peninsular Gneiss.

40, J2/824 | Acid Charnockite, Chamraj-
nagar, Mysore District.

41. | J2/822 | Intermediate Charnockite, |

Chamrajnagar, Mysore Dis-|
trict. |

42. | J2/722 | Basic Charnockite, Heggad-
devankotte, Mysore Dis-
trict.

43. | J3/631 | Hypersthenite, Nanjangud.

44. | J3/713 | Quartz-magnetite ore, Heg-

_gaddevankotte.

6:90 | | line itself is not abnor-
) | mally high in radium.

0-04 \

0:10

Ceae of intrusion doubt-

"12

These are considered to be
derived from the Char-
meckite magiias oo | magma.

| OF
el
io
ged

214 Dr. S. R. Milner on the Effect of
eu on Description. Radium. Remarks.
Group 7. Closepet Granite—later than Charnockite.
45. | J3/434 |Coarse grey porphyritic| 0°27 \
granite, Closepet, Bangalore | |In order of intrusion,
District. +| beginning with the
46. 33/430 | Grey granite, Closepet. 0°63 | | oldest.
47. | J3/429 | Red granite, Closepet. 2°14)
48. | Z2/621 | Dark quartz-felsparporphyry,| 2°42) | These porphyries are sub-
Yelwal, Mysore District. sequent to the Closepet
49. | Z2/648 | Pinkfelspar-porphyry, Kiran-| 1°37 granite and may belong
gur, Mysore District. to the same magma.
Group 8. Post-Archean dykes.
50. | J1/333 |Normal dolerite dyke, Kolar} 0°45 | Typical of a large series
Gold Field. of Post-Arehzan dykes:
which may be of Cud-
dapah age (Animikean).
XXIV. The Effect of Interionic Force in Electrolytes.

By 8. R. Mizner, D.Sc.*
ART dls

O the electrical forces which exist between the ions in

an electrolytic solution produce an effect on the ionie
mobilities ? This question has often been asked but has never
received a satisfactory answer, although it is of fundamental
importance in the theory of electrolytic dissociation. In
Part II. of the following paper an attempt is made to
provide an answer by establishing on the principles of the
kinetic theory a general proposition on the effect of inter-
ionic forces. Briefly stated this is that whatever effect such
forces produce on the osmotic pressure of the free ions in an
electrolyte in reducing it below what it would be were the
forces nonexistent, they will produce the same reduction in
the average velocity with which the ions move in carrying
a current. The bearing of this result on the theory of the
extent and character of the dissociation of strong electro-
lytes is then considered. In this part it is proposed to
consider in this connexion the well-known difficulty in the
theory of Arrhenius connected with the failure of the law
of mass action for strong electrolytes. The failure can,
I think, be shown to be of such a kind as to form an
objection apparently insuperable, not only to the original

* Communicated by the Author.

Interonc Force in Electrolytes. 25

theory, but to any theory of dissociation in which the
reduction of the molecular conductivity is ascribed solely
to a reduction in the number of the free ions, instead of to
a direct effect produced by the interionic forces on their
mobilities. In considering it we will confine attention to
dilute solutions of strong binary electrolytes in water,
where the essential facts to be explained by any theory are
these :—

(1) The molecular lowering of the freezing-point 7 is
found to be nearly, but not quite, twice as great as the theo-
retical value 7, valid for a non-electrolyte. Let us express
this fact thus :

2
sep Cea sk he. ay Mae a ete (a)
To ae

8; is in the first place a purely experimental quantity,
but by strict thermodynamic reasoning it can be identified

with the reduction in the value of a for the electrolyte

below the value which would apply to an eijectrolyte com-
pletely dissociated and obeying the gas law. In the theory
of Arrhenius it is further identified with the fraction of the
whole number of ions which, as the concentration is increased
from zero, have associated into molecules. Equation (1) is
usually written with the symbol y for the fraction of mole-
cules dissociated, but for the present purpose the use of
8, (=1—y) in the equation is more convenient. We are
dealing with strong electrolytes where at the most there
are only small departures from complete dissociation, and
of these according to the theory 8, forms the direct
measure. | |

(2) The molecular conductivity \ diminishes with increase
in the concentration, or we may write, if Apo is its extrapolated
value at zero concentration,

Xr
is «aes ° ° «6 ee (2)

B, can not be identified thermodynamically with anything,
but on the theory it also represents the fraction of the ions
associated.

(3) i= Poa aeoximatelys. / seat y. (3)

Much work has been done in testing the extent of the
agreement, the general result of which seems to be that

216 Dr. S. R. Milner on the Effect of

the equality is practically complete but the limit of experi-
mental error is still somewhat wide *.

(4) @, and ®, both vary with the concentration in a way
which requires to be accounted for.

Tbe original theory of Arrhenius of course explains per-
fectly the equality 6;=. by identifying each of them with
8 the fraction of the ions associated. The variation of 8
with the concentration C which it requires, namely

‘= 2

aa =o.
is, however, quite inconsistent with the experimental varia-
tion of 8; and B,. Numerous attempts have been made to
get over the difficulty by modifications in the mass action
equation (4), but this procedure, whether the modifications
have a theoretical basis or are purely empirical, will be
. found to increase rather than diminish the difficulty in which
the theory isinvolved. For the law (4) is a thermodynamical
result the truth of which is independent of any theory of
the mechanism, and rests solely on the assumption that thie
osmotic pressure of the ions and the molecules obeys the law
for perfect gases. If the phenomenon is one of pure dis-
sociation (and no successful explanations of the discrepancy
have been reached on other lines), the fact that (4) is not
obeyed is conclusive evidence that either the ions or the
molecules do not obey the gas law. But when this is’
the case, the experimental quantity 8, in (1) is no longer
the same as @ the true association. §; will in fact be
equal to 8 plus the additional reduction in the value of
PV/RT for the electrolyte due to the non-obedience of the
osmotic pressure to the gas law, and calculation shows that,
as far as can be estimated from the experimental curves, the
last term is many times greater than the first. Comparison
of 8, with a modified mass-action law is thus invalid.

An equally great objection applies to any attempt to
represent the 8, of the conductivity variation by means
of a modification of the mass-action law. For if @, is
actually the same as 8 the true association it must be
different from §), which is not the same as 8 except in tlie
single case when (4) is obeyed. This is, however, in conflict
with the experimental result (3). It may be thought that
the differences between (; and @, thus necessitated are small
second order ones concerning which experiment is incon-
clusive, but this is not the case. They are of the same order

* Cf. A. A. Noyes and K. G. Falk, Amer. Chem. Soc. Journ. xxxiy.

p. 484 (1912), where a systematic comparison of all the best data is
given.

(4)

Interione Force in Electrolytes. | DAT.

of magnitude as those which occur in the comparison of f,
with the unmodified mass-action law. The point may be
illustrated by a brief consideration of one of the most recent
positions reached in the development of the original theory.
Kraus and Bray”* have found as a result of a detailed
examination of a great many solutions in various solvents,
that the variation of X can be closely represented by thie
empirical formula
20
ion =K+D(Cy)”,

where y= ~ (=1-6,), and K and D are constants.
0

According to this formula the law of mass action (4) is
obeyed when the concentration is sufficiently small, the
term D(Cy)” being ultimately negligible. As the solution
becomes more concentrated the mass-action ‘ constant” K
becomes increased by a term depending on the concen-
tration of the ions. They conclude from their examination
that in all cases the conductivity ratio X/A, is a true measure
of the ionization, and that the simple law of mass action
applies if the solution is made sufficiently dilute. This view
has been mentioned with approval by Arrhenius ft, who,
however, lays stress on the fact that the fundamental
difficulty of the failure of the strict mass-action law is still
unremoved by the use of an empirical equation. Curve I.,
fig. 1, shows the agreement of the equation
ae 3() 4. De f 673 “= Xr
apr a 080 + 2°707%, Cy) &?, Y= [983°

with experiment in aqueous solutions of KCl (the constants
are given by Kraus and Bray, and the experimental numbers
by Noyes and Falk, oc. cit.). The curve represents well
the conductivities over a wide range except at the lowest
Xr
Xo
ionization, it is a simple matter to apply the general differ-
ential equation of mass action which is applicable in all
circumstances, to determine the variation of the molecular
freezing-point lowering with the concentration which is
thermodynamically necessitated by this assumption t. The

concentrations. Now assuming that — represents the true

* Amer. Chem. Soc. Journ, xxxv. p. 1515 (1913).
t Chem. Soc. Journ. cy. p. 1414 (1914).
t See Appendix.

Ve, Grane 1,1.)

4
°

“120

110

16

<=
(>),
Oo.

aS

A (For Curve 1)

ae
C

218 Dr. S. R. Milner on the Effect of

result is shown in Curve II. along with the experimental
variation of t/t) (Curve III.). The difference between the
curves is several times as great as the amount (@,) by which
the experimental curve differs from the theoretical value 2.

Fig. 1.
OO :
: ©)
aE ee 5
oy ie agg get
a at
. aL Sak eee
Tt ae +~2 32 iil. ;
™ +
OW
Pe
P '
=r! Be 9 ee 0:2 03 0-4 os
333 28.5) cee ae : r

This type of difficulty is inherent in any theory in which
8, 1s identified with B, 7. e., in which the reduction in the
conductivity with increasing "concentration is ascribed solely
to a reduction in the number of the free ions, their mobilities
remaining unchanged. We may say in general that with a
suitable assumption as to the osmotic pressure of the ions it
is always possible to derive a mass-action law which will
express the experimental results either for 8, or for B,
whatever they may be, but only at the cost of entailing a
theoretical difference between these two quantities which
does not actually exist. The view that the electrical forces
between the ions affect their mobilities (to a sufficient
extent to account for the greater part of the variation of X
in strong electrolytes) will of course dispose of this diffi-

culty; and so far as I can see it is the only view that will
do so.

Interionic Force in Electrolytes. 219
APPENDIX.

The problem to be solved is as follows :-—
Being given

2
(1) YC =K+D(Cy)”
i.
an empirical equation for an experimental fact when y stands
r
for “ag

(2) C=ctcet=p-+p,

where the capitals, undashed, and dashed letters mark the
total, molecular, and ionic, concentrations and pressures.

(3) v=o

an assumption which the result of the calculation shows to
be erroneous,

dp __dp' __dP
(4) oo an

a thermodynamic result indisputable in all cases if the
phenomenon is to be ascribed to dissociation,

() p=Eic, but 7 is not = Rie’,

an admission that the failure of the strict mass-action law is
connected with the non-obedience of p’ to the gas law (this
has to be made for either p or p’ if (4) is to hold),

T P
(6) a RTO?

a thermodynamic ; sees equation indisputable in dilute
== , oes

a a

solutions—express ~ — as a function of the measured quantities

qx: Peencend ; 2 & 0
C and
Xo

We have from (1), (2), and (3),

12

(7) —=K+De™,
and from (4) and (5),
dp' = —RTde,

220 Notices respecting New Books.

Substituting the values of c and dc in terms of ec! and de’
obtained from (7) we get

bi mDe'™

Integrating this equation we get pn’ as a function of ¢’,
i.e., of yO; substituting this in (2) we get P and from (6)
ce
— in the form
T9

T mY _Di(Cy)™
—=l+y--7 = d (Ory).
ne). KeDOye

In order to draw the curve of fig. 1 the integral was

evaluated graphically using the constants given by Kraus

and Bray.

XXV. Notices respecting New Books.

Researches of the Department of Terrastrial Magnetism. Vol. III.
Ocean Magnetic Observations 1905-1916, and Reports on Special
Researches. By L. A. Baver, Director, Washington D.C.
Published by the Carnegie Institution of Washington, 1917.
Quarto, pp. v+447, with 25 plates and 35 figures in the
text.

i) Las volume is issued also in three parts, dealing respectively

with the earlier magnetic observations taken at sea by the
‘Galilee’? 1905-1908, the later sea magnetic observations by
the ‘Carnegie’ 1909-1916, and the results of the observations
on Atmospheric Electricity taken on both vessels. There is a
very full account of the instruments, the methods of observation
and the reduction formule, and elaborate tables of results and
particulars of the errors in existing charts—American, British,
and German. The vessels and the instruments are illustrated in
the plates from a variety of points of view. ‘lhe observing vessel
‘Carnegie’ was specially built for magnetic work and is almost
free from magnetic material. This has proved a great simplification
in the reduction of the observations.

:
|
:

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND
JOURNAL OF SCIENCE.
[SIXTH SERIES.] “1B RSS
Lippe a
\ é al A oe ya

MARCH1918. “Nis, 19

XXVI. Rain, Wind, and Cyclones.
By R. M. Destry, MInst.C.£., F.G.S.*

Oboes G recent years our conceptions respecting the
conditions obtaining in cyclones have undergone very
considerable alterations. The old idea, which madea cyclone
consist of a lower spirally inflowing current of air directed
towards the centre of an area of low barometric pressure, an
internal rising current of warm air, and an upper stream of
air flowing outwards from the same area, requires very con-
siderable modification in view of modern discoveries.

That the cause of rain is due to the upward flow of masses
of air, which being cooled below the dew-point by expansion,
condense out some of their moisture, still holds true ; but the
distribution of temperature and the actual nature of the air
circulation prove to be very different from what was once
thought to be the case. ©

In all scientific work accurate observation and correct records
are essential. Here, however, meteorological records fail us
in a very important particular. The wind directions shown
on our charts are not always accurate. In Great Britain,
for example, wind directions are given to the nearest of
sixteen points of the compass. It thus comes about that
winds which are observed to differ in direction only one or
two degrees from each other, may be charted as differing
by as much as twenty degrees.

* Communicated by the Author.

Phil. Mag. 8. 6. Vol. 35. No. 207. March 1918. R

: oO c

222 Mr. R. M. Deeley on

Owing to the direct flow of the wind being prevented by
trees, buildings, hills, &., it is by no means easy to observe
the correct direction of movement. It often happens that a
wind blowing up a river will follow closely the river
windings ; passing clouds, especially rain-clouds, also pro-
duce, for short intervals, marked changes both in the
direction and force of the wind; but the difficulties that
are met with in observing the general direction should be
overcome by the exercise of extreme care so as to obtain a
result as accurate as possible. The observed results should
then be recorded in degrees from the true north, working
clockwise round the compass.

To render the points I wish to draw attention to clear, a
number of charts or diagrams of cyclonic disturbances &e.
have been taken from Shaw and Lempfert’s paper on “* The
Life History of Surface Air Currents”*. The wind direc-
tions are sometimes shown by continuous lines instead of
arrows, and the isobars -by dotted lines. In the original
charts the arrows often show many irregularities in the wind
directions of the same wind province; but these are due to local
causes such as hills, falling rain, &c. It is near the centres
of cyclones that the errors arising from the methods of obser-
vation now adopted appear to be most marked. Indeed, until
strict accuracy is arrived atin recording and plotting obser-
vations of wind directions, many features of atmospheric
circulation will continue to be obscure.

Shaw and Lempfert, in the paper already referred to, show
that the wind directions in cyclones, if they do not always
close in as regular spirals towards the cyclonic centre, do
travel in such a way that the air suffers horizontal contraction
and rises. That this is the case is rendered evident by the
consideration of the air trajectories they have worked out.

In the case of quite a deep cyclone the difference of
pressure between the centre and the margin seldom amounts
to more than one inch of mercury, and this is about the
change of pressure due toa rise of 400 feet. To give rise to
heavy rain, masses of land over which winds are blowing
and rising as they advance must be much higher than this.
In a cyclone the rain-producing effect resulting from the
fall of the barometer is very small indeed. Rain when it:does
fall nearly always results from the rise of air in the cyclone
or the mounting of one air current over another, the quantity
of rain depending upon the humidity of the rising air and
the height to which it is lifted.

According to the old view, the air rises spirally in the

* Meteorological Office Publication No. 174.

Rain, Wind, and Cyclones. 223

central portion of a depression. If such were the case, the
greatest rainfall would occur around the centre. Butit does
not often do so. The greatest rainfall is generally on one
of its sides, and considered with regard to the direction in
which the cyclone is moving, not always on the same side.
However, we may safely conclude that the area over which
rain is falling is that above which the air is rising, and this
fact, considered in the light of the directions of the wind on

Fig. 1.

‘

.@) { 2 3 a S00 FAILES. |
2 EE ES ae

i

e Rain falling.

eee a ele

a) ee

LA ER EE EN OEE CNR CL NE ECC LL CCL GLO LEE RCCL

eee

ns

ee ene re

the earth’s surface at the time, throws light upon some points
in the construction of the upper portion of the cyclone which
might otherwise be obscure.

A very obvious case of a rising wind producing rain,
without any marked barometric changes, is described by
Shaw and Lempfert*. The wind directions are shown in
fig. 1. Here we have a wind from the south-west blowing
against one coming from a little west of south; the line

* Meteorological Office Publication No. 174, p. 62.
R2

224 Mr. R. M. Deeley on

separating the two wind provinces was moving westwards at
about 29 miles per hour, whereas the wind following it up
was only moving at 20 miles per hour. Rain was falling
near to and on both sides the dividing line between the winds.
The upper portion of the westerly wind appears to have
been travelling more rapidly than the portion near the ground,
and was descending and forcing itself beneath the more
southerly wind. The latter being forced to rise precipitated
some of its moisture through the lower south-westerly wind
as well as along its westerly margin.

Fig. 2.

(¢) ) 2 3 4 S00HILES
ee eS

e Rain faliung.

All the weather charts figured are for the area of the
British Isles and portions of western Europe. The points
where rain was falling were generally on land, observations
out at sea not being available in many instances.

Occasionally cyclones, accomipanied by rain, do occur
which show, even as far as their lower levels are concerned, a
striking resemblance to the old theoretical cyclone*. Fig. 2
is such aone. Its centre lay over the sea between Wales

* Met. Office Pub. 174, Plate IV. fig. 3.

Rain, Wind, and Cyclones. 225

and Ireland. The wind directions show that the air was
circulating around the centre of the depression, and was
rising over the whole of it, with the exception, perhaps, of
a small area to the south-west. It was travelling from west
to east at a velocity of about 35 miles per hour, and the wind
velocities were high. The irregular nature of the wind
directions shown by the arrows is most probably due to the
flow of air from areas of heavy rainfall, resulting from the
mechanical effect of thefalling rain. To the east of the storm
centre, in the direction of Holland, a south-westerly wind

Fig. 3.

and westerly wind were in conflict, and rain was the result
there as well as near the centre.

With the exception, perhaps, of the portions near their
centres, the cyclones now to be illustrated do not show a
regular spiral flow of the lower air into the cyclone; but
are areas where several winds are approaching or receding
from an area of low pressure. These lower winds often pass
ever or beneath each other, and cause rain.

Fig. 3 shows the conditions in a cyclone which passed over

226 Mr. R. M. Deeley on

England on Nov. 13, 1901, based upon a chart given by
Shaw and Lempfert*. The wind over France, Germany, and
the south-east of England blew from the south-west. An
easterly wind from the Baltic Sea cut off abruptly the
south-westerly wind, and backing as it travelled round the
western side of the depression, finally became a northerly
wind. The conditions obtaining in this depression are
well described by Shaw and Lempfert. Its average rate
of motion from west to east was only about 17 miles per
hour, whereas in one locality the velocity of the wind was
59 miles per hour. The rainfall was exceptionally heavy,
four inches being recorded at several stations in Ireland. The
area of precipitation was a broad band stretching from west
to east along the line separating the south-westerly wind from
the easterly Baltic wind. Shaw and Lempfert remark, “As
might be expected, the air from these two sources was at
decidedly different temperatures, and... . suggest that the
process going on in the depression consisted in the warm air
from the south rising up over the top of the cold air from
the north-east.” The fact that rain was falling in Scotland
when the centre of the depression was well over the centre
of the North Sea, shows that the south-westerly wind, after
rising over the easterly wind, curved round and passed to
the north and west of the cyclonic centre.

A peculiar feature of the above described depression was
the area of westerly wind that prevailed over the Channel.
This small wind province was to the south-west of the cyclonic
centre and travelled with it to the east. Over this area the
south-westerly winds were interrupted.

Another cyclone is shown in fig. 4, which reproduces
many of the features of the one already described. It was
of the slow travelling type, moving from west to east at
about 11°3 miles per hour. It originated over the west of
the British Isles and grew deeper and deeper as it moved in
an easterly direction. Atthe time of its formation, October 7,
1903, there were south-westerly winds over the area and easy
pressure gradients. At 8 a.m, on October 8 there was a
well-marked but shallow depression over the south-west of
Hngland. An inflow set in from the Baltic area, the southerly
wind commenced to rise over it, and rain fell in the northern
counties of England. These features had become well marked
at 8 a.m. on Oct. 9, and the depression amounted to 0°6 inch
of mercury. Fig. 4 shows the conditions obtaining at this

* Met. Office Pub. 174, Plate VII. fig. 14.
+ Ibid. Plate IX. fig. 12.

Rain, Wind, and Cyclones. 227
time. The growth of the depression appears to have been
from above downwards. Re

Another condition of wind and pressure which oftenigives
rise to heavy rain is known as a V-shaped depression. Such
a one formed over England during the interval from Jan. 6
to 8in 1900. There was a low-pressure area over Iceland, and

Fig. 4,

re) i 2 3 4@ SOOMILES.
aS Ee eee eee ee |

@ fain falling.

a tongue of it was thrust in between a high-pressure area over
Scandinavia, and another in the region of the Bay of Biscay.
Fig. § shows the condition of affairs at 6 P.M. on Jan. 6th.
The depression subsequently extended in a south-westerly
direction, low pressure still holding to the north-west. It
shows a strip of southerly wind, with westerly winds to the
west of it, and easterly winds to the east. The southerly
wind is moving ata low angle across the isobars, from high
to lower pressures. The depression is thusa marginal feature
of the great Icelandic depression. The rain is considered by

228 Mr. R. M. Deeley on

Shaw and Lempfert to have been due to the mounting of the
southerly current over the northerly and westerly wind on its
westerly margin.

But there are other phenomena, of which rain and wind
are features, which are not the result of great differences of
barometric pressure, but arise from the variable heating

Fig. 5,

1 2 3 & SOOMILES
Se Se ee Ee See |
@ ayn fatling. '302 300 |
1292 29-4 59.6 ° 29- | } /

and cooling of the land and water areas by the sun. In
the Troposphere we have the atmosphere in a condition
approaching convective equilibrium ; due to the upward rise
ot heated air, such air cooling by expansion as it rises.
There is a generally held opinion that the full moon dissipates
the clouds. The fact that clouds often melt away after sunset
is probably the cause of this view. During the day in quiet
weather, the air over the land is more heated than is the case
over the surrounding seas, and it rises, thereby producing

Rain, Wind, and Cyclones. 229

clouds ; but during the night the land cools more than the
seas, the air descends and the clouds melt away. Thunder-
storms originate in this way, the air rising locally to such
great heights that heavy rain and hail result. |

The conditions shown on the diagrams that have been consi-
dered are such as occur near the earth’s surface. At low levels
the winds often approach and recede from the low-pressure
area much as comets approach and recede from the sun. But
the wind near the earth’s surface encounters considerable
frictional resistance, and some of it is drawn in and rises in
the cyclone. Indeed, if it were not for the friction of the air
and the ground, a cyclone once started should persist by
reason of its own momentum.

It must be remembered that the water content of the air
rapidly decreases with elevation, owing to the decrease of
temperature. Above 3000 metres the moisture content of the
atmosphere is very small owing to the low temperatures
prevailing at that level. Indeed, this lower region of the
atmosphere has been called the storm layer. In it the tem-
perature gradients are very irregular, for all upward currents
throw down moisture as they rise, and are prevented from
falling in temperature as much as they otherwise would do
by the liberation of the latent heat of condensation, whereas
descending currents undergo simple adiabatic changes of
temperature. Invery cold weather the temperature gradient
is often reversed by reason of the cooling of the air in contact
with the ground.

The diagrams of cyclones shown in the figures that have
been given make it clear that the winds near the earth’s surface
very often cross the gradients at high angles. Gold* has
carefully considered the relationship of the isobars and winds
at heights of about 3000.feet. Assuming that the isobars near
the earth’s surface hold true for greater heights, it is found that
the strength of the upper winds closely approximates, by calcu-
lation, to what would be expected from the gradients at the
earth’s surface. Gold remarks “It is to be noted, however,
that on the average, even for anticyclones, the tendency is for
the wind at 1000 metres altitude to blow slightly across the
isobars from high pressure to low.” Ina preface to the paper
referred to, Shaw says: “The general result of the investi-
gation is, in my opinion, to confirm the suggestion that the
adjustment of wind velocity to gradient is an automatic

* “Barometric Gradient and Wind Force,” Meteorological Office
Publication 190, p. 9.

230 Mr. R. M. Deeley on

process which may be looked upon as a primary meteoro-
logical law, the results of which are more and more apparent
as the conditions are more and more free from disturbing
causes, mechanical or meteorological.”

It was previously suggested that the surface winds often
followed directions resembling the eccentric paths of comets.
Their courses, however, at higher levels more closely resemble
planetary orbits. In the last paragraph of the preface to
Gold’s paper Shaw remarks: “The whole question of the
cause and meaning of the discrepancies between the gradient
wind and the actual wind is, of course, bound up with the
origin of pressure differences. To put the point in a crude
form, I do not know whether, in practice, the winds have to
adjust themselves to the pressure conditions, or the pressure
distribution is the result of the motion of the air.”

Perhaps the most promising way of ascertaining the
cause of the circulation of the winds is to be found by
studying the distribution of atmospheric temperatures. In
this direction a great deal has been done in the exploration
of the upper atmosphere by means of pilot balloons. In this
connexion we cannot do better than consult a paper by Shaw
and Dines* on “‘The Free Atmosphere in the Region of the
British Isles.”

Fig. 6 isa diagram showing the distribution of temperature
within a cyclone 2000 miles in diameter. It will be noticed
that the temperatures below about 10 kilometres are lower in
the centre than in the margin of the depression. It is clear,
therefore, that the temperature distribution below this level
is such as would cause a descent of air at the cyclonic centre.
The old theory that a cyclone results from the ascent of warm
air from the earth’s surface must be abandoned.

The dotted line B B shows the dividing line between the
Stratosphere and Troposphere. In the stratosphere the
temperature not only rises as the height increases, but over
the cyclone there is a local mass of heated air. I would
suggest that this heated air extends to the confines of the
atmosphere, and there is there formed a gravity gradient
for the flow of air outwards. The displacement of this air
reduces the weight of the air column, and the result is felt
at the earth’s surface.

Instead of the air moving into and filling up the low-pressure
area at once, it circulates round it, in a manner depending
upon the direction of the air-currents in the atmosphere at

* Meteorological Office Report, No. 2100.

Rain, Wind, and Cyclones. 23

the time of its formation. Dines* remarks: ‘“‘ The inference
drawn is that a cyclone is produced by the withdrawal
laterally of the air at a height of from 8 to 10 kilometres ;
for if we choose this height the observed and the theoretical.

Fig. 6.

ooo Mines. 500 | aa JOOO MILES,
variations of pressure and temperature agree, whereas they
would not do so if any other height were chosen for the
outflow of the air which undoubtedly flows in along the
earth’s surface.” My suggestion is that the air does not
flow outwards from the cyclone in any volume anywhere
except where a corresponding volume enters somewhere else;
there being, as regards horizontal flow to or from the centre,
above 2000 or 3000 feet—except at very great heights,—
a balance maintained between incoming and outgoing air.

* Met. Oftice Pub. 2104, p. 50.

232 Mr. R. M. Deeley on

The low pressure and inward flow of surface currents are
maintained by the buoyancy of the warm air of the strato-
sphere, in which there is also a cyclonic circulation.

According to this conception a cyclone is the result of the
outward flow of a volume of heated air in the upper portion
of the stratosphere, an inflow near the earth’s surface, and a
slight bodily lifting of the mass of air between.

A section of the atmosphere between the equator and the
pole shows the same temperature distributions as does a
cyclone. Fig. 7 is a generalized section from the pole to the
equator showing what would appear to be the temperature
distribution in the atmosphere, as revealed by observations
made with self-registering pilot balloons. The dotted iso-
therms are purely theoretical. A A is the earth’s surface
and the dotted line B B the lower surface of the stratosphere.
Fig. 7, it will be noticed, closely resembles the left side of
the cyclone shown in fig. 6. Over both low-pressure areas
there are masses of warm air in the stratosphere.

It is only possible to obtain an idea of the temperatures of
the upper portions of the stratosphere by a study of the atmo-
spheric pressure on the earth’s surface and the temperature
of the troposphere. The troposphere has been sounded in
many places, and its temperature is fairly well known at
several latitudes. In constructing the diagram fig. 7 the
temperatures of the troposphere, as ascertained for Batavia,
Milan, Pavia, England, and Pavlovsk, have been plotted,

and the isotherms drawn in. Comparing the variations in

weight of the air columns at these places resulting from their
different temperatures with the actual barometric readings, it
is clear that a ridge of cold air in the stratosphere must encircle
the earth at latitude 30° north and south of the equator,
and there is a trough of heated air in the upper atmosphere
over the equatorial regions, and a basin of hot air over each
pole. Isotherms have been drawn in the stratosphere to
illustrate this; but to obtain an accurate representation
of the variation of temperature further observations are
necessary.

The high-pressure ridge is rather a string of anticyclones
and the north polar basin has two minima, one over the North
Atlantic and the other over the Behring Sea. The South
Polar Continent is the centre of one great cyclone due to the
heated basin of air in the stratosphere. Were it not for the
powerful action of this basin, anticyclonic conditions would
prevail there.

3

0)

NI

“

»

=

fo)

>

> —< \

© — en this Jie GS Gh Gh amb ane dem oun
RS Eee Fe

5

" QS er ee se ee
3 Strong Westerly Winds.

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=

?

Rain

234 Mr. R. M. Deeley on

The general surface winds of the globe, like those of
cyclones, generally cross the isobars at angles more or less
acute. Itis probable, however, that all the winds of the
world above 1000 metres, or even lower, follow the isobars
very closely, and that the energy of cyclones both large and
small is dissipated mainly by friction at or near the earth’s
surface. Although the energy represented by such move-
ments is very large, it would be easy to overrate the strength
of the forces required to keep them going or to start them.
We must regard the circulation of the atmosphere much as
we do the movement of the planets, moons, comets, &c. of the
solar system ; but the retarding frictional forces are greater
in the case of the atmosphere than they are in the case of the
solar system.

I have ventured to show what I regard as the general
directions of the slow movement of the winds across the
isobars in figs. 6 & 7.

In the case of fig. 6, the arrows show the assumed very
slow movement to or from the centre of the cyclone during
its growth. The velocity of the wind along the isobars may
be high. In fig. 7 the arrows also show the assumed steady
movement to and from the centre ofthe cyclone. The velocity
along the isobars may also be great.

The temperature distribution in the atmosphere appears
to be the result of heating mainly in two ways. At the
upper surface there may be arrested many kinds of radiation
(undulatory and material) and also cosmic matter moving at
high velocities. It is thus heated at its upper surface, and
this heat passes downwards and gives us the conditions of
temperature found in the stratosphere. Light and heat rays
pass through the atmosphere until they reach clouds in the
lower atmosphere or the earth’s surface. From here the heat
rises and the temperature conditions of convective equilibrium
are established in the lower atmosphere.

I have suggested * that the great heating of the upper
surface of the atmosphere over the poles is primarily due to
the electrons shot out by the sun, which, being caught by the
earth’s magnetic field, are directed towards the poles, the air
in the neighbourhood of which they heat and probably ionize.
But we have to, account for the local heating of the upper
surface of the atmosphere required to produce cyclones. It
may be due to pencils of high-velocity cosmic matter; for the
optical properties of the upper atmosphere, as indicated by

* Phil. Trans. vol. xxxi., April 1916.

Rain, Wind, and Cyclones. 230

the varying sunrise and sunset effects, are otherwise difficult
to explain. |

The energy of impact of the cosmic dust need not be as
great as the energy of the cyclone produced ; for the winds
and isobars of a cyclone are to a large extent a modified
arrangement of existing isobars and winds.

Cyclones according to this theory must travel with the
winds of the upper atmosphere. That cyclones originate at
high levels and extend downwards would seem to be implied
by the following remark of Shaw and Lempfert*: “This
disturbance moved slowly in a north-westerly direction
and finally passed away to the North Sea. In the early
stages small ‘secondary ’ minima of pressure developed near
the primary minimum, and the process of travel appears to
consist in the formation of a ‘ secondary ’ in front of the storm,
and the filling up of the original minimum.”

The local heating of the upper portion of the stratosphere
which is considered to result in the formation of travelling
eyclones, is regarded as being produced rapidly and as dying
away slowly. In the case of the polar fixed cyclones the
heating of the upper atmosphere must be a continuous
process or nearly so; the heat is always passing downwards
as rapidly as the air moving into the lower portion of the
cyclone raises the upper surface of the stratosphere, and the
vertical distribution of temperature in the atmosphere remains
nearly constant.

One would expect the weather conditions to be much less
variable if they depended wholly upon the physical features
of the earth’s surface and the radiations received from the sun.
Indeed the Trade Winds, and even the general circulation
of the atmosphere, are fairly regular. It is to wandering
eyclones that our short-period weather variability is due, and
the want of regularity in the manner of their occurrence
would be only what might be expected if they were caused by
irregular streams of cosmic matter. Indeed, our atmosphere
probably protects us from a bombardment from space, not only
of matter but of many undesirable radiations. However,
the energy received from the sun is probably somewhat
irregular in its amount owing to the formation and dis-
appearance of spots on the sun, and some relationship
undoubtedly exists between weather condition variations
and sun-spot periods.

* Met. Office Pub. 174, p. 45.

236 Prof. R. W. Wood on

If the driving force of cyclones originates in the upper
portion of the stratosphere where it is locally heated, then
this heated air must be carried along with the stratosphere
wind, and the course of the cyclone on the earth’s surface
should indicate the direction of flow of the stratosphere wind
above. The generally easterly movement of cyclones favours
this assumption ; but the track of cyclones from south to
north in North America seems to require explanation. In
this connexion it is interesting to note that the dust of the
Krakatoa eruption adhered closely to the area of the equa-
torial trade winds, except over the Atlantic, where the dust
was carried to the north apparently along the American
Cyclone track.

XXVII. Resonance Spectra of Iodine. By R. W. Woop,
Professor of Experimental Physics, Johns Hopkins
University”.

[Plates VI.-VIIL7
INCE the appearance of the last paper on this subject
(Phil. Mag. ser. 6. vol. xxvi.), the stuly of these inter-
esting spectra has been continued without interruption, and
some new and very important relations have been brought to
light. As has been shown in previous communications, the
vapour of iodine in vacuo, when excited to luminosity by the

light of the Cooper-Hewitt mercury lamp (glass), emits a

spectrum consisting of a series of doublets, with a separation

of about 1:5 AU., very regularly spaced along the spectrum
and separated by intervals of about 70 AU. These intervals
increase gradually, however, as we pass away from the green
mercury line, at which point the doublet series has its origin,
until, in the extreme red, the distance between the last two
doublets observed is about 102 AU.,and the separation of the
components of the doublet has increased to 2°83 AU. By the
use of dicyanine plates the series has been followed to its
termination at wave-length 7685 and the wave-lengths of
the seven new doublets accurately measured. The doublets are
not all of uniform intensity, and some are missing entirely,
and it is the connexion between this circumstance and the
way in which the doublet series is related to the band absorp-
tion spectrum, that 1s the most interesting point brought out
by the recent investigations. By varying the conditions of the
experiment it has been found possible to excite by the green
mercury line not only the doublet series, but a simplified

* Communicated by the Author.

Resonance Spectra of Iodine. 237

system of fluted bands, few in number and regularly spaced
if the iodine is in vacuo, increasing in number and-com-
plexity if a gas of the helium group is mixed with the iodine,
or if more than a single iodine absorption line is excited by
the mercury lamp. It is probable that the lines forming the
doublets are themselves constituents of the fluted bands, and
the transfer of energy from one part of the vibrating system
to another, as a result of collisions between iodine and
helium molecules, enables us to build up, so to speak, the
complicated system of fluted bands shown in the absorption
spectrum, out of a number of simpler systems which can be
excited separately. ‘This constitutes a very great advance in
the analysis of band spectra, and brings usa step nearer to the
point at which we can picture some idea of the vibrating
mechanism.

In the more recent work, a method of illumination has
been employed which is distinctly superior to any previously
used, and as it is well adapted to purposes of demonstration
I shall describe it in some detail. The iodine tubes which I
now employ are of soft glass, about 40 cm. long and 3 cm.
in diameter. One end is blown out into a thin bulb, taking
care to avoid having the thick drop near the centre of the
bulb. This is best accomplished by drawing off the tube in
an oblique direction, which brings the drop—formed by the
melting down of the pointed end—well to one side. If this
is not done the drop is apt to form a small lens on the surface
of the bulb exactly on the axis of the tube.

The other end is drawn down, and a few flakes of iodine
introduced into the tube. Itis a good plan to provide the
tube with a lateral branch, by which the density of the vapour
ean be controlled, though this is not necessary for demonstra-
tion purposes. The iodine takes are now brought into the
bulb, or to the bottom of the lateral tube, and the tube joined
to a Gaede pump, interposing a U-tube immersed in liquid
air or solid COz,, or a tube filled with caustic potash, to keep
the iodine out of the pump. During the exhaustion it isa
good plan to heat the walls with a bunsen flame, except
where the iodine is located. Then allow the tube to cool
down to the temperature of the room, and heat the portion
where the iodine is located. The flakes will sublime rapidly
and crystallize on the cooler portions of the wall. The tube
is now sealed off from the pump and the drawn-down end
painted black for a distance of a few centimetres. For the
illumination I used a very simple modification of the “light
furnace ” described in the earlier paper.

The iodine tube is fastened alongside of and in contact

Phil. Mag. 8. 6. Vol. 35. No. 207. March 1918. S

238 Prof. R. W. Wood on

with a small Cooper-Hewitt mercury lamp (glass, not quartz).
The bulb should project a centimetre or two beyond the cap
on the positive electrode, and the drawn-down end should
reach not quite down to the negative electrode bulb. Two
small pads of thick asbestos paper should be placed between
the two tubes, which are then securely fastened together
with copper wire.

The Cooper-Hewitt lamp is supported in a clamp fastened
close to the negative bulb, just beyond the end of the iodine
tube, as shown in fig. 1.

Fig. 1.

A cylindrical reflector is now prepared by cutting off the
bottom of a beaker glass measuring about 12x 25 em., and
silvering the outside with Brashear’s solution. This can be
done with a minimum amount of solution by rotating the
beaker slowly in a glass or porcelain tray, tipped slightly on
its longer side. A preliminary trial with water shows at
once the minimum amount that can be used. It is sufficient
if the solution wets one side of the beaker from one end to
the other. After the silvering the inside of the beaker is
cleaned with a cloth dampened with dilute nitric acid, and
the hollow reflecting cylinder slipped over the iodine tube
and mercury lamp. The lamp is clamped at a suitable angle
for operating, say 5° from the horizontal, and started by
tipping the clamp-stand. The reflector should be supported
so that the tubes are centrally placed. The luminous iodine
vapour is viewed “end-on”’ through the bulb. Ifa prism of
about 8°, such as is used for mounting Lippman photographs,
is placed in front of and close to the bulb, it has the effect
of bringing the tube into the horizontal position, which is

Resonance Spectra of Iodine. 239

advantageous if an image is to be projected on the slit of a
spectroscope. ;

This is the arrangement which I[ have used in all of the
recent work, and besides having a very high efficiency, it is
easy to construct.

The fluorescence of the iodine is so intense that the
doublets excited by the green mercury line can be seen widely
separated in the first order spectrum of a large grating
with a telescope of three metres focus. In a small prism
spectroscope the complete resonance spectrum is extremely
brilliant.

The excitation of the iodine vapour results from the cir-
cumstance that the green mercury line and the two yellow
lines coincide with absorption lines of the iodine, of which,
as I have previously shown, there are between forty and fifty
thousand in the visible spectrum.

We will consider, first, the resonance spectrum excited by
the green Hg line. To obtain this pure, it is necessary to
screen off the light of the two yellow lines. This can be
done with a solution of neodymium chloride, or the double
salt of neodymium and ammonium, and as the use of a fluid
screen is impractical in the case of the method of illumina-
tion just described, it is necessary in this case to illuminate
the tube with a large condensing lens. As a matter of fact,
however, the yellow lines are comparatively feeble in the
ease of the Cooper-Hewitt glass lamp, and the doublets
excited by the green line are so intense that, in the greater
part of the work, no screen has been employed.

The more recent investigations have brought some ex-
tremely interesting phenomena to light, especially with
respect to the transfer of energy from the doublet series to
the band spectra, as a result of the admixture of helium or
other rare gases with the iodine.

On account of the complexity of the subject, it will be
necessary to touch briefly on some of the relations which
have been discussed in the earlier papers.

The band absorption spectrum of iodine covers the spectrum
range comprised between wave-lengths 5100 and 7700. It
is made up of exceedingly fine lines averaging 20 to the

ngstro6m unit in the green and yellow regions, or some
50,000 in all making the estimate on the above average. It
is covered on the short wave-length side by a band of con-
tinuous absorption in the blue-green region, which makes
the exact determination of its end impossible. In the red it
has been followed by means of dicyanine plates sensitive to
» = 9000, and its termination discovered at about 7 = 7700.

S 2

240 Prof. R. W. Wood on

A portion of this spectrum, in the vicinity of the green mer-
cury line, reproduced from an earlier paper, is shown by
fig. a, Plate VI. The entire spectrum, reproduced on the
same scale, would be about 80 metres in length. In the
yellow, orange, and red regions the lines form fluted bands,
or rather series of overlapping bands. In the green region
there appears to be so much superposition of bands that all
appearance of regularity vanishes. A good idea of the
general appearance of this spectrum in the region 5460—
5700 is given by fig. d, Plate VJ. This is in reality the
emission spectrum of iodine in a vacuum-tube, reproduced as
a negative, and with the dispersion employed could scarcely
be distinguished from the absorption spectrum, except for
the strong dark lines, which are iodine emission lines not
belonging to the band emission.

With this as an introduction, we will now take up the
remarkable spectrum emitted by the iodine when illuminated
by the green line of the Cooper-Hewitt lamp. This line
is shown in coincidence with the absorption speetrum in
fig. a, Plate VI. As will be seen, the main line falls nearly
midway between two of the iodine abscrption lines. It is
probable that only the left-hand absorption line is stimulated,
as the width of the mercury line is not quite sufficient to
enable it to reach the other. The short wave-length satellite
is also in coincidence with an absorption line, but, for the
present, we shall neglect the effect due to this. The reso-
nance spectrum excited by the stimulation of this absorption
line consists of a series of close doublets (doublet separation
about 1°50 AU.) very regularly spaced along the spectrum.
For convenience we will designate, as before, the one in co-
incidence with the exciting line as the doublet of 0 order,
those lying on the long wave-length side as +1, +2, 43,
&c., orders, and those on the other side as —1, —2, &e.

The doublet of 0 order is indicated on fig. a (Plate VI.)
immediately above the absorption spectrum. One line
(5460-74) is in coincidence with the iodine absorption line
covered by the mercury line, the other (5462-25) lies 1°5

U. to the right. The former is in reality a re-emission of
the absorbed energy without change of wave-length (Reso-
nance radiation), and I have accordingly named this line
the R.R. line. The other line we may designate the com-
panion line. On the scale of fig. 1 the next doublet (+1
order) would lie on the right at a distance of nearly two »
metres.

By means of plates sensitized with dicyanine, which were
prepared for me by Mr. Meggers, I have succeeded in

Resonance Spectra of Lodine. 241

photographing the doublets as far as the 27th order, with a

large plane grating and a Cooke lens of one metre focus.

| a permits of wave-length determinations correct to about
"1 AU. :

This marks the end of the resonance spectrum, | believe,
as the wave-length of the last doublet recorded on the plate
was 7685, and the plates are highly sensitive to well beyond
$500. Moreover, the absorption spectrum terminates at
about this point.

Photographs of the doublets are reproduced on Plate VI.
Fig. e shows the orders 0, +1, and +3. The doublet of
order +2 is missing, though a pair of faint lines appear
nearly in the position in whicl: it should be found.

Fig. b shows the doublets +6 to +13 inclusive, and fig. ¢
+15 to +22 inclusive; orders 9, 14, 16,19, and 21 are also

missing. The variable intensity of the doublets is also to be
noted.

Comparison Spectrum Neon Short Line.

The law governing the spacing of these doublets will be
discussed in a separate communication immediately follow-
ing the present one, in which only the general nature of the
phenomena will be treated.

Fig. d shows the doublets of order 0, +1, and +3 taken
with a large plane grating and an objective of 3 metres focus
(exposure 15 minutes), in superposition with the emission
band spectrum of iodine electrically excited in a vacuum-
tube. All of the photographs, with the exception of fig. a, are
reproduced as negatives. The resolving power employed in
the case of d was, of course, quite insufficient to completely
resolve the band spectrum, as can be seen by comparing the
width of the doublets with the width indicated in fig. a. It
nevertheless gives an idea of the relation of the doublets to
the band absorption spectrum. :

If we give a longer exposure, we find that the doublets
are accompanied by faint companion lines. These appear in
fig. e, which was exposed for an hour and a quarter. Some
of these lines are due to the excitation of other iodine absorp-
tion lines by the satellites of the green mercury line, but
others, I feel sure, result from the stimulation of the absorp-
tion line covered by the main line. The former come out
strong when the iodine is excited by the quartz mercury arc,
in which case the green line can be broadened until it covers
all of the seven absorption lines between the two arrows in
fig. a.

If now we give a greatly prolonged exposure, we find that

249 Prot RoW" Woudee

a band spectrum also appears. Fig. 7 is a 20-hour exposure
for the same region of the spectrum. The doublets have
fused to a wide band, owing to over exposure. Thecom-
panion lines, above referred to, come out strong, and in addi-
tion there is a fluted band to the right of the doublets of
order +1 and +3.

It will be observed that these doublets lie just within the
heads of the fluted bands, a circumstance which is better
shown by fig. 4, in which the heads of the bands are indi-
cated by arrows. In the case of fig. g the iodine tube,
instead of being highly exhausted, contained xenon at a
pressure of 15 mm. As is apparent, the effect of the xenon
is to reduce tremendously the intensity of the doublets, and
bring out strongly a number of fluted bands between the
doublets, of which scarcely a trace can be seen in the case of
iodine in vacuo. In the case of fig. h, we have the iodine in
helium at 4mm. The doublets are still further reduced in
intensity, the bands are stronger, and a new band appears at
the centre, no trace of which can be seen in fig. g. The heads
of the bands are not resolved, though on the original plate a
number of the component lines can be seen to the left of the
doublets. The doublet of the second order, which is missing,
would fall at a considerable distance from the head of the
band. There is in fact a group of lines at this point in
fig. f, but it is my opinion tnat they result from excitation
of the vapour by some of the satellite lines; at all events,
none of them fits into the series of doublets excited by the
main line.

If we compare fig. h with fig. d, we see at once that the
band spectrum emitted by iodine in helium with monochro-
matic excitation is much simpler than the complete band
spectrum. For example, there is in fig. da strong band-head
at A, of which no trace appears in fig. h. Moreover, fewer
of the bands appear in the case of iodine in vacuo than in the
case of iodine in helium.

If the excitation is by the quartz mercury are the bands
become more complicated, and in place of the doublets we
have groups of lines, which will be discussed more in detail
presently.

Relation between the Doublets and the Band Spectrum.

The absorption spectrum of iodine is made up of more or
less regular fluted bands, resolvable under high dispersion
into fine lines. The heads of these bands lie towards the
region of shorter wave-lengths, and there is considerable

Resonance Spectra of Lodine. 243,

overlapping which gives rise to considerable irregularity in
appearance, especially in the green region. The emission
spectrum of iodine, electrically excited in a vacuum-tube,
closely resembles the absorption spectrum, though they are
not exactly complementary, as has been shown recently
(Wood and Kimura, Astrophysical Journal, Sept. 1917).
Now the green line of mercury, which excites the series of
doublets, lies just within the head of a well-marked band in
the emission spectrum, and it will be observed that the
doublets of order +1 and +3 are similarly located. This
was ascertained by superposing the resonance spectrum on a
band emission spectrum. It is less well shown, except for
the doublet of the +3 order, by fig. d, Plate VI., which was
taken under conditions not well suited to emphasize the
heads of the bands, the line spectrum being too prominent.
The three bands above specified appear as emission bands
accompanying the doublets when the iodine is excited in
vacuo, as shown diagrammatically by fig. 2, in which the

Fig. 2.

mi ILI

doublets have been drawn a little longer than the lines
forming the bands. The band accompanying the doublet of
0 order is not as strongly developed as the other two, and
only its head shows in fig. /, Plate VI.

By comparing the plates of the resonance spectrum with
those of the band spectrum, it has been found that the
doublet of the fourth order also lies just within the head of
a band. Above this point the relations have not yet been
exactly determined, for the band spectrnm accompanying
the resonance doublets has not yet been photographed in the
red. Though the fourth order doublet, which is faint, les
near the head of a band shown on the plate made ot the
electrically excited vapour, it does not occupy a correspond-
ing position with respect to the band which forms a member
of the simpler system shown in fig. h, the spacing of which
is two-fifths of the distance between the doublets, 7. e., there
are five bands between the doublets of first and third order.

It will be necessary to trace this simpler band spectrum
throughout the orange and red region, before we can be sure

i

ANT

244 Prof. R. W. Wood on

that all of the strong doublets are located near the heads of
the bands, and the missing ones near the tails.

The doublet of the sixth order is very strong, and it lies
just within the head of a strong band shown by electrical
excitation, and the same thing appears to be true of the
eighth and tenth order doublets. The interesting point,
however, is that a simple system of fluted bands, spaced
apparently according to a law similar to that which governs
the spacing of the doublets, is excited by the stimulation of a
single absorption line.

Multiplex Excitation.

If, instead of the glass Cooper-Hewitt lamp, we employ a
quartz mercury are ( Westinghouse, Cooper-Hewitt) for the
excitation of the iodine vapour, we find complicated groups
of lines in place of the simple doublets. This is due to the
fact that the green mercury line has broadened to such a
degree that it covers a number of the iodine absorption lines.
This we may call multiplex excitation.

The first point of interest which we should note is that the
intensity distribution among the groups is practically the
same as for the doublets, 2. e., groups of strong lines are
built up around the strong doublets, weak groups around
weak doublets, and only a few very faint lines at the points
where the doublets are missing. This means that the
dynamics of the vibrating system excited is very much the
same in the case of the several absorption lines covered by
the broadened mercury line.

The complexity of the groups depends upon the width of
the green line which increases with the potential drop across
the terminals of the quartz arc, as has been shown in previous
communications.

If sufficient resistance is put in circuit with the are to
keep the potential down to 35 volts, the iodine emits the
doublets only, fig.7, Plate VII. With the potential at 60 volts
we have two new lines to the left of the doublets, as shown
by fig. &, Plate VII., while with a potential difference of
110 volts we have tie complicated groups shown by figs. /
and m, the latter showing the group of —1 order. These
groups are so similar in appearance that, until very recently,
J have considered that the lines corresponded to each other,
that is to say, that the fourth line from the left in each group
was excited by the same absorption line. I now feel certain,
however, that we must be a little careful about accepting
this conclusion, for reasons which will appear presently.

Resonance Spectra of Iodine. 245

In discussing the manner in which the groups are formed
by multiplex excitation, we must recall that in the case of
strictly monochromatic excitation, where a single absorption
line only is stimulated, we have a series of doublets, the
shorter wave-length component of the first doublet coinciding
with the absorption line.

It has been found that the doublets conform very nearly
to the following formula, in which 1/) represents the
frequency of the left-hand component of the doublet of
order m,

ne

+ = 183075 —2132m-+ ‘elt 13.

or, putting it in words, that (approximately) the distance
between the doublets increases by a constant amount as we
pass from each one to the one of next higher order. The
degree of accuracy with which this formula is followed will
be discussed in the communication following this one. The
circumstance that we have a group of lines formed around
the (unresolved) absorption lines which are excited by the
broadened mercury line, furnishes us with the clue as to how
the groups originate.

These groups originate in the following way: The seven
absorption lines which are covered by the broadened green
mercury line are simultaneously excited, and the vapour
emits these seven wave-lengths withont change. These lines
we may call the R.R. lines (resonance radiation). Hach one
of these is moreover the first member of a series such as is
expressed by the formula previously given. The R.R. lines
are not resolved by the spectrograph employed in photo-
graphing the resonance spectra and consequently appear
superposed. But each one is accompanied by one or more
companion lines, lying to tne right or left, and it is these
companion lines which form the group of 0 order.

The actual width of the group of seven R.R. liues is
only about 1/30 of the width of the group formed by the
companion lines.

Let ns now see how the groups of higher order are built
up. Snppose each of the seven R.R. lines to be the first
member of a series such as was represented by our formula,
and suppose that for each one we have the same values
of the constants. Suppose moreover that each member of
any given series 1s accompanied by a companion line. In
this case the group of 0 order will be exactly duplicated at
intervals along the spectrum. The centre of each group will
be composed of seven superposed lines (in reality separated

|
|

246 Prof. R. W. Wood on

by the same small intervals as the R.R. lines) each one of
which is accompanied by a companion line to the right or
left as the case may be. In an earlier paper I spoke of
the seven superposed lines as the “core” of the group. As
a matter of fact, the spacing is not exactly the same for the
seven series of main lines, consequently, as we ascend to.
higher group orders they begin to separate, even with the
resolving power employed in photographing the resonance
spectra. This accounts for the fact that the groups of higher
order differ in appearance from those of lower.

I have photographed the groups of 0 and +1 order with
the 7-inch grating and 3-metre objective in the fourth
order spectrum with an exposure of 48 hours. The lines.
were very faint but perfectly sharp. The appearance of the-
two groups is shown in fig. 3. The resolving power in this.

Fig. 3.
ia 4 3 45 6 7 *Si\5 pS
0 order |
group i |
iil |
jst order
group

case was but little less than that required to separate the-
iodine absorption lines, and we find that the centre of the
group of 0 order is a narrow band (line No. 6 in the figure)
made up of five barely resolved lines. It was possible to
count the lines by holding the plate somewhat foreshortened
under a magnifying-glass. These are the R.R. lines. The
other lines which form the group are the companion lines,
and the fact that there are more of them than R.R. lines:
suggests that probably some of the R.R. lines have two
companions instead of one.

Passing now to the first order group, we find that the
main lines which form the core (each one of which belongs.
to a series of which a R.R. line is the first member) are
more widely separated than in the 0 order group, the spectral
range having about doubled. This is due to the fact that
the value of the constant in the second term of our formula
is not the same for each series.

We will now consider the subject of the companion lines.

Resonance Spectra of Iodine. 247

In the case of the doublets excited by the Cooper-Hewitt
lamp the companion lines lie on the long wave-length side
of the main lines, at distances which gradually increase
with increasing group order. The widths of the doublets in
the various orders are given in the following table :—

Order. Width. Order. Width.
0 1:48 15 2°04
1 1°54 ee 2°19
3 1°64 18 2°25
5 ae 20 2°43
8 1°76 22 2°45
10 1°85. 23 ps
11 1:90 25 2-50
12 1°95 2 2°80
13 1°96

The increment is not quite regular, and it is my hope that
a new set of plates made with a more powerful spectrograph
will show no discrepancies. It is pretty clearly established,
however, that the distance of the companion line from the
main line increases progressively. If this is true of the
companion lines of the other main lines, this circumstance,
combined with the fact that the group of main lines widens
as we pass to groups of higher orders, explains fully the fact
that the groups gradually change in appearance as we ascend
the series.

Groups 4, 5,6, 7, and 8 are shown by figs. n and o of
Plate VII., the former excited by the Cooper-Hewitt lamp,
the latter by the quartz are at 115 volts. In the case of
fig. n we have faint series excited by the two yellow mercury
lines, one of which (5790) lies within the fifth order group
excited by the green line. The lines marked by small crosses
are the ghosts of the yellow mercury lines, and should not be
confused with the resonance lines.

In fig. o it will be observed that the doublets shown in
fig. n have become relatively weak, and that we have a new
series of strong doublets displaced towards the left with
respect to the old ones. This is due to the fact that, in the
case of the quartz are operating at 115 volts, the green
mercury line is strongly reversed and the excitation of
absorption line No. 4 (Plate VI. fig. a) becomes relatively
weak, as it coincides with the reversed core of the mercury
line. In the seventh order of fig. there are four lines.
Two of these, marked by dots, form the doublet excited by
the green mercury line; the other two belong to a series of
doublets excited by the yellow line 5790. The same con-
dition is found in the fourth order group, the dotted doublet
in this case lying to the right of the doublet of order —1

248 Prof. R. W. Wood on

excited by 5790. Similar complications, of course, oceur at
other higher orders.

If we could excite the iodine absorption lines one at a time
there would be no difficulty in finding out how the groups
are built up, but this is impossible with present facilities.

By varying the voltage at which the mercury lamp
operates, and by filtering the light through bromine vapour,
some clues have been obtained regarding the relations existing
between the absorption lines and the lines forming the groups,
but a complete analysis has not yet been made.

In fig. 4 I have given a diagram of the groups up to the

Fig. 4.
L pees 45.638 S Order.
: I i z: }: : ye
: : ee : ?
: n J | .
A al if ‘ He ts ;
: if a (ttt fj ri
i = R LE : 3
ante t | ba
1a - . F 3
I~ | , hae
t 1 t : 412
ia be 13

—_
eEeD
er
otticge
eD
ut

a | j 7

seventeenth order excited by the green mercury line of the
quartz lamp operating at 115 volts. The doublets (lines 6
and 7) excited by the Cooper-Hewitt lamp appear in all of
these groups, though they are relatively faint owing to the
reversal of the exciting line, and these doublets are brought
into coincidence in the diagram.

When the iodine is excited by the lamp operating at
60 volts, lines 2, 4, 5, 6, and 7 appear in the group of

Resonance Spectra of Iodine. 249

0 order, line 6 being of course the unresolved complex of
emission lines corresponding to the absorption lines covered
by the green mercury line. Line 7 is the companion line
which, together with the “R.R.”’ line corresponding to
absorption line 3, forms the doublet of 0 order. The
doublets of higher order lie immediately below, the increasing
distance between the components being very apparent.

Now line 2 is a companion line to the R.R. line corre-
sponding to absorption line 4, indicated also by line 6 in
the diagram. ‘These two lines form another doublet of zero
order. The higher orders do not lie immediately below, but
drift to the left, as indicated by the dotted lines. This is
due to the fact that the constant in the second term of the
formula is a little less than in the case of the first series of
doublets considered, in other words the doublets are closer
together.

In the group of the first order the main line of this series
of doublets can be separated from the main line of the other
series only in the fourth order spectrum of the grating. In
the third order group it is so far detached, that it was con-
fused for a long time with line 5 of the first order group.
lf we compare the orders 0 and 6 we shall see another case
of this kind: If it were not for this diagram arrangement of
the groups, we should probably assume that the first line to
the left of group 6 corresponded to line 1 in group 0, whereas
the diagram shows clearly that it corresponds to line 2.
Moreover, it appears in the 60-volt excitation, which does
not bring out lines 1 and 3.

In the construction of the diagram it is, of course,
necessary to leave blank spaces for the missing orders, .
otherwise the corresponding lines will not lie on a smooth
curve. :

It is a little difficult to explain in words just how this
diagram is to be interpreted, though it is clear enough if the
theory of the group formation which I have given is under-
stood. All of the lines with the exception of 6 in the 0 order
group must be companion lines, line 6 being made up of
the unresolved R.R. lines. In the case of the doublets, the
superposition of which forms the other groups, we must
distinguish between what I have called the main line and
the companion. As we run up the diagram the main lines
should lie on curves intersecting line 6, for example, the
dotted curve shown which belongs to the 2, 6 doublet.

I have not yet been able to identify certainly any other
main lines, though I suspect that the one corresponding to
companion line 9 descends from line 6 on a curve sloping

‘250 Prof. R. W. Wood on

to the left at a lesser angle than the dotted curves, 2. ¢., at
about the angle taken by companion line 3.

Various modifications in the conditions of excitation have
been made with a view of establishing which absorption lines
are responsible for the various doublets.

For example, it was found that the lateral emission and
the end-on emission of a Cooper-Hewitt lamp showed a very
different intensity distribution in the green mercury line, as
shown by figs. r and s, Plate VIIL., which were made with a
very fine plane grating by Dr. Anderson. If the iodine
vapour is excited by the lateral emission of the lamp, as with
the ‘‘light-fnrnace ” companion line No. 1 appears in addition
to the strong doublets. See 0 and +1 orders of fig. J,
Plate VII. After several failures I succeeded in obtaining
a record of the iodine resonance excited by the end-on
emission, and in this spectrum companion line No. 2 appeared
also. Now companion line No. 1 does not appear in the case
of excitation by the quartz arc operating at 35 volts, and the
short wave-length satellite of the green line is weaker, with
respect to the main line, in this case, than in the case of the
Cooper-Hewitt lamp, as is shown by figs. ¢ and wu, Plate VIII.
(¢ being the Cooper-Hewitt line and wu the quartz arc). This
makes it appear probable that companion line No. 1 arises
from the excitation of the absorption line which is in
coincidence with the short wave-length satellite.

Companion line No. 2 is probably due to the excitation of
absorption line No. 4. It comes out with excitation by the
““end-on” emission of the Cooper-Hewitt lamp owing to
the broadening of the main line which occurs under this
condition, and for the same reason it is the first line to appear
when the terminal voltage of the quartz are is increased. No
very definite conclusions have been drawn from the numerous
experiments which have been made with the exciting light
filtered through bromine vapour and nitrogen tetroxide.
With a potential of 90 volts on the quartz arc companion
lines 4 and 5 appear. If the exciting light is filtered through
bromine vapour contained in an exhausted bulb about 30 em.
in diameter, line No. 5 disappears in the groups of order 0
and + 1. In the third order group line No.5 is much
stronger than 4 and bromine filtration of the exciting light
equalizes the intensity. Line No. 4 must therefore be due
to the excitation of an absorption line which is not in
coincidence with a bromine line, and which is first covered
by the mercury line when the lamp operates at 90 volts.
‘This seems to be absorption line No. 5, while the other

Resonance Spectra of Iodine. 251

component, which is removed by filtration of the exciting
light through bromine, is probably due to absorption
line 6.

With a potential of 110 volts on the lamp, companion line
No. 3 appears, and this also is removed by the bromine
filtration of the exciting light, as is shown by figs. p and gq,
Plate VIII., in which g is the resonance spectrum obtained
when the exciting light is filtered through bromine. It
appears to be due to the stimulation of absorption line 7
which isin coincidence with a bromine line.

The difficulty in interpreting the results obtained is due to
the fact that the mercury line widens both to the right and
left as the voltage increases, so that two absorption lines
may be attacked simultaneously. If this happens, we can
differentiate between them only if one of them is in coin-
cidence with a bromine line and the other not. What is
most needed just now is one or more other filters similar to
bromine vapour, but I have not been able to find anything
with sufficiently narrow lines, though I have tried a number
of vapours which looked promising. What would be still
better would be to alter the wave-length of a narrow exciting
line so as to cause it to pass by degrees from one absorption
line to the next.

Excitation by the Yellow Lines.

The resonance spectra excited by the two yellow lines have
not been completely investigated as yet, though a large
number of photographs have been made. LHach yellow line
excites a series of nearly equidistant groups which resemble
roughly the groups excited by the green line. Six pairs of
these groups, from —l1 order to +4 order, photographed
with rather low dispersion are shown by fig. i, Plate VI. In
this case the excitation was by the quartz mercury arc
operating at 140 volts, the green line having been cut off
by means of a glass trough filled with a solution of eosine.
Some difficulty was found in securing the spectrum excited
by the Cooper-Hewitt arc, as the yellow lines are com-
paratively weak in this case, but satisfactory results were
finally obtained with the light furnace, the iodine tube being
wrapped around with a sheet of gelatine stained to a deep
orange-yellow.

In this case each yellow line excited a series of doublets,
but both series were much more irregular than the series
excited by the green line.

Dye Profs. Wood and Kimura on the

The separation of the components of the doublets excited
by the 57907 line varied in an irregular manner from 2:1
to 5:6 AU. In the case of the excitation by the 5769°6 line
we have also a series of doublets, though the companion line
is missing at the zero order, in other words the R.R. line
has no companion. The separation of the components of the
doublets is less irregular in this case, varying from 4°8 to
5'4 AU. The table of wave-lengths will be given in the

communication following this one.

XXVIII. The Series Law of Resonance Spectra. By Prof. R.
W. Woon, Johns Hopkins University, and Prof. M. Kimura,
University of Kyoto™.

‘le the previous communication a general account of the

results which have been obtained, up to the present time,
on the resonance spectra of iodine has been given.

The present paper will deal with the measurements of
wave-length of the lines in the groups, and the subject of the
series law which governs their spacing.

The wave-lengths in the lines in the groups of 0 and +1
order were determined from plates made in the fourth order
spectrum of a large plane grating with a telescope of 3 metres
focus. They are correct probably to 0°01 AU. The groups
+2, +3, and +4 were made in the second order spectrum,
and the higher order groups in the first order spectrum.

The series which has been most definitely determined, and
to which the greatest amount of study has been given, is
the series of strong doublets excited by the Cooper-Hewitt
lamp.

The two components of each doublet appear to be of equal
intensity, although, in the case of two or three, a different
ratio appears in the photograph as a result of absorption. It
was found, as has been stated in earlier papers, that the first
order group, which is usually recorded with the component
of shorter wave-length three or four times as intense as the
other, comes out with its lines of nearly equal intensity if
the lateral branch of the iodine tube is cooled to zero, while
the right hand component disappears entirely if the light
from the tube is passed througha large glass bulb containing
iodine vapour before it enters the spectroscope.

In studying the series law it has been found necessary to

* Communicated by the Authors.

Series Law of Resonance Spectra. 253

reduce all wave-lengths to vacuum, and convert them into
frequencies. ‘wes

We will take up first the study of the doublets, the wave-
lengths of which and their reciprocals are given in the fol-
lowing table, on the International Scale and reduced to
vacuum.

Doublets excited by Green Line of Cooper-Hewitt

Lamp.
| j
G 1 RYE. 1 Difference
es 7 (Obs.) D; © | x (Cal.) spoweeen
Hg or R.R. |
ae 5462-23 M895 183075 «50 183075 0
546374 183025
ee 5526°55 180945 51 180942 43
5528'10 180894
Benes. Missing
= 5658°71 176719 52 176715 204
5660°38 176667
oe 5726-59 174624 51 174621 +3
5728°25 174573
ee 5795°79 172539 bt 172539 0
5797-51 172488
MUS 5866714 170469 49 170470 1
5867°85 170420
CR MeRe Missing
ee 6010-66 166371 51 166370 +1
6012-50 166320
ee Missing
| 6160°63 162321 49 166322 mt
6162-48 162272
e... 6237-68 160316 48 160316 0
6239°56 160268 |
| eae 6216°16 ~ 158324 50 158324 0
6218°14 158270
Ge 6396-08 156346 49 156344 +2
6398°05 156297
14....... Missing
A): 6560°56 152426 49 152423 +8
6562°68 152377
i = 6645:0 150489 46 150481 +8
6647-0 150443 |
7 ae 6731:12 148564 48 148552 +12
6733°28 148516

Phil. Mag. 8. 6. Vol. 35. No. 207. March 1918. A

1 |

) |

254 Profs. Wood and Kimura on the

From this point on values determined from plates made
with telescope of 1 metre focus. They are correct only to
about 0°1 AU.

be SAMY 6618°63 146657 49 146636 +21
6820°01 146608

NS Bors Faint and masked |by mercury line

7 6998°96 142878 50 142842 +36
7001°39 142828

Qe iaee Missing

ya 7186:23 139155 48 139099 +h6
7188'68 139107

Pa eee 7282'39 137318 48 137247 ae
728492 137270

7p RASS Missing

2D.seee) LaOO Ee 133682 44 133580 +102
74829 133638

BAO. ss06 Missing

eet eee 7685°7 130110 50 129964 +146
7688°5 130060

The first point established by this table is that, while the
separation of the components of the doublets increases pro-
gressively from 1°51 AU. at 0 order to 2°5 AU. at the twenty-
seventh order, THE FREQUENCY DIFFERENCE BETWEEN THE
COMPONENTS IS A CONSTANT, 50. The extreme low values
46 and 44 found in the sixteenth and twenty-fifth orders
are undoubtedly due to the fact that the lines were extremely
faint, and the wave-lengths could not be very accurately
determined. The last doublet (the twenty-seventh order)
was fairly strong, and the frequency difference in this case
is exactly the same as in the case of the 0 order.

We will now consider the law governing the spacing of
the doublets along the spectrum, applying the calculations
to the first member of each doublet (shorter ) component).
If we confine our attention to the first few orders, it seems
as if the distance between the doublets increased by a con-
stant small increment. This:would mean a constant second
difference of wave-lengths. It was found, however, that this
condition held only for the first few orders. The reciprocals
of the wave-lengths were next examined, and it was found
that a constant second difference existed, at least over a
considerable range of the spectrum.

Series Law of Resonance Spectra. 255

If this condition held rigorously the series would be
represented by the formula .

1 m(m—1)
= 183075 — 2132 m + Lore p>

in which X,, is the wave-length of the doublet of the mth
order, 2132 is the frequency difference between orders 0 and
+1, 13 the constant second difference of frequency, and m the
order of the doublet. The most accurate value of the second
constant would be obtained by calculating it from a doublet
of high order, as a small error would be enormously magnified
by the term Nae the value of which is 351 for the
twenty-seventh order.

Calculating the constants 2130 and 12:2 from orders 0 and
3d gave calculated values of 1/X which differed from the
observed by the following amounts :—

Doublet Order. Difference. Doublet Order. Difference.
1 0 Ff + 2
3 +1 10 + 4
4 —l : is + 14
5 —] 18 65
5 0 23 +166
6 —l 27 +284

The large discrepancies in the higher orders are due to
incorrect determination of the constants. In spite of this,
though, the series is well represented up to the seventh
order. The following formula gave the best results over the
entire range :—
= = 183075 — 2131-414 = m — 1273)

The values given in the table were calculated by this for-
mula and, as will be seen, the agreement is good up to the
doublet of the fifteenth order. There is a small discrepancy
in the orders 1, 3, and 4, which appears to be inevitable if
the constants are so chosen as to make the formula cover a
wide range. Of course, the formula is not correct for the
entire series, and though we have tried formule involving
higher powers of m than the square, we have been unable to
develop anything superior to the one given. The discovery
of the fact that the frequency difference between the com-
ponents of tbe doublets is a constant, has been of assistance

T2

256 Profs. Wood and Kimura on the

in picking out other series of doublets in the series of com-
plicated groups excited by the quartz arc. |
For example, we may take the wider doublets shown
united by dotted lines in the previous paper (fig. 4).
The frequency differences for these doublets are given in
the following table :—

Order. Freq. Dif. Order. Freq. Dif.
0 161 8 158
1 159 10 157
3 159 12 157
5 158 15 152
6 157 iy 156

It will be remembered that the frequency difference of the
first series of doublets considered was 50.

The spacing of this series along the spectrum is only fairly
well represented by the formula
= = 183075 — 2119 m + ae

The observed and calculated values of 1/X for the com-
ponents of longer wave-length (the companion line is to the
left in this case) are given in the following table. It will be
observed that the doublets are missing in the fourth, eleventh,
and thirteenth orders, as well as in the orders in which the
doublets of the first series failed to appear.

Order: S (Obs.) + (Cal.) Dif,
0 183075 183075 0
1 180956 180956 0
3 176754 176757 3
A 172599 172600 1
6 170543 170556 13
8 166470 166487 17

10 162448 162470 99
12 158479 158505 26
15 152621 152655 34
17 148786 148820 34

In the following table are given the -wave-lengths and
their reciprocals, on the International Scale and reduced to
vacuum, of all of the lines in the groups between 0 and 17,
in the case of iodine vapour excited by the quartz mercury
are operating at 115 volts. \

The doublets excited by the Cooper-Hewitt lamp are
marked thus * and the other doublets which we have studied
thus t. This table corresponds to the diagram in the pre-
vious paper. In the fourth order group lines A and B were
added from an old series of measurements. The line between
them is the only one which appears on our recent plates, and
this line only is given in the table.

Series Law of Resonance Spectra. af

0 Order. Fourth Order. -
No. of . 1 No. of I
Line. ; i Line. ¥ nN
im). 5 +25719°62 174837
|e 5457°43 1832386 5722°05 174763

Vara 5458°33 183189 22-55 A747

Rt , 5460°88 183121 24:47 4688

Pipa 5461-07 183114 24-72 4681

eG a3 cs 5462-23 183075 +5725°18 4668
ee 5463-74 183025 25°35 A661

ese 5464:05 183014 *57 26°59 4624

Shee. 5466-04 182948 26°84 4616

: *5728:25 4573
Eh ata | 28:56 4564

1 SOSA 5520°30 181149 98:95 4552

: pe. $5521°33 181115

‘A 5522-92 181064 |

ee 5525°17 180989 Fifth Order.

Bie 5525°38 180983 +5788°45 172757
Se 5526-20 180956 91:03 2681
Toe 22 955 +5793-77 2599

6 } a ie 47 94-48 2578
i *5526°55 45 94-88 2566
‘71 39 *5795-79 2539
\ ‘80 Sasa | *5797-51 2488

(ia al *5528-10 180894. 98°70 2452

3 5528°39 180884

Bas, 5530-06 180830 Sixth Order.

Second Order. +585823 170700
5585-08 179048 60°8 625
86°33 179008 63-2 555
86°81 178992 15863°6 543
89-05 921 64:75 510
89:42 909 64:9 505
91-02 858 *5866:14 469
9117 853 *5867°85 420
91°38 847 68:1 413
93°33 784 68:1 395
93:77 770 :
Third Order. mee Naik Bolu
ie. 5651-06 176958 ;
ot. +5652°49 76913 +6001:38 628
Bey) 5654°55 176849 _ 04°88 531
+5657°57 176754 +6007-07 471

“ae *5658-71 176719 09°32 411

ae *5660'38 176667 anteiae we

oA... 566197 «176617 6010-66 371

566315 176580 pes Sis

: i 6012°50 320

and, in addition, faint lines as 12°80 310

follows:

5656-87 Tenth Order.

oT li 4614987. 162605

07°35 54:07 494

57°98 +6155'81 448

58°24 57°52 403

58°96 58:31 382

59°27 5914 360

59°50 *6160°63 321

60°70 *6162:48 272

61°02 63°65 241

258 Profs. Wood and Kimura on the

Eleventh Order. Thirteenth Order.
1 1
= Xr. es
aM: r A
6228°79 160545 6388°24 156538
31°44 477 92°14 4492
1 36°12 356 92°82 425
*6237-68 316 94°54 383
*6239°56 268 *6396°08 346
42:0 205 *6398°05 297
Twelfth Order. Fifteenth Order.
6294:°87 158859 16545°65 152773
99°40 745 +6552°19 621
+6303°71 636 59°05 461
08°57 514. *6560°56 426
16309°99 479 *6562°64 378
*6§316°16 324
*6§318°14 274 Seventeenth Order.
alk <i #671400 148923
+6721:03 787
*6731:09 564
*6733°27 516

Excitation by the Yellow Lines.

We have measured the wave-lengths of the lines in the
resonance spectrum excited by the yellow lines of the Cooper-
Hewitt arc, and the quartz are operating at 115 volts. The
values given in the following table are on the International
Scale and reduced to vacuum. They were determined from
plates made with the plane grating and Cooke lens of 1 metre
focus, and can be considered correct only to about 0-1 AU.
We have, however, made some measurements of the doublets
photographed with the 3-metre lens, which are correct pro-
bably to 0°02 AU.,and as the same irregularities were
found in the spacing, we have not thought it worth while
to measure the complete spectrum to the highest degree of
accuracy.

The wave-lengths are given in the following table. The
lines or doublets excited by the Cooper-Hewitt lamp are
marked thus *.

The series excited by the Cooper-Hewitt lamp in the case
of the 5769 line differs from that excited by the green line
in a number of respects.

In the first place, at the point of excitation we have only
the R.R. line with no companion. At orders 1, 2, 3, 6, and
9 we have doublets, 4 and 8 are missing, and at 5, 7, and 10
we have single lines.

Series Law of Resonance Spectra. 259:
Excitation by Hg 5769°6 (5771:2 Reduced to Vacuum).

—1 Order. Fifth Order.
L 1
X r
5701'S 175383 6139-0 162893
5705°2 278 ak an
8
0 Order.
5766-0 1734380 Sixth Order.
67:8 376
*5771-2 O74 eee onaee
*29-1 ria ae
+1 Order. *27-1 53a} 129
5834-4 171397
aiele ae Seventh Order.
*5847-2 022 \ 137 62049 158858
4
Second Order. *6303'5 641
cok ee Eighth Order.
*5915:8 039 :
ated iene 6374°3 156879
Third Order. Ninth Order.

5985-2 167078 Bee ree
Sacks Mite *67-9 609 ae
*95:1 803 } oe *73-2 459 ee

Fourth Order. Tenth Order.

60604 165006 6544'1 152809

64-6 164891 48°7 702
69:1 164769 *53-0 602

The 1/A difference, in the case of the components of the
doublets, is not constant, as in the previous case, but varies
from 143 to 126.

As to the spacing of the doublets along the spectrum,
we find that in this case the 1/X difference is very nearly
constant, as is shown by the following table :—

1 gies 1 iimiad
1 l Dit. = ev hatta
Order. x x 1 Order X x Dif
ey 175383 pais 5 162784 ae
0 173274 ae 6 160715 ae,
1 171159 7 158641
2120
- 169039 2094 8 2014
3 166945 “th 9 154612 lee
4 10 152602

260 The Series Law of Resonance Speetra.

The variation is irregular, and it is obvious that the series
is of a different type from the one excited by the green line, a
portion of which at least was well represented by a formula.

In the case of the excitation by the 5790°7 line, we ob-
tained different values of X in the case of the Cooper- Hewitt
lamp, consequently these values only are given in the table.
It is probable that in the case of the quartz lamp at 115 volts
reversal of the line causes the disappearance of the doublets
excited by the lamp running at a lower temperature :—

Excitation by Cooper-Hewitt 5790°7 (5792°3 Reduced
to Vacuum.)

1 1 1 1

r. ss it, r. ke" =a Sine
r A A X
— 2 Order. Fourth Order.
5658°6 176722 6084:'3 164357
5660°3 669 53 88:3 o49f «+108
—1 Order. Fifth Order.
57221 174761 34 6163'9 162235 58
5bj2Zo2 eae 66-1 1T7
0 Order. Sixth Order.
5792'3 172644 6242°3 160197
5797°9 476 } tee dba) aie 128 69
First Order. Seventh Order.
5871°3 170320 61 6325'8 158083 86
5873°4 259 29:2 157997
Second Order. Highth Order.
59368 168441 61 6404-0 156152 66
38'°9 380 06°7 086
Third Order.
6010°8 166367 69
13°3 298

In the case of the series excited by this line the doublets
are present in all orders, but the 1/X difference between
their components varies in a very irregular manner from a
minimum value of 34 to a maximum value of 168.

The spacing of the series along the spectrum is more
regular, however, tbe 1/X differences being as follows (the
last significant figure is omitted) :

196 205
211 211
232 193
188 212
207 200
203 209

On the Pressure Effect in Corona Discharge. 261

It does not appear to be worth while at this stage of the
investigation to give the wave-lengths of the lines in the
more complicated groups excited by the quartz arc operating
at various voltages, as the simpler series excited by the
Cooper-Hewitt lamp does not appear at the present time to
conform to any law. The cause of this may appear when
the relation of this spectrum to the band spectrum developed
‘when the iodine is in helium has been studied. This will
require exposures of many days, however.

XXIX. On the Pressure Effect in Corona Discharge.
By A. M. Trnpauu, D.Se., and Miss N. 8. Staruz, B.Se.*

~ | ade papers have been published within the past few
years on what has been termed “the ionization pres-
sure in corona discharge.” When a glow or “corona”
discharge starts between a cylinder and an axial wire in a
closed tube at atmospheric pressure, a sudden rise in pressure
is observed. Farwellf and Kunzt have contended that this
pressure effect is quite distinct from that produced by the
heating effect of the discharge, and have suggested that it is
due to the increase in the number of gas particles resulting
from ionization in the tube.
Kunz also has deduced the following formula connecting
the rise in pressure “ p,—p,” with the current “2,” potential
difference “‘e,” and the volume of the tube “ %?:—

- Uv,
‘= ~ (p1—Po)-

In support of this formula Warner § has shown experi-
mentally in a number of. gases that, if the potential differ-
ence is constant, “2” is proportional to “p,;—p,.” Arnold|l,
however, has suggested that the pressure effect can be
accounted for by thermal considerations alone, and has
pointed out that the small magnitude of the current excluded
the possibility of any appreciable contribution to pressure by
ionization. Recently Warner has replied to this criticism.
He has described a number of experiments on the pressure

* Communicated by the Authors.

t Farwell, Proc. A. M.I. E. E. vol. xxxiii. p. 1717 (1914).
t Kunz, Phys. Rev. vol. viii. p. 28 (1916).

§ Warner, Phys. Rev. vol. viii. p. 285 (1916).

|| Arnold, Phys. Rev. vol. ix. p. 93 (1917).

{| Warner, Phys. Rey. vol. x. p. 483 (1917).

~

262 Dr. A. M. Tyndall and Miss Searle on the

effect (1) due to discharge, and (2) due to a heating current
passed through the wire, and has re-affirmed the view that
the two effects are quite distinct.

Now in the past there has been general agreement among
physicists that in an ordinary glow discharge the fraction of
inolecules which are ionized is a negligible quantity. Itseems
therefore desirable to settle conclusively what is the origin
of the pressure rise.

The object of the present paper is to point out several
fallacies in the arguments of the advocates of the ionization
theory, and to present a quantitative verification of the
statement that this sudden rise of pressure accompanying
the start of corona discharge is purely a thermal effect.

It must first be pointed out that the formula given above,
upon which the advocates of the ionization theory base much
of their work, is necessarily incorrect. This may readily be
seen by applying the test of dimensions to the terms of the
equation. Before the dimensions of the two terms can be
equated, the left-hand term must be multiplied by a time “¢.”
The equation then takes the form it would have if the
pressure effect was due entirely to heat generated in a
vessel of constant volume, from which the radiation losses.
were always a constant traction of the heat supplied. If
reference be made to the proof of this formula in Kunz’s
paper, it will be seen thatin steps L and 4 a time factor has
been erroneously omitted.

But this does not dispose of a further argument which has
been brought forward: namely, that the pressure effect at
the start of the discharge is far too sudden to be accounted
for by heat generated during discharge, and that the rise
in pressure due to the latter is only appreciable after the
discharge has been passing for some time.

Thus Warner has published a number of curves showing
that the ‘“‘ corona pressure” reached its full value within
3 seconds of the start of discharge, whereas in the corre-
sponding effect produced by heating the wire this did
not occur until about 15 seconds after the heating current
was switched on. Moreover, the rise in pressure produced
by the dissipation of a given amount of energy in the wire
was much greater than that accompanying the same dissi-
pation of energy in the glow discharge.

But the argument that the pressure effects in the two
cases must therefore be different in origin entirely breaks

Se ee ee See

Pressure Effect in Corona Discharge. 263:

down if one takes into account the potent influence of the
electric wind in distributing the heat generated in the gas
and conveying it to the walls of the surrounding tube. It
may be presumed that the following process is in operation.
As in all cases of discharge from wires or points, the outgoing
ions set up movements of the gas to which the name electric
wind has been given. Heat generated in the gas by the
corona discharge causes a rise in pressure; but since the
wind distributes this heat throughout the gas, the rise in
pressure is quickly checked by the large cooling effect of the
metal wall of the tube to which the heat is being rapidly
conveyed. Witha large cooling surface and a small rate of
dissipation of electrical energy, the heat may be removed by
the walls as fast as it is generated, in which case the pressure,
after a small initial sudden rise, remains constant. If the
rate of generation is, however, too great for this to be the
case, a subsequent gradual rise in pressure will occur as
the temperature of the gas and of the wall of the tube rises.
This is well seen in figures 3 and 4in Warner’s second paper.

In the case of a supply of heat by a current in the wire,
no electric wind is present and the cooling effect of the walls
can only operate through the ordinary convection currents,
which are far less effective. Hence, for the same energy
supply the rise in temperature of the gas, and consequently
its pressure, is considerably greater and reaches its full value
later in time than when the electric wind is present. This is.
in agreement with experimental results.

Again, Warner states that the return to normal pressure
after a heating current in the wire is cut off is less rapid than
it is after the cessation of corona discharge; the numbers
quoted for the two cases under similar conditions being 25
and 18 seconds respectively. But this also receives a simple
explanation in terms of the electric wind, since the movement
of air does not instantaneously cease when the discharge
current is cut off.

It follows that computations in which conduction and
convection losses are neglected break down entirely when
the electric wind is present, however admissible they may be
in its absence.

Exception must also be taken to some other opipvions
expressed in support of the ionization theory.

First, the statement that near the wire every molecule
may be ionized and still the resultant current may be very
small is incompatible with the statement that the potential
gradient near the wire is very high. The latter is, no doubt,
correct, and the result is that the ions near the wire have a

264 Dr. A. M. Tyndall and Miss Searle on the

high ionic velocity. But if this is coupled with a huge ionic
density the currents would greatly exceed those whieh are
observed.

Secondly, Warner attempts to obtain support for the
theory from the variation of discharge current with the
initial pressure “yp,” for a constant potential difference
between the wire and the tube. Since, however, the
pressure “ p,” is not a constant and its variation with “ po”
is unknown, it is impossible that the curves he gives can
furnish us with any information on the subject.

Perhaps it may be well to show that the ordinary pressure
of the electric wind makes no appreciable contribution to the
effect. In the experiments cited above, the manometer com-
municated directly with a hole in the wall of the metal tube
receiving the current. Now the discharge sets up a radial
gradient of pressure between the wire and the tube, the
pressure at the former being below, and at the latter above,
atmospheric pressure. The following argument gives the
maximum value of excess pressure which the wind can set up
at the tube-wall. Circulation of air would decrease this
value.

Let the radii of the wire and of the metal tube be “a” and
‘“A” respectively. Consider a thin cylindrical layer of gas,
of thickness 67 and radius “7,” coaxial with the wire. Let
a discharge current of ‘i,’ per unit length of the wire, pass
uniformly through the layer. This will give rise to the
transfer of a small quantity of gas through the layer, and to
a difference in pressure, 6P, per unit area, between the two
sides of the layer.

Now 107r
a Qarrk ’

where & is the specific ionic velocity.

This difference in pressure e may be expressed as a rise of
pressure Op per unit area outside the layer, and a fall 6p’
Inside it, so that

d6P=dp+o6p!.
Then dp(A?—7?) = dp! (7? —a”) ;

_ t6r(a?—a*)
ay Qarrk( A?— a?)

* Chattock, Phil. Mag. {5} vol. xlviii. p. 401.

Pressure Effect in Corona Discharge. 265

The excess pressure “p” at the wall of the metal tube is
given by
2 ine a wwiog, A/G
p= dp= Wea Fah ka)
When “A” is large compared with “a,” this approximately
== (liad
Anrk
Putting K=1'4 em./sec., volt/cm.,

and io OE amps per CM,
p approximately ==0°003 cm. of water.

This is about 0°5 per cent. of the pressure actually observed
by the authors under those conditions, and is therefore a
quantity which may be neglected.

The authors have been able to verify quantitatively that
the corona pressure effect is solely due to heat generated in
the discharge, by supplying heat simultaneously from a
current in the wire and from corona discharge, instead of

experimenting separately with each source of supply. The

electric wind thus operates in rapidly conveying heat to the
wall of the tube from both sources of supply. Let it be
assumed that the heat from corona discharge is mainly
evolved close to the wire; then the wind will be almost
equally effective in distributing heat, whether it comes from
discharge or from a current in the wire. In other words, if
the observed pressure rise is due to heat, it should be approxi-
mately independent of the source of the heat when wind is
present. Hxperiments confirming this conclusion are herewith

described.

EHzperiments.

The discharge vessel consisted of a platinum wire of diameter
‘006 mm. placed along the axis of a horizontal brass tube of
diameter 2°21 cm. and length 28 cm. Toa side tube in the
latter, one limb of an oil manometer was attached. The brass
tube was closed by ebonite stoppers, through which the axial
wire passed. The wire was attached to one pole of a high-
potential dynamo giving voltages up to 5500 volts. The
current between wire and tube was measured by a Paul
microammeter, shunted when necessary. The potential dif-
ference was measured by a Braun electrometer previously
calibrated. Qne terminal of the galvanometer, the case of

266 Dr. A. M. Tyndall and Miss Searle on the

the electrometer, and one pole of the dynamo were earthed.

The wire also formed part of a separate heating circuit ; the
heat generated in the wire was measured by a voltmeter

anda milliammeter. The electrical arrangements included
a key so arranged that the discharge and heating currents
-could be switched on simultaneously or separately, as
desired.

It may be noted in passing that the tube for which
Farwell gives dimensions was 4°45 cm., and the wire
"019 mm. in diameter. This necessitated the use of higher
potentials, but these were not so convenient in the present

‘experiments.

In Table I. some results for positive and negative discharge

are given ; “‘ ,—) ’’ is the initial corona pressure effect due

to a current “2” and a potential difference “e.” The last
column gives the ratio of “ »,—” to “ze,” the watts dissi-

pated in the tube.

TABLE I.
Sign of U e Pie
Discharge. PIVGes Fs 19-2 amp. volts. 7e
a ‘08 (?) 6:2 4450 2°96 (2)

“13 16°5 4620 1°78

"16 22°¢ 4600 1°58

20 pire a 4780 1-50

24. 34:1 4780 1°51

35 555 5050 1:26

“46 76:0 5200 EEE

“52 87:0 5230 Lis

66 110-0 5350 ie

WA 120:0 5500 1:08

= 09 21°7 ' 4700 ‘90
16 35°4 4820 ‘91

-20 44:5 4770 93

"29 64:2 5030 ‘91

ror 81:0 - 5150 89

“42 94:3 5200 86

7 169:0 5400 "84

1:01 220°0 5480 "84

It will be noticed that in positive discharge the ratio shows

‘a marked decrease in value as the current increases, though

it appears to approach some limiting value at still higher
currents. This decrease may be attributed to the growing
strength of the wind as the current increases. Though a
decrease is also observable in negative discharge, it is much

-smaller.

Pressure Effect in Corona Discharge. 267

In positive discharge the value of the ratio is consistently
higher than that in negative. This may be simply ex-
plained in the following way. Ifa circulation of air is to be
set up, air dragged from the neighbourhood of the wire by
discharge must be returned to it by other paths. The greater
the uniformity of discharge along the length of the tube the
less easily can this occur. Now the uniform nature of
the positive corona implies that the discharge in this case is
sensibly uniform along the whole length of the tube; whereas
in the negative corona the glow is concentrated in a number
of small beads on the wire. Consequently, the circulation
is more restricted in positive discharge and the cooling
action of the wall of the tube less effective than in negative.
This is less noticeable at higher currents when the wind
pressure gradient causing circulation is steeper ; hence the
decrease in the value of the ratio under these conditions.

The degree of circulation, other conditions being the same,
will no doubt depend upon ‘the size of the tube, and this will
have its effect upon the value of the ratio ‘obtained and
its dependence upon discharge current.

The results of discharging from the wire, and heating it
simultaneously, are shown in Table II. (a) for positive and
Table Il. (6) for negative discharge. “w,” and “w,’’ are
the watts dissipated in discharge and heating coe re-
spectively; ‘‘ W ” is the total ee As before, “ p1—po”
the initial rise of pressure.

TABLE IT.
(P1—Po)/W.
Pi —Ps- Wy. W 5. W. Calculated
milk: ; : Observed. from Table I.
(a) AS + -O7eN tal 102 1:75 1:66
"26 "103 "084 178 1°46 1:44
38 "158 "137 "295 1:28 1-25
48 ‘204 "213 ‘417 Leas" 1:16
56 263 256 519 1:08 1°12
(b) 17-091 «= 093-—S “184 95 92
28 A157 138 "295 "94 ‘90
"52 240 338 ‘578 *92 "86
78 "242 573 815 96 84
(c) ‘Bay LE 045-045 55
ABI) (iste "094 "094 5k
GG, | 4258 137 "137 4:8
Lt eee 2 *300 *330 3°6
5 ae! 66 66 30

Change in pressure

:
i
i
i
Hl /
| )
} ,
}
2}
“|
0 10 20 30 40 50 60 70 80

268 Dr. A. M. Tyndall and Miss Searle on the

The values of the ratio of “‘ p,;—po” to “ W ” are givenina
separate column, and may be compared with the values of the
ratio for the same dissipation of watts with discharge operating
alone. The latter have been calculated from a smooth graph
constructed from the values in Table I.

The agreement between corresponding numbers in the two
columns is striking. In fact such a close agreement was
somewhat unexpected, since the strength of the wind is not
the same in the two cases which are compared. Such differ-
ences as exist between the observed and calculated values
are in general in the direction which neglect of wind variation
would cause. These results show conclusively that the
pressure effect is purely a heating effect.

Table II. (c) shows values obtained for the rise in pressure
produced by heat alone. It will be seen that owing to the
absence of the electric wind the values of the ratio are much
higher.

This is also shown in curves A, B, and ©, which are
typical of a.number which have been obtained. Curve A is
the pressure time curve when 0:091 watt was supplied by

Time tn seconds —

discharge alone. Curve B, that for combined supply of
0:094 by discharge and 0:091 by a heating current in the
wire. Curve ©, up to the point D, for 0-094 watt supplied
by current alone.

Pressure Effect in Corona Discharge. 269

At the time marked by the point D, in curve C, a discharge
current supplying ‘094 watt was switched on. The cooling
effect of the wind is well shown by the drop in pressure, the
final value of which was practically that which would have
been reached after combined action from zero time.

Lastly, mention must be made of an experiment by Warner,
by which he claims to have shown that instead of a heating
elfect at the instant of starting of the discharge from an
unheated wire, the gas near the wire actually cools. This
he deduced from the movement of the spot of a galvanometer
connected to a thermo-junction, placed in the tube at a
distance of about 4 mm. from the axial wire.

The authors repeated this experiment, and found that the
deflexion observed is not due to a thermo-electric current,
but to an electrostatic effect when the thermo-junction takes
up the potential of the air. Thus this deflexion is also
observed when the galvanometer is notincluded in the circuit
containing the thermo-couple, but is merely in electrical
contact with it. The cooling of a red-hot wire which Warner
quotes as occurring when discharge starts from it, is un-
doubtedly caused by the electric wind which is thus set up.

Summary.

When glow discharge starts between a cylinder and an
' axial wire, in a closed tube at atmospheric pressure, a sudden
rise of pressure is observed. It has been argued by some
that this cannot be due to heat generated in the discharge,
and an alternative theory, that it is due to ionization, has
been advanced.

In the above paper the authors criticise the argument for the ©
ionization theory, and the interpretation of the experimental
results upon which it is based. They also verify quanti-
tatively that the effect is purely thermal in origin.

Physics Department,
University of Bristol.
Jan. 7, 1918.

Phil. Mag. 8. 6. Vol. 35. No. 207. March 1918. U

p20 7

XXX. Some Problems of Evaporation. By Haroup
JEFFREYS, M.A., D.Sc., Fellow of St. John’s College,
Cambridge*.

‘| eae problem of evaporation is practically one of gaseous
diffusion: that is to say, if V denote the fraction of
the density of the air at any point that is due to water
vapour, V will vary from time to time and place to place
according to the equation

d~ Bal ae) * dy ("9y) * ela)

where & is the effective coefficient of diffusion, ¢ the time,
and d/dt denotes the total differential following a particle of
the fluid. In general V, as above defined, will be referred
to as the concentration.

Then if u, v, w be the components of velocity and V be
supposed expressed as a function of a, y, z, and ¢,

av’ OV: OV OV OV
aera ae igs ai a sph (2)

The boundary conditions are that the air in contact with
a liquid surface is saturated, so that V is there equal to the
concentration in saturated air at the temperature of the
liquid, and that at a great distance from any liquid V tends
to a finite value.

The equation (1) is identical in form with those that
determine the transference of heat and momentum ; and
when the transference is due entirely to turbulence the
quantity & has the same value in all three casés f. In contact
with a solid or liquid surface, on the other hand, the velocity
of the medium is zero, and i diminishes to the value it has
when there is no turbulence: that is to say, in the evaporation
problem & is equal to the coefficient of diffusion of the
vapour ; in the thermal problem it is the thermometric con-
ductivity of air; and in the equations of motion it is the
kinematic viscosity of air. These three quantities are of
the same order of magnitude, but are not equal. When the
air is at rest kis a constant in all cases and the problem is
simply that of solving the equation

ot kV, Qi ee

where \/? denotes the Laplacian operator.

* Communicated by the Author.
+ G. I. Taylor, “Eddy Motion in the Atmosphere,” Phil. Trans.

215 A, (1915).

Some Problems of Evaporation. 271

When the air is in motion, kat a distance from a boundary
is almost wholly due to turbulence, and is_ practically
independent of position. The velocity near a boundary
rapidly increases from zero to about half its amount at a
considerable distance; this transition is accomplished in a
thin layer of shearing, whose thickness in centimetres is
estimated at 40/U, where U is the velocity in centimetres
per second ata considerable distance *. In outdoor problems
it is therefore usually of the order of a millimetre or smaller.
The concentration and temperature also change rapidly
within this layer, and in most cases it will be justifiable to
assume that at the outer boundary of it each is constant and
midway between the values at the surface and ata great
distance. Outside this layer transference of heat and vapour
will take place according to equation (3), where k is now
put equal to the eddy viscosity.

If the dimensions of the liquid surface are of order J, and
the time needed for any considerable change of condition
over it is of order 7, we see that the terms like 0V/0dt,
ud V/dx, and kd?V/0#’ are relatively of orders

Las Ui andl bie:

Taking k to be of the order of 10? cm.?/sec., which is a
somewhat small estimate for outdoor problems, r=1 second,
and U=400 cm./sec., we see that the first term is small
compared with the last provided / isless than 10 cm., and for
slower variations this term will be small for stili larger values
of 1. Again, we see that the second term is small compared
with the last, ‘provided lis less than 2cm.; otherwise it must
be taken into account. In indoor problems U is probably not
far from zero, k=0°24 cm.?/sec., and the whole of d/dét can
therefore be neglected for a surface 1 cm. across provided
the saturation near it does not change considerably in
8 seconds.

When J is sufficiently small to satisfy both these conditions
the equation of transference reduces to V?V =0, subject to
the same boundary conditions as before. Without loss of
generality we can take the concentration at a great distance
to be zero. For if it is actually Vz, V— Va will satisfy the
same differential equation and will be constant over all the
wetted surfaces, and hence if a problem is solved for Vz=0
the solution for any other value of Vz can at once be found
by merely bei V—Vz for V throughout.

Now, V?V=0 is the equation of steady diffusion and

* Private communication from Major G. I. Taylor.

U 2

272 Dr. H. Jeffreys on some

is also satisfied by the potential in electrostatic problems ;}it
follows that, provided the initial change is not too abrupt
and the dimensions are not too great, the value of V at any
point is the same as the electric potential at that point when
the wetted surface is regarded as a conductor charged to
potential Vo, where V, is the concentration at the edge of
the layer of shearing when the air is moving, and the
saturation concentration at the boundary when it is at rest.
The charge needed for this is CVo. where C is the electro-
static capacity of the conductor. Now the rate of trans-
ference outwards from the boundary is

ov
P a
where Qv denotes the element of the outward normal, p is

the density of air, and the integral is taken over . the
boundary.

But in the electrostatic problem, if o denotes the density
of the electric charge on the surface,

av
Ov

ie = ser { ( cit Aa

dS,

=: —Aro,

and hence} DY
is { S

Hence the rate of transference outwards is

AmnkpCVo. . . . ° . . ° (4)

This determines the rate of evaporation, which is shown,
other things being equal, to be proportional to the linear
dimensions, since the electrostatie capacity varies in this
way for bodies of the same shape.

The above is practically Stefan’s* solution, which has
been experimentally proved to be correct subject to the
conditions stated {. When the velocity is large enough to
need to be taken into account, a general solution is no longer
possible, but for a large wetted surface the curvature may
be neglected, and the problem reduces to that of a wind
blowing over a flat surface. This is treated in the next
section.

* Wien, Akad. Ber. \xxxiii. Abteil 2, p. 618 (1881).
+ H. T. Brown & F. Escombe, Phil. Trans. 198 B. pp. 223-291.

Problems of Evaporation. 273

Steady wind blowing over a flat surface of water.

Take the plane of the surface for that of z=0. Let u be
the velocity of the wind in the direction of the « axis.
There is no wind in the directions of the other two axes.
Then outside of the layer of rapid shearing the equation of
diffusion when a steady state has been attained is

BY _,0°V

uae es a ea (5)
provided the distance from the margin is great enough for
o°V o’7V Me: : Pa Oar
a7? and ae to be neglected in comparison with ayer

The a oaiey, wis a function of z only, and k/u is supposed
constant, equal to h?, say. The value of V over the surface

when 2 is positive is Vy. Then all the conditions are
satisfied if

‘= =V.(1- 3h a) when # is positive. . (6)

V=0 when wz is negative.

Vv

ron’
Th 2 ee |
is makes ae Gel over the wet surface.

Hence the rate of evaporation is

kp ine = Vie oe per unit areas 2a (a)

Then the amount evaporated between 0 and 2 over a strip
dy in diameter is by integration

2pV (kua/m)2dy.

Finally, if the length of the strip from one margin to the
other be /, and parts near enough to the edges for the end
corrections at both ends to be important be neglected, the
amount’evaporated can be found at once to be

2pVo St) (ay . of sR grail)

taken over the whole area.

Hence for areas of the same shape and linear dimensions
proportional to a the rate of evaporation would be proportional
to at®. In particular, for a circular area of radius a it
would be

SOO KUL) 3) Coal haeh ist) 24 39)

274 Dr. H. Jeffreys on some

It has been assumed in this treatment that the linear
di o7V 0’V . ee
imensions are so large that —, and =, are negligible in
Da eae OF
comparison with ya

Now the value of V in (6) makes the first two quantities
of order Vo/a? and the last of order V,/h?a. The condition
required for the method to be justifiable is therefore that a
shall be large compared with h?; in other words, ua/k shall
be large.

It has also been assumed that k/u is constant. This
requires that the distance out to which the disturbance
extends shall be sufficiently small for the change in k/u from
its value just outside a thin surface-layer to be small compared
with the whole. This distance is of order 2ha?. Thus

2atdh/dz must be small. Now si is probably of the same

order as al But kp when z=0 is about ‘003pu?, for

each quantity is equal to the skin friction per unit area.
Thus ‘006(ua/k)? must be small. This method is therefore
correct provided ua/k lies between, say, 4 and 10+.

Consider first an outdoor case, with u=400 cm./sec.,
k=1000 cm.?/sec. Then a must be between 10 cms. and
250 metres. In a room with little draught, we may have
u=4 cm./sec., k=1 cm.?/sec., and a must be between 1 cm.
and 25 metres.

When wa/k is much less than unity, the velocity of the
wind may be neglected and the problem becomes one of
steady diffusion; the rate of evaporation is then proportional
to the linear dimensions. When wa/kis large compared with
104, the effect will extend upwards for such a distance that
in most of the volume concerned u and & will be nearly
constant and equal to their values at a considerable height.
The air up to such a height will be practically saturated,
and vapour will diffuse from the upper surface of the
saturated air precisely as before; the law that the rate of
evaporation is proportional to a!® will therefore still hold,
with different values of u and k.

The result that for bodies of medium dimensions the rate
of evaporation is practically proportional to a'® has been
discovered experimentally by Thomas and Ferguson*, who
also point out that this gives a fair representation of Renner’s t

* N. Thomas and A. Ferguson, Phil. Mag. xxxiv. pp. 8308-321 (1917).
+ O. Renner, Flora, 100. p. 474 (1910).

Problems of Evaporation. 275

results. Brownand Escombe also found that the actual rate.
of evaporation varied more rapidly than the radius but not
so rapidly as thearea. Thomasand Ferguson find that when
the surface of the water is below the rim of the containing
vessel and the evaporation is assumed proportional to a”,
where n is a constant for a given depth, n tends to 2 when
the depth is large.

The evaporation from a circular cylinder, wet at the
bottom and open at the top, may be considered here. So
long as the depth is either great or small compared with the
radius, we can assume that within the cylinder the layers of
equal concentration are parallel to the base. Let / be the
depth, a the radius, and R the rate of evaporation. Let the
concentration at the bottom be Vo, and at the top Vj.
Then inside the cylinder the condition for steady flow makes.

Moa

R=qa’*kp oY == wa*kp—>— :

Outside the cylinder, if the air is at rest, the vapour is
practically diffusing from one side of a disk at concentration
V,, and therefore the rate of evaporation is 4kpaV,* ; this
also must be equal to R, else the concentration at the mouth
would be changing. Eliminating V, we find at once

R=ma’kpV,/(l+ ita),

agreeing with a result of Brown and Hscombe (loc. cit.
p- 258).

If a strong wind is blowing over the top, the conditions.
inside will be unaltered, but at the top we shall have

R=3°95 pV, (kua?)2,

giving, after elimination of Vj,

l 1 \
— — a EES on erie eae ee |
A eo T 3-95 (kua®)2/?

which varies with / in the way found by Thomas and Ferguson.

Evaporation from the surface of a leaf.

The surface of a leaf consists of an almest impermeable
cuticle perforated by a large number of small holes, called
stomata, through which respiration and absorption of carbon
dioxide take place. It is a matter of some uncertainty

* The capacity of a circular disk is 2a/m, and therefore that of one-
side of it is a/z.

276 Dr. H. Jeffreys on some

whether carbon dioxide can by simple diffnsion enter these
holes as fast as is indicated by the observed rate of carbon
assimilation. If the rate of absorption were proportional to
the area of the holes it could not be great enough, but if
each hole absorbed at a rate proportional to its radius (this
being the correct law for an isolated hole as small as a stoma)
the total absorption would be much greater than is required.
Some points in the theory still require examination ; it is
not obvious that the surrounding stomata will not interfere
with the action of any individual to an important extent,
and a wind blowing over the surface, though unimportant
for a single stoma, may be important when there are thousands
of them spread over a considerable area. The problem is
mathematically the same as that of evaporation from the
stomata, by which it will be replaced.

First, consider the leaf to be in a steady state and
wind absent. Let the radius of a stoma be a, and the
number per unitarea n?. Then the average distance between
stomata is of order 1/n, and is large compared with a. The
value of V over the surface of any stoma is Vj. Then at a
distance r from an isolated stoma V is of order Voa/r. Now,
if the stomata acted independently of one another, consider
some particular stoma. By itself it would make V=V,
over it; the others will together add to this an amount

ie , which is not very different from vot n2d8, taken
over the whole surface of the leaf. ‘'Thisis of order 27V,an?l,
where / is of the order of the dimensions of the leaf. Now
Jorgensen and Stiles give * for a typical case 2a=0:00107
em. ;_ n?==33,000/em.? Thus the addition to V by the
neighbouring stomata would be of the order of 300 Vy for a
leaf of radius 8 em. This is of course impossible, for V
cannot be greater than Vy. The meaning of the result is
that the surroundings are enough to cause the air at any
point to be practically saturated, and only a small portion of
the vapour-pressure over any stoma is maintained by that
stoma itself. The total evaporation from the surface of a
leaf is therefore the same as would take place if V were
equal to Vy over the whole surface, and its amount is there-
fore 4a7kOVo, where C is the electrostatic capacity of the
whole surface of the leaf.

Looking at the matter in another way, the rate of evapor-
ation from a single stoma uninfluenced by its surroundings

* Carbon Assimilation,’ p. 63.

Problems of Evaporation. 277

would be 2arkeV, *, where ¢ is the electrostatic capacity of a
disk of the dimension of the stoma, and from all the stomata
on a leaf of area A it would be 2kncV,A, if they did not
influence one another. On the other hand, the evaporation
in unit time from the whole surface of a leaf when completely
wet is 47kCV>. The ratio of these two rates is of order
n*aA/2l, or practically n?al. Now, taking | to be 3 cm.,
and a=0-:00053 em., it follows that nal is unity if n? is as
large as 600. Hence, if there are more than 600 stomata
per square centimetre, the rate of evaporation from them
will be greater than that from the whole surface of a wet
leaf, which is absurd. If follows that evaporation must be
enormously restricted by the presence of other stomata.
Brown and Hscombe have stated that “the interference of
the density shells of small holes set at 10 diameters or more
apart is small, each hole beyond this limit acting almost
independently according to the diameter law.” The above
result shows that this is erroneous f ; in fact, the authors
themselves imply its error in the diagram they give of the
lines of flow through a multiperforate septum, when they
make them become approximately parallel at a short distance
from the septum. |

Again, taking n?=33000/cem.? and /=3 cm., we see that
nal is greater than unity unless a is less than 107° cm.
Hence we shall still have the result that the evaporation is
practically independent of the opening of the stomata as long
as a exceeds this limit ; in other words, until the stomata
contract to 5, of their original diameter, the rate of
evaporation will be practically independent of the diameter f;
when they have closed still further it will decrease, and will
finally vanish when they are quite closed. This appears to
contain the answer to Sir F. Darwin’s criticism § of some
results of Lloyd||. Lloyd has stated that the regulatory
function of stomata is almost nil, which Sir F. Darwin

* We must have 2 instead of 4, as evaporation can only take place
from one side of the stoma.

+t In Brown and Escombe’s experiments with multiperforate septa
the radius a was 0-019 cm., n? was 10U to 2°77, and J about 2 cm.,
making n’al equal to about 4 at the most, and in most cases much
smaller. The results are therefore irrelevant to the case of a leaf.

¢ Brown and Escombe, Phil. Trans. 193 B. p. 278 (1900), say that

stomata can close to 7 of their diameter without affecting assimilation.
§ “The Relation leeen Transpiration and Stomatal Aperture,”

Phil. Trans. 207 B. p. 413 (1915-16).

ee a E. Lloyd, “The Physiology of Stomata,” Carnegie Institution,

278 Dr. H. Jeffreys on some

regards as inconsistent with his further results that “ com-

plete closure..... reduces transpiration to or nearly to:

cuticular rate,’ and “‘when the stomata are open to their
utmost limit the highest rate of transpiration is the maximum
of which the leaf is capable.” A more satisfactory statement
would be that until the stomatal aperture is reduced to a

certain very small value the possible rate of transpiration is.
practically independent of the aperture, and nearly all of the

reduction to zero when the stoma closes takes place in the
last 2 per cent. of the reduction of aperture.

Next, consider the effect of wind. Suppose for simplicity
that the stomata are arranged in straight rows, the distance
between consecutive rows and between consecutive stomata

on the same row being b. Then b=1/n. Consider a square

column of air of side 6. To pass over a stoma it would take
a time b/u, and if it were unsaturated at the commencement
it would therefore acquire a weight of vapour 27rkpeVob/u,
if there were no mutual influence between stomata. Now
suppose the air to have moved forward a distance a, in time
x/u. Then the vapour in it will have spread out by diffusion
through a radius comparable with 2(ka/u)?; and if a/b is

great diffusion parallel to the surface of the leaf and across.

the wind will have practically ceased, and thus the vapour
will occupy half a flat cylinder of radius 2(kw/u)? and
thickness 6, its centre being of course at the point 2. Thus
the concentration in it will be of order cV)/z. Further, the
number of stomata much affected will be of order

An*b( ka]? =4n(ka/u)?.

Similarly, the number of stomata whose influence at this.

time will have affected the column of air when it has

travelled a distance between «—4b and #+4b is 4n(ka/u)?,

and therefore the total concentration produced by them is
AncV)(k/ux)?. This is then the concentration acquired by a
mass of air on account of what happened between times

(a+4b)/u previously, and the total produced by all times is.

to be found by summing the series for all such intervals.

Put z/b=r. The total is then 4xcVo(k/ub)2r—2, the summa-
tion being from r=1 to r=1/b, where / is the distance of the

mass from the stoma nearest the margin. When / is great

this is of the order of 8n?cV)(kl/u)?. Now bis about 0°05 mm.,

and thus is usually small compared with the thickness of the-
layer of rapid shearing *. A fortiori a, and hence ¢, are

* See p. 271.

Problems of Evaporation. 279:

smaller. Thus u, being the velocity within the region to
which the diffusion from a stoma extends, is much less than
the velocity outside. Similarly & is the true coefficient of
diffusion, about 0°24 cm.?/sec. The thickness of the layer of
shearing being 40/U, it follows that wis of order U*a/40.
Thus the quantity just obtained is of order 100n?V,(kla)?/U,
which is about 100 Vo. It follows by argument similar to
that used in the case of no wind that the eariier stomata
saturate the air before the later ones are reached. Thus the
total evaporation is not very different from that in the case
where the whole surface of the leaf is wet *, and is therefore
proportional to /+°, where / is now proportional to the linear
dimensions of the leaf.

This approximation will break down if n?ais much smaller,
for then the residual saturation from the earlier stomata may
be small compared with Vo, and the rate of evaporation will
then be the same as that obtained by summing the results
for the individual stomata, each being supposed isolated ;
this sum is 2an?kpcV,A. A similar result may be obtained
if u/k is much greater. It must always be noted, however,
that this formula can be applied only when the result it gives
is less than the rate of evaporation from a wet leaf with the
wind blowing over it; otherwise we should again have the
absurdity of the evaporation from a part being greater than
that from the whole.

The best method of determining whether it is better to
employ the sum of the possible evaporations from the
individual stomata, or to regard the whole surface of the leaf
as wet, is probably to calculate the rate of evaporation on
both bases and take the smaller of the two results as
supplying the correct upper limit to the amount of respiration
the leaf can perform. Similar remarks will apply to the
possible absorption of substances from the air.

It may be remarked that when the number of stomata is
so large as to make the problem reduce to that of a wet leaf,
the total evaporation is not a function of the number of
stomata, but that from any single stoma is inversely pro-
portional to the number. Thus increasing the number
diminishes the work thrown on any individual, which may
be of some physiological importance.

The above investigation concerns only the purely physical

* O. Renner, Flora, vol. 100. pp. 451-547 (1910), states on p. 485
that the evaporation from a leaf is the same as that from a water
surface

280 Some Problems of Evaporation.

side of diffusion. It does not preclude the possibility that a
reduction of the stomatal aperture may be associated with a
reduction of the rate of evaporation; but it does show
that in most cases the cause of such reduction is not the
mere extra mechanical obstruction to the passage of water
vapour, but must depend on the internal conditions. The
importance of these is obvious. For instance, in the
problem considered here the air has been supposed saturated
when in contact with a stoma and perfectly dry at a great
distance. Actually the concentration at a great distance
has the finite value Va; and that at a stomatal aperture is
probably somewhat less than the saturation concentration.
Let it be V,;. The latter question is further complicated by
the facts that the dissolved substances within the cells must
diminish the pressure of saturated vapour ; that the leaf
is normally at a somewhat higher temperature than its
surroundings, so that the pressure of saturated vapour will
on this account be greater than that at the temperature of
the surroundings; and that owing to internal restrictions to
the supply of water to the stoma the vapour-pressure may
be reduced. The effect of these changes is that in all the
formulee we must substitute V,— Vz for Vo.

Another complication arises from the fact that the
stomata are not usually mere pores with saturated air in
their planes ; in most cases they are pits sunk in the leaf-
surface. As long as their number is large this is not likely
to produce any great effect on the rate of evaporation, for in
exactly the same way as with flat stomata the earlier ones
met by the air will partially saturate it, and the air when it
meets the later ones will be nearly at the same saturation
as that inside them. When the number is small, on the
other hand, the formula for evaporation from depressed
stomata must be used. For circular cylindrical stomata this
gives for the rate of evaporation when the depth / is great
compared with the radius

mnvkpa?(V;—Va)A

(+ita

ee 28g
XXXII. General Curves for the Velocity of Complete Homo-

geneous Reactions between Two Substances at Constant
Volume. By George W. Toop, D.Sc.(Birm.), B.A.
(Camb.)*.

[Plate IX. ]

x HEN m molecules of a substance A react with n

molecules of a substance B to give one or more
resultants, there being no back reaction, the velocity of the
reaction is given by

SE ee

where 2 is the change in the concentration C in time ¢ and
k is the velocity constant. If & is known, the changes in
concentration for various initial concentrations of the reacting
substances can be worked out by integrating the above
equation, but the integration often absorbs valuable time.
By choosing suitable quantities it is possible to plot curves
which will apply generally to all reactions of a similar type.
The author has worked out some of these, and puts them on
record koping that they may save much time and labour.

Bi-molecular Reaction.

If the reaction is bi-molecular of the type A+B-1 or
more resultants, the reaction velocity is given by
dx
ai =k(a—a«)(b—2),
where a, 0 are the initial concentrations of A, B respectively.
The equation may be written

ie a 7b
ar = Ha (1-5) (2-2).

Putting “=X, where X= fraction changed,

dX
HKU) (p—X),
where K=ka and p=.

Take (i.) initial concentrations equal, 7. e. a=b or p=1,

then
Ki= ee le N
. |} cesses 1X7

* Communicated by the Author.

282 Dr. G. W. Todd on the Velocity of

The maximum value of X=1. Giving X values up to 1
we gel

SEA ih pe 2 3 “4 9) 6 af 8 I |
ME cies 0 ‘111 :250° -428 -666 1:00 1:50 233 400 900 o

These are plotted on fig. 1 (p=1), and the curve will
apply to any bi-molecular reaction in which the initial
concentrations are equal.

Take (ii.) one of the substances in excess, say ped
where p>1l. Then we have

ee { lop. uae

Li) Seer oD, > Gilichil nee Tie ‘94 +97 10
036 ‘077 ‘126 -183 °255 -344 °386 ‘549 -851 1:09 1:42 ow
These are plotted on fig. 1 (PI. IX.), and the curves will

apply to any bi-molecular reaction of the type A+B >1
-or more resultants.

l-p p-—xX
p=Vo.
ATE TER: iia | 12), Soman 8 6 7 8) 0 (aaa
Ké... 0 “O71 =*159° *267'' 401 572 -808 «11b 169 (297 Baie
p= a
OR | PAIR CSS: UN AM 8. 2)
Kz... O 054 ‘117 ‘194 -288 -405 559 ‘771 110 1°70 232 @
p=
HO
eee)

Ter-molecular Reaction.

Let the reaction be represented by 2A+B->1 or more
resultants, then the velocity of reaction is given by

& _h(a—2)"(0—2)

2
Mey *\{o. 2
= ha (1 “\(° =)

Bringing to the same notation as before gives

x dX
Kem | ax peKy
~where K=ka?.

Take (i.) equal initial concentrations, 2. e. p= 1, then
ak 2 Ke

MOS SS
o (2X)? 32 a—X)?
‘which gives
PAS persis Oot 2 3 4 5 6 wi Bo 2
Gio ala 0 ‘117 :281 -520 °889 1:50 263 506 120 50 o

Reactions between Two Substances. 283

These are plotted on figs. 2 and 3, and the curve applies to
any ter-molecular reaction in which the initial concentrations
are equal. b

Take (ii1.) B in excess, 7. e. p= sed 1. We have

ee (* dX aed flog PAX) ua
Ree yr) eae pas ee ee
p15.
ES icon La 2 3 “4 8) 6 ch 8 Or iaate@
Mass... 0 -077 °180 °323 -440 ‘848 1:38 2°37 462 12:48
p=2.
«5 Oryst “2 3 “4 Bb 6 lt 8 a 1-0
RBs 3 046.) - Les ere O0D 941) 56) 2907-3, bo
i
2 ae One dD 2 3 “4: 5 6 oh "8 SOP Fil
i 0 -088 086 151 ‘247 ‘373 ‘577 -932 168 401
p—4.
ae Ot “2 3 *4 ‘D 6 ‘7 8 OPO
a... Oe R23 O6t i linet, AIG 66D) LO 27 ea

These figures and the graphs plotted from them (fig. 2)
will apply to any ter-molecular reaction of the type 2A + B->1
or more resultants, B being in excess.

Take (iii.) A in excess, and let c= p>1, then we have
= aX 1 p-xX X(p—-1)
gE Eling PoE, BOD
\, (p -XPU=X)~ @=TPU Sp —X) © p(p—X)
where K=kb?.

p=li.
ee es 0 4 oT 8 ‘9 1-0
Wee o2 0c. 0k 0 ‘O76. 2:05 3°69 8°24 va)
p=1é.
ae it ene | a 73) ‘7 8 ‘9 94 1:0
Ws ses O>: O50. aetl4) 462 eris:.. 185, 354-4 6°02 ora)
g=2.
1 ee O° «I “2 'D "4 #3) zi 9 95 | 97 0
MGs ces O° 027) C62 010° “1627 3259. 3503) 1-29 Si 23) co
p=3.
2 ae yk “5 a | ‘9 * 95 ‘98 1:0

: CC PETe 0 012 093 "224 D381 ‘575 797 oo

284 Dr. G. W. Todd on the Velocity of

These figures and the corresponding graphs (fig. 3) apply
to any ter-molecular reaction of the type 2A +B->1 or more
resultants, A being in excess.

Quadri-molecular Reaction.

Let the reaction be of the type 3A + B->1 or more resultants,
and let the initial concentrations be equal. The general
equation is

aes
dt 2) (6-2)
3
ee (i- f © on *),
; ra |
when . —]
soe BA 1 ee
a (1—X)# 3 1 (1—X)3 “14
and K=ka?’.
=< aR Ms a O° “i ee fo “4 "5 6 65 4 32
IK Geen cte QO 128 °32 -64 121 233 487 7438 120 @

Now take B in excess, putting LSS | The equation
becomes i

x aX
k=) (—G=H)
P

Xx C i
= —A log.(l—X)+By—y + 2 | ax tf + Dea —¥?

where A, B, C, Dare constants depending on the value of p,

p=2, Tho A=1. B=-1. O=1) D==t

Lee 0 + “4 6 ‘7 8 1:0
KU. eneek 0 149 ‘501 1:68 3°49 9°09 00
f d 1 1
pa Then A=. a ae C=5. D=— ,-
IK Gicathie ate 0 3 *D of 8 9 1:0
Re iecscane 0 184 "564 2:06 516 22°7 ee)
1 1 1 1
p=—6. Then A=Joz- B= — 5. C= 5° D=— Fog:
cape 0 5 of 8 g 1:0
HKG too: 0 *265 ‘925 ‘25 9-52 00

These figures give the graphs (fig. 4) for a quadri-
molecular reaction of the type 3A+B->, the substance B
being in excess.

Reactions between Two Substances. 285

Now let the substance A be in excess, so that z= poi.
We have

ae = dX
where K=0’. oka at ek):

The solution is

—X Xx
Meee Aloo! = 2 Be
P p(p—%)
C us if
a tea are — Dlog,(1—X),
A, B, C, D being constants depending on p.
p=1k. Then A=—64. B=—16. C=--4. D=64.

ee 0 “2 “4 6 sie 8 10
ME es... 0 15 ‘47 151 3°01 11°49 oo
p=1i. Then A=—8 B=—4. C=--2. D=8.
a eee 0 ya D 6 =f 8 9 1:0
ae 0 082 “411 602 115 2°15 4°76 ee)
p=, Then A=—1. -B=—k,-C==1. D=h
eae 0 3 6 8 =) “95 97 1:0
1 ...... 0 065 ‘216 ‘O79 1-01 1°58 202 oo

The graphs for these are shown in fig. 5.
There is yet the quadri-molecular reaction of the type
2A +2B->1 or more resultants. In this case we have

Ki yi heh ana
T Dy Gh eae
where K=ka? and p=" =.

The solution is

ee |), Pe) ee ety

tp 1) Pee. (Pol) LOS )p SX). ap
pli
mee... 9 2 ‘4 6 7 8 10
_ a 0 21 ‘70 2-2 47 11:7 00
pT:
ae. Oa aE 3 5 ‘75 "85 ‘9 1:0
ee) QO, \03 + (a0: sam 1-465. 3°33), GOD.’
gS
te 0 5 ‘8 9 ‘95 10
i 0 ‘141 ‘707 1:80 4°14 Ps

The general curves for this type of quadri-molecular
reaction are shown in fig. 6.

Phil. Mag. 8. 6. Vol. 35. No. 207. March 1918. X

286 Pref. A. Anderson on the Coefficients of

Application of the Curves.

(i.) To find the velocity coefficient k.

When the order of the reaction and the initial concen-
trations are known we have only to measure the fraction
(X) changed in a given time ¢ in order to find Ké from the
curve and therefore k.

(ii.) To find the fraction (X) changed in a given time ¢.

This requires a knowledge of &, the order of the reaction,
and the initial concentrations. Then X for a given ¢ can be
read straight from the curve.

(iii.) To find the Order of a reaction.

Two or more determinations of X and¢ are needed together
with a knowledge of the initial concentrations. The
particular curve on which the points (X, ¢) best lie, determines
the order of the reaction.

London,
December 1917.

XXXII. On the Coefficients of Potential of Two Conducting
Spheres. By Prot. A. ANDERSON ™*.

i ie may be of interest to show how the coefficients of
potential of two conducting spheres may be obtained

directly without a previous determination of the coefficients

of capacity and induction, and without making use of electric

images. For this purpose the following elementary pro-

position, which is easily seen to be true, may be used.

If a conducting sphere whose radius is a have a charge H,
and if other charged bodies be brought into the field, the
potential V of the charge Ei at an external point P whose
distance from the centre of the sphere is 7 wil] be given by
the equation

rV =H+a(U—U’),
where U is the potential at the centre of the sphere of the
introduced charges, and U' their potential at P’, the inverse
point of P in the sphere.

Let A and B be the centres of two spheres whose radii
are a and 0, the distance AB being c. Let I, be the inverse
point of A in the sphere B, I, the inverse of I, in A, I; the
inverse of I, in B, and so on. Also, let J, be the inverse
of B in the sphere A, J, the inverse of J, in B, J3 the inverse
of J. in A, and so on. :

Let U be the potential of the sphere A at B, U,;, U., Us,
&ec. its potentials at 1, L, 1; &e., and Uy’, U,', U,', &e. its
potentials at J,, Jo, Jz, &c. Also, let V be the potential of
the sphere B at A, Vy, Vo, V3, &c. its potentials at I,, I, I,

* Communicated by the Author.

Potential of Two Conducting Spheres. 287

&c., and V,', V,’, V;', &c. its potentials at Jj, J, Js, &e.
Let the charge on A be unity, and let B have no charge.

ie 1!

We have, then, the two sets of equations,—
cU=1+a(V—Vj’),
Bay =. b(U = U,').
AJ,. U,'=1+a(V—V;’),
BJ;.V;/= 6(U—U,),
AJ,. U,{=1+a(V—V,’).

ye ees 11),
Ab ie (sv y),
BE 4 DS,

Al, e U,=1+a(V—V,),
BL .= b(U—U,;).

The first set gives at once
ab a? bh?

+e Gos BjOBI.. Ade. Al

U=(14aV)|1

a®b?
‘Dae RAISE A TS. Ad, Adie a
Bee et | ab
—abU | ey + By BEAT,
al?
+ 4) SBI aBIeSBI Ady. Any |
or cU=¢1 +aV)G;—abUF,,

288 Prof. A. Anderson on the Coefficients of

where G; and H; are the sums of the two series, the sub-
script 7 denoting that J’s enter into the expressions.
In like manner, the second set gives immediately

ab a*b?
ae Bee cseg Ki ki Bt, Be |

ab |
— BU av) lar + AL Al, Bi ee
or eV =bUG,;—b(1+aV)Fi.
Now pu=V+ = and pe—U;
hence (c+abF;) py.=aGjpnu,

i
and e(pi— “| + ab; pu= bGy pie

Hence i
Gee ) Sa,
mo Oe oc ate

a
P22

=; ab PPR ui

Since, GaN: Piz must be ue tO Po, it follows that
Gi=G,, ‘and, in fact, on examining the two series, it will be
seen that they are identical if

BJ, . 2 Al, ° BI,,
Ee Ad,—Al, «Bly
fie Vo e= > ie.

These equalities, though they can be proved without
difficulty, are not at all self-evident, and imply the further
relations

2
AT, BT ty ee aT, aM ee Aan
c

Thus we have a case where theorems in ptre geometry

are suggested by purely electrical ones.

Potential of Two Conducting Spheres. 289

We have, then, writing G for G, and G;,

1 eer
i oe er. abe Ab,
‘Ga te)(+ 2H) 26
C ¢ ¢
1 G
Pe arma ah ab
Pa; (ia) (1+ 8) - 4
ab
1 ee ts

rae Su nnn ale No ab Mab ia)
(1+ %R)(1+28)-Ge
It will be seen that
a
m=e(1+ “|

ab
FO oars ies G,

It has thus been shown that the values of the coefficients
of induction, capacity, and potential depend on the sums of
two series, |

1+ ab a*b?
Ce RD A RT a
a®h?
a hs RIT BE Rae:
which we have denoted by G, and
Be * ab 4 abe ia
fe Al. Be AG Ad As VAT. Bis. Bilge a0
which we have denoted by I’. Both of these series can be
expressed in terms of a, 0, c.
Taking the first series and denoting
Al, . Bi, by P12) Al, . Bay by P345 wal : BI, by P56» &e.. aes

it can be shown that

(? —a°— 0") pin — ab?

PAS Coie ;
P12
(ce? —a°—b*) p3,—a°b?
Doge Varer erie ye are
P34

&e., &e.,

290 Coefficients of Potential of Two Conducting Spheres.

or, if we write for c?—a?— 6”, x’, and for ab, p’,

P=’,
Ke ——
P3a= Ke ae 9
Ki —24
Ps ~~ ?
a K a
fe
The series is, therefore,
Pp a Kp
Letie: Stale wae BPE
Aap! (=p) (= 2p)
«8
* (=p (2 — Ip) (i — 3p) *
: a e’—a’—b?
or, denoting > OF ae by 4,
2
yey eae i a
a

e—1 @ ia?) | (@_- jee ae

Hence we have

aya) 4 il i il Pf a A )
Oa ( a a @-—1° (a@—l)(a’?@—2) “7

Similarly, for the second series, we can show that

Posie — a? —b?) —a*l?

P45 ’
P23
2 4
has) oe ay Y
Te ae 9
P23
2 4
_ Kk Pas P
vis 5 ?
Pas
&e...

a)

(?—b?+ac)(?— ae)
ce —

2__ f2
also 1 2 5 and P23

Let p23 be denoted by Ah’.

—

Notices respecting New Books. | 29]
Then pak,
Kh? — p*
Pale ar ia?

eth? — (x? + h?)p*
Per Pay Cae p ;
woh? — KK? + 2h?) p* + p®
Kh? — (x? + h?)p*

The series is therefore

Ps9=

é ab a?b? . ies BEN
a) E tet ei—pt* ae + Rp

a*h*
Ba Koh — KK? + 2h?) pt + p® coal ?

or
c Ak 1 i: )
re oe we ha ee Ee ee
ce — | | a (iz: 2(h?a—1) I: (ite? Naan is a) |
Ke

Thus
a*b? arb? )

Gy=at+ Sa | h2 che —p* Kh? — (x? + h?)p4 ce
For go. we must write > for a and a for 0, and A? will now
denote
(e?—a? + be) (c? —a? — be)
Si Cen eee

XXXII. Notices respecting New Books.

Centennial Celebration of the United States Coast and Geodetic
. Survey, April 5 and 6,1916. Washington Government Printing
Office, 1916. Large octavo, pp. 196.
a Coast and Geodetic Survey, under its present Superinten-
dent, Mr. E. Lester Jones, celebrated the 100th Anniversary
of its institution in April 1916. Addresses dealing with various
aspects of the scientific work of the Survey were delivered by
fifteen eminent Americans, and further addresses, including one
by the President of the United States, were delivered at a banquet,
which completed the proceedings. The volume contains these
addresses, and extracts from the principal Acts of Congress
relating to the Survey. A series of plates contain illustrations of
surveying vessels and apparatus of various kinds, and there are
full page photographs of the eleven gentlemen who have succes-
sively held the post of superintendent. There is much in the
volume that is of interest even to those who are not Americans.

XXXIV. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
| [Continued from p. 140.]

November 21st, 1917.—Dr. Alfred Harker, F.R.S., President,
in the Chair.

HE following communication was read :—
‘The Shap Minor Intrusions.’ By James Morrison, B.A., B.Sc.

The paper deals with the minor igneous intrusions occurring in
the triangular area between Shap, Windermere, and Sedbergh.

From their field relations and petrographic characters the in-
trusions are found to belong to one or the other of two well-marked
groups, a division which is regarded as connoting also an age-
classification.

The rocks of. the earlier set, characterized by the presence of large
orthoclase-felspars of the granitic type, are intimately associated
with the granite, to the immediate neighbourhood of which they
are practically confined. The rocks range from quartz-felsites to
lamprophyres. Of considerable interest in this group is a series of
hybrid intrusions, consisting essentially of rocks of a more or less
basic magma enclosing xenocrysts of a more acid (but allied)
magma obtained by settlement under intratelluric conditions. The
constitution of any given member of the series is determined by
two factors: the abundance of xenocrysts and the composition of the
matrix, an increasing basicity in the latter (due to original magmatic
differentiation) and a decrease in the former marking the successive
stages. The more acid have affinities with the porphyrites, the
more basic with the lamprophyres, the series ranging from modified
biotite-porphyrites to modified pilitic lamprophyres.

The later intrusions are typically free from the large orthoclase-
felspars, though quartz-grains may occur even in the basic members.
Associated centrally with the earlier set they are distributed over
a much wider area, overlapping the former in every direction. They
are the result of a further differentiation, and are assigned to a
later period when igneous activity was renewed on a more or less
regional scale. The rocks include acid felsites and spessartites.

The rocks of the earlier set agree in general direction with the
north-north-west fractures transverse to the strike of the country-
rock, while the later intrusions trend generally east of north.

Phil. Mag. Ser. 6, Vol. 35, Pl. VI.

ON

b th
eh |

Ms

. 6, Vol. 35, Pl. V1.

Phil, Mag. Ser

Woon.

§462°25

Ge
°
»
=
Ee)
a
jo}
a

5460-74.

c
o
vU
c
5
°

12 |4567

Baha a ad

7 Order.

6t order

c
o
=
me]
=

Vacuum

“4 hours.

“wut, UOUaX “Wl 4 Wni/e

ily
“Re

es

x.

oS oe

Yen

Css ata NO

OL rae

5 j - '
Z
‘ oe on —— ear i) 2 z ’ a gel ats
da” os 4 : % rites # oe Fibs 4 ., * he ake pan F Lz ‘me
fo Sei 4 , , Me ‘ oh : sia 3 : “ '
of 2 ‘ -
hy oi { ’ :
E * , . .
=
at hese

a 4 ii
aa) tA
; _— ; bY
a i
; eo bs
(ae) =
<
. (> ;
> ! \
5 ie) ; 2 i}
* . e 5 »}
; 8 r \
; » le
; on 1
’ cS ;
— ‘
“es 5Tt :
ey or
FY .
. FES
12)
ed : (@\7
ae arekee eS : aks ‘
7 “ E ‘ ; x . ¥ . , 4
a ‘ we ; pa” :
* . 3 P iw : & Y i it ey

pete eee ie | reo fay Conte PS EG

=
!
'"
«

-

Woop.

Oorder

bs
@

|
|
el eens S (ier usa
it

Phil, Mag. Ser. 6. Vol. 35, Pl. VII,

Wel eiaa tera nen Le

+3 Order

4”

ai + order ————-—— Eas a eh ae

i. Ve aay

am ee

nae LDN

Woop. Phil. Mag. Ser. 6, Vol. 35. P! VIII.

Lew N

oy ve |
eS WRG

om) 3
Or So

i

F c|

o 2

A 0) :
a a

= =|

bil &
i |

= Q |

2 iS ,
a w

c£

uJ

ne

Phil. Mag. Ser. 6, Vol. 35, Pl. IX,

Reaction Ter - molecular Reaction
b resultants DA +B — /or more resultants
re ess ; : | A is in excéss
ntration of B | _ _ Initial concentration of A
rtration of A | “initial concentration of B

| K= Kb?
| Fig. 3

Se | i

by.

aad ECR ee a cS a
rue oe a

Topp.

10

&i- molecular Reaction
= Molecular feactiol

" 8 j A+8 — /or more resultants
x A / When B is in excess
|/ _ Initial concentration of B
3 initial concentration of A
| H=ka
2 i) — Fig./.
J ial
: | ea |
0 i 2 4 5 6 7 8 9
Kt —
10

Quadri- molecular Reaction
Ie BOLIC)

3A+8— Jor more resultants
B is in excess

_/nitial concentration of B

initial concentration of A

k=ka?
fig. 4
0 u 4 5 | | | 9.
Kt >

x~—

©

10 —
90 —
Ps [Pe eS imi
80 =H
MV VA Pa lk
70 — + |__| ae
60 —- =I
BH | i, / ie Ter - molecular Reaction
I// 2A+8 — Jor more resultants
-40 HAZ |
I i B is in excess
30 WEEE |__| _thitial concentration of B
} i initial concentration of A
= fieB
20 H=ka
| Fig.2
0 ——
0
0 1 2 3 4 5 6 7 8 9
Kt >
10

Quadri= molecular Reaction
r= molecular Meactioy

3A+B— /or more resultants
A is in excess

initial concentration of A

initial concentration of B

K=k-69
Fig. 5

meen

Phil. Mag. Ser, 6, Vol, 35, Pl, IX,

a

iN

Ter - molecular Reaction

2A+8— Jor more resultants
A is in excéss

_ Initial concentration of A

inttial concentration of B

K=kb?
| Fig. 3
1 —
0 dl |
0 L 2 4 5 6 a 8 9
Kt —
10

Quadci- molecular Reaction

2A+2B — Jor more resultants
‘Bis in excess

initial concentration of B
“initial concentration of A

K=k-6%
Fig.6

7

Kt >:

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

Pa

4 ian ra <F

[SIXTH SERIBSG a |}
; a ; a 7 2 if .%

APRIL 1918

XXXV. The X-Ray Spectra and the Constitution of the Atom.
By L. Vecard, Dr. phil. University Christiania *.

at: CCORDING to. Rutherford ¢ the atom consists
of a positive nucleus and electrons circulating
round it. A number of different methods have led to the
result that the number of electrons in the neutral atom is
equal to the atomic number N, and that consequently the
charge of the nucleus is +Ne. The stability of the system
is not secured by the usual electrodynamic forces, but
Bohr i by his ingenious explanation of the series spectra
has been able to fix orbits of stability by means of the
energy quanta introduced by Planck.

The fundamental assumptions underlying Bohr’s theory of
atoms and their light emission are the following :—

I. In the normal state of the atom the electrons are
arranged in groups (rings) in such a way that for each

electron the angular momentum is equal to Qn OF

h

pe ae :
Weegee oe (1 a)
where m is the mass, » the angular velocity of the electron,
a the radius of the circular orbit, and 4 Planck’s constant.

* Communicated by the Author. The results of the present investi-
gation were communicated to the Kristiania Vid. Selsk. on October 23
and 30, and read before the Society November 23.

+ E. Rutherford, Phil. Mag. xxi. p. 669 (1911), and xxviii. p. 488
(1914).

} N. Bohr, Phil. Mag. xxvi. pp. 1, 476, 857 (1913).

Phil. Mag. 8. 6. Vol. 35. No. 208. April 1918. ¥

294 Dr. L. Vegard on the X-Ray Spectra

II. To produce the line of a series it is necessary that an
electron should be removed in some way or other from one
of the rings. The recombination of an electron towards the

broken ring may take place in steps between stability orbits
determined by the condition :

Moa" =T 5. eye) Ge te en (1 b)

III. To each stability circle of the recombining electron
corresponds a certain energy of the atomic system. When
the electron in one step passes from a stability circle 72 to
one 7, the energy must be removed from the system, and

Bohr assumes it to be radiated out in one quantum homo-
geneous radiation, or

hv=W.— W;, ol) ee le (2)

where v is the frequency and W the total energy. The
latter relation we may call Bohr’s frequency law. The orbits
corresponding to the normal state of the atom we shall call
primary orbits, and the stability orbits which the electron
may take up during its recombination we denote as secondary
orbits.

By means of these assumptions Bohr was able to deduce
the complete series spectrum of hydrogen and a spectrum of
helium which was emitted when a single electron recombined
to the isolated nucleus. He was also able to give an ex-
planation of the appearance in the series formula of the
universal frequency constant (R) of Rydberg, and his fre-
quency law explained in a simple manner the combination
principle of Ritz. But up to the present the atomic model
based on the assumption I. has not been able to yield any
theoretical deduction of the series spectra in general.

In the case of hydrogen and those spectra similar to that
of hydrogen, the theory of Bohr has proved to be very
successful indeed. Through the introduction of noncireular
orbits already Bohr * was able to deduce a formula for the
' Stark-effect which gave the right order of magnitude and

the right type of variation. Later on the question of non-
circular orbits has been much more completely treated by
Sommerfeld +, Schwarzschild {, and Epstein §.

Sommerfeld gave a generalization of the quant-conditions

* N. Bohr, Phil. Mag. xxviii. p. 506 (1914).

+ A. Sommerfeld, Sztz. Ber. d. Miinchener Akad. d. Wiss. 1915;
Ann. d. Phys. li. pp. 1 & 125 (1916).

$+ K. Schwarzschild, Berliner Akad. d. Wiss. p. 548 (1916).

§ P. Epstein, Ann. d. Phys. 1. p. 489 (1916).

and the Constitution of the Atom. 295

which proved to be of very great importance. He intro-
duces generalized coordinates g; (coordinates of position)
and ; (coordinates of momentum), and takes as the genera]
quant-condition for stable orbits :

( psdq=nh, rele] oh eco ena EC eb.)

where the integral is to be extended over one period.

This general quant-condition involves the new question as
to the choice of coordinates.

Sommerfeld carries out the calculation in polar coordinates
(mS) for a single electron moving round a positive charge.
The three quant-conditions then take the form

(* pydypanh, ( p.d,=n'h, (rao=na. . (4)
0 ry

If we merely consider orbits in a plane, only the first two
conditions will be wanted.
Carrying out the calculation in the case of hydrogen

m,, m' being the value of 7, n’ for the orbit from which the
electrons recombine.
In order to give the ordinary Balmer series it is
necessary that
N=n,+n'=2

and that M=m,+m’ is an integer number > 2. Further,
the number of combinations is limited by the conditions

m > 21,

m' Sn’.

This new form of the frequency formula is so far identical
with that of Bohr that it gives the same position of the
lines, but it is different with regard to the way in which we
may imagine the lines to be produced. Thus the H-line
may be produced in four different ways, as, e.g., by re-
combination from an elliptic orbit (mj=2, m’=1). The
case of circular orbits treated by Bohr is only a special
case of many, and corresponds to a recombination from
(m,=3, m'=0) to (n,;=2, n'=0).

The H-lines which are produced in these different ways
are only with certainty identical provided we may treat the
system as consisting of two a masses attracting each

2

296 Dr. L. Vegard on the X-Ray Spectra

other by a force inversely proportional to the square of the
distance. When other forces come into play, or if the
masses are not to be regarded as constant, the H-lines
produced in the different possible ways will no longer be
identical.

In fact, when the mass of the electron is supposed to vary
with the velocity according to the law given by Lorentz or
the principle of relativity, Sommerfeld finds that the lines
are split up, and carrying out the calculation he has been
able to explain even quantitatively the splitting up of the
hydrogen lines and to give a general theory of the formation
of multiple lines.

Further, we see that the frequency formula of Sommerfeld
gives us a possibility of explaining the Stark-effect, for when
a uniform electrostatic field is introduced into the system the
various ways in which a certain line may be produced will
no longer give the same frequency.

A complete determination of the Stark-effect was finally
given by Epstein*, by an ingenious method of selecting
the generalized coordinates by means of the equations of
Hamilton-Jacobi.

Application of Bohr’s Conceptions to the High Frequency
Spectra of the Elements.

§ 2. The law connecting the high-frequency spectra of
the elements which was brought out through the beautiful
experiments of Moseley tT was simply explained by the
atomic model of Rutherford{t and showed that the atomic
number played a fundamental part in the constitution of
atoms. In fact, we may say that the evidence gathered
from various sources leaves no doubt as to the correctness
of Rutherford’s conception of the atom. I think we may
safely take it as a fact that the normal atom has a positive
nucleus of charge Ne, surrounded by N electrons.

The atomic problem is then resolved into the following
two questions :—

1. The arrangement of the electrons which surround the
nucleus and the laws that govern this arrangement.
2. The constitution of the nucleus.

The nucleus is the seat of gravitation and the radioactive
transformations, and a possible theory of the nucleus would
have to gather evidence from these two phenomena. The

* P. Epstein, 1. ¢.

7 H. G. J. Moseley, Phil. Mag. xxvi. p. 1024 (1918), and xxvii.

p. 703 (1914).
t Loe. ct.

and the Constitution of the Atom. 297

outer electronic system is responsible for the light emission,
for the homogeneous X-rays, and for the chemical properties
of the atom.

Bohr * was able to show that the K-spectrum could be
approximately explained by assuming it to be produced
by the removal and recombination of an electron next to
the nucleus.

Moseley + found a better agreement by assuming four
electrons in the system next to the nucleus; but the way
in which he deduced his formula was open to criticism.
His formula may therefore be regarded as empirical,
although Nicholson { points out that it can be deduced by
a proper modification of Bohr’s frequency law to systems
of electrons.

Kossel § was the first to point out some very interesting
relations between the lines of the K- and L-series. Denoting
the frequency by v, the following relation very nearly
holds true :

Vyeg—Vx, =yp,. aie a0)

According to Bohr’s frequency law aun) ig frequency
is proportional to the differences of energy of the electron in
the initial and final state, and, as pointed out by Bohr ||, the
above relations would naturally convey the following con-
ception with regard to the formation of the high-frequency
spectra.

The electrons may be supposed to be arranged in rings
round the nucleus. When an electron is removed from the
ring nearest the nucleus, an electron from the next ring may
replace it and give rise to the emission of K,. If the electron
is taken from the third ring, we get Kg. When an electron
of the second ring is removed and replaced by one from the
third, we might get L,, and in this way Kossel’s frequency
relations should be explained.

Sommerfeld, following up this line of thought, has
been able to express a number of lines of the X-ray
spectra by introducing a number of “terms” peculiar to the
various X-ray series. Thus, e.g., he introduces a K-term

oth )\2
K= —— and a L-term L= ee and finds
VR=K— 198

* N. Bohr, Phil. Mag. (6) xxvi. p. 408 (1913).

+ Loc. cit.

t J. W. Nicholson, Phil. Mag. (6) xxviii. p. 562 (1914),
§ bh Kossel, Verh. d. D. Phys. Ges. 1914.

|| N. Bohr, Phil. Mag. xxx. p. 394 (1915).

q A. Sommerfeld, Ann. d. Phys. li. p. 125 (1916).

298 Dr. L. Vegard on the X-Ray Spectra

Further, he has been able to show that a number of lines
in the X-ray spectra appear as doublets. Thus he can show
that K, and K,’ form a doublet, and the difference of
frequency can be calculated from his theoretical formula.

The L-series consists of at least 13 lines. It is of
importance to adopt definite names for these lines.

Arranging them in the order of increasing frequency,
Siegbahn * and Sommerfeld f have the following somewhat
different denotation :—

Siegbahn RVelehainddele 0/0/01» | oy &19 8481 B2B3R57, 2934
Sommerfeld ......... edanvBy OC d3xwa

Sommerfeld finds the following doublets :
(al) or perhaps (28), (98), (en), and (2).

All four doublets give nearly the same value for the dif-
ference of frequency of its two components, thus :

B—a=6—y=n-—e=S—6.

Recently Debye t has attacked the problem in a some-
what different way. His method may be considered as an
application of Bohr’s frequency law to systems consisting
ot more than one electron.

Assuming that all atoms which give the K-radiation have a
ring next to the nucleus of p electrons, he is able to deduce
a theoretical formula for the K-line by putting p=3.

His deduction is based on the following assumptions :—

(a) The angular momentum of each electron in the normal
ring is that supposed by Bohr for the nermal state of the
atom (equation 1).

(6) When an electron is removed from the ring the
angular momentum of the remaining electrons is supposed
to be preserved.

(c) A line belonging to the K-series is supposed to be
formed when an electron recombines to the broken ring
from a secondary stability state (circle) corresponding to

U °
an angular momentum 7 oe where 7 is a whole number
greater than 1. a
(d) When an electron recombines in one step from a

* M. Siegbahn, “Bericht iiber die Rontgenspektren der chemischen
Elemente,” Jahrbuch d. Radioakt. u. Elektronik, xiii. p. 296 (1916).

ft Loe. cit. p. 138.

t P. Debye, Phys. Z.S, xviii. p. 276 (1917).

and the Constitution of the Atom. 299

secondary circle to re-establish a normal circuit, the dif-
ference of energy of the whole system (electron + circuit)
before and after reeombination is radiated in a single
quantum. 7 }

Let the effective charge of the nucleus be + Ne and the

ay,
angular momentum of each electron T 573 then, as shown

by Bohr and Debye, the total energy of a ring of p electrons
will be
ae ye
E=(—pRaG—eP)
where C is a constant, and as we have only to deal with
differences we put

N—Sp)?
W=phRO— SP) URAL TS ‘as

and we take simply — W asthe total energy. R is Rydberg’s
universal frequency. Debye puts R=2-°7337 .10% —, orin
wave-number per cm.

| R=109740 —.
Further,

SEY iB
t=1 S1In 21—
P

Let the total energy of the restored ring be — W,, that
of the broken ring — W., and that of the electron in the
secondary circuit — W.’', then Debye’s application of Bohr’s
frequency law gives

lv=W, —(W.+ W,’). - e ° ° ° (8)

Putting p=3 and t=2 he finds a frequency formula which
gives very good agreement with observations for values of N
smaller than 30. Above this value there is a considerable
deviation, which he shows to be due to the fact that for
higher atomic numbers the mass of the electron will increase
on account of increase of velocity.

Adopting a similar way of procedure to that followed by
Sommerfeld in his theory of doublets, Debye calculates the
energy on the supposition that the motion takes place subject
to the principle of relativity. The equations of motion under

300 Dr. L. Vegard on the X-Ray Spectra

these conditions can be derived from the ordinary equations
simply by putting

Th C) : (9)

In this case a fairly simple calculation gives for the total

energy
=e Ny p
w= 25 {1 1-2-8)" }
where
p= 2 =5:30.10- za

0
Forming from equation (10) the expressions for the
energies of the broken and unbroken system, the equation
(8) gives the frequency of the radiation.
Letting the electron recombine from a state corresponding
to t=3, Debye finds frequencies which are fairly near to
those of Kg.

Ambiguities with regard to the Determination of
the K-System.

§ 3. The question as to the system of electrons which
produces the K-series is a most important one, being the
first step towards a determination of the outer system of
electrons in the atom, and it should not be out of place
to see how far the most promising solution found by Debye
is the only one which may give a satisfactory agreement
with observations.

When we stick to the assumptions of Debye (a, 8, c, d)
and suppose the K-lines to be produced by the removal of
one single electron, his result that there are three electrons
in the inner ring is the only one possible.

We might, however, more generally assume that a group
of qg electrons were removed from the inner circuit of
p electrons, and we could assume that the lines were formed
when the system of g electrons reproduced the original
system under the emission of g energy quanta.

From equation (7) we get for the energies :—

For the normal ring,
W,=pAR(N—Sp)?.
For the broken system,
hR
W2=(p—@kR(N—S,_.)?-+95¢(N—p+9—Sq)

and the Constitution of the Atem. 301

Hence we get for the frequency:
v_p (N—p+q—Sq)?
OS Dee Go| Soo io
= 3/4N?+B,,N + Cpo, |

where B,, and C,, are functions of p and g. Independent of
p and g we get the undoubtedly right factor 3/4 of N’.

The combination y=3 and qg=1 represents the solution
found by Debye.

If q is different from 1, there is one and only one other
possible solution corresponding to

p=4 and g=4,

which give for the frequency :

R= 3/AN—0°9571)%, . RO LD

which is just the formula given by Moseley *.

When we take into account the variation of the mass
according to equation (10), we easily get from equation (11):

R=5 {a/1-2 w—sy—4/1-pN-8.) | (12 6)

or approximately
v/R=3/4(N —S,)?(1+ 5/16p(N—8,)”). . (12 €)
The accuracy of the formula (12c) will be seen from

Table I., where the values of = are calculated for a number
of elements.

TABLE I.

N v/R v/R Difference v/R Difference

te obs. Debye. jin Percent.| from 12 c. | in Percent.
i 2 1477 146°4 0:9 148°4 0-4
cs UY 0 ee 271°6 272°4 0°3 ~ 273°6 0°7
5h | aaa 435°4 437°3 0°4 437°7 0°5
2 eee 635°9 641°5 0:9 641°5 0-9
= TT eee 880°4 885°6 06 8859 06
Le a Ee 11560 1171-0 1:3 1172°0 14
IN ee 1482-0 1499-0 11 15020 1:3
UL. eS 1871:0 1877-0 03 _ 1876-0 03
if Le eee 2290°0 2297°0 03 2296°0 03
LM. Sas eee 2761°0 27770 0°6 2766°0 0-2

* Loe. cit.

302 Dr. L. Vegard on the X-Ray Spectra

We see that the formula (12c) satisfies the observations
with the same accuracy as does Debye’s formula. In fact
the two formulas give almost identical values. If we assume
the whole ring of four electrons to the normal state from a
three-quantical secondary, we should get frequencies nearly
equal to those of K. ‘The accuracy would be about the
same as that found by Debye.

If we adopt the frequency law in this more general sense
there are two possible solutions. The one would give four,
the other three electrons in the K-ring. Which of the two
is the right solution cannot be determined from a mere
numerical comparison with the observations, but we must
have regard to the physical consequences to which they lead,
and in this respect the solution of Debye has the advantage.

In the case of four electrons the whole ring of electrons
would have to operate intact. We should have to assume
that the system formed a kind of unity, in such a way that
they tried to keep the same angular momentum. I think
that there is reason to believe that as a component of the
atomic system the electrons are not to be considered as
independent unities ; but that they are linked together by
forces which are different from the ordinary central attrac-
tion and repulsion between the centres. The arrangement
of electrons in conformity with certain quant-conditions is
one aspect of these as yet unknown forces. But energy
considerations seem to show that this mutual attachment
cannot be so close as to prevent one single electron from
leaving the system.

The experiments of Barkla and Sadler * have shown that
in order to excite the K-radiation by means of Réntgen rays
the hardness of the incident rays must just surpass that of
the excited radiation. The frequency Ky of those rays which
are just sufficient to produce the K-radiation is accurately
determined by Wagner f and de Broglie t, and they also
find K, just a little greater than the frequency Kg.

This fact is simply explained by the quantum theory and
Bohr’s conception of the X-ray-and-light emission. An
electron will be expelled from the ring when the energy
quantum is equal to or greater than the energy required to
remove it from the atom. Debye’s formula involves the
assumption that one quantum is sufficient to excite radiation.
The assumption of four electrons would mean that four

* ©. G. Barkla and C. A. Sadler, Phil. Mag. xvi. p. 550 (1908).

+ E. Wagner, Ann. d. Phys. xlvi. p. 868 (1915) ; Phys. ZS. p. 482
(1917).

t M. de Broglie, C. R. elxiii. pp. 87, 354 (1916).

and the Constitution of the Atom. 303

energy quanta fyg, should be necessary to produce the

K-radiation. If so, an atom should have the power of
accumulating energy until it had taken up a sufficient
number of quanta to break the whole ring. In trying
to explain the mutual relation between y- and §-rays
Rutherford * has supposed that several quanta ean accu-
mulate to give energy to one @-particle ; but the assumption
of such an accumulation would, on the other hand, make it
very difficult indeed to understand why a certain frequency
at all should be necessary to produce the K-radiation.

Also the transformation of cathode-ray energy to that of
Roéntgen rays gives a similar result. The most important
investigations of Whiddingten + and the more recent mea-
surements of Webster { by means of the reflexion method
have proved, that in order to excite the K-radiation the
energy of one cathode ray must surpass the energy quantum
hyz,. If we suppose that a single cathode particle can

produce radiation, the assumption of four electrons is
therefore not justified ; but the assumption of Debye should
be in accordance with physical facts. The fact that we can
deduce two formule which both give a satisfactory agree-
ment with observations is still of some importance, as it
shows that we should be cautious in relying too much on a
mere numerical agreement.

The Explanation of the L-Radtation.

§ 4. Although the agreement between observed and
calculated values was less good in the case of Kg, there is
probably no doubt that also this line is caused by a recom-
bination to the inner circle, from a three-quantical secondary.
Kossel’s relation (6), if it indicates a physical connexion
between the K- and the L-lines, would indicate that L,
should be produced by an electron recombining towards
the K-circles, between the two secondaries corresponding
to7T=3 andt=2. This assumption would indeed give the

right factor (5-3) to N? in the formula for L,.

This assumption, however, does not agree with the
absorption phenomena. According to Barkla§ and Sadler||
an L-absorption takes place when the hardness of the

* E. Rutherford, Phil. Mag. (6) xxviii. p. 305 (1914).

+ R. Whiddington, Proc. Roy. Soc. 1911.

t¢ D. L. Webster, Proc. Amer. Acad. vol. ii. p. 90 (1916).
§ C. G. Barkla, Phil. Mag. xxiii. p. 987 (1912).

C, A. Sadler, Phil. Mag. xviii. p. 107 (1909).

304 Dr. L. Vegard on the X-Ray Spectra

incident rays surpasses that of the L-radiation of the sub-
stance; and Wagner * and de Broglie t have by means of
the reflexion method observed two absorption bands in the
L-region and accurately determined the sharp edges La,
and La,, which have just the sufficient frequency to give
absorption. The measurements of Wagner and de Broglie
show that the frequency La, is just a little greater than L,,
and La, just greater than Ly.

If we stick to the assumption that one quantum hy is
sufficient to excite radiation, an energy quantum Ay, or hye,
cannot produce L, if this line is emitted by an electron
recombining to a broken K-ring. Now absorption takes
place when the radiation falls upon atoms in the normal
state, and we cannot assume that atoms are found in the
normal state with a broken K-ring and the recombining
electron in some secondary circuit.

We might then naturally try to follow up the idea
suggested by Bohr to explain Kossel’s frequency relations,
and assume the electron removed from the K-ring to be
replaced by electrons belonging to some outer system. If,
however, we uphold Bohr’s first assumption that each
electron in the normal state has an angular momentum —
there seems to be no possibility of explaining the high-
frequency spectra by interchange of electrons from outer to
inner systems.

Suppose we have a ring of p electrons, and inside this,
systems which are made up of electrons surrounding the
nucleus. Outside the p-ring follows a ring of g electrons.
According to Bohr {, when the diameters of the rings are
not nearly equal, the forces due to the external systems can
be neglected, and the effect of inner systems is approximately
equivalent to a reduction of the charge of the nucleus equal
to the charge of the inner electrons.

Hence the effective nucleus charge of the p-ring :

N, = N-7;
of the q-ring :
N =N—-(p+r).
We now regard the system when one electron is removed
from the p-ring. The energy of the p- and q-ring is then :
W,= W,y-1+ W, = AR$ (p—I)Q,—8,4)? +9Q, Sa .

* Loe. cit.
+ Loe. cit.
¢ N. Bohr, Phil. Mag. xxvi. p. 476 (1918).

and the Constitution of the Atom. 305

The energy after an electron from the g-ring has regenerated
the p-ring will be :

W,= AR {p(No— Sy kg Y)Q.- S¢q-1)? t .
Applying Bohr’s emission law |
hv = W,-—W,

and inserting the values of W, and Wg, the frequency takes
the form : |

v

R= ay pat [a1 emer tee cor ay ts)
where A, , and By, are functions of p,g,andr. We see
that the factor of N? vanishes for all values of (p, q, 7) ;
whence we conclude that the above equation cannot express
any of the Rontgen-ray spectra, or the high-frequency
radiation cannot be explained by interchange of electrons
between rings of the normal atom, when each electron in
this state has an angular momentum oe

It can also be shown that from the assumptions of Bohr
and Debye we cannot get the right formula for L, by any
recombination to a system, where each electron has an
angular momentum of I On the other hand, the relation
between the absorption and emission frequencies leads us to
the assumption that the L-series is produced by a recom-
bination to some system which exists in the atom in its normal
state. If then we are to explain the L-radiation, we must
in some way alter at least one of the assumptions at first
made by Bohr.

In all cases where Bohr’s principles have led to a complete
or satisfactory determination of the spectrum, we have always
dealt with systems next to the nucleus. It might then be
natural to suppose that it is only for this inner ring that
Bohr’s assumption I. is fulfilled. The quant-condition to be
satisfied by circular systems in the normal state of the atom
might more generally be written

h
4
Lc Uk (14).
where n is an integer.

This would indeed seem to complicate matters, as we
introduce a new parameter m into the conditions which
secure the stability of the electronic systems. But still it
seems the right procedure to begin work with n as an

306 Dr. L. Vegard on the X-Ray Spectra

arbitrary whole number to be determined so as to fit in
with observations, because there seems @ priort no necessity
for putting n=1 for all systems in the normal state of the
atom.

Under these generalized conditions we shall calculate
the frequencies which are produced when an electron is
recombining to a normal system with g electrons which
have lost one electron.

Let the number of electrons between the q-ring and the
nucleus be p, then the effective nucleus charge is

N, = N-p;

and forming the energies from equation (7) in the same way
as before, we get:—

Bnergy of unbroken g-ring: W, = 4 (N/ —§,)?hR;

‘9 broken = ae (N,—S_-1)7AR ;
and energy of recombining electron in the secondary circle:
W,! = 5(N,—q+ DR.

If we still stick to Bohr’s frequency law, we get
hy W,—(W./+ W,”).

Hence

z —_ i a) N+ flr, gq, n, TN +f2(7, 7, 2, 7),

ne 7?
A(pgnt) = 3 (p+9-1)—75(P+ 81-1) — 73 SoS), 15)
Licey ae
A pqnr) = Gp +8, +84-1)(8,— 8,1) + (2)

Lf Pt ia

Cees
On the more general assumption (14) we might now also
possibly explain the formation of lines by assuming that an
electron removed from a ring is replaced by an electron

coming from one of the other normal ring-systems.

Let, as before, an electron be removed from the g-ring,
and let it be replaced by an electron from another ring with

and the Constitution of the Atom. 307

q ain Let the effective charge of the gq’-ring be
‘In aler to find expressions for the energy, we suppose
that the removal of an electron from one ring does not
change the energy of any of the other systems which keep
their electron number unaltered. This will mean that the
removal of an electron will change the radius of the external
rings if we suppose the angular momenta to be kept
unaltered.
Let the energy of the normal and broken
g-ring be W, and W,.-3.

ie =a .
59 > >3 » @-ring be W,and W,_3.

Then vh = Wo+ We-1— (Wo + Wo-1).

From equation (7) we get:

1 i |
= wee : q(N,—8,)?—-(¢-D (Na— Sy-1)"}

i ! /
— {55 9' Ne Sp) (4 - (Ny 8y-1)?t

n and n, being the quant numbers corresponding to the two
circles g and g’. The formula may also take the form :

v_(1_1)\ne_ 9 (filpg) _ bi(p'g’) 7m
ie) Cae) |

bo( Pq) $2(p'9')
where a n? Ri r . (16)
o1(pg) = 9(S~—Sq-1) +8y-1 +P, |
d2( pg) = 9(2pt Sot 8q-1)(Se—So_1) + (p +8y-1)?. J

Comparing the equations (15) and (16) we see that, if we
assume the normal atom to have rings with increasing number
of n as we pass from the centre, the assumption that the
recombining electron comes from one other ring with a
higher quant number leads to an equation of the right type.
Whether we shall assume that the X-ray spectra are pro-
duced by recombination from a secondary or a primary circle
is a question to be decided by the ability of the assumption
to give a satisfactory agreement with observational data.

To test whether the second hypothesis (eq. 16) may be
possible, let us try if the K-line can be explained by a
recombination from an L-ring.

308 Dr. L. Vegard on the X-Ray Spectra

In this case we put
n=1, m=2, p=9, P=4%
and get :

=3/4N?—2 (da (0g)— POD) N + gu(og) — 22090)

Pol ~

(17)

Ot
R= 3/4N?+ BN+C.
The factor 3/4 is an accordance with observations. The
essential further condition is that the factor B has the proper
value, and in order to fit in with observations it should be
nearly equal to —1°5.
Putting $(¢) = Q(S_—Sq-1) ia
we should have

46(q)—q—3 = $(q’).
This equation is to be approximately fulfilled for whole

numbers g and q’.
Values of ¢(q) for varying g are given in the following

Table :—

Tas eE II.
q. (9). 46(9)—¢—3.
1 0 —4:000
2 0500 —3-000
3 1-231 —2:307
4 2-097 1388
5 3:052 4208
6 4:082 7-328
7 5173 10-692
8 6305 | 14-220
9 7-512 | 18-048

First we have to select nearly equal values in the second
and third columns, and the corresponding values of q¢ give
the number of electrons in the K- and L-ring. As is seen,
the following values of g and g' might be possible:

9 | 4 | 5 | 6
¢|3s|e6|9

and the Constitution of the Atom. 309

If we now calculate the coefficients B and € of equation
(17), we find the values set up in Table III.

Taste III.
qi. q’. B. C.
= 3 —1°578 — 402
5 6 —1°563 —13°29
6

9 — 1-408 — 29°18

For the sake of comparison we can write down the corre-
sponding formule of Moseley and Debye, which give very
close agreement with observations for N < 40.

R= BAN?-15N+3 (Moseley),
R = 3/4N?—1-464N—0:125 (Debye).

The combination g=4 and g'=3 gives the best agreement
and leads to the formula :

A= 3/4N?—1:578N—4:02. . . . . (18)

For substances with fairly small atomic numbers the
Debye formula gives a much closer agreement. But if
we would correct for the variation of mass with velocity,
the corrected formula (18) would probably give better
agreement for the interval 35< N<55, where the Debye

formula gives too high values for = but as a whole

it must be said that the Debye formula gives by far the
better agreement. But still the difference is so small, that
we cannot decide from a mere numerical comparison which
is the right formula so long as the calculations are not
carried out with perfect exactness, and there may of course
also be effects of unknown origin, which taken into account
might put the assumption of recombination between primary
systems in a more favourable position. But so far as the
present investigation has been carried, it must he said that
the assumption of Debye that the K-spectrum is produced
by recombination from secondary circles has given the best

Phil. Mag. 8. 6. Vol. 35. No. 208. April 1918. Z

310 Dr. L. Vegard on the X-Ray Spectra

agreement with observations ; and we shall also see that the
assumption of an inner ring of three electrons very well
fits in with the chemical relations of the elements as they
are expressed in the periodic system. It may also in this
connexion be mentioned that Sommerfeld *, from his
expresssions of the frequencies by means of his “ terms,”
comes to the conclusion “‘ that none of the L-lines can be
derived from the K-lines by means of the principle of
combination.”

In our further investigation we shall then build on the
assumption that the electron producing the X-radiation
recombines from secondary systems in a way which is
independent of the systems exterior to the primary ring to
which recombination takes place.

On the Explanation of the Lea-Line.

§ 5. We now suppose the result of Debye as regards the
K-radiation to be true, and we ask whether it is possible

in equation (15) to give qg, », and 7 such values that the
observations are satisfied.

We put p=3, and from the formula of Moseley for

the L;-line,
a= =(5- R) ( N—7-4)2,

we see that we must put

2, Toe
If in (15) we further put
a 1.) 25 ore ‘

we find that g=7 gives :
Sip, % % T) = —2°0846,
Tol ps q, ”; T) ==(s oo 5

and we get for the frequency :

5
A= = 3,” —2‘0846N +9°3. °. . 3) (iis
The formula of Moseley put into the same form gives :
V ee 3° 2 ope .
R= ao —2°0556N+76. . . . (198)

* A, Sommerfeld, Ann. d. Phys. li. p. 155 (1916).

and the Constitution of the Atom. SEL

We see that the agreement between the two formule is
remarkably good. Curiously enough, we can alse in this
case find a good agreement when we assume the whole
L-ring to recombine from a secondary circle. If we now
suppose 4 electrons in the K-ring and 9 electrons in the
L-ring, we get

V 1S Rags
R= 36% 7°328)?,

in close agreement with Moseley’s formula.

If we take into account the variation of the mass of the
electron with its velocity, we can easily find the frequency
by means of equations (2) and (10).

Putting 3+8, = 53048 = a,
3+S, = 4°8274 = 3,

we obtain

Ra oa/ 1-2-8 P(N—b)? +a/ 1-8 (N-9}
-14/1-4(N—ay |; 200

and if expanded into series, we have approximately

v f 6 1
Bn Bt (gq N—2)*— Gg —B)'— 353 N—9)'), (200)

vo being the frequency calculated from the uncorrected
formula (19 a).
Observed and calculated values are given in Table IV.

TABLE LV.
; } )
| v/R |
y- | obs. (v/R) (v,/R)_. | (v/R),. dears i (ee
} ;
Ss a fie Aleph
aR Pike... | 738 71-0: “| aes 67-9 =F St = Se
Be Br l...... | 1086 105°8 | 102-5 102°8 Ser 53
eS) Ss: 53. | 1498 1476 | 1441 | 1448 =e ss
a | 198-3 196-4 1927 | 1939 aes
m Sn......... | 2535 2520 | 2483 | 2505 —06 | —1-2
ee O82... 32: | 315-2 3147 | 3108 | 3140 —02 | —04
60 Nd ...... | 394-6 3843 3802 | 3349 —01 0-0
aE D.....:. 4618 | 4608 | 4566 | 463-4 —02 | +03
7 5456 | 6443 5399 | 549°5 —02 | +07
ee i... | 7349 | 7321 | 7274 | 7455 —04 | +414
DTN... | 952-2 947°6 9427 | 9725 —05 | +21

Z 2

312 Dr. L. Vegard on the X-Ray Spectra

The column headed

(g) contains values from the formula of Moseley.
ARJ/ aw

(x) As » calculated from (20d).

V
(7 ae 29 99 oD) py) (19 a).

Py and Py give the errors in percent. for the formula of
Moseley and that of the writer.

The agreement between theory and experiment is seen to
be remarkably good. The variation of Py with the atomic
number may be due to influences of external systems which
we have not taken into account.

If our conception as to the production of the L,-line
is right, the relation of Kossel does not mean that L, is
produced by an electron recombining towards a broken
K-ring between secondaries corresponding to t=3 and
t=2. Nor does it involve that the lines are produced
in the way suggested by Bohr by recombination between
primaries, ‘The relation is more or less accidental, and due
to the fact that the first secondary K-circle has the same
quant number as the primary L-ring.

On the Origin of the other L-Lines.

§ 6. As already mentioned, Sommerfeld found” that a
number of lines in the L-series could be grouped into
doublets.

To L, corresponds Lg as a second component of the
doublet, and if we stick to the explanation given by
Sommerfeld, we should have to suppose that the normal
L-system has two stationary states—one circular and one
elliptic. Now the recombination to the elliptic state gives
the higher frequency, and we should suppose, as we have
already done, that L, corresponds to the circular and Lg to
the elliptic state. We are then led to the conception of one
elliptic state common to a whole system of electrons, which
would mean that the electrons forming one ring are in some
way mutually connected.

Now Wagner has shown that the absorption edges A; and
A, have the same difference of frequency as the doublet, or

Pas —Ya, = Vtg "4,"

This leads to the assumption that L,, and Ly, are due to

the transference of an electron from the circular and the
* A. Sommerfeld, Ann. d. Phys. li. p. 125 (1916).

and the Constitution of the Atom. 313

elliptic state respectively to the same final state, and that
both states are to be found in the normal atom. | .

Absorptien will set in when the quantum of energy hy is
great enough to overcome a certain amount of energy.
Wagner has adopted the idea that absorption sets in when
hy is just sufficiently great to bring the electron from its
place in the atom outside the sphere of influence of the
atom ; but in the case of the L-absorption the quanta hy,
and hy, are probably too small to bring the electron outside
the atom. Thus in the case of gold, A, =1°042, r,,=0°914;
but some of the lines belonging to the L-series have an even
smaller wave-length, or an electron recombining to the
L-circle should be able to radiate more energy than hy,,.

Neither does absorption set in when hv is just sufficient to
bring an electron to some secondary circle; for if so, the
absorption edges should give the same wave-length as some
of the emission lines.

Valuable information as to the relation between absorption
and emission lines would be got by studying the relative
intensity of the spectrum for velocities of the cathode rays
varying from just below to just above the critical value.

With regard to the origin of the other lines of the
L-spectrum, we might naturally assume the doublet (y, 6)
to be produced by recombination from secondaries corre-
sponding to r=4 to the two primary states.

Tf in equation (15) we put

n= 2, fe es = A;
we get 3 yee 69-9. oe (21)

ae ——
16

Valnes of v/R calculated from this formula and corrected
for variation of mass are given in Table V., which also
contains the corresponding values for some of the lines
of the L-series.

TABLE V.
pf R.

| | |
) iN: et Calor: Y 0. x
RODS ss Rf a
One al ORY: | 448 486 | 513
| 65 6| (GIG | 549 505, fh ego
y 370 «| On | ee 720 745
Da 1021 | 906 1054 1085
| 90 1297 | «(1148 1392 1435

«314 Dr. L. Vegard on the X-Ray Spectra

It appears that the calculated frequencies come out consi-
derably greater than those of L ; at any rate, for lower atomic
numbers they very nearly fall into the y region of Siegbahn,
being quite close to Sommerfeld’s lines 6 and y. As to how
this discrepancy is to be explained, I should not venture
to express any definite idea. In any case it shows that
equation (15) cannot be applied for values of 7 higher than 3
without the introduction of certain correction terms,—or the
secondary stability circles of higher quant numbers cannot
be determined as if the electrons outside the normal L-ring
were removed.

When proceeding to interpret the L-spectrum we should
also be aware of the possibility that an L-radiation can be
produced when an electron is recombining towards the
K-ring—to a secondary corresponding to r=2. A re-
combination from a cirele corresponding to t=3 would
ah 5 13

id 2 Pp 9)2
f= (N-2) {1476 2) . oie

This formula, which takes into account the variation of
mass, would give frequencies in the @-region of Siegbahn.

Whether lines of the L-series are produced by a recombina-
tion towards a broken K-ring, can be tested experimentally
by exciting the L-radiation with cathode rays which have not
sufficient energy to excite the K-radiation.

The l-Series.

§7. The softest line given by the ring of 7 electrons
should be the L,-line; but now we have in the L-series
the J-line discovered by Siegbahn, with a wave-length con-
siderably greater than that of L,; and to this line the
lines 9, €, 3 are probably closely related. ‘These lines we
shall call the l-series.

The explanation suggests itself that the /-series is due to a
second ring surrounding the L-ring and with the same quant
number n=2. This /-ring would have a radius just a little
greater than that of the L-ring, and the calculation of the
frequencies due to the /-ring would be very complicated, as
we can no longer suppose the effective atomic number to be
N —p, where p is the number of electrons inside the ring.

If, however, we take N—p to represent the effective
atomic number, and assuming 8 electrons in the U-ring,
we get:

Yo 5

p= gg N?-437N +308... . . (23a)

and the Constitution of the Atom. 315.
Corrected for variation of mass with velocity, we get:

Vv Vo 1,

a... So oN o'\t een 174. (23

where
a’ = 10+S8s = 12°805,
6' = 10487 = 12°305.

The agreement between observed and calculated values
will be seen from Table VI.

TABLE VI.

v/R v/R
N. obs. cale.
70 482 411
74 545 475
78 608 | 544
82 676 618
92 855 826

The calculated frequencies are too small, especially for
low atomic numbers, but, as already mentioned, we cannot
claim any great accuracy.

This /-ring should probably be responsible for the doublets
(en) and (¢3).

Weshould also expect the /-ring to give absorption bands.
An idea which might naturally suggest itself, is that the
absorption bands A, and A, are due to the two rings re-
spectively. Such a coordination is not impossible, but
still the assumption that A, and A, are due to different
states of the same ring seems, for the reasons previously
mentioned, to be the more probable. As the /-lines are
very weak, also the absorption ought to be weak and thus
may have escaped detection.

The M-Series.

We have seen that the typical features of the L-radiation
can be explained by the assumption of two rings with
7 and 8 electrons both with a quant number n=2. We
might naturally try to explain the M-radiation by means of
an M-ring with quant number 3.

316 Dr. L. Vegard on the X-Ray Spectra

The line M,, say, can be approximately expressed by the
empirical formula :

Ms : 2
ies Tee 237N+40. . .

If in equation (15) we would try to give p and gq such
values as to agree with this empirical equation, we should
find the best possible agreement by putting

Roe and .¢== 9108 i

Thus in order to explain the M-series we should have to
assume a number of electrons inside the M-ring just equal
to that found for the K- and L-rings. Thus the assumption
of an outer l-ring of 8 electrons is also necessary to explain
the M-radiation. 9 electrons in the M-ring give

v i

—— Tce e 4 .

R- iN 245N4+28; . . . (240)
10 electrons give

v 7

R= gg 795 N +31. . 2) oa

Although the agreement between observed and calculated
values is not so good as in the case of L,, the empirical and
theoretical equations are of essentially the same type, and
even the numerical agreement is surprisingly good.

The radius of the rings is given by the expression
e*n? 7
“= .
SER(N—p=s,y foe,

a

where p and g have the same meaning as in equation (15).
To correct for variation of mass we should have to multiply

with af 1—£ (N—p—8,)*.
The expression for the velocity is

wie jue N=p—Sq _ 2. if Pee . (26)
m n

7d

and is not changed when the variation of the mass is taken
into account.

The values of a for the K-, L-, J-, and M-rings for a
couple of elements are given in Table VII. and in fig. 1.

and the Constitution of the Atom. 317

TABLE VII.
|
@. | Zr (N=40). He (N=80).
ae Se | 134x107 1° em. | 0:63x107 em.
on a ena 1 (GS ae O89 )) oe
Cates scteeten a0 ea ack a 3:18 i"
mR CH NA: 4 25-4 | SHOU

I am far from claiming that the theoretical interpretation
of the L- and M-series given in this paper can be considered
as proved. As I have already stated, we must be cautious
in building too much on a numerical agreement; but if
at all we shall be able to proceed further in the direction
pointed out by Bohr, I think we can hardly avoid the
assumption that systems of electrons exist in the normal atom
with quant numbers greater than 1.

So long as we do not know ali the forces which are
engaged in forming the stability of the atoms, we have more
or less to grope in the darkness and feel our way forward.
The test of the correctness of the previous theory will be
whether it is in accordance with observation. At any rate,
I hope the attempt made to elucidate the laws governing the
X-ray spectra may prove to be of importance as a guidance.
in experimental research.

318 Dr. L. Vegard on the X-Ray Spectra

Lhe Constitution of the Elements based on the
X-ray Spectra.

The theory of the X-ray spectra involves the determination
of the electronic systems next to the nucleus, and may give
us valuable information as to the way in which the electrons
round the nucleus are grouped together.

The previous investigation has shown that the K-series
should be attached to a ring with quant number 1 next
to the nucleus and containing 3 electrons. The L-series
should be due to two rings with quant numbers 2 and con-
sisting of 7 and 8 electrons. The M-series should probably
have rings with a quant number 3 and 9-10 electrons.

If this theory is right, it would mean that if a ring is
formed for lower atomic numbers the same ring is kept
throughout the whole system of elements. Indeed I think
that this is to be considered as a necessary consequence to
be derived from the simple laws governing the X-ray
spectra, and is independent of any special theory which we
propose to explain the frequencies and the type of the
spectra. For a change in the number of electrons in the
K-ring, say, would necessarily involve a discontinuity in
the formula expressing the relation between frequency and
atomic number.

Now it might be legitimate as an hypothesis to take this
rule as a fundamental property of the atomic structure, and
quite general to assume that a system of electrons once
formed is kept also for elements of higher atomic numbers.
There is no reason why this rule should cease to hold because
we pass to lower frequencies.

Now if we would further build on the result of our present
theory as to the number of electrons in the K- and L-rings,
we should get a quite definite system for the first eighteen
elements, and from this start we should be able to see kow
the electrons are arranged in a series of elements forming
one period in the periodic system. If we have proceeded so
far, we can get further by the assumption that elements of
the same family, such as Li, Na, and K say, must have the
same number of electrons in the outer ring. I think this is
an assumption which is very well founded, because the
chemical properties must be mainly determined by the
outer electrons, and the assumption is independent of any
other special hypothesis with regard to the grouping of
the electrons.

Proceeding in this way, we assume in accordance with
Bohr that the strongest electropositive elements have

ee

and the Constitution of the Atom. 319

1 electron in the outer ring. Now the elements from Li
to Fl are assumed to maintain an internal system of
2 electrons and add one in the external ring for each step
in atomic number. This will make an external ring of
7 electrons. By Ne one electron is added which, however,
goes to the central ring, and hence forward we get the K-ring.
If so, the K-radiation should begin with Ne or Na; and
in fact this result is in agreement with experiments, for
Na is the first element for which the K-radiation has been
observed. Now the ring of 7 electrons is kept to form the
inner L-ring, and a new ring comes into existence for Na.
From Ar we have both L-rings with 7 and 8 electrons
formed, and the L-radiation might perhaps be expected to
begin with potassium ; perhaps some of the lines might be
traced to Na.

Now we come to the long period from Arto Kr. At first
a ring of 10 electrons is formed, completed by the elements
Fe, Co, and Ni with 8, 9, and 10 electrons in the external
ring respectively ; this should be the first M-ring with quant
number 3. At Cua new ring comes into existence, and we
get a monovalent electropositive element. During the next
long period from Kr to Xe the same process is repeated.

The next and longest of all periods which go from Xe
to Ra Em is peculiar because it contains the rare earths.
Now I think that the view here adopted with regard to the
constitution of the electronic systems may afford a very
simple and natural explanation of this peculiar group of
elements.

When we pass from Xe, a new external ring is formed,
with 1 electron for Cs, 2 for Ba, and so on until for Ce we
get a ring of 4 electrons. Passing now to the next elements
we assume the external ring to be kept, and that the new
electrons are forming a new internal ring. From our point
of view such an assumption is a quite legitimate one.
It would only mean that the new electronic system had a
smaller quant number than the external ring: for a smaller
quant number will, according to equation (25), give a
smaller radius of the ring. Thus the new electrons which
are taken up in the series of rare earths when we pass to
higher atomic numbers are, so to speak, soaked into the
atom, and the surface systems mainly determining the
chemical properties are kept unaltered. How these new
internal electrons are arranged we do not know. In the
graphical representation (fig. 2) I have assumed them to
form one system inside the surface electrons.

When at last the atom has become saturated as it were,

320 Dr. J. Vegard on the X-Ray Spectra

we pass fron: the rare earths ; new electrons are added as
before to the surface system, and we get systems of the same
type as those of the two long series.

The whole system here shortly sketched is graphically re-
presented in fig. 2. Along the horizontal axis the elements
are arranged in the order of increasing atomic numbers.
The principle adopted, that an electronic system once formed
is kept throughout the whole series of elements, makes it
natural to represent an electron by a horizontal line. These
lines are arranged into groups, and each group represents an
electronic ring system. The arrangement of electrons for
a certain element is got by drawing a vertical line from the
place of the element on the horizontal axis. The points
of intersection with the horizontal lines give the number of
electrons and their arrangement into ring systems.

On the Electron Affinity of the Elements.

When we pass from elements that follow an inert gas, we
begin with the strong electropositive elements, and as we
pass on they. become more electronegative. The transition
from electronegative to electropositive elements may either
take place by the passage through an inert gas or by passing
the groups Fe Co Ni, Ru Rh Pd, and OsIr Pt. In our
system the strong electropositive elements set in with the
formation of a new surface ring.

It might now be asked which quantity might rightly be
selected to express the chemical electro-affinity.

The idea would naturally suggest itself that the electron
affinity is measured by the energy necessary to remove an
electron from the external ring. ‘This, however, is identical
with the energy necessary to ionize the atom and is propor-
tional to the ionizing potential, which is no measure of the
chemical electronegativity *.

Nor can we take the energy which binds an additional
electron ; for the experiments of J. J. Thomson + on positive
rays have shown that the power of an atom to bind electrons
does not follow the chemical electronegativity.

I think the explanation of these facts may be found in the
following considerations. The electrons forming part of a
normal atomic ring system are not to be considered as free
electrons, but as linked together in some way, the nature of

* See J. Stark, ‘ Ionisierung der chemischen Jtlemente dureh Elektro-
nenstoss,” Jahrb. d. Rad. u. Elektronik, xiii.

+ Sir J. J. Thomson, ‘Rays of Positive Electricity, p. 40 (1913),
p. 395 (1916).

and the Constitution of the Atom. 321
Fig. 2.

Os IrPtAuHgTIPbBi Po - - ~ Ra -Th Ure

hare Earths -

i

ma Bll
i ie a
‘tt i |

—oe (ost
aE =e)

2S
=e A
=|

Faia ipsa

9:30 3) 32 33 34 35 36 37.38 39 40 4I 42 43 4445 46 47 46 49 80 SI 5253 5455 56 1 58 a 60 6) 62 63 64 65 66 67 6B 69 10 11 72 73 74 75 76 77 7879 BO OI B2 83 44 85 46 B7 BB BD 90 9) 9B, *

: : ii i j i = i Hk:
Break ae
|||
ae II! CN TE
ic

MoFe CoNiCUZnGaGe AsSeBrkr RbSr Y Zr NbMo ~ RuRhPd AgCS InSnSbte J XeCsBalaCePrNe - SafuGd TbDyHoEr - AdCp ~ TaW

(LE 12 13.0% 15 16°17 18 19.20 21 22 23 24 25 26 27 26

NaMgAl Si P_SCI Ar K CaSeTi VaCr

322 Dr. L.. Vegard on the X-Ray Spectra

which we do not know. Now the forces which are engaged
in the chemical binding of elements do not act on a single
electron as in the case of an ionizing agency, but much more
on the ring asa unity. From this consideration it might
be more natural to take the energy necessary to remove an
electron when all the other electrons of the ring were
removed simultaneously. According to Bohr this energy
is equal to the kinetic energy of the electron, and thus
elements with the more slowly moving electrons are the
more electropositive. Equation (7) gives for this energy

w=hR OO Se
n i

where approximately: N'=p and p is the number of electrons
in the surface ring ; AR is the value of the energy wy for
hydrogen.

eee "pay

WH nN

. 2

Let us first consider the variation of o for elements which
have the same number of ring systems and only differ with
regard to the number of electrons in the external ring.
Suppose, e¢. g., that we consider the elements from Na
to Cl. For such a series the quant number n is constant
and o consequently proportional to (p—S,)*, and we can easily
see that o increases with increasing values of p by forming

Op41— Sp= {2p +1—(S,11—-Sp) 11 — (Spyi— Sp) }-

As both factors on the right side are positive, op,1 > op.

If o could be taken to represent the electronegativity, the
elements in each such group would pass from electropositive
to more electronegative as we proceed towards increasing
atomic numbers.

Let us next consider elements which are chemically
related ; such elements have the same value of p. As we
pass from lew to high atomic numbers, the quant number n
will increase and the value of o will diminish. Thus elements
of the same chemical family should be more electropositive
as we pass towards increasing atomic numbers, which is
indeed a well-known property of the elements.

The Electric Conductivity.

There can probably be no doubt that the electric conductivity
in some way or other is related to the energy which binds
the electrons of the surface system. Introducing a quantity

is

and the Constitution of the Atom. 323

which he calls the atomic conductive capacity, Benedicks *
has given a curve which most beautifully brings out the
periodic character of the electric conductivity. This curve
is shown in fig. 3. :

wa
:
i i BE?) . #
, ort! ee ;
Ca Cu Rs
i \ f\ . aq Dx
Os RL Mo f “45 ’ i be (
. i S , Hackl V 4 x Va
0 D (|
es. Loe J el Re dae Ee?)
re) 100 420 324 180 180 200 2280 ‘

Co 46 60 8

Now it is to be expected that a great atomic conductive
capacity corresponds to a small value of c. For, taking the
view of Benedicks, the atomic conductivity should be propor-
tional to the frequency v of the atom and equal to cv, where
c is the conductive capacity. But c¢ must be proportional] to
the probability that an oscillation shall result in a free
electron. Now this probability, and thus c, ought to increase
when o diminishes, and thus we might expect the conduc-
tivity to show a similar variation to 1/c.

Fig. 4 gives the variation of o and 1/o when for a given
value of n the number of electrons in the surface ring
increase from p=1. From this curve we should expect
the conductivity suddenly to take a high value each time
a new surface ring is commenced. This is exactly the
type of variation which is brought out in the curve of
Benedicks.

* C. Benedicks, Jahrb. d. Rad. u. Elektronik, xiii. p. 362 (1916).

324 Dr. L. Vegard on the X-Ray Spectra

For the same value of p the conductivity ought to
increase with the atomic number on account of the increase
of n. Also this variation is clearly exhibited by the eurve
of Benedicks. At any rate the rule seems to hold without

Fig. 4.

t z $ + 5 6 7 8 9 fo

Number of electrons in surface ring

exception for elements with atomic numbers smaller than
those of the rare earths. Above these elements there is
again a drop in the conductivity, as if there had been a drop
in the quant numbers.

The Electron Systems and Radioactivity.

The constitution of atoms here proposed gives us also
the simplest possible conception with regard to the changes
of the electronic system which accompany a transformation
process. The elements formed only have to add one or drop
two electrons from the surface ring, according as the product
is formed by a §-ray or an e@-ray transformation, and the
element comes into its right place in the periodic system.
No further arrangement of the external system should be
necessary. |

According to the view put forward in this paper the quant
number may vary as an integer from the value n=1 nearest
to the nucleus. Now the idea suggests itself that passing
inwards from the K-ring the electrons which partake in the
constitution of the nucleus might have quant numbers smaller
than unity.

and the Constitution of the Atom. 325

As I have already shown in previous papers *, an electron
moving with an angular momentum which is a fraction of
h/2m would possess a very great kinetic energy and move
in an orbit with very small diameter. Now according to
Rutherford f and his collaborators the radioactive elements
give y-radiations which are very much more penetrating than
those of the K-series, and we might ask whether these
y-radiations could be due to electronic systems moving
inside the K-ring and with an angular momentum

nh

m 2a’
where n and m are whole numbers and
m>n.

This assumption, however, would meet with the difficulty
that for elements of high atomic numbers the velocity would
soon exceed the velocity of light. The velocity of an electron

moving inside the K-ring with an angular momentum 5 a
vis

would be
v=2.10°(N+1)m.

For uranium N=92, and if v is to be smaller than the
velocity of light, we must have

m <1°6.

It might, however, be argued that electrons bound up in
the atom may acquire velocities greater than that of light,
because they are not setting up any radiating electromagnetic
field when moving in astationary state. Of course, when an
electron is set free in the form of a @-ray, the ordinary
electromagnetic field would be active; the electron would
meet a sudden retarding impulse which at once would reduce
its velocity to less than that of light. Or we might say, at
the very moment of release the electron moves according
to the principle of relativity, which makes the light velocity
an upper limit.

The line A=0-072.10-* cm. observed by Rutherford and

* L. Vegard, Phil. Mag. xxix. p. 651 (1915) ; Ann. d. Phys. liii. p. 27
(1917).

+ E. Rutherford and E. N. da C. Andrade, Phil. Mag. xxviii. p. 263
(1914).

Phil. Mag. S. 6. Vol. 35. No. 208. April 1918. 2A

326 The X-Ray Spectra and Constitution of the Atom.

Andrade *, which forms a mixture of RaB and RaC, might
be accounted for by assuming an electron to recombine
4 3A

t les with angul Ek omy
between circles with angular momenta 6 On and 6 On

Let more generally an electron pass from a circle with
angular momentum ph to one with a momenta «ame

m 2c m 27

If we take into account the variation of mass with velocity,
we get

RaW +1ym(5—- 5) {12607 (28)

Putting N=82, m=6, n,=3, n»=4, we get

a |: 4
R 1°23) ae
while the observed wave-length gives
Paps 7s 4
hm 1:26. 10%,

The numerical agreement is good enough, but I think we
must be very careful in drawing conclusions from a single
coincidence. I merely put it down as a suggestion which
might be worth consideration.

Recent experiments of Barkla and Miss White have given
indications of a homogeneous J-radiation more penetrating
than the K-radiation, which Barkla calls the Y-series. From
the absorption coefficient of the rays just hard enough to
excite the J-radiation they find for Al a wave-length

A=0°37 .107° cm.

If this series really exists it can hardly be explained by

electrons belonging to the external system, but should be

produced by the electrons forming part of the nucleus.

The equation (28) would give nearly the right frequency

when. we put
MA ny =1, and j2,—2.

Physical Institute, Christiania.
December 14, 1917.

* E. Rutherford and EK. N..daC. Andrade, Phil. Mag. xxviii. p. 263
(1914).

per]

XXXVI. Relativity and Electrodynamics. By G. W.
Waker, VA. FAS, AR.C.Sc., formerly Fellow of
Trinity College, Cambridge™. |

[Plate X. ]
IR OLIVER LODGH’S recent papers in the Philoso-

phical Magazine have brought into prominence once
more the difference of attitude of the protagonists in “ Rela-
tivity Doctrine” and “Newtonian Dynamics.” That Sir
Oliver’s equation of motion for a moving planet requires
some amplification in order to take full account of the
special features of electrical inertia, will be recognized,
and Prof. Eddington has suggested a method of dealing
with the problem. Unfortunately, Hddington’s method
introduces an assumption which is frequently made by
relativists in dealing with electrical inertia, and which in
my opinion is inconsistent with the fundamental equations
of electrodynamics. In former papers I have drawn atten-
tion to this assumption, which is closely linked with the
“‘ quasi-stationary principle,’ and I had not intended to raise
the point again. But Sir Oliver has suggested to me that
an exposition of my views as to the parting of the ways
between the logical development of electrodynamics and the
doctrine of relativity would be of value, and I have agreed
to his request. My remarks must, however, be confined to
electric inertia, and I do not propose to enter on the gravi-
tational and astronomical developments of Hinstein’s hypo-
thesis.

‘The main point at issue may, I think, be put very concisely.
Relativists assume that “the kinetic energy of a moving
electrical system is a function of the resultant speed only
and is independent of the direction of motion.”’

My thesis is that this assumption is not consistent with
the fundamental electromagnetic equations for the ether
(supposed immobile), and that ‘the energy, or preferably
the modified Lagrangean function, depends on the accele-
ration as well as on the speed of the system and involves
also the relative direction of these.”

While the above appears to me to be the main point, there
is no doubt that subsidiary considerations arise. Theory and
experiment have interacted in a curious way, and I think
the discussion should proceed by taking notice of the his-
torical development. Sir Joseph Thomson was the first to

* Communicated by Sir Oliver Lodge.
2A2

328 Mr. G. W. Walker on

prove theoretically that a moving electrified system would
possess inertia, which Heaviside showed would depend on the
speed with which the system moves. A later calculation by
Thomson referred to a particular form of nucleus and to the
momentum which it would carry with it in virtue of a
uniform translation. It is extremely important to realize that
the character of the nucleus determines the manner in which
the speed enters in the expression for the momentum or for the
energy. It is also vital to realize that while the momentum
or the energy can be calculated for a particular form of
nucleus moving with a uniform speed, it has not so far been
found possible to give a complete solution when the speed is
variable.

M. Abraham extended Thomson’s calculations, and he
assumed that while the nucleus was still a sphere it was a
perfect conductor, and he consequently obtained a value for
the momentum in a state of uniform translation which differed
from that found by Thomson when squares of the speed were
retained. He emphasized the distinction between the effec-
tive inertia for acceleration along and perpendicular to the
direction of motion.

But finding that he could not obtain the exact solution for
a variable speed, Abraham made use of what is called the
“‘ quasi-stationary principle,’ which amounts to saying that
if we can calculate the momentum, or if we preter it the
Lagrangean function, for a uniform motion we can infer the
equations of motion for a small departure from this state in
the ordinary way. My contention is that we can no more
do this logically for electromagnetic systems than we can
for ordinary dynamical systems. We know quite well that
we do not get the correct equations for small departures
from a steady state, when the steady motion values are
inserted in the Lagrangean function before the differential
equations of motion are formed. The steady motion values
may be inserted after the equations have been formed from
the general Lagrangean function.

Abraham calculated expressions for longitudinal and trans-
versal electric inertia by means of the quasi-stationary
principle. Experiments on transverse inertia became possible
with the discovery of the Becquerel rays, and of the minute
negatively charged particles projected from radium with
speeds only little short of that of light.

The matter was taken up first by W. Kaufmann, and I
have a special personal interest in this since I was working
side by side with him in the laboratory at Gottingen while
his experiments were in progress.

Relatinty and Electrodynamics. 329

Kaufmann deflected the particles by crossed electric and
magnetic fields, by which the particles are sifted out accord-
ing to their speed, so that the ends of their trajectories form
a curve on a photographic plate.

Kaufmann considered that his measurements proved that
Abraham’s expression for transverse inertia was correct, and
that the inertia of the particles was purely electromagnetic
in origin.

We must now retrace our steps to consider the important
contributions to the theory of moving systems made by
Prof. H. A. Lorentz and Sir Joseph Larmor. They proved
that a mathematical correlation held between an electrical
system at rest and a certain system maintained in uniform
translation. “If the moving system has a uniform speed
ke (where ¢ is the velocity of light) in the direction x, and
the «x linear extent of the moving system is (1—k?)? of the
linear extent of the fixed system, and the variables time
and distance w in the fixed system are transformed to ¢’ and
x' in the moving system by a certain linear transformation in-
volving &, then the state of the fixed system in terms of ¢and &
is the same as that of the moving system in terms of t' and 2’.”

It is reasonable to inquire if the contraction in the pro-
portion (1—£*) actually takes place when a system at rest
is put into uniform translation, for if so it provides an
explanation of the Michelson-Morley experiment.

Now it is quite certain that the mathematical transforma-
tion is not true when & is variable, and therefore not true
at any intermediate stage by which the system at rest might
conceivably pass‘to the correlated system in uniform motion,
if it ever does so at all. But relativists have assumed that
the correlation proved by Lorentz and Larmor for a uniform
translation only is true, and that the change actually takes
place, when the speed is variable. It appears to me that if
the primary equations are correct, the assumption is not
merely not permissible, but is not true; and, on the con-
trary, if the assumption does represent actual truth, then the
primary equations are wrong and must go. We await
proof, which so far has not been offered.

The longitudinal and transverse inertia of a “ contracted ”
electron have been calculated by the quasi-stationary method,
so that there is a double source of error in the result.

Experiments by Kaufmann, Bestelmeyer, and others have
been offered as experimental proof that the formula for
transverse inertia of a contracted electron on relativity
doctrine is correct. My contention is that while the expe-
riments do not conflict with the relativity formula, the

—

330 ~ Mr. G. W. Walker on

formula is inconsistent with the electrodynamic equations,
and that several other formule correctly deduced from the
primary equations agree with the experiments equally well.

I doubt if many people in this country realize the very
meagre character of the experimental results, and I therefore
give a full-sized reproduction (Pl. X.) of the photographic
plate from which Kaufmann made his measurements. The
electric deflexion is across the paper and the magnetic
deflexion up the paper, and it may be pointed out that if the
inertia of the particles were quite independent of speed, the
small curved arcs would be parabolas, and that it is only in
so far as these arcs differ from parabolas that any depen-
dence of inertia on speed can be made out at all. Further,
the highest speed particles are those for which the deflexion
is least.

I now return to the theoretical treatment of electric
inertia. In order to avoid the error of the quasi-stationary
principle, I developed some time ago a method of obtaining
the longitudinal and transversal inertia directly from the
primary equations by Newtonian methods. The method is
rather tedious, but its correctness has not been called in
question. Its application is general, but to get definite
results the character of the nucleus must be specified.
Various systems may be examined provided they do not
violate any fundamental restriction imposed by electro-
dynamic conditions. In this way I examined the nucleus
assumed by Sir Joseph Thomson and was able to confirm his
result for transverse inertia, but obtained a different result
for longitudinal inertia. On the other hand, with the

nucleus assumed by Abraham I was able to confirm his_

result for longitudinal inertia, but not that for transverse
inertia.

Again, recently I examined the case of a contracted con-
ducting spheroid which agreed in form with Lorentz’s con-
tracted electron for the uniform speed, but did not alter its
form when acceleration was imposed *. The results for both
longitudinal and transverse inertia differ from those adopted
by relativists.

The differences that arise in these examples only become
important when squares and higher terms in the speed are
retained, and they arise from the fact that when acceleration
is imposed, additional electric forces are set up which have
to be allowed for in utilizing the boundary conditions at

* The restriction is unnecessary, as I now find that my results are

not altered when has surface deforms under acceleration as Lorentz
assumes.

Relativity and Electrodynamics. 331

the surface of the electron. Thus, for instance, with a
conductor there is a redistribution of the charge, which
depends on the acceleration, the speed, and the direction of
the acceleration relative to that of the speed. So the
Lagrangean function must involve these things, and I am
doubtful if it is the resultant speed that alone enters.

The results I have obtained for electric inertia by my
direct, if pedestrian, method, can be shown to prove that the
energy cannot be expressed as a function of resultant speed

only. For if
T=/@)

we find that in the direction of motion, say along 2, we get
a at
dt dé
and at right angles to this, say along y, we get

d dP
died!

— &(2f" + 4y?f""),

therefore longitudinal inertia =m,=2/'+ 4v7/",

and transversal inertia =m,=2/'.
Hence, since

kg Myv= = 2uf'=2f' +47",

dv
wa dmv
dv
must be satisfied. But it does not follow that T=/(v’) if

ne is satisfied.

m=

My results for the cases mentioned do not satisfy this
condition, and unless it can be shown directly from the
primary equations that arithmetical error has entered into my
calculations, it follows that the kinetic energy of a system in
variable motion is not expressible as a function of ithe
resultant speed only.

The conclusion is that Eddington’s proposed treatment of
the astronomical problem is invalid, and I see no help for it
but to start with the equations in the tangential form,

dv
ds"

M_v*/p = aN.

Mv

where m, and mz, are different functions of v?, which can be
calculated when the electrical system is fully specified.

332 Mr. G. W. Walker on

Starting thus with longitudinal and transverse electric
inertia given by m, and m, as functions of the resultant
speed, and employing what I hold to be the correct pro-
cedure in forming the equations of motion, viz., resolving

along and perpendicular to the resultant direction of motion,
we can proceed as follows :—

Let the origin be the sun moving in space with com-
ponents of velocity u, v, w, which are constant, and let
a, y, 2 be the co-ordinates of a planet relative to the sun
and referred to axes through the sun. The components of |
relative velocity of the planet are

Bs Ds
and of velocity in space ¢+u, y+, z+w, with resultant

say V.
The components of acceleration are

The acceleration along the resultant direction of velocity is
{(¢-+u)é+ (9 +») ++w)2}/V,

and if the components of force are X, Y, Z, the component
along the direction of V is

{X(@+u) + Y(y+v)+Z(2+w)}/V.
Hence the equation
mil (é+ulit (ytv)9jt+(z+w)z} |
=X(#+u) + YV¥(y¥+v)+Z(z+w)
=S .Say.<), +s ee ee

Resolve along any direction X, p, v, at right angles to the
direction of V,

then Mo{rAG+ py +vz}= XA+ Yu Dp,
and (@+u)rA+ (y+tv)w+(ze+w)v=0.
Hence m= X+ (#+u)k,

my = Y + (y+ v)k,

where & is some quantity to be determined. Multiply these

Relativity and Electrodynamies. 333

in order by = (@+), etc., and add and use (1). Then we
hy |
find

S= “184 *rv2,
Mo Mop

or

p=") gry2 where V?=(e+u)?-+(y-+0)*+ (4-0)?

aay
Hence the Cartesian equations of motion are

meé= X + ee — {X(@+u)+ YV¥(y+v)+Z(z+wv)},

m= V4 = Ma) I) Xe 4 u) + Veg te) + ZE+ud },

(m_,—m,) (z + w)

Myz=L+ lin SWE +Y(y+v) + Z(z +w)}.
1
Also for a central attraction
X=—pa/r, Y=—py/r", = — pe/r*.

From these it appears that a planet’s orbit cannot remain
a plane orbit during transference through the ether, except
in the special case when the direction of the sun’s motion is
in and remains in the plane of the orbit.

We can readily transform the equations to polars since
the disturbance is along the direction of the resultant
velocity in space.

For the example under recent discussion by Sir Oliver
Lodge and as a first approximation, we might take

mM, =m,(1+aV?/c?), Mo = m(1+bV?/c*),

and neglecting squares of V?/c?, treat the problem as one of
a disturbed central orbit.

Astronomers doubtless know the best mode of dealing
with this, and could obtain a solution without difficulty.

It is important to remember that these equations are first
approximations only to the general equations connecting
moving material systems with the state of the ether.

A special case amenable to elementary treatment is that

334 Mr. G. W. Walker on

of the orbit of a particle, having electric inertia, round a
fixed centre which is at rest in space.

0

Let v=resultant velocity at any point of the orbit.

p=radius of curvature.
y=radius vector.
p=perpendicular on tangent.

p/r?=attraction to the centre.

m, = longitudinal inertia=m,(1+ kv*/c’).

m,= transverse inertia =m(1+4,v*/c?) neglecting

squares of v?/c?.

The orbit is plane, and resolving along and perpendicular
to the path we get

dv pe dr :
7. ae aie (1)
and m3v"/p= = - (2)

Integrating (1) we get

mMo(v + $kyv'/c?) = zs = ”

where a is a constant.
ppdr dr
7 4/2) — Mp @ . ee
From (2) mMo(v? + kyv*/c?) = = de since Paes
Let w=1/r and w=1/a,
*, mo(v? + $h,v4/c?) = 2uu— pr,

aaa mo(v? + kyv*/c?) = — up ~

Relativity and a a 335

; 3 :
Ps approximately Mov" = = 2mu— Puy — a (2uu— Huy)? ’

and 4 (azanly 3 du
(2uu— pup) 4+ aca ae (2 — pup)? = — pp + dp’

or putting x= 2puu— pu,
and N= (kg —$hy)/myc?,
we get A dz
@+Ar*=—Z
=P To,
or Hodpus ey ae |
os =ae} 2 1+Arx J’

*, integrating we get

Ey thd
Fi a ee =2—)2” neglecting squares of A,
where 0 is a constant of integration.

Now 1 Aes
p “ae (79

a 2
b? u?+ (a) i = (2pu— pup) —A(Qpu— pro)’,
“() = (2p — pto) —A(2pu— pg)? —

so that the integral is of the form
a2 >

lu=l+e cos[ 4

‘0-2

where 7 is an arbitrary constant and / and e are determinate
constants in wu, a,b, and». This solution implies an elliptic
- orbit slowly revolving in its own plane. The eccentricity

does not change, but the apses will advance in the direction
of description of the orbit by

2 ve
Tey for each description of the orbit,
1+ t
1. €. since A is small,
by _ Amp’nr

Meee

Be ae 2/0) SNPS

. so that

336 Mr. G. W. Walker on

Expressed in terms of the semi-major axis Rp, eccentricity
e and periodic time T, the progress per revolution is

27rRo } *(Lky — ke)
Tae) a Aor.

The apses therefore progress or regress according as
(44,;—k,) is positive or negative.

We have no knowledge as to the proper forms of m, and
m, for matter in bulk, but the following are results for
hypothetical single nuclei.

For the contracted electron using relativity methods,

m,=m(1+3v?/c”), Mg= Mo 1+ 4v?/c?),
so that 44;—k,=+ (or progression).
From the primary electromagnetic equations my results*

are :—
For the contracted conducting electron :

ily Al
My =m(1+ 10°!) My=M(1+ BoP le),

so that 1k, —ky= — = (or regression).

For Thomson’s electron with special surface condition :
My =m(1 + a w/c), mg=m (1+ = v"/c?)
See 10 ’ eal 10 ’

ghy—hg= eI (or regression).

20

For spherical conductor which does not change in shape :
19
m= m(1 + $y? /c”) ) m= m1 + 60 v*/¢?),

cet thy—ke= = (or progression).

This last case is numerically almost the same as that for
the contracted electron by relativity methods. This is im-
portant, because it shows that so far as inertia enters in the
astronomical problem we can get practically the same result

* Proved only for disturbance from a steady state.

Relativity and Electrodynamics. 337

as a logical sequence from the fundamental equations as has
been obtained by relativity doctrine.

Our results for the apsidal progression per revolution are:
Contracted electron by relativity method,

27Ry ais
Td Sey c*
Spherical electron by orthodox method,

f 27Ry } 2 17 a
UTd=ays Be
Now Einstein obtained

27 Ry 2 67r
Til-e)F fe?

which numerically is in close agreement with observations
on “ Mercury.”

This result is obtained by assuming that the attraction
depends on the velocity. It is easily seen from our analysis
that if

= po (1+ kyv*/c”),

we get for the apsidal progress

3 —hy),

{ 27Ry 2 4a k
Tae} ;

1 =
(1—e?)? Ce ($4,+

In order to get the observed value for “ Mercury” k;
would have to be 5/2 if 3k,—k, is 1/4.

It is important to recognize that it is only by introducing
either explicitly or implicitly this comparatively large de-
pendence on speed, of the attraction between bodies that
Kinstein can get the numerical agreement. Such depen-
dence based on the known forces between electrical currents
has been recognized before now in the theory of electro-
dynamics, but is hardly acceptable in gravitational theory.
On these lines it appears that orthodox electrodynamics
is quite as capable of providing an explanation of this astro-
nomical feature as Hinstein’s theory. It is, however, im-
portant that endeavour should be made to determine, if
possible, the numerical value of $4,+3k;—k, for matter in
bulk.

There still remains the question of the effect of transference
in space as suggested by Sir Oliver Lodge. Hddington’s
conclusions on this problem may be modified considerably

338 Dr. H. S. Allen on Molecular

when what I hold to be more correct equations of motion
are used. iad

Notr.—At_ Sir Oliver Lodge’s request I have calculated
m, and m, for Bucherer’s electron which has the same form
as Lorentz’s electron but keeps its volume unchanged. My
results are

17 ol
my = m{ 1 + 3% v?/ e), ia mal 1 T 66 v?/ a),
so that ee, — 1

90°

XXXVII. Molecular Frequency and Molecular Number. By
H. Srantey Aen, 1.A., D.Sc., University of London,
King’s College*. ;

Part I.
§ 1. Molecular Number.

A work of Moseley on the high-frequency spectra of

the elements has established securely the importance
of the “atomic number ” of an element: that is, the number
which determines the place in the periodic classification and
fixes the charge carried by the central part of the atom. It
is now certain that the atomic number is more fundamental
than the atomic weight. Recent investigations of the atomic
weight of lead of radioactive origin have shown that the
value obtained for this quantity depends upon the source
from which the material is derived. An interesting account
of these researches has been given by Soddyf, who points
out that the atomic weight as ordinarily understood is not
the unique quantity hitherto supposed. In the future in-
creasing importance will be attached to the atomic number.
It is the conviction of the present author that this will prove
true not only in connexion with the properties of the chemical
elements but also in dealing with compounds. In the latter
case it is convenient to introduce the term “molecular
number ” to signify the sum of the positive charges carried
by the atomic nuclei contained in the molecule. Thus when
a molecule contains a atoms of an element A, 6 atoms of B,
c atoms of C, so that its chemical formula is A,B,C, the
molecular number N=aN,+6Nz+cN¢-, where Ng, Nz, N. are
the atomic numbers of the component elements. For

* Communicated by the Author.
+ Royal Institution Lecture, ‘ Nature,’ vol. xcix. p. 414 (1917).

Frequency and Molecular Number. 339

example, the molecular number of water (H,O, hydrol) is
10*, for:the nuclear charge of hydrogen is 1, and of oxygen
is 8.

It may be remarked that the molecular number is usually
even. This arises from the fact that when the valency is
odd, the atomic number is usually odd also. But in the case
of an element such as copper, which may be either univalent
or divalent, or in the case of some of the metals of the eighth
group, the molecular number may be odd.

In former paperst it has been shown that simple relations
exist between the atomic number of an element and the
characteristic frequency deduced from observations of the
specific heat in the solid state. In the present communication
similar results are found in connexion with the molecular
number of a compound and its characteristic frequency.
So far as the writer is aware, this is the first attempt to
establish a relationship involving molecular number, previous
work in different branches of physics having been restricted
to considerations of atomic number only.

§ 2. Characteristic Molecular Frequency.

At high temperatures the law as to the specific heat of
compounds enunciated by Joule{ and verified by Kopp§
shows that, as the specific heat is then mainly additive, the
heat energy arises for the most part from the vibrations of
the individual atoms|]. At sufficiently high temperatures the
vibrational energy of each atom approaches the value 3RT.
At low temperatures, on the other hand, Nernst supposes
that the vibrations of the molecules play a more important
part than the vibrations of the atoms in the molecule. In
the case of regular monatomic solids Debye has deduced an

* This fact is probably at the bottom of the remarkable numerical
telations involving powers of 10, pointed out by the author in a paper
read before the Physical Society of London (Proceedings, vol. xxvii.
p. 425, 1915). It wasshown that there must be a numerical connexion
between the unit of length and the unit of mass in the C.G.S. system,
‘“‘and there is no reason why it should not involve ‘the number 10.”
This negative statement may now be changed to a positive one. There
zs a reason, in the constitution of water itself, why the number 10 should
be introduced.

+ H. S. Allen, Proc. Roy. Soc. vol. xciv. p. 100 (1917); Phil. Mag.
vol. xxxiv. p. 478, p. 488 (1917).

t Joule, Phil. Mag. [3] vol, xxv. p. 334 (1844),

§ Kopp, Leb. Ann. vol. iii. pp. 1 & 289 (1864).

|| Cf. Sutherland, Phil. Mag. [5] vol. xxxii. p. 550 (1891).

q Nernst, Vortrdage tiber die Kinetische Theorie, p. 79 (1914). ‘The
Theory of the Solid State,’ p. 81 (1914).

340 Dr. H.S. Allen on Molecular

expression for the specific heat, C,, which is reduced, at
sufficiently low temperatures, to a simple law of propor-
tionality between C, and T*. That this is also true for certain
regular polyatomic substances has been shown experimentally
by Eucken and Schwers* in the case of fluorite, CaF, and
pyrites, FeS,.. Thus it would appear that near the absolute
zero the forces uniting the atoms in the molecule are sufii-
ciently great, as compared with the forces uniting the
molecules, to compel the individual atoms to follow the
movements of the molecule of which each forms a part. At
low temperatures the specific heat can be represented by
Debye’s formula assuming a single characteristic frequency.
At higher temperatures Nernst introduces one or more
Hinstein terms, with appropriate characteristic frequencies,
to include the vibrations of the atoms in the molecule. It
will be shown that the characteristic frequency, v, for the
molecular movement conforms to the relation previously
found to hold for the elements, viz.

Nv=nv, or Nv=(n+4)r,

where N is now the molecular instead of the atomic number,
nis an integer, and v4 is a fundamental frequency having a
value very near to 21x10”sec.~'. The term “ frequency
number ” is suggested to denote the numerical factor, n or
n+.

It would, of course, be possible to avoid the introduction
of the fraction } by introducing a fundamental frequency
which is 4 that just quoted, but as the number of cases
requiring the fractional value is comparatively small, it
seems better to retain for the present the larger value for v,.

§ 3. Characteristic Frequency from Specific Heat.

For a small number of compounds low-temperature mea-
surements are available, and the characteristic frequency
can be deduced from the specific heat. In 1912 Nernst and
Lindemann f published observations on the specific heat of
rock-salt and sylvin at temperatures down to 22° K. For
NaCl the characteristic frequency, v, was determined by
the equation By=287°3, whilst for KCl By=217-6, where
B=4:78x10-". From these results we find the value of
Nv for rock-salt to be 8 x 21:0 x 10”, whilst for sylvin it is
nearly the same, 8x 20°5x10. In his address on the
Kinetic Theory Nernst gives different values for the Debye

* Eucken and Schwers, Ber. deutsch. phys. Gesell. vol. xv. p. 578 (1913).
+ Preuss. Akad. Berlin, p. 1160 (1912).

Frequency and Molecular Number. 341

term which is predominant at low temperatures. For NaCl
he finds Bv= 229, which would require Nv=64 x 20°6 x 10”.
As Lindemann’s formula gives a smaller value (215) for Ay,
it may be suspected that the true value for Nv at very low
temperatures is 6X21x10”. This is supported by the
value for KCl, for which Nernst gives Bv=166, so that
Nv=6 x 20°8x 10!7- The change in the frequency number
from 8 to 6 must be attributed to the introduction of the
Einstein term, and points to the relation Nv=nv, being
obeyed by the corresponding frequency. This is actually
found to be the case, as is shown in a separate paper by the
writer.

Nernst also gives values for the characteristic frequency
of chloride of mercury and chloride of silver. For HgCl
SBv=115, from which we find Nv=11 x 21:'2x 10", and for
AgCl Bv=102, giving Nv=64x 21:0 x 10”.

Experiments by Hucken and Schwers* which are believed
to be very accurate give the characteristic frequency for
two compounds containing three atoms in the molecule. For
fluorite (CaF) Bv=474, resulting in Nv=18 x 20°9x 10”.
For pyrites (FeS,) @v=645, and as N=58 the product Nv
is so large (782°6x10™”) that it is difficult to be certain of
the value to be assigned to n. If n= 37 the product
Nv=37x21'2x 10"; if, as is more probable, n=36 the
product Nv=36 x 21°8 x 10”.

EKuckent found for carbon dioxide (CO,) in the solid state
Bv=119, giving Nv=24x21:9x10". This is of interest as
indicating that the rule applies in the case of non-metallic
compounds.

§ 4. Characteristic Frequency from Lindemann’s Formula.

For the majority of compounds no measurements of the
specific heat at low temperatures have been made, and in such
cases itis necessary to have recourse to some more or less
empirical formula such as thatof Lindemann. This formula
gives

y= 3:08 x 10%\/(T,/MV9),

where T, denotes the melting-point, M the molecular weight,
and V the molecular volume. 7
It is easy to understand that the frequency calculated in
this way may not be identical with the frequency deter-
mined from the specific heat at low temperatures, for the

* Eucken & Schwers, D. P. G. V. vol. xv. p. 578 (1913).
+ Eucken, D. P. G. V. vol. xviii. p. 4 (1916).

Phil. Mag. 8. 6. Vol. 35. No. 208. April 1918. 2B

342 Dr. H. 8. Allen on Molecular

molecule of the solid at or near the melting-point may not
have the same constitution as the molecule near the absolute
zero of temperature. The frequency as given by Lindemann’s
formula must be taken to represent the characteristic frequency
the substance would take at low temperatures, on the assump-
tion that the molecular structure remained unchanged in
cooling from the melting-point to the absolute zero. If
either polymerization or dissociation occur in the process of
cooling, then a change in the characteristic frequency is to
be anticipated.

A definite decision as to whether the relation between N
and v is exact or only approximate cannot be reached until
further accurate determinations of the specific heat of com-
pounds at very low temperatures are available. Hmpirical
formuls, such as that of Lindemann, may be employed for
the evaluation of v, but it must be remembered that such
formule generally give only approximations to the true
value, and therefore cannot furnish a decisive test. It has
been suggested that to render Lindemann’s formula accurate
an additional factor is required depending upon the relation
between the molecular volume at absolute zero and that at
the melting-point™. It may be anticipated that the formula
in its present form should give comparable results for
chemical compounds of similar constitution. This would
be shown by agreement between the values of Ny, or by
concordant values of v4. It has, in fact, been shown pre-

viously that such agreement exists in the case of similar
- elements, and evidence is now put forward that relations of
the same kind hold for similar compounds.

It must be borne in mind in considering the results not
only that Lindemann’s formula is merely an approximate
one, but that the numerical factor is purely empirical.
Further, it is to be noticed that the molecular volume is
usually found from determinations of the density made at
ordinary temperatures. It might be better to employ the
density of the solid near the absolute zero of temperature or
at the melting-point.

The data employed for the calculation of the characteristic
frequencies in the following paragraphs have been taken
mainly from Nostrand’s ‘ Chemical Annual’ for 1913, which
gives Tables convenient for this purpose. Kaye and Laby’s
‘Physical and Chemical Constants’ (1911), and the Smith-
sonian Physical Tables (1914) have also been made use of
for certain compounds. In cases where a range of values

* Cf. Sutherland, Phil. Mag. vol. xxx. p. 318 (1890); vol. xxxii.
p. 524 (1891) ; and Griineisen, Ann, d. Phystk, vol. xxxix. p. 298 (1912),

Frequeney and Molecular Number. 343

is given for the melting-point, the highest value quoted has
been used ; similarly the largest value of the density has been
taken in calculating the molecular volume.

§ 5. Inorganic Compounds (Lindemann’s Formula).

It may be pointed out in the first place that the product
Ny frequently has the same value for compounds which are
similar in their chemical constitution and behaviour. Thus
in the case of the alkali metals we find for the chlorides
of sodium, potassium, and rubidium the values :—

NaCl, 123:2x 10"; KCI, 125°8x10"; RbCl, 124:7 x 10",

Similarly for the iodides of the same metals :—
Mat t452 x 102; KI; 143-1 x10"; Rbl, 145-4 x10".

Further it may be noted that the product for the iodide
exceeds that for the chloride by an amount which is approxi-
mately constant, and equal to about 20x10. Similarly
the difference between the product for sodium chloride
(123°2 x 10") and that for lithium chloride (101°6 x 10) is
21:6 x 10", whilst the difference between the product for
sodium iodide (145°2x 10) and that for lithium iodide
(124:7 x 107) is 20°5x10'%. All such relations, and their
number is far too great for them to be fortuitous, may be
included in the formule

Nv=nvy, and Nv=(n+3)p,,
where v, is approximately 21 x 10” sec.7}.

These formule have now been tested for those inorganic
compounds for which the necessary data are recorded, and
it has been found that the number of cases in which one or
other of the formule cannot be applied issmall. It is hoped
to publish details of these results later; at present it will
suffice to quote the figures for twe series of compounds. In
Table I. are given the results for all the lithium compounds
for which data are available. This element has been selected
on account of the small atomic number (N =3) rendering the
product Nv comparatively small; consequently it is not
necessary to employ Jarge values for n, and a more satis-
factory test of the new relation can be obtained. The
frequency number in the last column of the Table falls
between 34 and 8, whilst the extreme values for yy, are
20°1 x 10 sec.-! and 21°3x 10” sec.-1 Application of the
theory of probability to the figures shows that there is only
1 chance in 282 that these results should occur by accident,

2B2

344 Dr. H. 8. Allen on Molecular

TaBLe I.
Lithium Compounds.

Name. Formula, N. | VX10-12,| Ny x10-22,
LitHIuM
AMIGO. cece tce eee LiNH, 12 6:073 2x 20°8
bromide .5.2..<-sncaske LiBr 38 3°234 6 x20°5
carbonate ....:2..2.56 Li,CO, » a6 3°435 6 xX206
chloride (5/2. seer LiCl 20 5079 5 x20°3
MUOLIAG. ..20csh eee Lik 12 9:207 53x 20°1
HOMIGS cov ccnecneese ieee Lil 56 2°227 6 x20°8
MIETAGE {sles ho eee LiNO, 34 2°821 42x 21'°3
perchlorate::...... 0... LiClO, 52 1°742 43x 20°1
phosphate .....ss.... Li,PO,.H,O| 66 | 2345 | 74x206
Biligate”™ 2. 52.chacsee Li, SiO, 44 3°756 8 x20°8
sulphate (acid) ...... LiHSO, 52 1°636 4 xX21°3
sulphate (normal) ...| Li,SO, 54 2°680 7 X20°7

Mean value of v, =20°66x 10.

In Table II. are recorded the figures for the chlorides of
the alkali metals, and the monochlorides of copper and silver.
No data have been obtained for aurous chloride. The pro-
bability calculated in the same way as for Table I. is
about 1/18. It will be noticed, however, that all the
frequency numbers in Table II. are integers. The pro-
bability that this should be the case is about 1/2000.

TasueE IT.
Monochlorides of the Metals of Group I.

Formula. N. vx10-}, Nvx 10712,
LiCl... 2 eee 20 5-079 5x 20°3
Na) ccsreeeene 28 4401 6x 20°5
RCI epcbeneeeee 36 3°493 6x 20°9
RIDOL Aeon 54 2°310 6x 20°8
CsCl): coveeeee: 72: 2:065 7X21:2
CuCl 22 46 2°726 6x 20°9
AoOls 2 eceereee 64 2°343 7x21°4

Mean value of v4 =20°86 x 10!2.

In view of the fact that so many other inorganic com-
pounds as well show fair agreement with the proposed
relation, it is hardly possible to doubt that it must give at
least a close approximation to the truth.

* The melting-point of lithium silicate is given as a standard tempe-
rature (1201°) in the Smithsonian Physical Tables,

Frequency and Molecular Number. 345

One point of interest may be mentioned. Inorganic com-
pounds which contain water of crystallization conform to the
general rule. In some cases the frequency number of the
dehydrated salt is the same as that of the hydrated compound.
Thus for sodium sulphate (Na,SO,) the value of Ny is
8x 20°4 x 10”, for the hydrated salt (Na,SO,.10H,O) it is
8x 20°6x 10. In other cases there is a change in the
frequency number. An example is afforded by calcium
nitrate—the anhydrous salt (Ca(NO3).) gives for the product
Np the value 64 x 21°3 x 10, whilst the hydrated compound
(Ca(NO,),.4H,O) gives 4 x 21°5 x 10”.

A comparison between the results obtained from the specific
heat at low temperatures and those found by calculation from
Lindemanno’s formula is only possible in a few cases.

TaB_e III.
Compound. | Nv x 10722.

/ Specific heat. . | Lindemann’s formula. |
Ae

a | 8 x21-0 | hes |
| +x 206 | 6X 205

es ed 2 At ) eed / 6x 20°9

20 2 ere 63 x 21-0 | 7X21°4

Sia: -n2. 23. 18 x20°9 | 9x 2071

os eee 36 X21°8 | 10x 21°7

It is curious that there should be such a large difference
between the two values of the frequency numbers for calcium
fluoride and iron disulphide. Interpreted according to the
theory of Nernst, this may indicate that for these compounds
the contribution to the specific heat arising from the internal
vibrations of the molecule forms an important part of the
whole.

§ 6. Organic Compounds (Lindemann’s Formula).

Chemists have not, as a rule, devoted great attention to
the determination of the density of organic compounds in the
solid state. On examining such a Table as that given in
the ‘ Chemical Annual ’ it will be found in general that where
the density of the solid is recorded, the melting-point is
wanting and vice versa. Amongst the results available up
to the present time are to be found many suggestive cross-
relationships between the values of Nvx10-, in which
a number approximating to 21 or 4x21 is of frequent
occurrence. This is illustrated for some aliphatic derivatives
in the following Table.

346 Dr. H. 8. Allen on Molecular

TABLE LV,
Name. Formula. N. |Nvx10-,
Maleic anhydride ...| <(CH. 00),>0 50 60°17,
Succinic anhydride...) <(CH,CO).>0 52 70°56
Malic acid (¢) ......... CO,H.CW,.CHOH .CO,H 70 82°11
Oxalic acidv320) 08, CO,H .CO,H+2H,0 66 82°24
Maleic acid ............ CO,H.CH:0H .CO,H 60 | 82-42
Citrie arid aster (CO,H .CH,),C(OH)CO,H+H,0 | 110 93°71
Tartaric acid ......... CO,H[CH(OH)],CO,H 78 93°79

Here the characteristic difference is found between the
two anhydrides ; the three dibasic acids, malic, oxalic, and
maleic, have a common value for the product but different
values for N; and citric acid (a monohydroxy tribasic
acid) and tartaric acid (a dihydroxy dibasic acid) have the
same value for Nyx 107%, but a value exceeding the previous
common value by the characteristic difference.

Data are available for a larger number of aromatic deri-
vatives, and amongst these compounds many interestiug
correspondences occur. In Table V. are given the results
for a number of hydroxyl derivatives containing the benzene
ring, and in Table VI. some of the halogen derivatives of
benzene in which the two substituted groups occupy the
para position.

TABLE V.

Name. Formula. N. Nvx1o—!,
Phenol o.écis. te. seers C,H,OH 50 63:29
Cresole (p) ............ OH, .C,H,OH 58 64°23
Mylenol sii) isass eee (CH,),.C,H,0H 66 72:84
Hydroquinone (p) ...| C,H,(OH), 58 82:07
Pyropallol. cocci escsee C,H,(OH), 66 82°52

TasBie VI.
Name. Formula, N. Nvx107),
Chlor phenol ......... Cl1C,H,OH 66 68:40
Dichlor benzene ...... C,H,Cl, 74 69°63
Chlor nitrobenzene ... ClO,H,NO, 80 78°86
Brom phenol ......... BrC,H,OH 84 79°41
Dibrom benzene ...... ,H,Br, 110 88°64

Brom nitrobenzene ... BrO,H,NO, 98 89°95

Frequency and Molecular Number. 347

Data are available for a number of ketones of the benzene
series.

TaBLE VII.

| —12

Name. Formula. Ne Ne LO
Methyl-pheny! ketone OH, . 0O.0,H, 64 63°16

(aceto-phenone) ...

Kthyl-phenyl ketone ...| C,H, .0O0.C,H; 72 64-48
Methyl-benzyl ketone .. CH, . co. CH, ‘oh HH, 72 65°03
Propyl-phenyi ketone ...| C, H, COG? i, 80 65°81

Here the values of Nv x 10-™,though not differing greatly
for the various compounds, tend to increase slightly with the
complexity of the chemical molecule, indicating the presence
of a constitutive influence.

An attentive examination of the results just recorded for
organic compounds will have shown that in a large number
of cases the value of the product Nv can be expressed in
the form already employed for inorganic compounds. In
other cases, however, the simple form of the relation cannot
be applied successtully. The two Tables following (VIII.
and IX.) contain the figures for a number of well-known
aliphatic and aromatic compounds for which the relation is
found to hold good.

It is noteworthy that the frequency number for organic
compounds is usually small, varying from about 3 to about 5.
This fact may be correlated with the low melting-point of
these compounds and the small atomic numbers of the con-
stituent elements.

Tasue VIII.
Aliphatic Compounds.

Name. Formula. N. | vx107”- | Ny x107

Butyl carbinol (tert.) .. | .| (CH,),C .CH,OH 50 1-242 3 X20°7

WUPORIE iia ncedese cs NH,OO, .C,8, 48 1-307 3 X20°9
U¥eatis22).. heb? ESSE | CO(NH,), 32| 2°243 34 x 20°5
Ethylurea .............4. C,H,NH . CONH, 48| 1:500 33 X 206
Kthylene iodide ......... | CH,I.CH,1 122} 0-672 4 x20°5
iC as CHI; 166) 0:549 4°-X 20°3

Carbon tetrabromide ...| CBr, 146| 0-703 5 X20°5
| Oxamide po casthteon steel CON H, COMMyh®, | 46). 2208. | 15, 203

SSS

348 ‘Dr. H. 8. Allen on Molecular
TABLE IX.
Aromatic Compounds.
Name. Formula. N. | »x10-™. | Nex iat
Xylidene (1:4:2) ...... (CH,),0,H3. NH, 66| 0:954 3 x210
Chlorquinoline (py 2)... C,H,CLN 84} 0°842 34 X 20:2
Chlorquinoline (py 4)...| C,H,CIN 84| 0-858 34x 20°6
Naphthalene.. ............ Coa O,H, 68} 1:062 33 X 20°5
Triphenyl phosphine ...| (C,H,),P 138| 0-592 4 x 204
Hydrazo benzene......... C,H,.NH.NH.O,H,| 98} 0842 4 x20°6
Camphoryiiiic (2k store Oe 0 84/ 0:989 4 x20°8
Nitraniline (p) «0.0.0.0... NO, .C,H, . NH, 72| 1173 4 x21:1
Tetrachlor benzene (s)...| C,H,Cl, 106} 0°874 4% x 20°6
Camphorie acid (d) ...... O,H, ,(CO.H), 108} 0-859 43 Xx 20°6
Camphoric anhydride...| C,,H,,0, 98; 0949 435X207
Anthraquinone ......... C7, = (CO), :C,H, 108) O86 5 x207

It has been mentioned already that for a censiderable
number of organic compounds the proposed relation, in its
simple form, does not hold. One possible suggestion as to
the reason for this failure will be considered in the following
section.

§ 7. Molecular Association in Solids.

A question of great importance, which can only be con-
sidered briefly in this paper, is the determination of the true
value of the molecular weight of a solid compound. Nernst
has shown how the constitution of the molecule may be
inferred from the correspondence between the molecular heat
at low temperatures and the value calculated by Debye’s
formula. For example, he concludes that the molecule of ice is
not hydrol (@v= 227) but dihydrol (@v=155). On the first
supposition we find Nv=47:5x10", which cannot be ex-
pressed as a multiple of v,, but the second supposition gives
Ny=64°8 x 10¥=3x21'6x10", showing good agreement
with the new relation.

If, in the next place, we apply the formula of Lindemann
to ice (Ts=273%1 K, p=0°917), we find that neither H,O
nor (H,O), gives satisfactory agreement. When, however,
we assume that the ice molecule at the melting-point is
trihydrol, (H,O);, we find Nv=24x 21:4 x10". The method
does not give a unique determination of the degree of asso-
ciation, for equally good concordance is obtained by assuming
that the ice molecule contains 9 groups of H,O, which makes
Nv=3x21'4x 10".

On general grounds it seems most probable that ice at the
melting-point is pure tribydrol. This was the conclusion

“g

|

Frequeney and Molecular Number. 349

reached by Sutherland*, who emphasized the fact that water
erystallizes in the hexagonal system, whilst trihydrol can
be represented by three linked oxygen atoms at the corner
of an equilateral triangle with the six hydrogen atoms
arranged symmetrically round them. .

It is easy to examine the effect of association on the
frequency as determined by Lindemann’s formula. If M
denote the molecular weight of the simplest molecule, that
of the associated molecule may be written «M. The molecular
volume V will also be increased x times. Hence the frequency,
determined by the equation

AV ®)

will be divided by a*+?=2*, The atomic number will be
inereased x times, and consequently the value of Ny will
be multiplied by 2*. It is, then, a simple matter to find an
integral value for aso as to satisfy the relation Nv=ny,. As
in most cases there is no independent check on the degree
of association of a solid, it does not seem desirable in the
present state of our knowledge to attempt to apply this
method in detail.

§ 8. Conclusion.

In this preliminary survey of the subject of the relation
between molecular frequency, v, and the molecular number,
N, it has been proved that the product Nv frequently shows
related values for analogous compounds. There is considerable
evidence for the validity of a formula of the type Nv=nv, in
the case of most inorganic solid compounds and of a number
of organic compounds. To what extent the formula is to be
considered approximate can be decided only when further
data as to the specific heat of solids at low temperatures are
available. The physical significance of such a formula has
been discussed in an earlier paper. Perhaps the simplest
interpretation that can be suggested for the “ frequency
number,” n, is that it is related to the number of valency
electrons concerned in imparting to the solid its crystalline
structure. Further investigation on these lines may serve
to throw more light on the nature of the forces connecting
the molecules, and the problem of molecular association, in
crystalline solids.

- * ache: Phil. Mag. vol. 1. p. 460 (1900); Faraday Soc. vol. vi.
(1910).

1

f. 350]

XXXVIII. On Transpiration through Leaf-Stomata.
By Sir Josepn Larmor, F.R.S.*

ieee acute and valuable paper by Dr. Harold Jeffreys

(Phil. Mag. for March, pp. 270-280) on evaporation
and diffusion shows, by inadvertence, less than due apprecia-
tion of the investigations of H. T. Brown and EF. Escombe on
transpiration through the stomata of leaves. The question is
so important in plant-economy that misunderstandings should
not be allowed to persist. The statement held to be erroneous
on p. 277 (for which originally I had some degree of respon-
sibility) still appears to me to be quite correct, in its proper
context. The diffusional suction to or from each stomatal
opening is local ; thus, when the openings are as much as
ten diameters apart the interference between the adjacent
shells of diffusion is surely very slight, as stated. But
Dr. Jeffreys finds that this would make the transpiration of
vapour from the stomata very many times greater than the
evaporation from the entire leaf when wet. Whence, then,
the discrepancy ? His calculation compares a system of
actual stomata, each of diameter 107? cm., and at about
5.107? cm. apart, with a single giant stoma the size of the
whole wet leaf and so of diaineter 6 cm. But the calcula-
tion for the latter case implies that the air is absolutely still
throughout the shells of diffusion, which then extend to
several diameters from the leaf, so that most of this mass of
air remains highly saturated with vapour. Under natural
conditions, where the atmosphere and the leaf are not quite
still, the evaporation will be at the very least hundreds of
times greater.

The natural comparison is that discussed later by
Dr. Jeffreys (p. 279), where he passes on to consider the
effect of wind or movement of the air. But here again, by
reasoning similar to that described above, he seems to arrive
at misunderstanding of Sir F. Darwin, and holds that when
the stomata become constricted ‘“ until the stomatal aperture
is reduced to a certain very small value the possible rate of
transpiration is practically independent of the aperture and
nearly all of the reduction to zero when the stoma closes
takes place on the last 2 per cent. of the reduction of aper-
ture.” The reasoning on which this statement is based should
be capable of some Jess paradoxical form of conclusion. If I
am not mistaken, the underlying idea may be developed in
altered form as follows. We may imagine partitions erected
perpendicular to the surface of the leaf, so that each stomatal

* Communicated by the Author,

%
“

a

ee

ee

2 Sage ae 4 ee eS ee

On Transpiration through Leaf-Stomata. 351

aperture is isolated from the others and the transpiration occurs
within its own cylindrical tube of diffusion ; this will not
sensibly affect the course of the phenomena. The electric
idea of conductance of the diffusion-current along this path
is now the appropriate aid to discussion. We can imagine
the tube prolonged on the other side beyond the stoma, and
thus consider the parallel case of electric flow along a con-
ductor having a sharp local constriction representing the
stoma, whose area is much smaller than the cross-section of
the conductor. The resistance of the whole tube is propor-
tional to its length, provided the latter is increased by a
constant correction in order to include the extra resistance
arising at the constriction*. The methods by which this
correction may be practically estimated were developed by
Lord Rayleigh in 1870: ef. ‘Theory of Sound,’ ii. ch. xvi.
The current being the same all along the tube, the resistance
in any segment of it is proportional to the fall of head (of
potential, or of density of diffusing substance) between its
ends. When the constriction by a transverse barrier is, as
here, to less than one-fifth of the radius of the tube, it will
not be far wrong to estimate the fall of head across the con-
striction as if the enclosing tube were absent. This procedure
leads, on the same lines as in Brown and Escombe quoted by
Dr. Jeffreys (p. 275), for a circular constriction of radius a,
to a correction to the length of each half / of the doubled
tube, of amount equal to the area of the section of the tube
divided by 4a.

Now Dr. Jeffreys considers that under natural conditions a
layer of air, as much as 1 mm. thick before the disturbed motion
beyond is reached around the leaf, may be regarded as still.
For the dimensions of stomata quoted by him, a would be
4.1073 cm. and the area of section 3.10~* em.?, while J would
be taken as 1mm. The correction to / would then be } mm.
As this is a small fraction of ] the main resistance to tran-
spiration would arise in getting across this highly saturated
layer of air, as much as 1 mm. thick around the leaf. The
stomata would transpire into it, and the vapour would not
get away as rapidly as it could be supplied; the case is
analogous in a lesser degree to a leaf enclosed in a bottle
with narrow, open neck, in which the air would soon become
nearly saturated with vapour and the transpiration would be

* For the problem in two dimensions of space the exact solution is
known, and a diagram of flow is given by Prof. Lamb, ‘ Hydrodynamics,’
§ 306. In that case the correction to be added to the half length J (infra)
in order to obtain the effective half length, when the resistance of the
constrictions is included, proves to be — loge sin 37k, where k is the ratio of

the area of the straight stomatal strips to the whole area. But this result
is hardly applicable even to illustrate the actual problem of local stomata.

352 Dr. 8. R. Milner on the Eject of

reduced to avery small amount. Hach stoma is fully efficient
in proportion to its radius; but the output will be diminished
for all sizes of stoma because it has to transpire into a
nearly saturated space. When the stoma becomes constricted
it operates more feebly, but under better conditions ; ifa
circular stomatal aperture of dimensions as above is con-
stricted to + of its radius, its own resistance will be half the
total resistance to transpiration instead of only 3; if it is
constricted to =!,, the resistance of the stoma itself will be
% of the whole, and the layer of moist air outside will hardly
count. The question is whether we can assume a still layer
of air anything like | mm. thick. As moist air is sensibly
lighter than dry, such a layer could hardly be established if
the leaf itself is not quite still, unless possibly to some degree
on the lower surface of a horizontal leaf.

The problem whether a cause like this, which I take to be
the essence of Dr. Jeffreys’s important suggestion, has really
intervened in observations such as those quoted by Darwin
from F. &. Lloyd (p. 277) could only be probed by further
experiment under suitable precautions.

The question whether the sap-current in trees is reduced
on still days is much simpler, for the whole region of the
tree-top may become nearly saturated.

These considerations, which apply to the transpiration of
vapour from the stomata, are pertinent, of course, equally
to the diffusion of carbon dioxide into them. They seem
directly to confirm, from a different aspect, the conclusions
of Brown and Hscombe that the stomatal cavities are
capable of much more absorption than they are called upon
to perform.

Cambridge, March 9.

XAXXIX. The Hfect of Interionie Force in Electrolytes.
By 8. R. Mirygr, D.Sc.*

Part II.
Ionic Mosinity AND Osmotic PRESSURF.

A lists first attempt to determine the. effect of interionic
force on the ionic mobility is due to Sutherland f,
whose method is based on the following idea :—The effect of
the forces will approximate to what would be obtained if the
ions were regularly distributed throughout the liquid, say
at the centres of equal cubes. When they are displaced

* Communicated by the Author.
+ Phil. Mag. xiv. p. 1 (1907).

Interionic Force in Electrolytes. 353

from these positions by an applied electric field, the interionic
forces act as restoring forces in a way which would give rise
to a sort of rigidity. of the ionic configuration were. it not
for the fact that the “actions which produced the original
uniformity ” (thermal motions ?) will cause the rigidity to be
continually breaking down. The process of breaking down
originates a special type of viscosity, which acts in addition
to ordinary viscosity when conduction is taking place. A
second type of viscosity due to the polarization of the medium
is also discussed, and the conclusion is reached that when
these viscosities are taken into consideration the diminution
of X, with increase in the concentration, can be accounted
for without any association of the ions into molecules taking
place.

Sutherland’s calculation does not bring the conductivity
variation into any relation with that of the freezing-point ;
and it is based on several speculative hypotheses which are
not always convincing. This is particularly the case in
regard to the assumed configuration of the ions ; the special
type of regularity of this is a feature which is inconsistent
with the general theory of the distribution of ions to which
the kinetic theory leads.

Effect of permanence of the distributzon on the mobility.—The
method of calculation adopted here is based on the assumption
that the distribution of the ions remains undisturbed in the
interior of the electrolyte when a current is being carried.
Suppose we have a mixture of positive and negative ions
contained in a volume, and in the first place suppose that
they are subject to no interionic forces. We must assume
that they are distributed at random throughout the volume,
for there are no data for assuming anything else. Now
suppose that an external electric field is applied which gives
each positive ion a velocity to the right, and each negative
ion one to the left. In the interior of the volume the
random distribution will not be disturbed, as is easily seen
whether the velocities are all equal or whether they vary
arbitrarily from ion to ion.

When interionic forces are present the distribution is no
longer random. It becomes modified in such a way that the
chance of a positive ion being found in a given position will
depend on the mutual potential energy which it possesses in
that position with the other ions. ‘Tf we now imagine all
the positive ions displaced to the right and the negative ones
to the left—with the same or with arbitrary velocities—the
distribution will be disturbed. It will, in fact, tend to be
converted into a random distribution, Consequently we see

354 Dr. 8. R. Milner on the Effect of

that, if the distribution is to remain permanent when the
electrolyte is carrying a current, the velocity with which
each ion must be supposed to move under the influence of
the applied electric field must be a function at each instant
of the mutual potential energy which the ion possesses with
the others.

The way in which the random distribution will be modified
when the ions are subject to interionic force is given by a
theorem due to Boltzmann. Let us suppose that we take
a large number of instantaneous views of a certain region
of the liquid, and that in each view we observe the positions
and signs of all the ions which are present in it. We
will confine our attention in the first place to those views
alone, m in number, in which the region contains m ions,
A,...Am, and no more, and we will suppose that these are
all so far away from the ions outside the region that the
forces between the ions inside and outside are negligible.
This will simplify the statement of the argument without
affecting the generality of it in any way, since the region
may, if necessary, comprise the whole of the liquid. In a
certain number, say v, of the n views, the m ions will be
found in small equal volumes dv...dv,, situated at the
points Py...Pm. For shortness we will call this the P confi-
guration, and speak of the ion A, as “‘ occupying the position ”
P,, the uniform size of the elementary volumes being under-
stood. In another number v’ of views the ions will be found
in positions P,'...P,,'(P' configuration). Ona purely random
distribution we should have

p=v',

but in the modification caused by the presence of interionic
force,

le Tew a eer (5)
v/v' here stands for the probability of the P configuration
relative to that of the P’. ¢ and @’ are the respective
mutual potential energies of the ions in each configuration,
2. e. the work done by the system when the ions are moved
to infinite distances apart. When the forces are attractive
and ¢’ are negative quantities. *T= % x average translatory
kinetic energy of an ion.

Equation (5) can easily be transformed so as to represent
the absolute probability of a given configuration (estimated
under the special conditions attached to the total number of
views n) by writing it in the form

pe aes hoy, setae di ss ae fete.)

Interionic Force in Electrolytes. 355

The essential feature of the ionic distribution represented
by (6) is of course its permanence, that is, that it is un-
disturbed by the thermal motions of the ions. Let us suppose
that the m views were taken at certain times ¢,....¢,. If we
were to take another set of views at times t,+T,....t, FT,
where 7 is a very small time, we should find the same number
of views in which the ions have the configuration P ; the
actual views will not be identical in the two cases, in some
of the n views at t the ions will have left the volumes dv,
but in an equal number of other cases ions will have come
into them. |

Now suppose that the electrolyte is under the influence of
an external electric field when these two sets of views are
being taken. In addition to the thermal displacement
which each ion undergoes in each of the times 7, it will be
dragged by the electric field a certain distance to the right
or left. We can effect a considerable simplification by
observing that, since the thermal displacements do not affect
the distribution, we can imagine them to be non-existent
without affecting in any way the result of reasoning con-
cerning the effect on the distribution of the displacements
due to the electric field. With this simplification the
problem to be solved becomes this :—A set of views at times
ty,...tn, and another set at t;+7,...t,a+7, being taken of a
system of ions existing in configurations the probability of
which is given by (6) and at rest during the intervals 7 so
far as thermal motions are concerned, but in which each ion
is dragged during the intervals by an external force to right
or left according as it is positive or negative, what is the
average velocity with which an ion such as A, in a given
position must move in order that the distribution may not
be disturbed ? By the average velocity is meant the average
for all the views which show the ion in the given position,
but as it makes no difference whether it varies from view to
view or is uniform, we may in the calculation treat it as
uniform.

Let w,...U¢m be the velocities of eachion Ay,...A,,. Describe
small cylinders of area a and of arbitrary lengths dx,...dxm,
near each of the points P,,...P,, to the right or left of the
points according to the sign of the ion. The number of
cases in which an ion is to be found in each of these cylinders
will be the same in the views at¢+7 as it is in the views at ¢.
Consequently the number y, of views in which an ion
will enter each cylinder during the intervals 7 is the same
as the number y, in which an ion will leave each cylinder.
But », coniprises all those views in which the m ions are

356 Dr. 8. R. Milner on the Effect o,

situated in the infinitesimal volumes aujT,...aupT at P53: ee
or
— /KT
v= Ke ayt.. dt.

A similar expression holds for v, except that both @ and the
w’s are infinitesimally different :

Vo= Kn heat ++ Leda, a+ fe

i
du, dum
a ( + Ta 1)? a tin + Tate) 7
Hquating v, and v, we get

Gb ig ern Gs ig OE i
ia, *° Mab =..= 7 fe OFT at =0. * (7)

Now, if we write ¢ in the form

p= hit oi =$2t go’,

where ¢, is the mutual energy of A, with all the rest, o,/ the
mutual energy of all the m—1 ions other than Aj, ¢, that
of A, with all the rest, &., we see that the only part of ¢
which is affected by d/da, is $;, and similarly for each of
the other ions. Consequently, integrating (7), we see that
the conditions which must be satisfied in order that the
distribution may not be disturbed are
ue?" = const.,
use” °2"* —const., &e.

The constant is independent of 2, that is, of $1, do, &e.,
and is equal to the velocity with which an ion will move
when it is so far away from the others that its mutual energy
with them is zero. In these circumstances the velocity will
be conditioned simply by the friction of the water, and it is
clearly the same for all ions of the same sign. Calling it up
in unit field, we shall then have for the mobility u,, which an
ion must be reckoned to possess when it exists in a place
where its mutual energy with other ions is qy,

Uy = uperuet . e ° e (8)

Effect on Osmotic Pressure.—In the method of the kinetic
theory of considering the pressure of a gas as the rate at
which momentum is transferred through a unit plane within
it, a careful distinction must be made between “internal ”’
and “external” pressure. Consider a gas in which—say by
impressed mechanical forces—the potential energy of a

Interionic Force in Electrolytes. 357

molecule when in a certain region R is @ (a negative
quantity) and zero elsewhere. ‘The distribution will be such
that the chance of a molecule occupying a position inside R is
to that of its occupying one outside as e~®”": 1, and, in fact,
the densities in the two parts will adjast themselves in this
proportion. If we imagine a unit plane situated inside R,
the momentum transferred through it per second will be the
total pressure inside R, but it is only a certain fraction—
e*?**__of the molecules passing through the plane which
are capable of transferring their momentum outside the
region. We can thus divide the pressure in R into two
parts—the external pressure, which is due to momentum
eapable of being transferred outside it, and which is, in fact,
in equilibrium with the pressure outside, and the internal
pressure, which in this case will be exerted on the mechanical
constraints which cause the increased density in R.

A similar state of things occurs when we deal with a group
of ions existing momentarily in a liquid. The whole
momentum passed per second by the ions of the group
through a unit area drawn in the interior or the group will
not be delivered to places of zero potential energy, and the
fraction of it that is so delivered will be in statistical
equilibrium with the pressure exerted by those ions which
are in positions of zero energy. We may call this fraction
the external pressure p of the ions in the group or the
pressure of the “free” ions—understanding by free ions
those which momentarily have no mutual energy with any
others. Let us inquire how much of the momentum of an
ion existing in a group such as that considered above would,
on the average, be capable of being transferred to a place of
zero potential.

Consider a single ion of mass m moving in a random
direction with velocity v. The average rate at which it
transfers, parallel to a given direction, the component in
that direction of its momentum is 4mv’, or, if we take into
the average all the possible velocities which it may have,
dmv? or kT. The sum of this quantity for every ion in the
mixture gives the total (2. e. internal+external) pressure
x volume, PV, of the electrolyte. The contribution of each
ion to the total PV is thus a scalar quantity 4mv? associated
with the ion—these contributions will therefore obey the
same law of distribution as do the ions themselves.

Consider now two configurations P and P’ of the group
of m ions dealt with above. In P the ions occupy the

Phil. Mag. 8. 6. Vol. 35, No. 208. April 1918. 2C

358 Dr. 8. R. Milner on the Effect of

positions P,...P,, the mutual energy is 6=¢,+¢,’, and the
number of views in which it is found is

y= KneW oe Ou dy. Ane a

In P’ let Ay...A,, be in the same positions as before, but
let A, be in a place of zero mutual energy. The mutual
energy of the group is now ¢,/ and the number of views in
which this configuration is found is ,

-_ "Kk
v'=Kne P11 Tdv,...dVm.

In the series of » views taken one after the other at
arbitrary times the configuration of the system is changed
between one view and the next by complex thermal motions.
Omitting all the other views, let us confine our attention to
the}y views of the configuration P and the v’ views of P’.
These are observed at certain successive times and are all
the views of these configurations which are observed in the
series, In the intervals between them the configurations
change over one into the other as the result of thermal
motions, but not indiscriminately. It is only in a fraction
v'/v of the v views of the P configuration that a change by
thermal motions into the P’ configuration will occur. Ona
random distribution the fraction would be unity. The change
in the configuration considered consists simply in the trans-
ference by thermal motion of the ion A, from a position of
mutual energy dq, to one of zero mutual energy. We see
that, given the ion in this position, the probability that such
a transference will take place is not the same as it would be
on a random distribution (7. e. in the absence of interionic
forces), but v’/v or e*"" times as great *.

On a random distribution the whole of the scalar property
1

Lmv® associated with each ion is capable of being transferred
from one position to another. This gives a random, or on the
large scalea uniform, distribution of the pressure throughout
the volume. It follows from the preceding proposition that,

* The proposition is so far only proved for the case in which the
m—1 ions other than A, remain fixed during the transference of Aj.
If these also undergo displacements we shall have a simultaneous
alteration of gd, and ¢,'. In so far as these displacements affect ¢, only,
it is immaterial to the argument whether A, reaches a state of zero
mutual energy by its own displacement or by suitable ones of the other
ions, the expression for v'/y being the same in either case. If, on the
other hand, they alter ¢,', the expression (9) shows that this is an event
independent of the change in A,’s mutual energy, the probability of the
simultaneous occurrence of the two events being the product of the two
probabilities. The truth of the proposition is thus unaffected by all
other ionic changes which may proceed simultaneously with the trans-
ference of the ion A, to a position of zero energy.

Interionic Force in Electrolytes. 359

when interionic force is present, the average amount of the
property which is capable of being transferred from a position
in which the mutual energy of its ion is ¢, to a position of
zero mutual energy, is

WE ess). ah ey eee (LO)

This quantity represents the contribution to the external
pV made by the ion when it forms part of a group. The
summation of it for each ion in the electrolyte would give
the external pV or, strictly speaking, its instantaneous
value in the view observed. The external pressure of a
system of ions subject to interionic force determined in this
way is a perfectly definite thing everywhere in equilibrium
throughout the volume. It is identical with the pressure of
the free ions as defined above. Superposed on it in the
interior of groups is the internal pressure (got by summing
Lmv?(1—e*") for each ion), which is exeried against the
mutual forces and is not effective on the walls.

A comparison of the result (10) for the contribution of
an ion to the external pV when it is in a position of mutual
energy with others with that (8) for the mobility of the ion
in the same circumstances, shows that both are affected
by the mutual energy in exactly the same way. Suppose
now we follow in imagination the history of an individual
ion in an electrolyte for a long time. We shall observe that
its state as regards the mutual energy ¢, which it possesses
with other ions is continually varying. If we take the average
value over a sufficiently long time of the quantities

ue * and kT. 6%",

we shall get in the first instance the average value of the
mobility «u, of the ion (which is of course the same as
that w of any other ion of the same kind, but up differs if
the kind differs), and in the second instance a quantity
which, when multiplied by the total number N of ions in the
electrolyte (combined or not) gives the product pV of the
pressure of the free ions into the volume. It is clear that,
whatever the effect of ¢; may be on the actual values, we
shall always have
uty
pV N&T
Hence in an electrolyte, while alterations with the concen-
tration of w and pV may, and indeed will, be produced as
the result of the presence of interionic forces, the alterations
will always be such that the ratio of u to pV remains
unaffected,

= G0lisb: esas CL)

202

360 Dr. 8. R. Milner on the Effect of
Part ITI.

The theorem thus proved can be applied to throw some
light on the nature of the interionic forces in strong electro-
lytes, but before applying it it is necessary to identify u,
and p clearly with measured physical quantities. As regards
uo there is no difficulty : it represents the mobility of an ion
in a region where it is free from interionic force. wp is
there conditioned only by the friction of the water, and can
thus be identified with the experimentally determined
mobility at zero concentration to which the same condition
applies. The case is different for p the free ionic pressure.
In what relation does this stand to tlie measured osmotic
pressure? It will be useful to consider this point in
connexion with three possible theories of the constitution of
electrolytes. .

(1) In the original theory of Arrhenius electric interionic
forces are neglected, and an ion is assumed to be definitely
either associated (when it contributes nothing to the free
lonic pressure p) or free. Although p is not susceptible
to direct measurement, a clear conception of it can
be got.

The general theorem is of course independent of the law
of force between the ions and applies to this theory equally
with others. Interionic forces (of the kind referred to
below as “chemical” forces) produce an increase in the
frequency of occurrence of ions in an associated state with a
consequent reduction in the pressure of the free ions in the
ratio of 1—@:1. The average mobility of an ion, taken
over a period long enough to include its being combined as
well as free, is reduced in the same ratio. This agrees with
the experimental requirement of an equality in the freezing-
point and conductivity variations, for in calculating the
complete osmotic pressure P, allowance must be made for
the molecular pressure of the fraction @/2 of associated ions.

The reduction in P is thus in the ratio (1- B+ e) : 1, and
consequently

Toe Nici Mu
ee a Le B,

in agreement with the experimental results (v. Part I.).

The difficulty here, as already mentioned, is the failure of
the mass action law (4) to represent correctly the variation
of 8. When we consider the dynamical assumptions on
which (4) is based, we find it essential that the forces which
tend to produce association of the ions must fall off very

Interionic Force in Electrolytes. 361

rapidly with the distance*. The law indeed forms a limiting
case for which the field of force surrounding each ion is
infinitely strong but confined to an infinitely thin shell.
This means that the mass law (4) applies solely to asso-
ciation and dissociation which is the result of what we may
call “ chemical ”’ forces, using this name to distinguish forces
of this type from the electrical forces of the free ionic
charges which fall off very slowly with the distance. It
association is produced by these (4) will not apply to it.

(2) The preceding conclusion suggests that the failure of
the mass law may be due to a mistaken view of the nature
of the interionic forces which cause association. The
apparent association may be partly—or wholly—due to the
electric forces. In attempting to investigate this view the
first difficulty is to know exactly what is the law of force
between the ions. Are we to assume that every + ion
attracts every — ion and repels every + ion in the liquid
according to the inverse square law? If the mixture of ions
were a gaseous one this assumption would presumably be a
sound one, but its validity is more doubtful in the case of an
electrolyte, where the forces between the ions are affected by
the intervening water molecules. As the first step, however,
it seems the most straightforward assumption to make, and
in previous papers ft I have worked out by what is, I think,
a strict method, the approximate effect on the osmotic pressure
of a mixture of ions in which interionic force of this charaeter
is assumed to exist. The calculation, as might be expected
from the complexity of the forces, is lengthy and need not
be further referred to. The net effect of the interionic
forces was found to give a reduction of the osmotic pressure
which appears to be in accurate agreement with the experi-
mentally found results for dilute aqueous solutions of strong
binary electrolytes. Inthese cases therefore there is ground
for believing that “chemical association,’ if existent, is
extremely small, and that the effects observed are due
entirely to the electrical interionie forces. Before we can
apply to this view the proposition of the present paper, it is
necessary to settle the relation in which the free ionic
pressure p stands to the measured osmotic pressure P.
Unless they are different from each other, an znequality

* This is the kinetic aspect of the thermodynamical stipulation that
the osmotic pressures of the ions must obey the perfect gas law. The
assumption is formally made in Boltzmann’s original deduction, and it
can easily be shown that any other will result in a law different
from (A).

+ Phil. Mag. xxiii. p. 551 (1912); xxv. p. 742 (1913).

362 Dr. 8. R. Milner on the Effect of

between the freezing-point and conductivity variations will |
result from the proposition. For if 8 is the fractional
alteration of pV, and therefore of u, and if further p=P,
we shall get

Tt) Via i. Gada

eT Ree Rp 20-8) ; x =1—8,
which is in conflict with the experimental result (3)
(Part I.). |

p and P, however, cannot be the same, as is evident from
the following consideration :-—Consider a pair of ions which
happen to be fairly close together and under the influence
of each other’s attraction, and let a number of representative
views be taken. In a certain fraction—e®/“"—of the cases
the ions contribute to the free ionic pV, in the remainder of
the cases the ions act as though bound together and contribute
nothing to pV, but in these cases the pair will make to the
measured PV exactly the same contribution as if the ions
formed an actual molecule. ‘The effect of electrical forces is
in this respect exactly similar to that produced by chemical
association. The electrical bond it is true persists when the
ions are well separated from each other, while the chemical
bond acts only at very small distances, but this difference is
immaterial in considering the molecular pressure which a
pair of bound ions will exert. For the type of force con-
sidered, however, the bonds are not confined to single pairs
of ions, but each pair must be considered as forming part of
a large group, with the rest of the ions in which it possesses
a certain mutual energy, and so is not free to exercise its
full molecular pressure. While a difference between p and
P may on these lines be inferred to exist, it is difficult to
settle exactly what it is*. It seems doubtful that it would
be such as to give the exact equality between ®, and B,
which experiment suggests unless the mutual energy between
each pair and the rest of the group is negligible.

(3) A theory which is to some extent intermediate
between (1) and (2) has much to recommend it. We must
infer from the preceding comparisons that the interionic
forces must extend over considerable distances, as it is only

* The straightforward way to settle this point is to calculate p by a
method in accordance with its definition on p. 357 and compare it with P,
The calculation can be carried out strictly by exactly the same method
as that by which P was originally determined (Joe. ct.). It has been
done, but unfortunately the numerical results in both cases can only be
obtained in an approximate form which is not sufficiently accurate to
determine definitely what is the difference between them.

Interionic Force in Electrolytes. 363

in this way that a satisfactory explanation of the failure of
the mass action law can be got. On the other hand, the
idea of the molecular pressure of a pair of associated ions
seems also necessary to obtain an accurate agreement with
the experimental equality of the freezing-point and con-
ductivity variations. Both conditions will be satisfied if
the law of interionic force be such as practically to confine
the electrical attraction to pairs of nearest ions. Now, the
view that in an electrolyte each + ion attracts every — ion
and repels every + ion is undoubtedly a highly artificial
one. How artificial it is is made evident by observing that
the mutual energy. of an ion with others would have to be
expressed as a sum of hundreds of terms before any close
approximation to its value could be obtained. These repre-
sent the mutual energies with the nearest ion, the next
nearest, the third nearest, &c., and form terms which partly
cancel each other as the successive ions are + and —.
It is unlikely that this state of things represents a physical
reality. Indeed, the assumption on which it is based, that
the action of the water molecules can be simulated by that
of a continuous medium of S8.1.C. the same as that of water in
mass, is hardly likely to be true. It is probably a good deal
nearer to the truth to imagine surrounding each ion a
number of polarized water molecules which tend to form
chains linking together pairs of temporarily nearest oppo-
sitely charged ions. Such an action would not be the same
as that of a uniform medium; it would be more analogous
to the action of iron filings in forming chains between two
magnetic poles. The general effect would be to increase the
attraction between an ion and the nearest one to it of unlike
sign at the expense of the attraction of more distant ones,
which latter might well be negligible in consequence.

The view of the constitution of an electrolyte which is
thus attained will satisfy both the requirements mentioned
above, which are essential to a satisfactory theory. On it
we may imagine all the ions divided into pairs formed of ions
which are temporarily nearest together, the individuals of
each pair undergoing continual change. Between the ions
of each pair electrical force exists, which in many ways is
similar to a chemical bond, but is different in others. Thus,
with chemical association, an ion is either free or combined,
it cannot be both together, but here it possesses simultaneously
characteristics of both conditions. The ions in each pair,
for instance, will be separated spatially from each other
nearly as widely as if they were quite free; they are free
(e. g. to carry current or exert ionic pressure) in a fraction

364: Prof. G. N. Watson on Bessel Functions

of the cases in which the pair is observed, while they act as
combined (exert a molecular pressure) in the remainder.
It seems to me that it is a theory on lines similar to these
which will ultimately succeed in reconciling all the difficulties
connected with strong electrolytes.

SUMMARY.

(1) A critical discussion of the way in which the law of
mass action fails for strong electrolytes leads to the conclusion
that the reduction in the molecular conductivity with
increasing concentration must be ascribed mainly to a
reduction in the mobilities of the ions, and not to a reduction
in their number by association into molecules.

(2) A theoretical investigation of the effect of interionic
force shows that identical variations with the concentration
will be produced in the conductivity and in the osmotic
pressure of the “free” ions (as defined on p. 357).

(3) The application of this result to strong electrolytes
shows that the variation in the conductivity and the freezing-
point can be best explained by a modification in the view we
take of what constitutes association. According to this ions
in strong electrolytes are not associated into molecules; they
are neither completely associated nor completely free, but
pairs of ions which are temporarily nearest together, in
consequence of the electric forces between them, will, in
a fraction of cases, act as if bound together, and in the
remaining cases as if free.

The University, Sheffield,
December 1917.

XL. Bessel Functions of Equal Order and Argument. By
G. N. Watson, 1.A., D.Sc., Assistant Professor of Pure
Mathematics at University College, London*. -

1. PPROXIMATE formule for the Bessel function

and its derivate, J,(n) and J,'(n), (when n is

large) have been discussed in numerous papers during the

last few years} ; several of these papers have appeared in
this Magazine.

* Communieated by the Author.

t Debye, Math. Ann. lxvii. pp. 5385-558 (1909). Rayleigh, Phil.
Mag. Dec. 1910. Nicholson, Phil. Mag. Dec. 1907, Aug. 1908, Feb.
1910. Watson, Proc. London Math. Soe. (2) xvi. pp. 150-174 (1917) ;
Proc. Camb. Phil. Soc. xix. pp. 42-48 (1917). Various numerical results
have also been given by Airey in a series of recent papers in the Phil.

Mag.

of Equal Order and Argument. 365

The associated function

1
i} Jn(nz)dx,
0

which does not occur in many of the physical problems in
which the other Bessel functions present themselves, appears
to play a prominent part in connexion with various series
arising in the theory of Hlectromagnetic Radiation, and
consequently Professor Schott has asked me to determine
whether there is any approximate formula analogous to the
results

Eg) 3°L(8)
Jn OO 2 aa 19 J 3 °
My mT 2338ns ya m2iné
This note, in which I prove the remarkably simple result
that

(*S.eadeood, os Vi atiiaene aah

is the outcome of his inquiry. A closer approximation is
given in $4, but it involves the gamma function of 1/3 ;
this more precise result is

(5 (ndaneeee eee
Joe ieee B85 nT Dh

In order to obtain this approximate formula I propose to
employ not the elementary methods which I have used
elsewhere * in connexion with J,(n) and J,'(n), but the
methods which depend on the contour integrals of Debye ;
the latter methods yield the desired result with a much
smaller expenditure of labour.

2. We take the well-known contour integral

(0+)
J,(n@) = :

5 pe), - Babs

Gin which the contour starts from —, encircles the origin
once counter-clockwise, and then returns to —), and on
integrating under the integral sign we get

‘i Le hree Re 4n(t—1/t) t—"dt
\ J,(nz)da= nineties se 1! Fa]

iy. dt
== St eae v9 1}

nit Y_« t?7—1

1 (1-1
+f" oni

* See the last of the papers cited.

366 Prof. G. N. Watson on Bessel Functions

In the last integral of all, we deform the contour into a
circular are of indefinitely great radius, starting and ending
at —o. Since n is supposed to be positive the integrand
is O(t~*) on the deformed contour, and so the integral is zero.
We thus obtain the formula
*O+) dt.

1 ,
) Jn(na)da= al gage) 1
0 —o t —l

i
nT
Now that the large variable x only occurs in a single

term of the integrand, we proceed to apply the methods of
Debye by choosing a contour on which

$£(t—1/t) —logt
is purely real.

Writing t=re", we see that the contour has to satisfy the
condition (r+1/r) sin@—20=0;
this equation is satisfied if @=0 or if

r= 6 cosec 0{1+,/(1—6-? sin? 9}.
Taking the upper sign, so that

r= 6 cosec 0{1+,/(1—0-? sin? @)}, . |, (8)
we obtain a contour of the required type if we take @ to
vary from —7 to 7.

The contour, which is symmetrical with respect to the
real axis, passes through (1, 0) and has an abrupt change of
direction at that point ; its direction immediately above the
real axis is inclined 4m to the positive direction of the real
axis *.

We thus get
- pie i e —nFE(6) t if dr :

where 7 is given as a function of @ by equation (3), t=re®
and f
F(6)=4(r—1/r) cos O— log r.
Now wat 1
t#—1° (r—1/r) cos 0 +2(r + 1/r) sin 6
_ (l/r) cos 0—1(7 + 1/r) sin 8
ie r* + 1/r?—2 cos 20

* This is easily proved; it is suggested by the fact that $(¢—1/t) —logt
has a triple zero at t=1. Various properties of the contour are given
in the first of my papers to which reference is made in§1. The curve
obtained by giving the lower sign to the radical is the inverse of the
curve obtained by taking the upper sign.

+t In my previous paper, it was convenient to call this function F(6, 1).

of Equal Order and Argument. 367

Since 7 is an even function of 6, we easily deduce that

‘Tt “or
{ Jy(na)da= = = {1a
0

ae + 1/r) sin 0(dr/rd@) — (—1/r) cos 6
y? + 1/7? —2 cos 20
Now it is easy to show that
(r+1/r) sin 0(dr/rd@) —(r—1/r) cos @
7 + 1/7?—2 cos 20
led ,27 sin 0

= ee 1—?7

jo sin 6 ie en sin. G4)
=3 a0 l+r asd 1—rcos@ j *
But the contour starts from (1, 0) in a direction making
an angle 27 with the line joining this point to the co ;
and so tan-1{rsin@/(1—rcos@)} decreases from 37 to 0
as @ increases from 0 to 7, while tan~'{rsin 6/(1+7 cos @)

increases from 0 to 7 in the same circumstances.
We therefore write equation (4) in the form

a i 9 e
\ J,(nxz)da= aa, ji—e*} a }tan™ a —n} dé
ae! =<[{1- enna} { tan- He ee -o} |
rs =e ae {n— oad ae sind F’(6)d0
To 1—r

on integrating by parts ; that is to say

1 7 ;
if Jn(na)dz= s ere { tan =r F’(6)d@, (5)

it being observed that F(0)=0, F(7)=, so that the
integrated part vanishes at each limit.

Now F’(@) is positive* when 0=6=7, so that F(@) isa
steadily increasing function of 6. Moreover, we can show
that m—tan~1{2r sin 6/(1—7r’)} is a steadily decreasing
function of 8. For we have

1d -2rsin

eG + ‘a. sin in O(de /rd0) — (r—1]r) cos 6
7 + 1/r*?—2 cos 26
By 6+ sin @ cos @— 26 cot 8 iu ‘
~ 2(6?— sin? 6 cos? @),/(0?— sin? 6) ae to

* Proc. London Math. Soc, (2) xvi. p. 153.

“d6. (4)

368 Prof. G. N. Watson on Bessel Functions

Now 6+ sin 8 cos 0 — 26? cot 8 vanishes when 6=0 and has
the positive derivate 2(cos 0 —6 cosec @)? ; hence it is positive
when 0<@<7. The function

7 — tan~' {2rsin 0/(1—7’)}

therefore has a negative derivate, and the desired result is
roved. °

Taking F(@) as a new variable, I, and writing $(F) for
the function m— tan! {2r sin 6/(1—7”)}, we have

- 1 5 V\ .—2k
( Jn(na)dae= — ( b(F)e "dk,
wOt te Le Fi)

where $(F) is a positive decreasing function of F such that
P(O)=37. . vs Mae

Hence, since the integral on the right is uniformly con-
vergent for large values of n*,

1 ies)
Lim [nf Ty(nse)de | = Lim =| h(u/n)e—“du
n> 0 n~—>o 7)
= (0)/r=3.

That is to say,

a)

{ Jn(na)da os ne

; 3n’
which is the result stated.
From the formula

Ty oO
n nn dae = a) h(u/n edu
0 To

it is evident that the function on the left steadily increases
(for all positive values of n) as n increases.

3. When v is an odd integer we can express the integral
as a sum of Bessel functions, since we have

wa nN
n| Tn(na)da= | Jn(y)dy

e/ 0-

In(Y) =In—2(y) — 25 m—1Y)

and

whence we find
1
nf Jn(nx) da
Q

=| 105 (y) —2J3_'(y) —— 2S 4'(y) eiates Soa 2J,-1(y) dy
0
=1—$Jo(n) + 232(n) + 23n(n) +... + 2d n-a() f.

* Bromwich, ‘Infinite Series,’ pp. 434, 436.

of Equal Order and Argument. 369

When n is an even integer, we find that

DY J. nzjde— \ "Jo(y)dy —24Fi(n) +I;(n) +... +In_aln) f-
20 0

The integral on the right does not séem to be expressible
by elementary functions, but we have

( Jo(y)dy =1— | Joly ay
sin (}7 —n)
/ (a7)

by integrating the ordinary asymptotic expansion of J(y).
The following table indicates the mode of increase of

ool —

1
nf J,(nx)d# to its limit (namely 4) as n increases through

Jo
the odd putegral values: L, 35 eae! .

:. th, J,(nx)de. | nN. ng n(mzjdz.
1 02348023 | 13 0:3175245

3 02878604 =| 15 03190390
har 5 03020018 =|, 03202397
Pare 0:3087657 || 19 0:3212187
| 9 03128134 || 2 0:3220347
ae 03155424 | 283 0:3227271

4, We can obtain a closer approximation to the integral.
in the following manner :—

When @ is small we have
9+ sin 6 cos O—26? cot dw 86°/45,

6? — sin? 0 w 64/3,

6? — sin? @ cos? 6 ow 464/3,

F(0) ww 469/(9,/3).
Hence from (6) we find that m—tan7!{2r sin 6/(1—77) }

is a function of @ approximately equal to
37 —O'/(5\/3),

and the complete expansion of this function involves even
powers of @ only.

370 Notices respecting New Books.
Since F'(@) is an odd function of 0, it follows that $(F) is

expansible in a series of ascending powers of F%, convergent
when F is sufficiently small, in which the first two terms are
given by the formula

b(F) coda — 2238 3/20,
and hence the integral of J,(na#) possesses an asymptotic
expansion in which the ratio of consecutive terms is of
order n*, the first two terms being given by the formula

: LY 2888
J,(nw)dwoo5- — - =I
{ mi. on 20nnt
a 23
wo — ——
dn 385nsT\(4)
This approximation gives the value of the integral correct
to four places of decimals when n= 23.

201

XLI. Notices respecting New Books.

The Electron. Its isolateon and measurement and the determination
of some of rts properties. By Ropurt ANDREWS MILLIKAN,
Pp. xii+268. The University of Chicago Press. Price 7s. net.

~TARTING with a brief historical account of the rise of the
electron theory, the author soon reaches the question of the
determination of the electronic charge e, and, after describing
briefly the early work of the Cavendish school, and the difficulties
encountered, he devotes special attention to the experiments
carried out by himself and his students in the Ryerson laboratory
of the University of Chicago. In these experiments, in place of
Wilson’s cloud, a single oil drop was observed; and Professor
Millikan gives a most interesting account of the method by which the
capture of single electrons by a drop was observed, the corrections
to Stokes’ law necessitated by the small radii of some of the drops,
and the final determination of e to be (+'774+:005) x 10-1” electro-
static units, and N, Avogadro’s constant, to be (6°062 +°006) x 1023.
Healso details experiments carried out on the Brownian movement
in gases to determine N.

Not long before the war Ehrenhaft published an account of
series of experiments which he considered to demonstrate the
existence of a sub-electron, or charge very much smaller than the
electron. Few English physicists found the work convincing,
but, nevertheless, it aroused some attention. Professor Milhkan
devotes a chapter of his book to discussing this question of a
sub-electron, and brings forward very strong arguments, based on
his own experience, for supposing Ehrenhaft’s results to be a
development of experimental errors and uncertainties. There
seems no doubt that, according to the best experimental evidence
at present available, the electronic charge e is constant and indi-
visible, as was universally assumed.

Notices respecting New Books. 371

The last two chapters are devoted to the structure of the atom,
and the nature of radiant energy. The author gives a clear
account of Moseley’s famous experiments on the characteristic
X rays, and of Bohr’s very successful atom model of Rutherford
type. In discussing the quantum theory he describes his expe-
riments on the initial velocity of the photo-electrons, and the
consequent determination of #. He concludes with a sketch of
the still unexplained difficulties which beset the construction of a
satisfactory theory of radiation.

The style of the book is very clear and pleasant, all but the very
simplest mathematical considerations being dealt with in a little
series of appendices. The Cavendish work, so familiar to English
physicists, is dealt with briefly, while Professor Millikan’s own
work, which has been less described in text-books, is exposed at
greater length. The short accounts of the most recent theories
are excellent. The book will be read with pleasure and profit by
the physicist, while at the same time its clarity and directness
render it available to any man of general scientific training.

Napier Tercentenary Memorial Volume. Edited by C. G. Knorr.
Published for the Royal Society of Edinburgh by Longmans,
Green and Co. Quarto. Price 21s. net.

THis volume of Essays and Addresses, contributed to the Ter-
centenary Congress, forms a worthy memorial of the publication
of Napier’s wonderful discovery. It is appropriate, too, that on
this occasion special attention should be drawn to the extensive
logarithmic tables calculated by Edward Sang, which have too
long remained unpublished. ‘* What more fitting outcome of the
Napier Tercentenary could there be than making accessible to
the civilized world the fundamental part of these great tables,
calculated in the very city where John Napier invented the
logarithm and gained undying tame as a benefactor of his
race ?”

In his inaugural address, Lord Moulton traces the different stages
in the development of the discovery of logarithms, whilst other
papers deal with the life and work of Napier. The question of
Napier’s claim to priority of discovery is fully discussed, and valuable
papers on logarithms and logarithmic computations form an im-
portant part of the volume. The Essays more directly concerned
with mathematical tables will be of most interest to calculators,
whose labours will be considerably reduced thereby. Prof. Andoyer
contributes an interesting and suggestive paper on fundamental
trigonometric and logarithmic tables. The arrangement of tables,
the reduction of the number of entries and of the mean error, and
a method of extending the accuracy of tables by improvement of
differences, include only a few of the numerous and important
essays contained in the volume. Mathematicians interested in
history and methods of calculation are greatly indebted to the
Editor, Dr. Knott, and to the experts who have made such valuable
contributions to the study of this subject.

372 Notices respecting New Books.

Modern Instruments and Methods of Calculation. A Handbook of
the Napier Tercentenary Exhibition. Edited by HE. M. Horspuren.
Geo. Bell & Sons and the Royal Society of Edinburgh. Price 6s.

Tae Handbook is a mine of information on all questions relating
to calculations, and the aim of the Editor and the Committee to
make this volume useful to those engaged in computation has
been fully realized. Special mention should be made of the list of
mathematical tables, including tables of logarithms and other
functions and of the chapters devoted to calculating machines and
mathematical laboratory instruments. Logarithmic computation
is being extensively supplemented by that of mechanical calculators,
and the increasing number of those who use these machines will
find the descriptive article on this subject of much interest. The
chapter on instruments gives a detailed account of integrometers,
planimeters, harmonic analyzers, and other mechanisms required
for special purposes. Other chapters deal with slide-rules, ruled
papers, and mathematical models. Ihe Handbook forms a fitting
companion to the Memorial volume, and will be a valuable addition
to the library of every student of mathematics.

Elliptic Integrals (Mathematical Monographs, No. 18). By Pro-
fessor Harris Hancock. Pp. 104. New York: John Wiley
& Sons; London: Chapman & Hall. 1917. Price 6s. net.

Tis excellently produced volume is one of a series of mathe-
matical monographs now. appearing in America. It contains
an account of the three elliptic integrals, the integral of the
third kind, however, receiving only: passing notice. Starting
with the definition of elliptic integrals as the integrals of ex-
pressions when cubics and quartics occur under the root sign, the
author gives the reduction to Legendre’s normal form, and illus-
trates some of the more obvious properties of the functions with
excellent graphs. After treating the sn, cn, dn, functions and the
Gudermannian he deals in detail with the reduction of various
types of integrals of the first kind to Legendre’s form, and then
passes on to the methods of numerical computation of the first
and second kind of integral. This part of the subject is developed
at some length, and illustrated with detailed numerical workings.
After a few well-selected examples and exercises the book closes
with five place tables of elliptic functions of the first and second
kind, taken from Levy’s well-known work.

The book is well-planned and clearly written, and can be under-
stood by any student well-grounded in the elements of the calculus.
Tt forms an excellent introduction, and will, we think, be weleomed
by those who, without having much time to devote to the subject,
wish to possess a concise and accurate account of the fundamental
properties of elliptic integrals,

WALKER. Phil. Mag. Ser. 6, Vol. 35, Pl. X.

oe

Pm.
Ir

a!
7

THE
LONDON, EDINBURGH, ann DUBLIN
PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

ee eK

—

[SIXTH SERIES.]¢ ~~

fA PF 3
x i ’ F

\ -
th ae
®,

MAY 1918.

XLII. On the Scattering of Light by a Cloud of similar
small Particles of any Shape and oriented at random.
By ord Rayueien, O.M., P.RS.*

OR distinctness of conception the material of the par-
i ticles may be supposed to be uniform and non-magnetic,
but of dielectric capacity different from that of the sur-
rounding medium; at the same time the results at which we
shall arrive are doubtless more general. The smallness is, of
course, to be understood as relative to the wave-length of the
vibrations.

When the particles are spherical, the problem is simple, as
their orientation does not then enter t. If the incident light
be polarized, there is no scattered ray in the direction of
primary electric vibration, or if the incident light be un-
polarized there is complete polarization of the light scattered
at right angles to the direction of primary propagation. The
consideration of elongated particles shows at once that a want
of symmetry must usually entail a departure from the above
law of polarization and may be one of the causes, though
probably not the most important, of the incomplete polariza-
tion of sky-light at 90° from the sun. My son’s recent
experiments upon light scattered by carefully filtered gases t
reveal a decided deficiency of polarization in the light emitted

* Communicated by the Author.

+ Phil. Mag. vol. xli. pp. 107, 274, 447 (1871), vol. xii. p. 81 (1881),
vol. xlvii. p. 375 (1899); Scientific Papers, vol. i. pp. 87, 104, 518
vol. iv. p. 397.

t Roy. Soc. Proc. Feb. 28, 1918.

Phil. Mag. 8. 6. Vol. 35. No. 209. May 1918. 2D

tA

‘1919 3

SAP @PAee

374 Lord Rayleigh on the Scattering of Light by a

perpendicularly, and seem to call for a calculation of what is
to be expected from particles of arbitrary shape.

As a preliminary to a more complete treatment, it may be
well to take first the case of particles symmetrical about an
axis, or at any rate behaving as if they were such, for the
calculation is then a good deal simpler. We may also limit
ourselves to finding the ratio of intensities of the two polarized
components in the light scattered at right angles, the prin-
cipal component being that which vibrates parallel to the
primary vibrations, and the subordinate component
(vanishing for spherical particles) being that in which the
vibrations are perpendicular to the primary vibrations.
All that we are then concerned with are certain resolving
factors, and the integration over angular space required to
take account of the random orientations. In virtue of the
postulated symmetry, a revolution of a particle about its own
axis has no effect, so that in the integration we have to deal
only with the direction of this axis. It is to be observed
that the system of vibrations scattered by a particle depends
upon the direction of primary vibration without regard to
that of primary propagation. In the case of a spherical
particle the system of scattered vibrations is symmetrical
with respect to this direction and the amplitude of the
scattered vibration is proportional to the cosine of the angle
between the primary and secondary vibrations. When we
pass to unsymmetrical particles, we have first to resolve the
primary vibrations in directions corresponding to certain
principal axes of the disturbing particle and to introduce
separate coefficients of radiation for the different axes.
Hach of the three component radiations is symmetrical with
respect to its own axis, and follows the same law as obtains
for the sphere *.

In fig. 1 the various directions are represented by points
on a spherical surface with centre O. ‘Thus in the rect-
angular system XYZ, OZ is the direction of primary
vibration, corresponding (we may suppose) to primary pro-
pagation parallel to OX. The rectangular system UVW
represents in like manner the principal axes of a particle,
so that UV, VW, WU are quadrants. Since symmetry of
the particle round W has been postulated, there is no loss
of generality in taking U upon the prolongation of ZW.
As usual, we denote ZW by 0, and XZW by ¢.

The first step is the resolving of the primary vibration Z

in the directions U, V, W. We have
cosZU=—sin@, cosZV=0, cos ZW=cos@. . (1)
* Phil. Mag. vol. xliv. p. 28 (1897) ; Sci. Papers, vol. iv. p. 305.

Cloud of similar small Particles of any Shape. BY

The coefficients, dependent upon the character of the par-
ticle, corresponding to U, V, W may be denoted by A, A,
C; and we seek the effect along the scattered ray OY,

Fig. 1.
Z,

U

perpendicular to both primary vibrations and primary pro-
pagation. The ray scattered in this direction will not be
completely polarized, and we consider separately vibrations
parallel to Z and to X. As regards the former, we have
the same set of factors over again, as in (1), so that the
vibration is Asin?@+Ccos?@, reducing to C simply, if
A=C. This is the result for a single particle whose axis is
at W. What we are aiming at is the aggregate intensity
due to a large number of particles with their positions and
their axes distributed at random. The mean intensity is

u

fi 4 A+ (C—A) cos? 6}? sin ado | sin 0d0
e 0 0

0. ead |
=A? 4 ACT AS 4, Co AY = (8A? +3074 440). (2)

This represents the intensity of that polarized component of
the scattered light along OY whose vibrations are parallel
to OZ.

For the vibrations parallel to OX the second set of re-
solving factors is cos UX, cos VX, cosWX. Now from
the spherical triangle UZX,

cos UX =sin (90°+ @) cos 6=cos 6 cos ¢g.
Also from the triangles VZX, WZX,
cos VX=cos VZW =cos (90°+ ¢) = —sin ¢,

cos WX=sin @ cos ¢.
2D2

3876 Lord Rayleigh on the Scattering of Light by a

The first set of factors remains as before. Taking both sets
into account, we get for the vibration parallel to X

—Asin @cos 8 cosf+ CU cos 6 sin @ cos ¢,
the square of which is
(CA? sin? 0 cos? 0 cos’ @. ) . a

The mean value of cos? is $. That of cos?@ is 3 and that
of cost is 4, as above, so that corresponding to (2) we have
for the mean intensity of the vibrations parallel to X

KC-A)A—-H=7(C-A. . . . @
The ratio of intensities of the two components is thus

(C—A)?
GAP+3074 400”

Two particular cases are worthy of notice. If A can be
neglected in comparison with CU, (5) becomes simply one-
third. On the other hand, if A is predominant, (5) reduces
to one-eighth.

The above expressions apply when the primary light, pro-
pagated parallel to X,is completely polarized with vibrations
parallel to Z, the direction of the secondary ray being along
OY. If the primary light be unpolarized, we have further

to include the effect of the primary vibrations parallel to Y.

The two polarized components scattered along OY, resulting
therefrom, both vibrate in directions perpendicular to OY,
and accordingly are bota represented by (4). In the case of
unpolarized primary light we have therefore to double (4)
for the secondary vibrations parallel to X, and to add to-
gether (2) and (4) for the vibrations parallel to Z. The
latter becomes

qs (9A? +4074 2AC),
and for the ratio of intensities of the two components

2(0— A)? (6)
OP 4-107 OAM eae

When A=0, this ratio is one-half.

For a more general treatment, which shall include all
forms of particle, we must introduce another angle > to

sil

Cloud of similar small Particles of any Shape. oy hy

represent the inclination of WU to ZW produced, fig. 2.
The direction cosines of either set of axes with respect to

Fig. 2.
"a

the other are given by the formule *

cos XU =—sin ¢ sin W+cos ¢ cos f cos @
cos YU = cosd¢dsiny+sin ¢ cos vf cos @
cos ZU =-—sin 6 cosy

cos XV = —sin ¢ cos W—Ccos ¢ sin > cos 0
cos YV = cosdcosw—sin ¢ sin cos 0
cosZV = sin@siny

cosXW= sin@cos¢d }

cos YW= sin @sing i Sth ot) Ce eRe dy as HD)
cos ZW= _ cos Y |

Oy a)

» + (8)

ae ae) Se ee se)

Supposing, as before, that the primary vibration is parallel
to Z, we have as the first set of factors

cos ZU = —sin @ cos yy, i

cos ZV =sin @ sin yp, '

cos ZW =cos 6 J

For the vibrations propagated along OY which are parallel
to Z, we have the same factors over again with coefficients

(10)

* See, for example, Routh’s ‘ Rigid Dynamics,’ Part I. § 258, 1897.
y and ¢ are interchanged.

378 Lord Rayleigh on the Scattering of Light by a

A, B, C as before, and the vibration is expressed by
A sin? 0 cos’ + B sin’ @sin? y+ Ceos?6; . (11)
while for the intensity
T= A’ sin‘ @ cos* a + B? sin* @ sin* + ©? cost 8
+ 2AB sin‘ 6 cos? W sin? + 2BC sin? @ cos? @ sin?
+2CA sin? 6 cos? 6 cos? Wr. . . . . .) {i
This is for a single particle, and we have now to take the
mean for all orientations. The mean value of sin*ty, or
costa, is 23 that of sin?ycos*y is }; and that of sin?’y
is 3. The averaging with respect to > thus yields
[= 3(A? + B?) sint 0+ C? cost @+1AB sin* @
+ (A+ B)C sin? @cos?6.". .. .
Again, the mean value of sin‘ @ is 48, that of cos*@ is }, and
that of sin?@ cos?@ is 3%. Thus, finally, the mean value
ot I over the sphere is given by

mean I=}, {3(A?+ B?+ C?) + 2(AB+ BC+CA)}. . (14)
This refers to the vibrations parallel to Z which are propa-
gated along OY.

For the vibrations parallel to X, the second set of factors
is cos XU, cos XV, cos XW, as given above, and the vibra-
‘ion is expressed by

— A sin @ cos y(—sin ¢ sin p+ cos ¢ cos cos @)
+ B sin 6 sin y(—sin ¢ cos f—cos ¢ sin cos 6)
4+Ccos?'sin@icor>. . .. (ss oye
Accordingly for the intensity
[= A? sin? 0 cos? (sin? $ sin? yr + cos? d cos? y cos? 0
—2 sin ¢ cos ¢ sin cos cos @)
+ B? sin? 6 sin? y(sin? ¢ cos? yr + cos? d sin? cos? 6
+2 sin ¢ cos ¢ sin ~ cos yr cos 8)
+ C? sin? @ cos? @ cos? h
—2AB sin? 6 sin cos ¥(sin? ¢ sin W cos p
— cos’ d sin cos cos? 8+ sin ¢ cos ¢ sin? cos 8
—sin ¢ cos ¢ cos? > cos 6)
+2BC sin? @ cos @ sin cos 6(—sin ¢ cos
—cos ¢ sin y cos 0)
—2CA sin? @ cos 8 cos cos ¢(—sin ¢ sin p
+cos@coswcos9). . (16)

Cloud of similar small Particles 0) any Shape. 379

In taking the mean with respect to ¢, the terms which are
odd in sing, or cos ¢, disappear, while the mean value of
sin? @, or cos? ¢, is}. We get for the mean

T=1A? sin? @ cos? (sin? w + cos? v cos? 8)
+ 3B? sin? @ sin? wv (cos? y+ sin? cos? @)
+43? sin? @ cos? 0
— AB sin? @ sinyy cos . sin W cos W sin’ @
— BC sin? @ cos 6 sin > « sin y cos 6
—CA sin’? @ cos Ocosy.cosycos#@. . . . (17)

The averaging with respect to y now goes as before, and
we obtain

4(A?+ B?) sin? 6(£+ 2cos? 6) + $C’ sin? 6 cos? 8
— iABsin‘@—3(A+B)Csin?6@cos?6;. . . . (18)
and, finally, the averaging with respect to @ gives
A?

andl g pees CF_ AB_ (A+B)
16 oa 15 1d 15

= J {A?4 B? +6 eb BO—CA 0 2 .(19)

mean [=

This represents the intensity of the vibrations parallel to X
dispersed along OY, due to primary vibrations parallel to Z.
It vanishes, of course, if A=B=C; while, if A=B merely,
it reduces to (4).

The ratio of the two polarized components is

A? + B?4+@—AB—BC—CA (20)
3(A?+ B? + C’) + 2(AB4- BC+ CA)’ . . 4

reducing to (5) when B=A.

If the primary light travelling in direction OX is un-
polarized, we have also to include primary vibrations
parallel to Y. The secondary vibrations scattered along
OY are of the same intensity whether they are parallel to Z
orto X. They are given by (19), where all that is essential
is the perpendicularity of the primary and secondary vibra-
tions. Thus, in order to obtain the effect along OY of
unpolarized primary light travelling along OX, we have
merely to add (19) to both components. The intensity of

380 Lord Rayleigh on the Scattering of Light by a
the component vibrating parallel to Z is thus
151 3( A? + B74 C*) + 2 AB+ BOT CAT
+ |, {A’ + B?+C?—AB-—BC—CA}
= {4(A?+ B?+ 0?) 4+ AB+BC4CA};. (21)

while that of the component vibrating parallel to X is
simply

{A?+ B?+C?—AB—BC—CA). . . (22)
The ratio of the two intensities is

(A? 4+ BE C?—AB—BC—CA) s

4(A?+B?+0)+AB+BC+0a’ ~ &
reducing to (6) when B=A.

It may be observed that, since (21)=(14)+(19), we
obtain the same intensity whether we use a polarizer trans-
mitting vibrations parallel to Z and no analyser, or whether
we use an analyser transmitting vibrations parallel to Z and
no polarizer.

If neither polarizing nor analysing apparatus is employed,
we may add (21) and (22), thus obtaining

zi, [ 6(A?+ B?+C?) -AB—BC—CA]. . . (24)

When the particles are supposed to be of uniform quality,
with a specific inductive capacity K’ as compared with K for
the undisturbed medium, and to be of ellipsoidal form with
seml-axes a, 6, c, we have

: K/—K K'’—K K'—K

Sta. Hr eee eT, 5 oe See a ee
Att Oh ee aK L:1+ rea M:1+ re N,
where 2
ey dn .
L=2mabel (a? +r)? (0? + r) 72(c? +r)? - (26)

with similar expressions for M and N.

If the ellipsoid be of revolution the case is simplified *.
For example, if it be of the elongated or ovary form with
eccentricity ¢,

a=b=cV/(l—e’);. Vitehtide (hE Te hee) aa
Lt) Tse
Tanke “log +h yen tm it

e 2e° l—e

Layee!
L= mati

visitas | pee
N=4n}.,-1} | slog eit. (29)

* See the paper of 1897.

Cloud of similar small Particles of any Shape. 381
For the sphere (e=0)

4 RE cp
tL) ee)
In the case of a very elongated ovoid, L and M approximate
to the value 277, while N approximates to the form

a? 2c bs
N=4nS (log { —1). Haale Maes

vanishing when e=1. It appears that, when K'/K is finite,
mere elongation does not suffice to render A and B negli-
gible in comparison with C. The limiting value of C: A is
in fact 3(1+K’'/K). If, however, as for a perfectly con-
ducting body, K'=x, then C becomes paramount, and the
simplified values already given for this case acquire
validity™.

Another question which naturally presents itself 1s whether
a want of equality among the coefficients A, B, C interferes
with the relation between attenuation and refractive index,
explained in my paper of 1899T. The answer appears to
be in the affirmative, since the attenuation depends upon
A?+B?+C’, while the refractive index depends upon
A+B+GC, so that no simple relation obtains in general.
But it may well be that in cases of interest the disturbance
thus arising is not great.

The problem of an ellipsoidal particle of uniform dielectric
quality can be no more than illustrative of what happens
in the case of a molecule; but we may anticipate that the
general form with suitable values of A, B, C still applies,
except it may be under special circumstances where reso-
nance occurs and where the effective values of the co-
le may vary greatly with the wave-length of the
light

* But the particle must still be small relatively to the wave-length
within the medium of which it is composed.

Tt An equivalent formula was given by Lorenz in 1890, ‘Ciuvres
Scientifiques, t. 1. p- 496, Copenhagen, 1898. See also ’ Schuster’s
‘Theory of Optics,’ 2nd ed. p. 326 (1909).

ps2 4

XLII. Note on Fou Talbot's Method of obtaining Coloured
Flames of Great Intensity. By G. A. Hemsautycu, Ho-
norary Research Fellow in the University of Manchester*.

| ies RAYLEIGH has very kindly called my attention
to a short but interesting paper by Talbot, in which
the latter describes an exceedingly simple device of pro-
ducing intensely luminous metal vapours in the flame of a
spirit-lampt. In view of the historic interest attached to
this paper as recording an example of the endeavours made
by the early spectroscopists to improve upon the efficiency
of the light sources at their disposal, and also in consideration
of the fact that the method is not mentioned in modern works
on spectroscopy, it may perhaps be of some use to recall
Talbot’s experiment, and to give the results of a few
personal observations on the same subject.

In Talbot’s own words the method is as follows :—“ It is
only requisite to place a lump of common salt upon the wick
of a spirit-lamp, and to direct a stream of oxygen gas from
a blowpipe upon the salt. The light emitted is quite homo-
geneous and of dazzling brightness. If instead of common
salt we use the various salts of strontium, barytes, etc., we
obtain the well-known coloured flames, which are charac-
teristic uf those substances, with far more brilliancy than by
any other method with which I am acquainted.” It is to be
noted that Talbot precisely states that the stream of oxygen
is directed upon the salt. Noexplanation is, however, given
to account for the result achieved, and it is not certain
whether the greater luminosity obtained in the manner set
forth is due to an actual combustion of the salt or to a more
effective dissociation of the latter, caused by a rise in tem-
perature of the flame. The following series of experiments
was made with the object of elucidating the mode of action
of the oxygen in Talbot’s experiment and of applying his
method to the air-coal gas flame.

1. Observations with a spirit flame.

With a small piece of salt on the wick the flame appeared
feebly coloured yellow and showed the D-lines only. A
current of oxygen gas escaping from a glass nozzle was then
directed towards the salt, and the latter immediately burst
out in a brilliant light, emitting the D-lines most strongly,

* Communicated by Sir E. Rutherford, F.R.S.
+ H. F. Talbot, Philosophical Magazine, vol. iii. p. 35 (1833).

Fox Talbot’s Method of obtaining Coloured Flames. 383

as alsoa fainter line on the more refrangible side of the
former, probably the pair at 25685. The greater part of
the flame, although more intensely coloured than without
the oxygen, remained, however, relatively feeble as com-
pared with the vivid brightness exhibited by the immediate
vicinity of the salt. Thus the light distribution in the flame
is not very uniform, there being a most pronounced maximum
near the salt.

In addition to the solium lines the bands of the Swan
spectrum were likewise very prominent near the base of the
flame where the oxygen passed through. In fact the path of
the oxygen through the flame is rendered visible by the
more vigorous combustion of unburnt hydrocarbons, which
gives rise to the emission of a greenish light (more bluish
and fainter when the oxygen passes through the upper parts
of the flame, where less unburnt gas prevails).

When the oxygen was passed through the coloured region
of the flame situated just above the salt there was hardly any
increase in the intensity of the light emitted, as though the
free radiating sodium atoms remained unaffected by a rise in
temperature. (The fact that the temperature of the flame
is “aa raised by the oxygen will be demonstrated
in § 2.)

Similar results were obtained with calcium chloride. But
in this case there was a notable brightening of the spectrum
bands when the oxygen was sent through the middle region of
the flame well above the salt. This may be due to undissociated
particles of the salt being carried upwards through the flame
and broken up as they enter the presumably hotter zone of
the oxygen path.

Thallium chloride showed no line when placed on the wick
of the spirit-lamp. But as soon as oxygen was blown against
it, the green line appeared not only in the vicinity of the
salt but also in the upper part of the flame.

Thus it is evident that Talbot’s method constituted an
important improvement in the days when the spirit flame
was practically the only means in general use of vaporizing
substances for spectroscopic purposes.

2. Observations with an air-coal gas flame.

When oxygen is passed into the flame given by an ordinary
Bunsen or Méker burner, there is a great tendency for the
former to strike down the burner-tube, owing to the increased
velocity of the explosion. This inconvenience was completely
obviated by placing on the top of a Méker burner a perforated

384 Mr. G. A. Hemsalech on Fox Talbot's Method of

brass plate, } inch thick, having 49 one-millimetre holes per
square centimetre. Experiments were made with the salts
of sodium and calcium. The greatest intensity of the metal
vapour emission was also in this case obtained when the salts
were held near the base of tlie flame or placed on the per-
forated plate. When the oxygen was passed through the
luminous vapour in the region above the salt, about half way
up the flame, the brightening along the oxygen path was
relatively feeble; but when now the salt was raised and held
in the path of the oxygen, the intensity again was fairly
great, though much less so than when both the stream of
oxygen and the salt were close to the explosion region. On
the other hand, near the tip of the flame, where combustion
of the gases is nearly complete, no luminous effect was ob-
served when oxygen was blown against the salt. Thus the
oxygen, in order to be effective by this method, must be
directed upon the substance to be vaporized in the presence
of unburnt gases (hydrocarbons and hydrogen). This fact
is further illustrated by the following observations, which at
the same time prove that the light effect obtained in Talbot’s
experiment is not due to mere combustion of the material
placed on the wick: when the salt is held in the flame near
the border, and the oxygen enters from the opposite side
(fig. 1) in such a way that it traverses a long zone of flame

Fig. 1.

[—Explosion region

Maximum Effect.

containing unburnt gases before encountering the salt, the
luminous effect is a maximum. If, however, the salt be held
near to the point of the border at which the oxygen enters,
no effect is observed (fig. 2).

obtaining Coloured Flames of Great Intensity. 385

That the temperature of the flame is raised when oxygen
is passed through near the base is easily shown in the
following way:—A piece of iron wire 1 millimetre thick is
held in the fame about half way up from the burner-plate.

j / | \
(oe tae
| ie ba )
x ; i i sed

\ acai eel

|

No Effect.

In this position it will be raised to a bright red heat. If now
a stream of oxygen be directed through the flame just above
the explosion region, but well below the wire, the latter
immediately becomes very much brighter. The same heating
effect is observed when the oxygen is blown against the wire
within the flame.

Hence it is evident thatthe increased intensity of the light
emission in Talbot’s experiment is of thermal origin, at least
in so far as the rise in the temperature of the flame entails a
more vigorous chemical dissociation of the salt. This con-
clusion was further corroborated by directing the high-
temperature flame from an oxy-coal gas blowpipe upon the
salt on the wick of the spirit-lamp. The salt in this case gave
out an exceedingly brilliant light, in fact much more so than
with the oxygen alone, anda large volume of luminous vapour
rose into the spirit flame above.

3. Application of Talbot’s method to flames coloured by
means of a sprayer.

The foregoing experiments were made in accordance with
Talbot’s original procedure of blowing the oxygen against a
lump of salt held in the flame. Butthe method is equally well
applicable to flames into which the material to be vaporized
is introduced in a very finely divided state, such as is provided
by a sprayer. Also in this case it was found that the greatest

386 Mr. G. A. Hemsalech on Fox Talbot?s Method Oj

effect is obtained when the oxygen enters the flame near its
base, and the effect vanishes near the top. The reason for
this is of course the same as before, namely, the accelerated
combustion of the unburnt gases which predominate near
the base. It is, however, well to remark that the intensity
of the coloured flame produced in this way falls appreciably
short of that obtained when the oxygen is thoroughly mixed
with the air and coal gas before these gases reach the burner
plate ™.

In order to observe the effect of varying the relative pro-
portions of the gases in the air-coal gas mixture, an electric
sprayer was used in connexion with burner No. 1 described
ina previous paper. With this arrangement the funda-
mental condition for the successful working of Talbot’s
method, namely, the presence of unburnt gases, is most
strikingly demonstrated by the following experiment:—The
glass nozzle through which passes the oxygen is held close
to the edge of the flame about 4 inch above the burner-
plate. The velocity of the oxygen is such that its curved
path can be distinctly followed to the opposite edge. When
the air-coal gas mixture is so adjusted that the explosion
region just begins to rise from the burner-plate, otherwise
stated when there is an excess of coal gas in the mixture
with consequent deficiency of combustion, then the portion

Fig. 3.

WN
Path of Oxygen

\
/

LL;
ie

Hixeess of Coal Gas.
of the coloured flame situated above the sharply defined
oxygen path is most notably increased in luminosity, as
indicated by the shaded portion of fig. 3. If now more air

* Hemealech, Philosophical Magazine, vol. xxxiv. p. 243 (1917).
+ Hemsalech, ibid. vol. xxxiil. p. 6 (1917).

obtaining Coloured Flames of Great Intensity. 387

be admitted and the combustion of the coal gas thereby
rendered more complete, the luminosity of the flame above
the oxygen path gradually decreases and finally becomes
less than that of the region below, as sketched in fig. 4. In

Excess of Air.

fact, when viewed in the direction of the oxygen path, the
luminosity along and above it is found to have been com-
pletely arrested, the reduced light as seen sideways being
merely due to the thin borders of flame rising up on either
side of the oxygen stream. That the extinction of the
luminous vibrations in this case is caused principally by
actual cooling of the vapour and not merely by oxidation, is
shown by the fact that when a stream of nitrogen is passed
into the flame the same extinction is observed. On the other
hand, a stream of ammonia does not reduce the intensity of the
light emission in the upper zone; the feebly luminous flame
with which the ammonia burnsas it passes through the air-coal
gas flame seems to be just sufficient to keep up the tempe-
rature of the latter. Also when a high temperature oxy-coal
gas flame is directed through the air-coal gas flame under
these conditions, the portion of the latter situated above the
path of the oxy-coal eas flame remains always a little
brighter. All the above observations were made on the
luminous vapours of sodium, strontium, lithium, calcium, and
barium.

Summary.

The series of observations recorded in this note clearly
shows that the rédle of the oxygen in Talbot’s experiment is
to increase the rate of combustion of the unburnt gases in

388 Prof. A. Anderson on the Problem of Two and

the flame, thus causing a rise in temperature of the latter.
The temperature is highest along the path of the oxygen, and
when the latter flows in the direction of the substance to be

vaporized, the full effect of the temperature is, as it were,

concentrated upon it. The effect is greatest near the base
of the flame, where combustion is only beginning, and least
near the tip. It is consequently essential that the oxygen,
prior to reaching the substance, should pass through a region
of the flame containing unburnt gases.

In conclusion, I wish to thank ‘Lord Rayleigh for having
provided me with the opportunity of paying a modest
tribute to the work of one who came so near to “discovering
spectrum analysis.

Manchester, Feb. 18, 1918.

XLIV. On the Problem of Two and that of Three Electrified
Spherical Conductors. By Prof. A. ANDERSON, J.A.*

\ 7 HEN an insulated conducting sphere of radius a is

charged to potential A, the potential V due to the
charge at any external point, P, whose distance from the
centre of the sphere is 7, is given by

rV=aA.

If now, charged bodies are brought into the field, this
equation no longer holds: we have, instead,

rV +aV'=aA,

where V' is the potential that the introduced bodies have
at P’ the inverse point or, as we may call it for shortness,
the image of P in the sphere. A has, of course, altered in
value and V is, as before, the potential due to the charge on
the sphere.

This equation may be used to find the coefficients of capacity
and induction of two conducting spheres.

Let the potentials of the spheres be A and B, their centres
O, and Os, their radii a and 6, and the distance apart of their
eentres c. Also, am fig. 1, let I, be the image of QO, in B,
I, the image of J, in A. ia the image of I, in ce and so on,
raha in oie sphere being the image of [, in that sphere.
We have thus a series of points L. ie I;, I,, &c. inside the

* Communicated by the Author.

lee

that of Three Electrified Spherical Conductors. 389
sphere B, and a series I,, Iy, Ig, Ig, &e. inside A. Let A,,
B,, denote the potentials due to the charges on the spheres A
and B at the point I, and By the potential due to the charge
on B at the centre of A.

Fig. 1.

an

We have ,

cBy +0A,=0B,
Ori ° Ay + aB, = aA.
C a aA
| BO Ouageean OGEP
and, likewise,
Oot. aA
ae B,— OT: B,=B- 0,1,’
Ont, a e. aA
b ° B,- Ow . B,=B-—- 0,1. .

Oss, a aA
| = ite Bo, ona a Bo, 9= BD O; 1. aie

Hence we have

aA ab
,: Bo=B oF “te OL Oi) OT, 01 1c Oulg
aA a*l?
+(B— 57) qnemp., 0s Ou,
ao cA ash?
+ 7 Oks eT... 0,1; . Os Oe 1041,
RE,
+(B- ad oi) ahr
Osten. 0,140 Onley, Ogls,
St

Phil. Mag. 8. 6. Vol. 35. No. 209. May 1918.

2K

390 Prof. A. Anderson on the Problem of Two and
But if E is the charge on A, A=By+ a and therefore

E ee — a Bo-
Thus

ab
aie {a+ (ort 20,1, . Oe
ab?
7 0,1, 0), 04: ee
nn
isk 3 O iy e eee Ox Tena. ele ee me OS a - :

-B2 fit a 7+ azb? ‘
O Le ° Gy . Oye Te ° On;

ab”
echo ON MMONKE cE OH Fes 07, ae

Thus 91, and giz have been found, and, of course, also qos,
by a simple application of the above equation.

The same method is applicable to the case of several
conducting spheres. for three spheres whose centres are
at the corners of a triangle the work is necessarily much
longer than that for two, but it is possible to find the values
of the coefficients of induction and capacity to any degree of
approximation.

We require for the solution of the problem a set of points
1, L, 1,... inside B, and a set I,, 1, I¢,... inside Aj asm
the problem for two spheres, and corresponding to these,
for A and C, a set Ky, K3, K;,... inside C, and a set K,, Ky,
K,,... inside A. But other points besides these are needed.
Take one of the points, say I;, inside B. We take its image
in C and denote it by L;,, the image of this in A by Lg,
the image of L3. in B by L33, and so on, going round again
and again in the positive direction.

For the I points inside A we do a similar thing, the image
of I, in © being Ly, and that of this in B Ly, and so on,
going round in ‘the negative direction. We do the same
thing for the K points. Thus the image of K; in B is M;,,
and the image of this in A is M;., and so on. But this does
not exhaust all the points required. Starting from any
Lor M point, we reverse the direction and find an infinite
series of points for it. Thus, taking the point M5, its
image in Cis M;3, but we also take its image in B and call
it Ms, the image of this in C, M59, and so on. A few of
the points are shown in fig. 2.

that of Three Electrified Spherical Conductors. 391

The centres cf the spheres are the points A, B, C, and we
shall use these letters to denote also their potentials; «, 8, y
are the lengths of the radii of the spheres, and a, 6, ¢ the

lengths of the sides of the triangle. The potential due to
the ; sphere A at any point, say @., will be denoted by Ag,,,
the potential due to B at Ls; by B;,,, and so on. The object
of the problem is to find B,+ Cy.
We have
c. Be +@(A;i,+Ci,) =@B,

b . Caty(Az,+ Bz) == ry).
eu ca eene | B Yay) ie Cian
B,+C,+ Aa F pant 7 Un a 5 Pn = ~P ee C.
Thus the first two terms of the exprassion for B,+C, are

Pp+to.
c b

392 Prof. A. Anderson on the Problem of Two and

Again, AT, . A; +a(B,, +0;,) =A,
AK, . Ay, +2(B;,4+ C,) = ;
hence
fk ea ene a8
BatCat Vat + Bag oe ay, (Bi + C; )- Bn wi (Bz, + C;,)
ieee. YA
ees (% Ale ws AK, )a.
Also, BK, 2B, + 8( An. + Ca.) =e:

CI. Ci, +9(An, is B,,) ae YC,
and Aly . Ar,+ «(Bz,. Se Ci,.) a= aA.
AM,, . 3 ae + aC ben > Cri) = aA,

from which we get, by substitution,

Bz ac Cy.- acl (B,. ie CJ-

1 By 1
c. AL, rs TAK, (Bx, + C;, )— +. Be Jims .
ee bel | (Bye a
Seay One ee bh. BK. AM,, my 2 ny»)
Pics os oe

+ golly, 6 + Se)

Be oe

AL,
eae, Py | ss ee
bh e A Sonat ie B Kee -B- Gis OL : C

1
+ ary i a AMG,” ¢. CL, Alig 5

Thus, as far as the sixth term,

DAA B
Boe (A WR fe

+B(P-

an ae

that of Three Hlectriped Spherical Conductors. 393

Proceeding in this way, we find that as far as the fourteenth
term,

i Biss 1
By+ C= —a[ A, ae AK — ay ( ; BK as
1
BAPTA ar) a
+B [fe - By af" a: Bry
b. BRoe Cc. Al, 6 Bisy Dx AK,. Bk,
Bry |
De Ces Bis
Vee ay" 1 eey 1)
Sees SUCK ee CK, 7 ook sO,

4. YB ]
DBR SOME, |

By continuing this process we may find any number of
terms of the expression for B,+ Cy.
Now, if E is the charge on A, we have
sprue lt
A=B,+Ga7 =,
a

or K=Aa—(B,4-C,).

oe ea a pa

vy b ° BK, .AM,, Gas Cl, . Alina tek

aS) aby) ea
=B|* h. BK, Ge BT,
ea apy
oy epee cL BL |
ea a ee By ary?
= co. Cl, | Ape CK;
By _ay*B sf:
TAL, Clee bi: PBK: . CMe

Thus the first terms of the expressions for q,, 912, 913
have been determined and the corresponding terms of qa,
933) 923 may be got by changing the letters. We can get
the formula for two spheres. from this by making b, AK,,

BK,, AL,,, BL, all intinite and C=0.

394 Prof. A. Anderson on the Problem of Two and
Thus

aa ee 22 B a3 223?
B=A [a+ "ey +...|-B[S 4

f c? — 8? a
or, since Al,;= B’, and BI,=c—— —;,
c e—p
a’ B ap a2?
nma[a+ 8, 4..]-3[24 7,
c"— 8? oe c e(c?— 8? — a”) %
As an example, suppose we have three small equal con-
ducting spheres, whose centres are at the corners of an
equilateral triangle the length of whose side is c. The terms

3
we have found will give 911, Giz, Gis correctly to (*)

2af Dee
p= qn=qu=4( 1+ pment =),

a” 1 a Qa?
Vin = 913 = 923= — = ‘ah aie =);

from which we obtain the coefficients of potential

a Pa £ 2a
PT fee 8 (1— Po TF =) ?
i dat at) hae
Pie Pies 5 (1+ a a 20 ee
The energy of the system, each sphere being supposed to
have unit charge, is

a ; 3
G+)

Be. @ Cc

and the force acting on one of them is

= 15) poe
J3(5 ea =);

ct
the force being oe in the case of point charges.

By the above method the potential due to two charged
conducting spheres at an external point can be written down
easily. Let the centres of the spheres be A and B, their
radii a and 6b, the distance between the centres c, and the
potentials U and V. Let the image of any external point
P in B be P,, the image of P, in A, P,, the image of P, in B,
P;, and so on. Also, let the image of P in A be Q,, the

image of Q, in B, Q:, the image of Q, in A, Q;, and so on.

that of Three Electrified Spherical Conductors. 395
The potential at P is
Uae = ped — ‘ie eee
Beebe AP, APSR) ZAG.
a*b? arl?

=BP.AP,.AP,.BP, AP. BQ,.BQ,.A0..AQ

4

a®h? 4 )
fon OAL, Ab. ompaees BPs:

4V 2) hf ab ab?
(ep AP< BQ, BE. AP). BP,
a2b? a2h3

PAP. 6O,. 60,40) BP.AP, AP,. BP, BP,

a®bh?
ao no eee a)

If, now, the image of A in B is J, the image of I, in A, I,,
the image of I, in B, I;, and so on, and if the image of B in
A is J,, the image of J, in B, Jo, the image of J, in A, Js,
and so on, it is easy to show from similar triangles that

BP. AB=6.PI,,
AP. BQ, A@e=e. PT,. Aly.
BP /AP(( BP,. Meee Pl, . BE! AT,
AP. BQ,.AQ.BQ;.AQ,=c. PI,. Al;. BI,. Al,

Hence the potential is
a ab ab
U(xp -ePE ¢. Presse
a7b? a®b?

elPl BE, An MeaeL, Al, BI, AT,

abe HS )
i e,P1,. DE, AREAL, 5

b ab ab? a*b?
a V(ap Saeed, * ¢ POEL 2. PJ, Alga +) ;

which shows that the potential outside the spheres has the
same value as that due to a series of point charges at

396 Mr. W.G. Bickley on Two-Dimensional Potential
A, B, i, Ji, L, Jo, Is, J3, I, Ju, &e., equal respectively to

Ua. vp. Usb Mab Ua Nalt Se
Tee Gs? eT,” Gola eine eer

Va?b? Uae? Va2b3
Soceses

6h Ads Boy? CEL BI, . All? ets eae eee

which are the image charges in the usual way of treating
the subject.

Note.—The above paper was written before the one that
appeared in the March number of the Philosophical Magazine
on the same subject. It is, perhaps, unfortunate that the
word ‘“‘image” has been used for “inverse point.” The
method has, of course, nothing to do with electrical images.

XLV. Some Two-Dimensional Potential Problems connected
with the Circular Are. By W.G. Bicxury, B.Sc.*

§ 1. iL this paper a method of dealing with potential

problems in two dimensions, depending on the

use of functions of a complex variable and of the method of

images, is applied to the solution of problems connected with

un infinitely long lamina, the section of which is a circular

arc. The results obtained are interpreted in terms of
electricity and hydrodynamics.

§ 2. The first stepin the investigation is the determination
of the transformation by which the two sides of the are in
the z-plane become the real axis in the plane of an auxiliary
variable € (=&+). The arc is taken as that part of the
circle z=—ve® for which —«=@<za, so that the angle sub-
tended at the centre is 2«. For any point on this circle the

ratio (1+.e2)/(¢+4) is purely real, its value being cobs, sO

that when @ lies within the above limits, the values of

tne 4 / tate ow (=) —] os

are purely real, but become complex when @ lies outside
them. Also, when <>—1, the expression (1) tends to 0 or #
according as the root is taken negatively or positively ; and
when 0=-+a, the expression has the values +1.

* Communicated by the Author.

Problems connected with the Circular Are. 397
Hence the transformation
sin (1+u2) + y/ sin? (1+12)?—cos?4 7 us
pe ee)
a
COS 5 (+2)

transforms the two sides of the z are into the real axis of
the &-plane, the extremities of the arc into the points f= +1.
This relation may also be written’

bo

Ccoss + 2uesing + cos—
Jape aga Se Li heey!

Maem? 96 aie? Me
¢ COS 5 —20€sin 9 + C085

2

It is easily found from (3) that z=2 corresponds to

c=1(sing +1)]cos5.

only one of which is in the upper half of the €-plaue.
§3. Suppose that in the &-plane there is a charge at the
point

R

_ & a
—— (2 + sin 5)) COs 5,
and that the real axis is a conductor. This will correspond
to a charge at infinity in the <-plane, with the are a conductor,
or, what is practically the same, to the charged conducting
are in space. Letw,as usual, denote the potential d together
with ¢ times the conjugate function y. Then in the €plane,
the method of images gives

“

{ 1+ sin; 1+ sin , (
w= lov €—, —__— | || ¢+ .—_— a a)

coo) i a
OSs COS .
i | 2

[bOI R

or, what is the same thing,
ye
1+ sin;
c= 7 Ree | Pa rarer . . . ° (9)
2
The elimination of & between (3) und (5) then gives, after

some reduction,

as
1+e-”’sin 5
z=—t a) GY 5)
if Oe
1l+e”sin=

2

398 Mr. W. G. Bickley on Two-Dimensional Potential

or
ub paneer nee a eh 8
w= —— log | 5 e cosees { (e+ t) +4/ 2? + Quz cos 2—i} |. (7)

‘With regard to the branch of the two-valued function to
be taken, the arc may be considered as the branch line
joining the two branch points (its extremities) and that
branch is to be taken which, inter alia, is equal to +4 when
z=0(0, and tends to +z asz>u.

Equations (6) and (7) then give the potential due to the
charged arc. Separating the real and imaginary parts of
(6), after writing —(¢ +s) for w (since the real part is to
be negative) we obtain

2 sin; cosh g sin y + sin? 5 sin 2p |

L£=

|
1+ 2e-% sin = cos w+e7? sin? = | f
2 2
i | ae |
1+2 sin 3 cosh cos r+ sin” 5 COs 2p | ;
Ls Camm a }
20-% sin— =p
1+2e sing coswp +e sin 9 J
Fig. 1. |
aml
‘i NY
Wee He
7 a rants
Vo

Using (8), the equipotentials and lines of force are easily
plotted. In fig. 1 they are drawn for the case of a semicircle,

Problems connected with the Circular Are. 399
with intervals °25 of @, and 73 of. These show distinctly

the physical features, which of course were anticipated in
a general way. The charge resides for the most part on
the outer face, the surface-density becomes infinite at the
edges, and at a distance the equipotentials are approximately
circular. One point of some interest appears in the figure,
namely, the approximate uniformity of the field on the axis,
just beyond the centre.

If we expand (6) on the supposition that | w| (and conse-
quently |z|) is large, we obtain

_ ae ing +e" os?)

z~ —ve-" (sing +e" cos’ ) ,
ere e+ (CO ie sing, Be PELE he i)
showing that the field at a great distance is approximately
that due to an equal charge at the point —1eos?5. It is

seen that this point, the “centre of charge,” is the mid
point of that portion of the central radius cut off by the
chord of the are. This result may also be obtained by
integration from the expressions for the surface-density now
to be obtained.

§4. On the lamina, w=ap, while z= —ve’, and from (6)
or (7) we obtain the relation

: GN ia Je
sin (— 5)sing =P SUNS eas 24 (3 (10)
. . . dp
Now the surface-density is proportional to Wa” and so

cos i
+ 1}convex face,

a

)
wae vi ha |
2 2

matt. <UL)

ox [_ si a concave face. |
Ta _ sin? J

The values of c, multiplied by the factor sin 5 (which makes

ox

the mean of the values at the pole equal to 1, and so

|

(1-000 | -913/|1-087 {1-000 |! -741 1
1-020 | -933|1-107 {1-020 | -761 |1-2
£091 1-003 |1-177 |1:090 | -828 1:3
(1-250 [1-162 |1-336 |1-249 || -981 |1°5

1-9

Plane.| a—10°. a—30°.

1-667 |1:577 {1-758 |1-664 1-383

400 Mr. W. G. Bickley on Two-Dimensional Potential

facilitates comparison), are tabulated for the values 10°, 30°,
90°, and 170°, of «, for the central point and for four more
dividing the semi-arc into five equal parts. The fractions of

charge on the two faces are added, and also the values for

a plane lamina. The third column in each case gives the
mean of the first two, and even for «=30° shows no great
difference from the values for the plane.

Table of Surface Densities.

ago ||) + cea

|

1-000 |

ba

000 | -293

"240 404

| 472. +528 aT 585 25 | +47
| a |

These may be compared with the values obtained by
Lord Kelvin for the analogous problem of the spherical
bowl.

§ 5. Of course, the above results may also be interpreted
as applying to the motion of an incompressible fluid cireu-
lating round the iamina, but this is passed over as of relatively
small importance. The more interesting problem of the
flow past such a lamina when placed in a steady stream, or
the motion in an incompressible fluid due to a motion of
translation of the lamina, may be solved in a similar manner,
by taking a double source in the €-plane. In view of the
final hydrodynamical interpretation it will be convenient to
make w purely real along the real axis and to choose the
strength and inclination of the source, so that w->—ze-* as
SPD,

This gives the value of w as a function of €:—

/
5 | 0278) -9782

¢ eB eB
2 sin $ (1+ sin$) | cts vie
i 2h ee 1+ sIn 5 1+ sing L
re Pe ane 1 €—-e——— + e-—--—— f
COS 5 | Gan wae NG
L 2 By

409 sin 5 sin eC, (iz

where the last term has been added to give the final result a

i |1-707 -0038 |1-9962 | 1-0000

1-020 '-305 |1-720 1-018 |:0042 |1-9962 | 1-0002
6 1-087 |*358 |1°772 1-065 |-0055 /1:9966 | 1:0010

1 1868

1 2:163

me |

|
|

‘1:161 |-0097 |2°0017 | 10057:
642 °749 |2°163 |1:456 ||-028) |2°0198 | 1024 |

*

Problems connected with the Circular Are. 401

neater form. The elimination of € between (3) and (12) is
rather lengthy, but not difficult. We obtain

° - a
ezw” + uw(z + )(ze~® —ue*) + (240)? — sin? 5 (ze~"F + ce’8)?=0, (13)
from which we have as the value of w

(z+4)(ze—*? — se") + (ze? + 0e*),/(2? + 2uz cosa — a1)
2Z

w= _—

(14)

where the same branch of the multiform function is to be
taken as in $3 above. The real and imaginary parts of w
give the velocity potentialand stream function of the motion
due to the disturbance of a uniform stream by the lamina,
the undisturbed velocity of the stream being unity and its
direction inclined at an angle 8 to the axis of x The
corresponding values for the motion of the lamina in a
liquid at rest at infinity are obtained by adding <e-‘* to the
above value of w. Owing to the computation of square roots
of complex. quantities being a laborions process in practice,
the stream-lines have not been plotted, though of course,
(13) could be used if it were required to do so. In the
above the radius of the are, and the velocity, have been taken
as unity. The only modifications to include the case of a
velocity U and radius a are that in (14) a factor U must be
introduced on the left-hand side, and z, w replaced. by </a,
w/a respectively.

§ 6. To obtain the “impulse” of the second of the above
motions, we have only to expand in terms of 1/z and take
—2mp times the coefficient of 1/z (p=density of liquid).
The required coefficient is found to be

—4{e8(1— cos a)—4 sin? ae-*},

so for the components of impulse we have
* a i : a
sap “ Ct pce Ne La: yo ca cee
X = 2arp cos 8 { sin 5 — sin aI =2p cos 6 sin 33 }

Y=2npsing | sin” + jsinta b =2npsin Bsin®s (1+ cos" “<).

On epee the velocity U and radius a, these become

(15)

X = 2arpa’? sint eu cos8, X=2zrpa? sin?S ( 1+ cos?) Sie Bat (ioe

We may proceed to the limiting case of a plane lamina of

width 2/ ‘by making a> and ios in such a manner that
asina->b. We ohtat

=U eer 0 U sie 8, 6. 3) LB)

Pp

m

402 Mr. W. G. Bickley on Two-Dimensional Potential

a known result. Also by making «=7, we have the case of
a complete cylinder, supposed to be filled with liquid of the
same density as surrounds it :—

AX=2rpea7U cos 8, ‘Y=2rpa7U sin 8, 2 eee

another known result. For the kinetic enerey, we easily
derive from (15’)

i eetdlo ce" U)~ | cos? sin*5 + sin? @ sin? : (1 + cos? ah ao hoe

Equations (15') and (18) show that the lamina behaves as
if it had a mass depending on the direction of motion,
compounded of masses m,, my, the coefficients of U cos 8,
U sin B, respectively in (15’).

§ 7. Next to determine the resultant of the pressures on
the lamina. Substituting z= —ve in (14), w is found to be
real (as it should be) when | 6! <a, and so we have

ii 7
“tT

o=—2 { sinSeos{ 5 - ) + <in(§ —8)4/ sin? — sint§ | seh sin’s b, (19)

the ambiguous sign referring to opposite faces of the lamina,
+ to the concave side. The velocity g at any point is given

O¢
by — 50° so that

DS

g= < cos(O—B) + cos (5-8) sin! ic: af

fond

0
sin "cosy sin( = met
==
a/ sind? Fst sin? =

But p=const—4pq?, so denoting the excess of pressure on
the concave face at any point Ap, we have

(20)

2 cos (0@—8) { cos (5 —)(sin®5 =~ — sin? a) — sin = 5 5 0085 sin {5 (5-8)

ye 29
x / sin?” 5 sin 5)

(21)

Problems connected with the Circular Are. 403

Let the total thrust, which evidently must act through the
centre, have components P,,, Py, then

P= | Ap sin 0 d@=m7p sin? a cos « sin 8 cos B,
nite . (22)
P= —{ Ap cos 0 d@= —7p sin’ a ( cos? B sin? 5 + sin? 8 cos? 4)

These results are apparently in contradiction with the
known fact that the resultant of the pressures should reduce
toa couple. The explanation is that the velocity at the edge
becomes infinite, and consequently the pressure also, and
though the thickness of the lamina is assumed to be infinitely
small, yet considered as the limiting case of a thin lamina,
with finite velocities and pressures, the resultant pressure at
the end tends to a finite limit as the thickness decreases.
To see this, consider the case of the flow round a semi-infinite
plane, given by w=Az*. Suppose the fluid inside one of
the parabolic stream lines is solidified. The motion of the
remaining fluid is unaffected, and when we calculate the
resultant thrust exerted by it on the solid, we obtain a force

PA parallel to the axis, tending to drag the solid further

into the liquid. The magnitude of this force is independent
of the particular stream-line selected. Turning now to the
problem in hand, the flow in the vicinity of the edge is of
the type just mentioned, for on substituting e’*(—7e+ ¢) for z,
where € is small, we obtain from (14)

w=const—1/2 sin «. sin : —B)e + higher powers of €. (23)
Hence there is a force along the tangent at the end, of

; at ii
magnitude Tpsin a sin®(S ~B). and consequently at the

other end, one of magnitude 5 P sin asin?(5 +8). The

resultant of these has components

a 5m gin a { sin’ (5 —8) — sin alae +8)} COS a,

=—mpsin?acosasin 8 cos 8;

Py = : Tp sin a { sin? (5 —8) + sin (5 +8)\ sin a, oe (24
J

. . a
= 7p sin? a { cos? 8 sin? 9 + sin? B cos?s ne ;

AOL Dr. H. S. Allen on Molecular

and passes through the point —zssec ae. Nowcombining (22)
and (24), the resultant of all the pressures is seen to reduce
to a couple of moment —47rpsin’asin 228.

Introducing the velocity and radius, we have

Couple = —47pa?U? sin? asin28. . . (25)

This becomes zero in the case of the complete cylinder,
a==7, and reduces, in the case of a plane, to

—trpob?U* sn 28, . 2. 7 ae

which are a m to be correct results.
§8. In (23) above, it follows that if B=5 ~, there is no

term in @. This corresponds to the case in which the
stream-line yw=0 divides at the edge of the lamina. It this
may be applied to the case of an aeroplane wing, itis seen to
lead to the conclusion that the entering edge should point
downwards, at an angle equal to the angle of attack of the
chord of the (eambered) plane. Observation and _photo-
graphs would. seem to show that in practice such a condition
has been found most favourable.

P.S.—Since the majority of the above results were
obtained, a paper has appeared (Dr. J. G. Leathem, Phil.
Mag. (6) xxxv. Jan. 1918) in which it is shown by a general
method, that (14) above is deducible from (6), and, in fact,
when applied, does give the result (14) obtained indepen-
dently above.

Loughborough,
Feb. 5th, 1918.

XLVI. Molecular Frequency and Molecular Number. Part L..
The Frequency of the Longer Residual Rays. By
H. Svantry ALLEN, d1.A4., D.Sc., University of London,
King’s College*.

METHOD of studying the frequency of vibration of

the atoms in compounds is afforded by the “ residual

rays”? obtained by repeated reflexions from the surfaces of
solids, and studied by Rubens and his collaborators. It may
be assumed that the frequency of such infra-red radiation
corresponds with the frequency of vibration of an electrically
charged ion, and, in certain cases at least, it is to be expected
that the ion in question may be identified with one of the

* Communicated by the Author.

Frequency and Molecular Number. 405

atoms contained in the compound. The writer has shown*
that the characteristic frequency, v, of the atom of a solid
element is connected with the atomic number N by the
relation

Nv= ap, or Nv=(n+4)y,

where is an integer, and vy, a constant frequency having a
value not far from 21x10 sec.-1. Seeing that the forces
which control the vibrations of the atoms in solids are the
same whether it be a question of the specific heat of the solid
or the reflexion of radiation, it might be anticipated that the
same relation would hold in connexion with the residual
rays.

Be costa to the theory developed by Nernst + the heat
energy of a compound in the solid state is made up of the
energy due to the motion of the molecules relative to one
another and that due to the vibrations of the atoms in the
molecule. The first contribution is calculated by the formula
of Debye, the second by the formula of Hinstein. Hach
calculation involves a knowledge of the corresponding
characteristic frequency; the first, »,, is given, approxi-
mately at least, by Lindemann’s formula; the second, v9,
is found from the residual rays of Rubens. Nernst has
pointed out that in the case of KCl and NaCl, the value of v,
is almost 0°75 times the value of v.; and in other cases,
although the molecular frequency is slower than the internal
frequency, the two are not very different. Indeed, in
his earlier work Nernst obtained fairly good agreement by
calculating the values of the atomic heat on the assumption
of only one frequency, v. The writer has shown{ that the
molecular frequency is connected with the ‘“ molecular
number” by a relation of the same form as that already
quoted for the atomic frequency and the atomic number. The
collective results suggest that the internal frequency may
conform to the same relation. In the following pages the
experimental results are examined for a number of com-
pounds, and it is found that there is a strong presumption in
favour of the establishment of the relation suggested.

NaCl. In the case of rock salt§ two rays of different
strengths were observed, a stronger at 53°6m and a
weaker at 46°9. The corresponding frequencies are
5°60x 10" and 6°40x10" sec.-1, Assuming the former

* H. 5. Allen, Roy. Soc. Proce. vol. xciv. p. 100 (1917); Phil. Mag.
vol. xxxiv. p. 478 (1917).

+ Nernst, ‘The Theory of the Solid State ’ (London, 1914).

t H.S. Allen, Phil. Mag. vol. xxxv., April 1918.
§ Rubens and Hollnagel, Preuss. Akad. Berlin, vol. iv. p. 26 (1910).

Phil. Mag. 8. 6. Vol. 35. No. 209. May 1918. 2F

406 Dr. H. 8S. Allen on Molecular

to correspond with the atom of sodium (N=11), and the
latter with the atom of chlorine (N=17), we find for the
sodium atom Nv=3 x 20°5 x 10” and for the chlorine atom
Nv=5 x 21'7x10!*. If, however, we associate the chlorine
atom with the stronger line and the sodium atom with the
weaker, we obtain for the products 95:1 x 10 and 70°4 x 10”,
which are not integral multiples of 21 x 10”.

KCl. For sylvin{ similar results are found, but in this case

the stronger line (62°0y or 4°84 x 10" sec.~1)imust be associated ©

with chlorine, the weaker (70°3y or 4°27 X 10! sec.~!) with
potassium (N=19). In this case we find Nv=4x 20°6 x 10”
and Nvy=4 x 20°3x10” respectively. The products obtained
when the atoms are interchanged are 91:9 x 10” and
72°6 x 10”, which are not integral multiples of 21 x 10”.

KBr. The stronger line* is at 86°5 w (v=3'47 x 10” sec™?),
the weaker at 75:64 (v=3°97x10"sec.-1). The former
must be associated with the potassium atom, giving
Nv=3 x 21°9x 10”, the latter with the bromine atom, giving
Nv=64 x 21:'4x10". The occurrence of 4 in the latter
case is noteworthy, as it is supported by independent evi-
dence in connexion with specific heats.

It may be noted that for these three compounds the
stronger line is associated with the element of smaller
atomic number.

KI. The residual radiation has not been separated into
two rays of different strengths, but it is stated by the expe-
rimenters + that such a constitution is possible. In sucha
case we may call v, and v, the frequencies corresponding to
elements having atomic numbers N, and Ng, and put

Nyy, =NjVa, Nove =NoVa.
By addition, Ys
N,v, + Nove= (ny +72),
But if vy, and vy. are not very different, we may replace either
frequency by the arithmetic mean $(v,+ 172), and so obtain

VPia-V5\ =
(N,+N,) ae = (n+ M2 )V 45

The observed wave-length for KI is 96°7y, so that we have
4(v, +2) =3°10 x 10” sec.—!.

For N, +N, we may substitute N the “ molecular number.”
Taking N;=19 and N,=53, the product is found to be
11 x 20°3 x 10", which is in satisfactory agreement with the
suggested relation.

* Rubens and Hollnagel, loc. cit.
+ Rubens and Hollnagel, loc. cit.

Frequency and Molecular Number. 407

AgCl. The average wave-length cf the residual radiation*
was found to be 815m (v=3'68x10"). The value of
(N, + N,)v=11 x 21:5 x 10”.

PpCl,, The average wave-length. in this case* was
91-0 (v=3'30 x 10”), giving (N,+2N, jv=18 x 21-2 x 10”.
When such large integers are involved, it is always possible
to find an integer Which will give a Govcontant value for v,.

HegCl. The average wave-length observed* was 98°8u

(v=3°04x10"). Taking N,;=80, N,=17, we find
(N, + N.)v=14x 21:0 x 10”.

AgBr. For this salt* A=112°-7p (v=2°66 x10"), and
(N,+N.)v=102 x 20°8 x 10”.

CaC0;. The case of cale sparf is interesting as it contains
three elements, whilst only two bands have been recorded, a
strong band at 93:04 (v= 3°23xX10”) and a weaker at
116*lw (v=2°58x10"). It is found that the strong band
must be assigned to calcium, giving Nv=3~x21'5x 10”,
the weaker to oxygen, giving Nv=1x20°7x 10”.

In a later paper Rubens and Wartenberg{ have given the
wave-lengths for two ammonium salts and three compounds
of thallium.

NH,Cl. The observed wave-lengths are very nearly the
same as those for NaCl, being a4 Ou v=d 56x10”) and
46-3y (v=6°48 x 10”) respectiv ely. This is what might be
expected if the group NH, be regarded as a compound
radicle replacing Na, for the “ molecular number” of am-
monium (7+4) is the same as the atomic number of
sodium (11). ‘Taking the first line as associated with the
NH, group, Nv=3 x 20°4x 10", whilst the second line gives
for the chlorine atom Sv=5 x 22:0 x 10.

NH,Br. Here again two wave-lengths have been ob-
served, 62°34 (v=4°82 x 10”) and 55°34 (v=5°42 x 10”).
The former line must be associated with bromine, giving
Nv=8X21:1x10; the latter with the ammonium group,
yielding Nv=3 x 20:0 x 10”.

TICl. The mean wave-length recorded is 91*6u (v=38'27
x10). The atomic number of thallium being 81 and that
of chlorine 17, we find Nv=15 x 21°4 x 10.

T1Br. For the bromide the mean wave-length is 117-0u
(v=2-°56x10'). In this case we find Nvy=14~x 21:2 x 10".

THI. For the iodide the mean wave-length is 151°8u
(v=1-98 x10”), giving for Nv the value 13 x 20'4x 10.

* Rubens, Preuss. Akad. Berlin, vol. xxviii. p. 513 (1918).
+ Rubens, D. P. G. V. vol. xiii. p. 102 (1911).
t Rubens and Wartenberg, Beas Akad. Berlin, p. 169 (1914).

408 Dr. H. 8. Allen a Molecular

Thus for the halogen derivatives of thallium the interesting
result is found that the “frequency numbers” (15, 14, 13)
diminish by unity in passing from chloride to bromide and
from bromide to iodide.

It must be stated that the suggestion which attributes the
two absorption bands of NaCl, KCl, KBr to separate atoms
is not anew one”. Nernst is of opinion that it was a mere
coincidence—“ a very curious and misleading one indeed ”—
that calculations of the specific heat on that supposition gave
quite good results. Rubens was unable to find two bands
in the case of AgCl and PbCl,, and came to the conclusion
that the two apparent bands were due simply to water vapour,
which has a great number of absorption bands. Ifthis con-
clusion be accepted the results quoted above will require
modification in the sense that the “ molecular number ” must
be employed instead of the atomic number; but the pro-
posed relation will still hold good. Thus for NaCl we find
Nv=8 x'20°6 x 10%, for KCl Nv=8xX20°1x10", angie
KBr Nv=9x 21°6 x 10”.

It would appear probable that the relation here discussed
applies only in the case of the longer residual rays, having a
wave-length greater than say 20m. For shorter wave-
lengths, corresponding to a higher frequency, the value of
the product, Nv, is so large that no real test of the proposed
relation can be obtained. It is unfortunate for our present
purpose that although the residual rays from quartz in the
region of 9uand 13 have been measured with considerable
accuracy, the longer waves have not as yet been determined
accurately; we knuw only that quartz shows strong selective
absorption for the region between 60 and 80uy.

It may be worthy of mention that water vapour has an
absorption band at 14:3, which is the wave-length corre-
sponding to a frequency 21:0 x 10"sec.~1, that is the
frequency here denoted by v,. Further, the vapour of carbon
dioxide has an absorption band at 14:1yu. It would be of
interest to know whether other substances show absorption
in the same region.

The possibility of deducing the wave-length of the infra-red
radiation from the elastic properties of the solid has been
discussed by Madelung and by Sutherland. By considering
a cubical space-lattice the former obtained for the wave-length
the expression

oe MM, 7.3 D)

* Cf. Nernst, ‘The Theory of the Solid State,’ p. 80 (1914).

Frequency and Molecular Number. 409

where M,, M, are the masses of the atoms, K is the com-
pressibility, and D the density. This may be compared
with Hinstein’s formula. Rubens and Wartenberg have
shown that this formula gives results in moderately good
agreement with their observations.

They have also obtained fair agreement by employing a
modified form of the equation of Lindemann, viz.

ue MM, *)
a= Ca/ (at PME)

where V is the molecular volume and T, the melting-
point.

In both formule the constant must be determined em-
pirically.

The results of this and preceding papers support the
following conclusions:—

(1) The forces binding the atoms in the molecule are
similar in character to those which bind the molecules of
the solid, that is the forces of chemical affinity are of the
same nature as the forces of molecular cohesion”.

(2) There must be something of a discrete character in
the nature of these forces, in order to account for the
occurrence of integral values of x. The simplest hypothesis
is to assume that the forces arise from the presence of
valency electrons. As it is probable that these forces act
only in definite directions, it is a plausible suggestion that
the linkages between the atoms are constituted by Faraday
tubes of force, which would then be regarded as physical
entities. The fundamental frequencies, v, and vz, would
depend on the properties of the unit tube of force. It has
been pointed out by Prof. Nicholson} that such a view seems
to be required in order to explain the relations between the
frequencies of spectral series. Attention may also be drawn
to an important article by Sir J. J. Thomson on the Forces
between Atoms and Chemical Affinity, in which chemical
valency is discussed from the same standpoint.

* Compare Nernst, ‘The Theory of the Solid State,’ pp. 4-9 (1914) ;
Langmuir, Ann. Chem. Soc. Journ. vol. xxxviii. p. 2221 (1916).

+ Nicholson, Phil. Mag. vol. xxvii. p. 541, vol. xxviii. p. 90 (1914).

¢ J. J. Thomson, Phil. Mag. vol. xxvii. p. 757 (1914).

Arlo:

XLVI. On Wood’s Criticism of Wien’s Distribution Law.
By Haroup Jerrreys, VW.A., D.Sc.*

| ta the February number of the Philosophical Magazine,
pp. 190-203, Mr. F. E. Wood offers a criticism of
Wien’s law of the distribution of energy in the spectrum
of a radiating gas. If 2 denote the wave-length and 6 the
absolute temperature, Wien’s law is that the energy of the
part of the radiation with wave-leneths between > and
V+ dn is
b(A, 0)dx= # Paice hs

where C and & are constants. Wood shows from the same
assumptions as Wien that the omission of a factor by Wien
in his first equation led to an error in the final result and
that the correct formula based on these hypotheses is

(A, 0)drA= ne an.

With reference to this law it must be noted that the total
radiation is obtained by integrating ddv from zero to infinity,
and as this must by Stefan’s law be proportional to 0*, it
can easily be shown that Ck-** in Wood’s equation is not a
constant, but is proportional to 6-3. As Wood’s argument
is involved and in places somewhat obscure +, I offer an
alternative proof which is shorter and appears to be equally
satisfactory subject to substantially the same assumptions.

If N be the total number of molecules in a mass of gas,
the number whose absolute velocities lie between v and
v+dv is

dN =4Nor 2a ee dy,

where a is a velocity whose square is proportional to the
temperature. Then Wien’s first assumption is that the
wave-length and intensity of the radiation emitted by a
molecule depends on v alone. If € be the rate of emission

* Communicated by the Author.

+ The argument at the foot of p. 197 and the top of the next page
implies that the number of molecules with velocities between v and
v+dv is kv? * egy instead of kv?6~ 26 9 dy,

The introduction of an ideal type of gas on p. 198, with the notion of
corresponding velocities, is unnecessary ; so is the use of the meaningless
law AO=constant, which requires such careful interpretation that it is:
more difficult to apply than the second and third hypotheses of the
present paper, expressed in equations (5) and (14), which are equivalent
to it.

On Wood's Criticism of Wien’s Distribution Law. 411

of energy by a molecule of velocity v, that emitted by all
the molecules whose velocities lie between v and v+ dv is

dR=¢(d, 0)dX=4Na 2% ede, . . (23

where e is a function of v only.

Now the total radiation for all wave-lengths together is
obtained by integrating this from v=0 to v=o, and must
by Stefan’s law be proportional to 6*, and therefore to 2°.
We have thus an integral equation fore. Assume that e(v)
can be expanded in powers of v for all values of v, so that

eo) Sao, cr Dai elt ane ieee
0

n+2o-"/@? can be integrated term by

eo
and that the series Sa,v
term. :
Then R can be expressed as a power series in a, thus:
i fea) @o —
Ra? _ S a ook La s eo... (26 1) ta 4
4N ar 2™ °* “2K—1 ei 9xtl 2K—2
By hypothesis R is equal to Ne’a®, where ¢ is an absolute
constant, and therefore all the a’s are zero exceptag. Hence

ae

Ne? 94 Nag,
and therefore
_ 64m tag 3 10 ,-v%/a?
div= 945 Ne?a~*y%e FL RAN i gic Eaam C:

Thus the distribution of energy with regard to the
molecular velocities is completely found from the first
assumption alone, and all that is now required is to determine
the relation of the velocity to the wave-length.

At this stage a further assumption is needed; and this
will be that the graph of ¢ against » for any temperature
can be derived from that for any other temperature by
homogeneous strain in two dimensions. In other words, if
the temperature be changed from @ to k°0, or, what is the
same thing, if « be replaced by ka, two numbers a and 6 will
exist, so that

pan, ee sea, 8): Ss pe) 4)
for all values of X, where a and 6 are functions of k alone.
Now it is evident from (4), by putting v?=/(d), that
(A, @) is of the form
we eee ie oon See)? ¢6}
where F and fare at present unknown, but are connected
by a differential relation.

a) ae |
\
7
’ =.
=
a

412 On Wood’s Criticism of Wien’s Distribution Law.
Hence we must have, by (5),

Now k, a, 6 and X are perfectly independent of «, and
therefore this transformation cannot be possible for other
values of the temperature unless

Jian) =f) \.)'.0
for all values of 2X.

Hence f(aa'r) =k2(a')f (ar) = H2(a')k2(a) fir),
and also = h?(aa’) f(r).

Therefore k(aa’)=k(a)k(a'), and thence it is easily shown

that k(a) must be equal to u log k(e), where e is the base of —

the Napierian logarithms, and k(e) is an unknown constant.
Log k(e) may therefore be put equal to a further constant h.

Therefore fiar)=a"{r), .. (2. 3

and fen =...

whatever a may be, showing that

F(A)/rx™ is an absolute constant.

Hence vis proportional tor, . . . . (11)
making
64-2 dv
-3 es 24—3,10 2U
aT BN) = 945 Ne?a rr
== NO-29N Ts
where g is another absolute constant.

Altogether ;
br, 0)=Ngo-I ee

where g, h, and J are unknown constants.
Wien introduces the further assumption that the number
a is equal to k~?, making

ha — Sov Mela er

and v” proportional to 1/A. In this case the distribution
Jaw reduces to Wood’s form as modified in the first paragraph
of this paper, namely

13
b(r, 0) =NgO-in- 207, 2... (15)

a ee eal tLe.

Gite

XLVI. The Resolution of Mixed Colours by Differential
Visual Digfusivity. By Herpert E. Ives, Ph.D., Captain,
Saget... USA

. Introduction.

. Experimental procedure.

. Results with purple light.

. Results with monochromatic and compound yellow.
. Discussion.

po

Ort 29 bo

1. Introduction.

NDER the title “ Visual Diffusivity,’ the writer de-
scribed some time agoft experiments on the lagging
of different colour impressions relatively to eachother. These
were predicted on the basis of a theory developed to explain
the behaviour of the flicker photometer t. An essential part
of this theory was the ascription to each colour impression
of a characteristic rate of transmission along the visual
channel. In developing the theory it wasassumed that each
colour acted quite independently.

In two instances it appeared necessary to modify this last
assumption. One was the case of alternated colours exposed
for unequal lengths of time. Here the flicker-photometer
theory apparently called for much greater effects of dis-
symmetry than experiment showed. ‘The other was the
failure of an attempt to resolve a purple into its constituents
by moving it across the field of view. It was concluded in
the light of these experiments that some mutual action of
colours in the act of transmission must take place.

Later work on the flicker-photometer theory showed that
when the complete equations were employed, dispensing with
the earlier assumption that the effect of varying speed could
be neglected, the case of unequal exposures did not require
the assumption that the individual colours affect each other
in transmission. There remained then to be explained the
failure to resolve purple in the manner described. In view
of the very satisfactory quantitative success of the flicker-
photometer theory as demonstrated by the more recent work
with the polarization design, it appeared highly desirable to

* Communicated by the Author.

+ “ Visual Diffusivity,” Ives, Phil. Mag. Jan. 1917, p. 18.

1 “Theory of the Flicker Photometer,” Ives and Kingsbury, Phil.
Mag. Noy. 1914, p. 708, and April 1916, p. 290. “A Polarization Flicker
Photometer and some Data of Theoretical Bearing obtained with it,”
Ives, Phil. Mag. April 1917, p, 360. “ Hue Difference and Flicker Photo-
meter Speed,” Ives, Phil. Mag. August 1917, p. 99.

414 Dr. H. E. Ives on the Resolution of Mixed

go more fully into the question of visual diffusivity with
mixed colours. The outcome of this further study has been
to establish the experimental conditions pre-requisite to the
detection of the resolution of mixed colours by the different
visual diffusivity of the components. With the proper con-
ditions, not only has it been possible to resolve purple into
red and blue, but observations have been made on both
monochromatic and compound yellows, with results of
interest in colour-vision theory.

2. Experimental procedure.

In making the experiments recorded in the paper on
‘“‘ Visual Diffusivity,” full use was made of the fact that the
eye is very sensitive to a break in the continuity of a straight
line. The two coloured strips under observation were set
slightly out of line, and then passed before the eye at a speed
such that they appeared to form a continuous instead of a
broken line. In attempting to resolve a purple into red and
blue this peculiarity of the eye was not taken advantage of.
A purple strip was moved across the visual field and evidences
of widening or duplication were looked for. To this the eye
is much less sensitive.

On reconsidering the problem recently it was realized that
the criterion of linear continuity should, if possible, be used
to judge of a resolution of the kind expected. The arrange-
ment cf coloured areas finally decided upon was that of a
continuous slit—red at one end, blue at the other, with purple
in the middle, the purple being the mixture of the two end
colours. This arrangement is shown in fig. la. When this
strip was moved across the field it was hoped that not only
would the red and blue be displaced, in agreement with the.
previous experiments, but that it would be evident that the
displacements overlapped, in the manner shown in fig. 1 6..
That is, instead of looking for the duplicity of the portion p,
which could not be detected previously, attention would be
directed to a possible widening which would continue the
straight line of both parts, r and 6.

The production of such an overlapping red and blue
strip, which could be placed in the disk apparatus formerly
used, was first attempted by photographs on an autochrome
plate, but the purity of the resultant colours was not sztis-
factory. The final apparatus used was a simple colour-
projection scheme, the requisite motion of the colour patches
being provided for by the movement of a mirror before the
eye. A plan of it is shown in fig. 2, where ais a box con-
taining two point-source tungsten lamps, controlled by separate-

Colours by Differential Visual Diffusivity. 415

resistances >. The light from these lamps passes through
diaphragms c and coloured glasses d to the matte white
screen ¢, which is viewed through the slit f by the eye at h,
after being reflected from the mirror g, arranged to rotate

Fig. 1.

La Lb

1a. Stationary slit coloured red (7) at one end, blue (4) at the other,
and the purple formed by their mixture (p) in the middle.
16. Appearance of slit when in motion in the direction of the arrow.

about a vertical axis. At are movable screens of short
vertical length, which can be so placed that either top or
bottom of the screen e is illuminated by one colour of light
alone. With this arrangement it is possible to have the
slit / appear of either colour given by the glasses at d, half of
one colour and half of the other, or eise part one colour, part
the other colour, and part the mixture of the two as in
fig. La.

3. Results with purple light.

Four possible behaviours of the purple mixture region of
the strip shown in fig. la may be imagined. On lateral
movement either (1) the purple may move bodily as the red
constituent, or (2) as the blue constituent, or (3) it may
exhibit an intermediate shift, or (4) it may spread out into
red and blue. Hither (1), (2), or (3) would be possible

interpretations of the previous trial.

416 Dr. H. E. Ives on the Resolution of Mixed

On making the experiment with the new arrangement, it
was found that the appearance presented was unmistakably
that of fig.1 b—that is, the mixed colour when passed across the
field of vision was resolved into its components, just as it would
have been if viewed through a weak prism. By careful

Arrangement of Apparatus.
a, box containing two point-source tungsten lamps.
b, resistance for controlling current through lamps.
c, c, diaphragms.
d, d, coloured glasses.
e, matte white surface.

fy slit.

g, mairror.

h, observer's eye.

jy, movable screens of short vertical length.
k, monochromatic illuminator.

i, opal glass.

adjustment of the intensities the effect was made so clear-cut
as to be immediately apparent, and was verified by a number
of observers. This effect is shown very strikingly if the
slit image is oscillated by the proper mirror movement. In
this case the two colours appear to slide over each other at
their overlapping portion. The appearance here is as though
a purple slit of moderate length were viewed through a prism
set with its edge at right angles to the direction of the slit,
and then rotated back and forth with the line of sight as
an axis.

Yolours by Differential Visual Diffusivity. 417

It appears from this that we have here a new method of
colour analysis, dependent on the properties of the eye and
not on those of an inanimate physical instrument. It is of
interest to apply this new method to other mixed colours—in
particular to yellow.

A. Results with monochromatic and compound yellow.

Yellow is of peculiar interest in the study of colour vision
because the same sensation may be produced in two different
ways. One is by the pure yellow of the spectrum, the other
is by the mixture of red and green light. Unlike the blue-
_ green of the spectrum, which may be matched similarly by a
mixture of the colours to either side, yellow gives a sensation
totally distinct in character from its components. So defi-
nitely is this so that the psychological elementary colours,
not recognizable as mixtures, are red, yellow, green, and blue.
MacDougall, Schenck, and others consider it probable that
the red and green sensations have been developed from a
more primitive yellow sensation, the latter one of the two
sensations (warm and cold), into which primitive mono-
chromatic vision first separated.

Trial by this new method (provided it proved sensitive
enough to give definite results) offered the possibility of
answering the following questions, suggested by the nature
of yellow light, as just discussed :—

(1) Does a mixed yellow become resolved into its con-
stituent red and green ?

(2) Does a pure yellow become resolved into a red and a
green ?

If the answer to both these questions is affirmative, it
might be interpreted as meaning that the action of both
kinds of yellow light is to break down two substances, red
and green, at the surface of incidence, which then travel back
at different speeds to a point where the double product either
combines into a yellow substance or is interpreted as yellow.

If the answer to both these questions is negative, it might
be interpreted as meaning that either kind of yellow light
breaks down a yellow substanee, which travels back with its
own individual velocity, 2. e., that colour fusion takes place
at the surface of incidence. A third possible answer—namely,
that the mixed colour is resolved and the pure is not—might
be interpreted to mean that distinct red, yellow, and green
substances are broken down by the light, travel back with
their appropriate velocities, and that red and green fusion
occurs after this transmission. The answer to these questions
can be obtained only by experiment.

A18 Dr. H. E. Ives on the Resolution of Mixed

For the yellow light tests the apparatus as above described
was provided with red and green glasses of narrow spectral
transmissions for the mixed yellow, and a monochromatic
illuminator & was added, illuminating an opal glass /, which
could be viewed through an opening in the screen e. By this
means either red light or green light or their mixture, or a
monochromatic yellow matching the mixed yellow may be
obtained, disposed in any desired way along the slit 7.

The slit was first arranged to show half red and half green,
and the intensities adjusted until on movement of the mirror
the lag of green behind red was well shown. ‘This occurs at
a rather low intensity, and care must be taken to have ab-
solutely no stray light in the room. (In these experiments
the slit f was placed over a hole ina door between two rooms,
so that an assistant could make the necessary manipulations,
without any stray light reaching the observer.) When this
condition was found the screens j were placed so that the
middle of the slit showed the compound yellow. This com-
pound yellow was clearly resolved by lateral motion across the
field of view. In fact, the resolution was even more satis-
factory than it had been with purple, because the two edges
of the centre patch were seen on several occasions to be red
and green. ‘This greater success is probably due to the very
distinct difference in appearance between yellow and its
constituents.

The next point taken up was the behaviour of the pure

yellow, adjusted to be a subjective match with the compound
yellow, and arranged to exactly take its place between the
red and green. It wasat once apparent that pure yellow
does not separate into red and green. ‘This fact is strikingly
shown by arranging the slit so as to be all compound yellow,
-excepta small portion of pure yellow. When stationary the slit
appears alike thoughout its whole length in brightness, hue, and
definition. But upon moving the image sideways, or oscil-
lating it, the compound yellow immediately broadens out and
becomes ill-defined, the pure yellow remaining narrow and
sharp. The appearance is identical with that produced by a
weak prism, and again demonstrates that this phenomenon
provides a new, if rough, method of spectrum analysis.

5. Diseussion.

The results obtained with purple light are of chief interest
as justifying the assumption made in the flicker-photometer
theory, that each colour is transmitted with its own cha-
racteristic speed, irrespective of whether it occurs alone or

me!)
a

Colours by Differential Visual Diffusivity. 419

with other colours. The results from the yellow light expe-
riments are, however, of great interest from the standpoint
of theories of colour vision.

If we consider these experiments without reference to any
other work on colour vision, it is evident that the explanation
called for is one in which wave-length composition rather than
appearance or colour-sensation analysis determines behaviour.
Recently Houstoun has suggested a theory of colour vision
according to which colours do maintain their physical wave-
length identity in transmission to the brain. He supposes
the spectral colours cause retinal vibrators “‘ to set up waves
in the nerves and that the nerves carry these waves to the
brain,” the dominant wave-length of the transmitted dis-
turbance being that of the incident light. The results of the
present paper might be interpreted as supporting this theory.
A place at which the new phenomena might at first sight
appear to fit into Houstoun’s theory may be noted as suggestive.
If the writer understands the theory correctly, it calls for
the production of fluorescence at the retina, which illuminates
the brain through the transparent nerves (for the “‘ waves in
the nerves ”’ must, because of their frequency, be light waves).
Now it is one of the elementary facts of physical optics that
a reduced rate of transmission of light waves occurs in a
medium of high refractive index, and that a different rate of
transmission of colours, in the order of wave-length, occurs
in a medium possessing, as most transparent media do, an
absorption-band in the ultra-violet. Thus the velocity of light
in carbon bisulphide is less than in air, and red light travels
faster than blue. If, then, to the transparent nerves, which
appear to be a part of Houstoun’s theory, we ascribe a high
refractive index varying in the usual manner through the
spectrum, we have a system which would give the differential
speed effects found by the present experiments. Unfortu-
nately the necessary refractive index to reduce the speed of
light to the order of magnitude such that even a hundredth
- of asecond lag of blue light would be possible in any distance
available in the human head, is enormous and fantastic.
Apart from this quantitative difficulty, the parallelism between
the behaviour of light in the eye and in the highly dis-
persive medium is close. There is, however, no obvious
explanation along this line of the variation of speed of
transmission with the intensity of the stimulus.

Whether or not the visual diffusivity phenomenaagree with
the Houstoun theory, the latter requires, in the writer’s
opinion, much more support before it is profitable to attempt
to harmonize new observations with it. The objections

420 Dr. H. Ei. Ives on the Resolution of Mixed

presented to it or any “‘ wave-length” theory by the facts of
colour mixture and colour blindness are very great. In
particular, the production of white by various pairs of com-
plementaries of different centres of gravity, and the neutral
points in the spectra of the two types of red-green blind,
present serious difficulty of interpretation.

A great deal of evidence points to the probability of
vision being a process of photochemical decomposition, in
which either the decomposition products or electric currents
set up by the process pass along the nerve tracks at the
speeds characteristic of nerve impulses. Apart from the
facts of colour mixture and colour blindness there is no
reason why the number of separate decomposable retinal
substances should not be very large. There is nothing in
the present experimental work to indicate that the number
is restricted to the red, yellow, green, and blue experimented
with, although the method is so crude as a means of spectrum
analysis that it would be difficult to show conclusively that
a much greater number of speeds of transmission than four
are exhibited in the length of the visible spectrum. As the
method stands, it is inferior as a means of colour analysis to
the single prism and wide slit first employed to exhibit the
spectrum. Ifor an answer to the question of how many
reacting substances of different velocities of transmission
exist, the resolving power should be more nearly that re-
quired to show the Fraunhofer lines. What is required to
effect an improvement: comparable with the instrumental
advance quoted is some drug which will slow down the rate
of transmission of the visual impressions, or the “time
machine” of Mr. H. G. Wells. .

Actually the mixture phenomena point to the existence of
three, and only three, such reacting substances—a red, a
green, anda blue. The question of paramount.interest then
is, how can the different behaviour of the spectrally pure
yellow and its subjectively equivalent mixed yellow be
harmonized with three-colour theory? It is evident that
the “three different types of nerve fibre”’ postulated by
Young, or the “ three different, independent, and mutually
unopposed elementary activities”? in the statement of the
theory by Helmholtz, must be located (if existent) at some
distance along the conducting path between receiving surface
-and brain. By this new method of analysis we can actually
see red and green being transmitted a certain distance at
different speeds* before combining to make yellow, and we

* Or possibly with the same speed for different distances. In this

connexion see Koenig, ‘Ueber die lichtempfindliche Schicht in der
Netzhaut des menschlichen Anges,” Ges. Abh. p. 333.

Colours by Differential Visual Diffusivity. 421

can see yellow being similarly transmitted, as yellow. Red
and green light acting together do not cause a yellow
reaction or the breaking down of a “yellow substance” at
the surface of incidence. Neither does yellow light cause at
this surface the breaking down of a red substance and a
green substance. Jf a trichromatic mechanism is necessary, as
wt appears to be from the phenomena of colour mixture and
colour blindness, then the present experiments call for the
additional complication of an antecedent transmitting process
where colours retain their physical (spectral) individuality.

Along the lines of the Schenck modification of the Young-
Helmholtz theory the diffusivity results might be interpreted
as showing that the primitive yellow substance is still present
in the retina, as well as the red and green substances which
have developed from it. Only upon the red and green sub-
stances reaching a certain depthis their equivalence with the
yellow substance established. It may be imagined that this
isolation is due to their going along separate channels. This
interpretation practically amounts to assuming four reacting
substances in theretina. The fourth is superfluous from the
standpoint of colour mixture, and so appears in contradiction
to the bases of the three-colour theory, which the Schenck mo-
dification is not. This throws us back, if we hold to the three-
colour theory, upon the above suggestion that the diffusivity
phenomena take place before the trichromatic mechanism is
reached. Unless, therefore, a theory appears which is in
accord with all the facts of colour mixture, but calls for a
wave-length basis of colour vision, these new facts appear
merely to add complications to an already too complex
problem. Unfortanately, this has been the almost uniform
history of research in vision.

Physical Laboratory,

The United Gas Improvement Company
Philadelphia, Pa., Sept. 1917.

Note added on correction of proof.

The transmission of colour impressions in order of wave-
length, which is indicated as taking place antecedent to the
trichromatic mechanism, has every appearance of being a
purely optical effect. It could be accounted for by the
assumption of strong chromatic aberration in annular focussing
elements in the retina, concentrating the incident light on
photo-sensitive fibres lying in the axes of the elements. The
different periods of transmission of the various colours would
then be due to the different distances the resultant decompo-
sition products had to travel along these fibres. Further dis-
cussion of this possible explanation is deferred for the present.

Phil. Mag. 8. 6. Vol. 35. No. 209. May 1918. 2G

Be Peay

XLIX. An Astronomer on the Law of Error.
By Professor F. Y. Epcewortu, F.B.A.*

ies law of error has been disputed by the Astronomer
Royal for Scotland + on two distinct grounds.

I. He finds that the law is not perfectly fulfilled by
astronomical observations, not even by those made by
Bradley which Bessel tested, still less by others. This
verdict does not disconcert the statistician who (after
Laplace) grounds his expectation of the law on the inter-
action of numerous independent causes. Where that
condition is imperfectly fulfilled there is no reason to expect
the Jaw of error to be realised perfectly, any more than we
expect a body attracted to another according to the law of
gravitation to move in a perfect conic when there is a
resisting medium. In the case of astronomical observations
we have reason to believe that the conditions will be some-
times badly, sometimes fairly well fulfilled t. The like is
true of physical observations generally, with which may be
classed shots aimed at an object§. Similarly the neigh-
bouring class. of statistics not grouped about an objective
thing (e.g. statures of a population) fulfil the law more or
less perfectly. Greater perfection may be expected in those
classes of phenomena to which Laplace and Poisson confined
the application of the law, namely magnitudes each of which
is an average—or more generally a linear function—of
numerous observations or statistics of the two classes above
mentioned and (exemplifying such functions ||) occurrences
at games of chance. The most perfect fulfilment is pre-
sumably presented in a molecular medley by velocities
considered as the resultants of innumerable compositions.

* Communicated by the Author.

t+ “On the Law of Distribution of Errors.” By R. A. Sampson.
Fifth International Congress of Mathematicians, 1912, vol. i. p. 163
: P Halaiment of the law is not to. be expected where the observations
are affected by a few dommant sources of error, as pointed out by
Morgan Crofton in some instructive remarks on the nature of errors
in astronomical observations,” Philosophical Transactions, 1870, p. 177.

§ In the class of shots may be included guesses e.g. as to the age of
an individual and even estimates of a less objective magnitude, such as
the worth of an examination paper; as to which see Phil. Mag. August
1890. As to the classification of phenomena obeying the law of error,
see article on “Probability,” Encyclopedia Britannica, 11th ed., § 117
et seg. Cp. as to imperfect fulfilment of the conditions, § 157.

| £.g. a great number of dice being tossed, the frequency with which
certain faces turn up. If the dice are not perfectly symmetrical (as

Weldon found, Phil. Mag. vol. 50. 1900, p. 168), the data have some
affinity to statistics not representing an objective magnitude.

"|, aq. -t

An Astronomer on the Law of Error. 423

TI. The ground for expecting that the law of error will be
fulfilled more or less approximately in the preceding. cases
would be cut away if Dr. Sampson’s objection to the proof
of the law were held valid. His attack (loc. cit. p. 167) on
the proof given by Poisson after Laplace strikes at all the
applications of the law *; it cannot be limited to the par-
ticular class of astronomical (or more generally physical)
observations—to which indeed Laplace+ and Poisson did
not propose to apply the law. The objection here combated
is not based on the want of that independence which the law
postulates ; whether as between the (total) errors of succes-
sive observations, considered as not “‘ accidental”? (Sampson,
loe. cit. p. 168), or between the “small errors” (p. 166) the
components { of the entities which may be expected to fulfil
the law §. It is here conceded that so far as such inde-
pendence is not present perfect fulfilment of the law is not
to be expected ; the case will present imperfection of the
kind admitted under head [. But Dr. Sampson in his
attack on the theory of Laplace and Poisson (loc. czt. p. 167)
does not dispute the initial stages of the proof in which
this independence is implied. It is implied that if one
component error is distributed according to the frequency-
function /;(x)|| and another according to /,(a2), the proba-
bility that the respective component errors x, and x, should
concur is proportionate to /{(2;) X fo(ze). What the point of
Dr. Sampson’s objection is may be shown by a free version
of his argument in a simplified case. Let us suppose that

* Including the method of sampling which is becoming so important
in social statistics; as to the theory of which see Bowley’s Presidential
Address to Section F of the British Association, 1906, and as to the
practice ‘ Livelihood and Poverty,’ 1915, by Bowley and another.

+ Glaisher more than once remarks on the fact that Laplace did not
employ his theorem to establish a presumption that observations them-
selyes—as distinguished from averages thereof—iulfil the law of error.
See ‘Memoirs of the Astronomical Society,’ vol. xxxix. pp. 104, 106;
and ‘ Monthly Notices of the Astronomical Society,’ vol. xxxiii. p. 397,
par. 3 (1878).

t{ Where the entities are averages their components consist of the
original data, observations or statistics of the kinds described above
(divided by the number thereof).

§ For an example of the fist sort of interdependence, see Article on
*‘ Probability’ (Zinc. Brit.), §157. Where the figures grouped were
each an average of aset of consecutive observations of the kind instanced,
the materials would illustrate interdependence of the second kind.

|| Meaning that the number of observations which occur between
x and r+Ar=Nf,(x)Az, where N is the total number of observations:
=f,(x)4z=1; Az is small, so that for the purpose in hand it may be
replaced by dz.

2G 2

4AD4 Prof. F. Y. Edgeworth: An Astronomer

the law of frequency is one and the same for all the com-
ponents, say f(z). Let this function be symmetrical about
a point which is taken as the origin. Further, let the
weights, called by Dr. Sampson after Todhunter* y, be equal.
Then in the place of p; in Dr. Sampson’s (and Todhunter’s)
notation we have

A= ( J (a) cos axda,

e~-a

where a and —a are the limits which the errors cannot
exceed t. Then, since

( _Ke)de= 1,

ee —

p= 1—42? | a? f(a)da + 5 athe

a ml

—]—olk,+...,

if we put k, for the mean square of deviation from the mean,
that is twee Dr. Sampson’s h? §. If s is the number of the
components, we have |

R=p'= (l—a?ik?+. : Nee log R= —arsth?+. aia

‘“‘Then approximately,” as Todhunter has it in his version
of Poisson’s reasoning ||,

Re=e 2c, where | e=2sk.

Accordingly the required expression for the frequency of
observations between assigned values of the abscissa is given
by multiplying R upon a certain function of « and the said
values of the abscissa, and integrating the expression thus
formed with respect to « between limits © and 0. To this
procedure it is objected by Dr. Sampson that “the terms

* See Todhunter, “‘ History of the Theory of Probability,” Art. 1002.

+ In Todhunter’s version respectively a and 6.

t This condition seems to obviate Dr. Sampson’s objection. “If the
arbitrary distributions f;(2) have any zero—and this is not excluded by
the process of demonstration—I do not see how they can fail to re-
appear as zero in the product R”’ (doc. cit. p. 168, par. 1).

§ Loc. cit. p. 167, par. 3; “A” is used differently elsewhere, p. 167,
par. 1, pp. 169--172.

|| Op. cit. p. 565; Todhunter’s Y corresponding to Dr. Sampson’s and
our R, and Todhunter’s x? to our 3@ and to twice Dr. Sampson’s =h;?
(when the A’s are identical).

on the Law of Error. 425

in a‘ in the separate factors have been omitted” * ; meaning
no doubt that had the terms been taken into account the
coefficient of a* in R would not have proved approximately
equai to the corresponding coefficient in the expansion of e—2¢.

Now this is exactly what was maintained by the present
writer in the Philosophical Magazine for 1883 f. There, and
elsewhere subsequently, he has instanced forms of 7(2) such
that when any number of components are superposed in the
manner of Laplace, the compound does not contorm to the
law of error §. Dr.: Sampson, then, is right so far as he
teaches that the independence || above postulated is not
sufficient by itself and without any additional conditions to
secure the fulfilment of the law of error. But he should
have added that commonly and practically such additional
conditions are present.

What those conditions are may be shown by continuing
the expansion of R. Put sé, for the mean fourth power
of deviations for one of the components from the mean
(identical with Todhunter’s £’’ now that the mean is coin-
cident with the origin). We have then for the logarithm
of p,

1
— asa +a! (Fb Bhs?)

(kf. as before denoting the mean square of deviation for any
one of the component elements). Whence

log = —a?shkyo+ a (kg okey)

In the coefficient of a* in the expansion of R the remainder
after the first term tends to be negligible in comparison with
that term as s increases. For the first term contains s? con-

* So the quesitum is defined by Todhunter after Poisson. But it
seems much simpler en the lines of Laplace to investigate the proba-
bility of an observation occurring az (or in the immediate neighbourhood
of) a particular point ; as to which conception see “ Law of Error” (by the
_ present writer), Camb. Phil. Trans. vol. xx. p. 181 (1905). Cp. p. 40,
et passim.

+ Vol. xvi. pp. 304 & 307.

} “ Law of Error,” Camb. Phil. Trans. vol. xiv. p. 140 (1885).

§ The functions are of the family specified below (p. 429) as ‘‘repro-
ductive.” They may be expanded in ascending powers of z—by a series
of integrations with respect to a between limits o and 0.

|| Absence of interdependence or correlation, both between the several
component small errors which make up a total or composite error of
observation and also between successive composite errors; these terms
being used in a wide sense so as to cover the case where the compound
is an average (or other linear function) and the components are (errors of)
observations or the same divided by m the number of observations

(ep. above, p. 423 note) ; and where the “error” is not a deviation from
an objective magnitude.

4.26 Prof. F. Y. Edgeworth: An Astronomer

stituents of the order of magnitude k, (or k,”). Whereas the
remainder tends to contain only s constituents of that order;

since the central term in the expansion of ky, viz. 6hok2 ae
is cancelled by the main term in the expansion of 3(sk,)?*.
It must be postulated that the mean second and fourth
powers of deviation are finite: a postulate which may be
taken for granted when the range of the component
frequency-functions is finite. The reasoning may be ex-
tended—if the postulate also is—to further terms in the
expansion of R. Accordingly Laplace was quite right
when, referring to symmetrical identical frequency-functions
with limited range, he affirmed that “taking hyperbolic
logarithms, we have very approximately (@ trés-peu prés),
when s is a large number” + for the (Napierian) logarithm
of R (in the notation above used) an expression equivalent
to that which has been given above.

The conclusion may be extended to the case of frequency-
functions not identical ; provided that their mean powers of
deviation are of the same order. The conditions have been
stated elsewhere by the present writer with more precision f.
Tt is needless to reproduce that statement here, since the con-
ditions (for even trequency-functions) have been adequately
stated in a treatise to which Dr. Sampson has referred,
Poincaré’s Calcul des Probabilités§. Poincaré supposes
“the functions even, or in other words no_ systematic
errors”’|| (Calcul, p. 184) ;“‘that the errors are independent”

* For a fuller exposition, see the present writer's article on the “ Law
of Error” in the Transactions of the Cambridge Philosophical Society,
vol. xx. p. 42 e¢ seg. (1905).

+ Théorie analytique des Probabilités, liv. 2, Art. 18, p. 336, National
edn. 1847.

t In the 1905 Paper referred to in the penultimate note.

§ The proof of the law of error by way of approximate identity
between the mean powers of the representative function and those of
the actual locus was put forward by the present writer in 1905 without
acknowledgment, because without knowledge, of Poincaré’s similar

roof for the case of even functions, published in 1896. Priority may
still be claimed for the essay of 1905 as having extended that proof to
odd functions, having proposed several collateral proofs, and having
carried the approximation beyond the stage at which it was left by
Poisson (the “ second approximation ” referred to below note p. 428).

|| It may be worth while to recall that ‘‘symmetrical” and “ svste-
matic” errors are not necessarily coincident. The centre of asymmetrical
group may not coincide with the true point, and the centroid of an
unsymmetrical group may. In the case of symmetry as well as of
asymmetry something must be known or presumed as to the relation
of the true point to the group as a whole. Cp. Phil. Mag. vol. xvi.
p- 873 (1888) and Journal of the Royal Statistical Society, vol. Ixxi.
p- 500 (1908).

on the Law of Error. 427

(op. eit. p. 182); “that the individual errors, though
not following the same law, are practicably (senszblement)
of the same order of magnitude, and each contributes little
to the total error” (op. cit. p. 183). Recapitulating these
conditions he concludes that “in this case the resultant
error suivra sensiblement la lot de Gauss” *.

The reasoning whereby the (proper or normal {) law of
error has been found as an approximation to the actual locus
(which results from the composition of several independent
elements) may be extended to obtain a closer approximation
by taking into account the first of the terms which have
been neglected.

Put = S92, S denoting summation with respect to all
the elements and /, the mean square of deviation (not now
identical for all the elements). And for S(4,—34,”) put
K,. Then R may be written

= ee (1+ 7 +— (= Ke: 3) cos axdu.

~0

The first and main term of the integral is the normal
error-function
ar

if
<p — 5» Say
ofa G oo

The second term, the sought correction, is

( enter A 4K, cos arda;

0
which integral may be replaced by ee ES ia process
may be extended to terms of Baier one so as to form a
descending series, the “ generalized law of error” { for even
functions.

The transition to odd components is effected by obtaining
likewise an odd descending series; the whole of the odd
series—like the part of the even series which remains after
the first main term—tending to vanish as the number of the

* Op. cit. § 4, Legon xvi. Cp. end of § 11, Legon xiv.

+ The term “law of error” in this Paper may be understood ac-
cording to the context either as the “generalized law,” or only the
first and main term of that approximation, the so-called “ Gaussian,” or
“ normal,” law.

t Camb. Phil. Trans. Joc. eit. (1905). Cp. Journal of the Royal
Statistical Society, 1906, “ Law of Great Numbers,” by the present
writer.

428 Prof. F. Y. Edgeworth: An Astronomer

components is increased. Thus the first term of the odd
series may be written

1 ae

~ BY Stage”

where K,, =S3*, is the mean third power of deviation for any
element from its mean value; the mean value being taken as
the origin for each of the elements, (and for their sum) f.
When the differentiation is performed it is seen that the
term is affected with a coefficient which diminishes as the
number of components, say s, is increased. For example,

if each element is a binomial assuming the value 0 or a with
respective probabilities g and p,

ks=3(q—p)sa*/ (4spq)*a°,

that is of the order 1/,/spg. Thus not only Laplace’s proof
of the law of error for even frequency-functions, but also
Poisson’s proof thereof for odd functions, together with his
determination of a second approximation {, are found to hold
good upon certain conditions. One who considers that those
conditions are very generally realized will regard as mis-
leading the Astronomer Royal’s statement : “The conclusion
is therefore unwarranted and there is no proof at all that
peculiarities of the functions f efface themselves in the final
result” (loc. cat. pp. 167-8).

III. It remains to notice two other proofs § of Laplace’s
law || which have been likewise approached and missed by
Dr. Sampson.

There is first Morgan Crofton’s proof—either by way of a
partial differential equation], or more directly **—that the
continued superimposition of frequency-curves of any form
will result in the normal law of error. Dr. Sampson
applies Morgan Crofton’s method of composition to certain

* It will be noticed that the subscripts of the %’s correspond to
powers (of the component errors). The subscripts of the K’s correspond
to corrections of the normal function.

+ Mutatis mutandis, if the elements are weighted.

t Given by Todhunter, op. cit. pp. 567-8.

§ Two among several variant proofs which are referred to in the
article on “ Probability” (Ency. Brit.) §§ 104-111.

|| Laplace rather than Gauss deserves to be the eponym of the law of
error when, as throughout in this paper, it is considered as resulting
from the random combination of numerous independently fluctuating
constituents.

q Cited in the article on “ Probability” (Zncy. Brit.), § 109.

** See Phil. Trans. 1870.

on the Law of Error. 429

frequency-curves obtained by grafting an oscillating element
on the normal law, as thus

1

Oe) SS
#2) WV 7(1+ A)
He shows that when curves of this type are compounded the
divergence from the normal law tends to disappear. This
curlosum may have some bearing on astronomical obser-
vations. But it may be doubted whether observations
extending to infinity—-other than the law of error—have

much concrete significance.

The result of continued composition may often be more
advantageously contemplated by means of Laplace’s method
above indicated. Let the frequency-function tor one element
be (2), for another ¢.(x), ranging between given limits,
which may be infinite. If the functions are evenf,
put pi=¢,(2x) cosav. And let | p,da (between extreme
limits) =0,(«). Let 6,(a) be formed likewise from ¢,(2).
Then for the compound of the two frequencies we have

| 0,(a) X 0,(a) cos aada.
0

This method may be employed to obtain an answer toa
question which Dr. Sampson has raised, but has not rightly
answered: namely, what frequency-curves enjoy the property
that when two of a family are compounded the result belongs
to the same family. Dr. Sampson appears to think that the
normal law of error is the only curve which possesses this
property in perfection, “with any generality ” (loc. cit.
p. 170). He is not aware that the property appertains to
a wide class of which the normal law is a species: namely,
the class for which 6(«) (as above defined) is of the form
exp—at. There is here disclosed a variant proof of the law
of error. If there is a final form resulting from continued
composition, it must be reproductive ; and reproduction is the
proprium of curves for which @(a) is of the form exp—e‘.
A further condition (borrowed from Morgan Crofton) which
must be fulfilled by the sought form limits ¢ to the value 2.
Using the value of (a) thus obtained, viz. exp—a’?, we
obtain the normal law f.

e—Wx"(1 + acos hax) *.

* A simple case of the more general type proposed by Dr. Sampson
(loc. cit. p. 170). Note that a is less than unity, and A is taken so that
the integral of the function between limits +00 and —o is unity.

As in the case above instanced. For the case of odd functions
see Camb. Phil. Trans. loc. cit. p. 53 (1905).

t~ See Camb. Phil. Trans. 1905, “ Law of Error,” Part 1. § 4, and

Appendix, § 6.

430 An Astronomer on the Law of Error.

Dr. Sampson is thrown off the track by a little slip in a
mathematical operation such as the best may incur when
not put on their guard by prior knowledge of the subject.
Dr. Sampson experiments on frequency-functions of the

ee (or the more general form which is presented

(0+
when we take a for the parameter of #*). Here it may be
observed O(x) is of the form e—4 (or e—24), and accordingly
we know a priort that the continued superposition of a com-

a
ponents of the era) will have for result 4a
Dr. Sampson, employing Morgan Crofton’s method of com-
position by way of integration, finds this result true for two
components; but not true in general, for any number of
components. That is, in our notation, if the frequency-.

a the

: M : :
function —-~——~ is compounded with alata

m(1+ 2”)

1 eS: when a=1:
m (a+1)?+ 2’ Le tos
but not for larger values of a.
In Dr. Sampson’s own symbols, © the abscissa, and ® the
form of the resultant frequency-curve,

result is

i ae Ween yg 0 d)
bg at a4A7'14+(@—A')

an 2am 30?
~ Of+ 207(a? + 1) + (4-1

This result is evidently untenable; since it imports that if
we put together two frequency-curves, each symmetricak
about a central point and thence descending continuously,
the central ordinate for the compound will be zero! But if
we resolve the expression above equated to ®(®) into two
rational fractions according to the general rule for inte-
gration, we shall find that while the denominator of the
result is rightly given by Dr. Sampson, the numerator ought
to have been (a +1)|@?+(a?—1)?]; the expression in square:
brackets being a factor of the denominator. Dividing out
we obtain, as we ought, for the result

1 a+l1
er @’+(a+1)"

ri (loc. cat. p. 170).

* Substituting in the last-written expression for x, z/a and dividing:
the expression by a (substituting for dr, dx/a).

On Transpiration from Leaf-Stomata. 43?

This is not the only passage in the paper under con-
sideration where an inappropriate conception has led to
inexact work*. In connexion with his peculiar notion as
to the nature of an error of observation, the writer appears
to hold that if the extent of an error varies with the time
according to the law y=sint, the frequency of error between
the extreme values which it can-assume is constant tT. Not
so, surely ; the number of errors per unit of time being -
constant, the number of errors per unit of space { is propor-

tioned to a , that is gyi ED

dé V1—#
not a straight line but a symmetrical curve of the fourth
degree with an infinite ordinate at the centre and contact
with the abscissa at the points distant ¢ from the centre in
either direction.

In this connexion it is remarked by Dr. Sampson, “it
seems probable that Laplace and Poisson were on the wrong
lines.” To those who have followed the preceding com-
ments it will seem more probable that Dr. Sampson is on
the wrong lines.

The frequency-curve is

L. On Transpiration from Leaf-Stomata.
By Haroup Jerrreys, W/.A., D.Sc.§

wes most of Sir Joseph Larmor’s note in the April

Philosophical Magazine I am in agreement. It is
clear that the amount of water evaporated from the stomata
ofa leaf ina given time must be less than that from the whole
surface of a wet leaf of the same size; it must also be less
than the amount that could be evaporated from all the
stomata if their total number were the same and all were so
much separated that they could be treated as isolated.
Hitherto only the latter criterion has been used to indicate
the upper limit, and the important question is to decide
whether the former imposes a further restriction or not.
When the air is at rest the question is simple, for both limits

* The following criticism is due to Professor Bowley.

+ “For y=sin¢ .... the distribution is represented by n=1 for
—1<£<1 and n=0 beyond those limits (fig. 3).” Fig. 3 and the
context bear out the interpretation here given.

¢ The number which multiplied upon Ag a small fraction of (the
unit of) the abscissa, £ gives the (proportionate) number of observations
occurring between 2 and ++Aé (or a+3Az and 2—348).

§ Communicated by the Author.

432 On Transpiration from Leaf-Stomata.

can be well determined; the wet-leaf limit is then found to
be the smaller and therefore gives the more useful criterion.
When the air isin motion the question is much more difficult,
for while the evaporation from isolated stomata remains prac-
tically constant with ordinary wind velocities, that from a
wet leaf increases rapidly with the velocity and may come
to exceed the isolated-stomata limit. In most particular
cases it is probably best to find the wet-leaf limit by direct
experiment, for its theoretical value depends on the wind
velocity and the amount of turbulence, which are more
difficult to determine. The isolated-stomata limit can be
calculated without difficulty from the formula I gave.

I stated in my paper that to decide which limit was the
more important in a wind depended on the particular cireum-
stances considered. When a wind is blowing over a fixed flat
surface the air behaves differently in two regions. In contact
with the surface the velocity is zero, and within a certain
small distance, estimated by Major Taylor as 40/U centimetres,
where U is the velocity ata great distance measured in centi-
metres per second, the motion is purely laminar. The
velocity at the edge of this layer is a considerable fraction
of U, say3U or 4U. Within this layer heat is communicated
by heat-conduction, momentum by ordinary viscosity, and
gaseous constituents by ordinary diffusion. Outside of it
the motion ceases to be laminar and becomes turbulent.
The turbulence causes masses of air to be transported bodily
for considerable distances in all directions, and their mixture
with surrounding air causes a great increase in the ease of
transference of heat, momentum, and gases. The effect of
this is to add to the coefficients of kinematic viscosity,
thermometric conductivity, and diffusion the same quantity,
called the eddy conductivity, which is much greater than any
of their values in non-turbulent motion. Now the problem
of evaporation consists of two parts: first, the diffusion across
the layer of laminar motion, and second, through the tur-
bulent region. Sir Joseph Larmor neglects the resistance
to diffusion in this region, thus practically taking the eddy
conductivity as infinite, which appears to be justifiable in the
ase he considers. In the notation I used previously the
rate of evaporation on the isolated-stomata hypothesis is
2arn*kpcV A, and for a circular leaf of radius / and circular
‘stomata of radius a this becomes 47rn*kpal?Vo. The wet-leaf
hypothesis gives 3°95pVo(KU/*)?, where K is the eddy con-
ankalt :

- With
(KU)3
n? =33000/cm.?, a=5 x 10-* cm., =5 cm., k=0°24 cm.*/sec.,

‘ductivity. The ratio of these is practically

On Transpiration from Leaf-Stomata. 433

K=1000 em.?/sec., U=400 em./sec., this becomes 0°05
roughly. Thus the isolated-stomata law gives much the
smaller limit and is therefore correct. For indoor expe-
riments, however (and most transpiration experiments have
been done indoors), K and U are both much smaller than is
above assumed, and the limits will occur in the other order,
so that the wet-leaf law will apply and evaporation from
individual stomata will accordingly be restricted.

Consider then a leaf in air moving sufficiently slowly for
the wet-leaf law to hold. What happens when the stomata
contract ? At first they are capable of sending into the air
more water vapour than the turbulent air can carry away;
and as long as this remains true it seems to me that vapour
will stay in the layer of shearing, and practically saturate it;
then the rate of evaporation remains nearly constant, for it
depends almost entirely on the outer region, which is in the
same state throughout. As soon as the contraction reduces
the possible supply of vapour to below the maximum amount
that the turbulence can remove, the rate of evaporation will
diminish, finally reaching the limit zero when the stomata are
quite closed. Thus most of the reduction to zero will take
place in this last stage. I do not see in what respect this
result is paradoxical, and it does appear to be well confirmed
by experiment.

The amount of transpiration possible on the wet-leaf
hypothesis seems to be quite adequate. Thus, consider a leaf
of 5 cm. radius, in an atmosphere containing 0:04 per cent.
of carbon dioxide by volume. Then even in still air the
volume of CO, absorbed per hour could be 13 ¢.c., or 0°17 c.c.
per sq.cm. ‘Thoday finds experimentally that Helianthus
annuus in the open air can at the utmost absorb 0°14 c.c. of
carbon dioxide per sq. cm. per hour. On the question of
evaporation I have seen no quantitative statement amenable
to numerical calculation except Renner’s definite assertion
that the evaporation from a leaf is the same as that froma
water surface of the same size.

Note by Sir Josern Larmor.

Dr. JEFFREYS agrees that the standard discussions on this.
subject are not in error, though there isa question of how
closely they conform to the conditions of any particular
experiment. The idea of an eddy conductivity may prove
useful with due limitation; but I find it difficult to.

434 On Transpiration from Leaf-Stomata.

attach any definite value to K. It may be of use to sum up
in other terms the broad essentials, which seem to include
all the calculation that the facts are definite enough to
warrant.

The sensitive surfaces of a leaf are protected from injury,
by being the walls of interior chambers which are connected
with the outer air by narrow necks called stomata. Their
efficiency as absorbers of carbon dioxide or as transpirers of
water vapour depends of course inversely on the narrowness
of these necks. The reciprocal of this efficiency is the quantity
that can be discussed directly ; for it is proportional to the
total resistance of the channel of diffusion, which is made up
by addition from that of the chamber, that of the neck itself,
and that of the tube of diffusion outside extending to where
the atmosphere becomes normal as regards its carbon dioxide
or its vapour. The length that may be assumed for this
latter tube depends on circumstances inv olving the degree of
stillness of the air: and it is this that Dr. Jeffreys proposes
to elucidate by introducing the modulus K. Perhaps with
moderate motion of the leaf it has practically no length at all,
the air sweeping over the leaf and being renewed even close
to its surface. Ifa large proportion of the wall of a chamber
is fully efficient, the air in the chamber is, except near the
neck, practically free from carbon dioxide or saturated with
vapour, as the case may be: this follows from the usual
electrostatic analogy. If both these conditions hold good,
the one just stated on the inside and free sweep of air close
to the leaf on the outside, the total resistance is that of the
meck alone including the corrections usual in electric problems
for its ends, there being none for its outer end if the air is
renewed constantly close over that end. Ifthe stoma were
a mere aperture ina thin plate, its resistance would then
reduce to this correction for the inner end alone, as

Dr. Horace Brown remarks: it would be inversely as the —

linear dimensions, being for a circular aperture measured by
the reciprocal of its diameter, multiplied by the specific
resistance for air. This is the reason that a large number of
very small stomata can permit almost free interchange
‘between the interior of the leaf and the outside.

<

ee

ee

a

page |]

LI. A Method of obtaining General Reaction- Velocity Curves
for complete Homogeneous Gas Reactions at Constant
Pressure. By Georce W. Topp, 1).Sc. (Birm.), B.A.
(Camb.) *. |

HE writer has already worked out general curves for
homogeneous complete reactions between two sub-
stances when the volume of the reaction space remains
constantt. In many binary gas reactions a continuous
change of volume takes place during the reaction at constant
pressure. The concentrations of the reacting gases therefore
not only depend on the amounts transformed but also on the
volume of the products. Since many technical gas-reactions
take place at constant pressure, sets of general curves for
such reactions should prove of great value. The paper gives
a method of obtaining the curves.

Bi-molecular Reactions.

Take a reaction of the type A+B-1 or more resultants.
The resultants need not necessarily be gaseous. We will
assume that B is in excess.

Let a gm. mols. of A react with b gm. mols. of B.

Let 2 gm. mols. of either reactant be changed in time ¢.

Let v be the the total volume of reactants and resultants.

The velocity of the reaction is given by

CAE RTL —2 b—x
di\v) ~ <a
where k= velocity constant, or
ldx xdv
fat. wie dt = 5 (a—a)(b-2). < ; = (1)

Pato -=X, 1. e. the fraction of A changed.

If the pressure is maintained constant esha! the
reaction, then the volume at any time will be proportional
to the number of gas molecules present.

The initial volume when ¢=0 is

v) « 2a+(b—a),
(b—a) is the excess of B present.

* Communicated by the Author,
+ Phil. Mag. vol. xxxv. p. 281.

436 Dr. G. W. Todd on Reaction- Velocity Curves for
The volume after a time ¢, when the fraction of A changed
is X, is
v « 2a(1+aX) + (d—a),
where « is a constant for the particular reaction.

If the volume increases a is positive, if the volume
decreases « is negative.

We get la 2a(1+«X)+(b—a),
: 2a+ (b—a) f
and dv Zavox aX

ai 2a+ (6a)? dt
. Equation (1) now becomes

adX aX2anedX ,a’ b
ia Aa a eee
or dX

a

which is independent of «. Hence a volume-change at
constant pressure does not affect fa} so that the general
curves given in the paper previously alluded to will do for

the case of bi-molecular reactions at constant pressure as
well as at constant volume.

Ter-molecular Reactions.

Many binary gas-reactions are of the type 2A+B-1 or
more resultants. Let us work out this case assuming first
that B is in excess.

Let 2a gm. mols. of A react with 6 gm. mols. of B.

At time ¢, 22 gm. mols. of A and x gm. mols. of B will
be changed.

Let v be the total volume at time f.

The velocity of reaction is given by

d(x ah we) (=*)

as) = .( DLN a re

: Liv rd k

- = a = g(a—2)*(b—-2). | ia

(N.B. The concentration of the gas A has been taken as
half the number of A molecules per unit volume.)

* If there is also initially present m molecules of an inert gas, then
2a(1+aX)+(b—a)+m
Qa+(b—a)+m ”

but this does not affect the final expression for Te

=Vo

Homogeneous Gas Reactions at Constant Pressure. 437
Put —=x, the fraction of A changed. The volume at

any ees being proportional to the number of gas- -molecules
present, we have

vu « 3a+(b—a),
and v « 3a(1+aX)+(b—a),

aa( a(l+eX)+ (b— oe
3a+ (b—a) j

whence v=vt-

and dv Baug dX
dt 3a+(b—a)° dt

Substituting in equation (2) gives

F()

dt 14. "10d ay ae
2a+b
or, putting = Ps |
dX _) = aCe
dt re G ;
2+p™
da

Xx

je SE sidan
2 x 2+
7 law = EE i aR) a
i J0 (1— x) ( p— X) Co

This is the general equation for B in excess.
* If in addition to the 2a molecules of A and the 6 molecules of B,
there are also initially present za molecules of an inactive gas, then

3a(1+aX)+(b—a)-+na
3at(b—a)+na ‘

cae Vo

so that

a ,(1—-xy? eal
dt =h (5 a SE es
at 2a+b-+na

and equation (3) becomes

3a E
ay? (Xt at pt
x (“) i oe AX.
») =) T=X¥@-X)

Phil, Mag. 8. 6. Vol. 35, No. 209. May 1918. 2H

438 Dr. G. W. Todd on Reaction- Velocity Curves for

When A 2s in excess we have by similar reasoning,
since % « 3b4+2(a—bd),
and v x 3b( Co

x 1+
1+2p 7:
dX, a
Kc) See X) 4)
p in this case being —

If nb molecules of an inactive gas are also present in the
initial mixture, equation (4) becomes
3a

k(--) al Ot Tp tn ox

Y% 0 (—XPU-K)

Reactions with B in excess.
Let the volume-change when neither reactant is in excess
be given by
2 vols. + 1 vol. -> 2 vols.

5 —— 1
te a a= 3°

Equation (3) becomes

nf
i
K(2) t= ee
U 9 1=XP(p— a
iL 2 1—X +1 X
are Mog Dieu? ) i Bat
pt+2 \(p—1)? p—-X ~ p-11-X

For p=1
Ries mie Ta a (Se 9 10
z (2) hae 0 271 815 1:333 2253 4147 9333 3600 o

0

p=],
Se eee 0 3 5 6 7 8 9 10
e(S)¢ iA 0 Sh 98 158 270 530 170

Vo

p=2.
RE el Gees 5 7 8 5 16
u(Z) « ene 0 -296 -547 #+%+41368 #245 590

Vo

Homogeneous Gas Reactions at Constant Pressure. 439

6:0

VELOC/TY OF THE REACTION

- 2ZAtB —->2C
al
Ne

p=3s.
BoP. 0 5 7 8 9 ‘95 10
id eae ee 0 349-839 «1-470 «3-406 7-389 «sao
0.
p=4
Be 0 6 3 9 95 10
a(S) ¢ Wes 0 389 1060 2424 5180 Pa
0
2—6.
7 Sa 0 7 8 9 ‘95 10
& (2) « ee 0 295 507 1148 2-43 s
0
Fig. 1.
Lo 5 1-0 5 2:0 2:5 278-0 35 40 45 5:0 5:5
:
_ SERRE 2 eee eee
jee.
(EG eee eee oe
tot s ine e=. Pt
mela at i eee
HAA ay pease eee
Ca Ce Ee
A a oeea
6 7 al
UL ae

Binvexcess, a=> 4y

X= fraction of Aconverted intime t

SRR eebaw
pe ee OPN Ae

2 initial concentration of A

_ initial concentration of B

b
“initial concentration of A a

2 a) tS k=velocity constant
Seas ee
See | | | ere
Sea |
0 =< ‘0 15

20 2a 35. 40 4) 50 Ss 60

These figures have been plotted in fig. 1. The curves
obtained are applicable to all ter-molecular gas-reactions of
the type 2A+B- at constant pressure, the volume-change

440 Dr. G. W. Todd on Reaction-Velocity Curves for

being such that a2=—1},and the gas B in excess. Hxamples
of such reactions are

2H,+0,>2H,0 oxygen in excess.
2CO0 +0, 2C0, a
2NOAOs 2NO3 oe

Now let the volume-change, when neither reactant is in
excess, be given by

2 vo. + 1 vyol.-sa wol:

1 = —2
Ue &. a= 3°

93

99

Hquation (3) becomes

5 eee
79%
i( Yi=(" Jp Lame
Vo (1— — X)' (p— X)
i 1 { ee nb ss: —X Ne Xx
pre Vad pk) pee
For p=! :
ROR Oe reniesey (°° oR. CUBR ip, Meenas 9 10
; 2
#(S) Vas 0 -261 -751 1167 1-874 3235 6666 22°50 o
0
p=1}.
Gis GAS, Kes |e 7 8 9 10
2
z(=) Bos eus O +358 -864 1242 217° 399 1OOkM ae
0
p=2,.
BO VNR en a we
2
z(S) ¢ Bane 0 -214 -500 750 1166 2-000 4:500 9500 o
0
pS,
Bi 8 Cs My Hie «6 eee | ge
2
z (2) FU 0 218 -485 747 1265 2-797 5830 oc
0
p=.
saad Tame et an ae = 9 95-87
2
z(S) ane 0 +195 -444 -758 1-667 8465 5868 o
0
p=s.
a A, OH eg 8 9 95 ry A
2
z(<) ¢ ete 0 -281 -475 1055 2205 3731 @
Wer EA

Homogeneous Gas Reactions at Constant Pressure. 441

These numbers give the curves in fig. 2. The curves are
applicable to all ter-molecular gas-reactions of the type

Fig. 2.

150. (55. 60

ane
ACK
\
a
|

D>.

ea
‘a
a
a
g,
aj

VELOCITY OF THE REACTION
2AtB —> C

Binexcess; @a@=- %

N= fraction of A converted in time t
= = initial concentration of A

_ intial conéentration of B - 2
inttial concentration of A a

P

ae | Ae

kh =velocity constant

Ale *p—

=

een 35° 426 Tee Boe Son len

gett NAN

yy
ot

2A + B-> at constant pressure when the volume-change is
such that «= —2 and gas B isin excess. Such a reaction

is 2NO+0O,->N,O, oxygen in excess.

Reactions with A in excess.

Let us reconsider these reactions in which a=—1 and
a= —2 when the A gas is in excess.

Take first a= —4,

Hquation (4) becomes

| ae
(a (" ax
a Jake Se

1 { 2p p—-X p+l xX \.

—— OD if EE eb BASS | eeu ess
1+2p ((p—1)? “*pG—X) p(p—1) p—X

442 Dr. G. W. Todd on Reaction- Velocity Curves for

For p=1.
Bai igi pee re) te Rete mere Pye payee 9 10
2
u(>) « jae 0 -271 ‘815 1:333 2-253 4147 9:333 3600 ©
0
p=lyo
ihe eee. aie 6 7 5% 8 10
2
“(-) Patan 0 Sl) 97 164 O74. 37) aoe
0
p=lt.
Maes ba: OF a8 5 "7 8 9 10
b 2
z (=) Nah ae 0 1909 -558 (145 262° “ogee
0
p=li.
Tue at DO. vee 6 7 8 9 1-0
x ( =) e ate 0 394 657 995 1578 “SeeLe
0
p=2,
Ae See Pe ey i eo 9 9 19
2
a(>) sae 0 -294 -319 -455 679 41115 1706 o
0

The curves obtained from these numbers are given in

fig. 3.

Now take a= —2.
Hquation (4) becomes

-ie a

eo) (5 a poe x
26- 2p-1), p—xX I x}.
= hp (p—1)' “©"p1—X) pl) poe

For p=1
bgt ean aU Oe neds) | SB eee 9 10
R(o)e aes 0 -261 751 1167 1:874 3-235 0°666 22:50 o
pHlis
Gea ney 0:3) a ee 6 7 8 9 10
eS) Agee 0 36 86 140 222 409 96

Homogeneous Gas Reactions at Constant Pressure. 443

p=li.
meee. Dees nae Yun Queene 1D
b 2
z (=) ¢ Bi 0 -271 641 -956 1465 2395 473
0
Fig. 3.
Sas 20. eo ee ss 4p SAS SSS 260

°o

@ ,

CE pee Tt a
aenee= Soe SEE
Ze ie ee

erte® Cee
Bz ate pt |

Cer ee ee
ELE He alias

iw)

Ss)

Me Neem
ae

at VELOCITY OF THE REACTION

ct ee 2ATB Ze

Ainexcess, a=-%

i7aiias = fraction of 8 converted.in time t
ey a

|

é = inttial concentration of 8

f
We hi p= iitial concentration OFA ,, 2

Wutial concentration of 8

ee k = velocity constant
et St ef
__TEaSRee

0 5 FSI 40 ee SO Ss) 60

p=1i.

= 4 ae a Wea 7 8 9 105.) 11:0
5 2

z(-) Be, Tp ec Wa ey cs
0
p=z2.

at | 0. “4n\ ee +9 95 9 10
b 2

«(=) ae 0 +147 -292 -592 -938 1318 1616 o
i?)

These numbers are plotted in fig. 4.

Curves for reactions in which « has other values can
easily be worked out from equations (3) and (4). Values of
a other than —} are uncommon. When the resultants of
the reaction are solid or liquid, e= —1,

444. Notices respecting New Books.

General Curves for Reactions of Higher Orders.

Gas-reactions of higher orders are rarely met with (except
in the case of combustion of many hydrocarbons), therefore

Fig. 4,

VELOCITY OF THE REACTION
2AtB — C
Ain.excess; a=- %
N= fraction of B converted in time t
é = initial concentration of B

p= intial concentration ofA _@
initial concentration of 8/ ~ b

k= velocity codstant

5 0 “tt 20°25 30 35 40. 4:5) sagen

examples have not been worked out. General curves for
any particular type of reaction can be obtained by following
a procedure similar to that used in this paper.

London, February, 1918.

LIL. Notices respecting New Books.
Annuaire du Bureau des Longitudes pour Vannée 1918.
Pp. viiit+869. Paris: Gauthier-Villars. Price 2 francs.

MP\HIS excellent almanack contains, in addition to the usual astro-
nomical data, many other tables of interest relating to meteoro-
logy, terrestrial magnetism, and kindred subjects, together with a
very useful collection of physical constants. Besides the tables there
are special articles, written by good authorities, on the following
subjects: Sundials, the Egyptian Calendar, Time at Sea, the Sun
and Terrestrial Magnetism, and an obituary notice on Professor
Gaston Darboux by M. Picard.
The book is embellished with star maps and figures, and is a
very useful and wonderfully cheap publication,

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SERIES.]/ _. >

wv?

JUNE 1918. \% ~o |
ee. NG ie
LITT. Molecular Frequency and Molecular Number Part EIT:

Inorganic Compounds. Lindemann’s Formula. By H.

STanteEy ALLEN, M.A., D.Sc., University of London,

King’s College*.

N the present communication are given the values of the

characteristic molecular frequency, v, calculated by
Lindemann’s formula for those inorganic compounds for
which data are available. The formula in question may be

written
ik
i uh Mv?

where T, is the absolute temperature of the melting-point,
M is the molecular weight, and V the molecular volume.
The coefficient & has been assumed constant, and Nernst’s
empirical value, 3°08 x10", has been employed.

For each compound is tabulated the value of the molecular
number, N, and of the product Nvx10-¥. In the majority
of cases it has been found possible to express Ny either in the
form av, or in the form (7+ 4)v,, where mis an integer and vy
is a constant frequency having a value about 21 x 10” see.-1.
In the few exceptional cases where neither of the above forms
is applicable the product has been expressed as (n++2)y, or
(n+2\v,, but no special significance is at present attached
to such results. All the results here given must be con-
sidered in the light of the general considerations affecting
Lindemann’s formula put forward in Part I. of this papert.

* Communicated by the Author.
t+ Phil. Mag. vol. xxxv. p. 338, April 1918.

Phil. Mag. 8. 6, Vol, 35. No, 210, June 1918. 21

446 Dr. H. §8. Allen on Molecular

Compounds containing water of crystallization have not
been included, as in such cases it is often difficult to decide
whether the recorded temperature is a true melting-point or
the temperature at which the solid dissolves in the water
of crystallization.

Groupl. .,

Of inorganic compounds for which the density and the
melting-point are known, a larger proportion belongs to the
first group than to any other group in the Periodic Table.
With a small number of exceptional or doubtful cases, the
results show good agreement with the suggested relations. It
is noteworthy that when the temperature of the melting-point
is known accurately, the agreement is better than when this
temperature is uncertain.

For several of the lithium compounds the melting-point
has not been determined with great accuracy, yet the values
of v, do not differ widely from the mean value which is
20°7 x 10! sec.~' The mean value for the elements, given
in a former paper *, was 20°9 x 10'sec.~1, when the same
factor was employed in Lindemann’s formula as that here
used. It has been pointed out previously} that the chance
of these results for the lithium compounds (including
Li;PO,.H,O) agreeing accidentally was about 1 in 280.

Grove I.
Lithium Compounds (N=8).

Name. Formula. N. Vv x10-12, Ny x10-12,
LitHrum
AMNVABL, Yee eeeees exec oeee LiNH, 12 6:073 33 X 20°8
browmitde, Nye ss.a eee LiBr 38 3°234 6 x20°5
GATHONAEO oieweeeeeunr Li,CO, 36 3°435 6 X206
chloridle (4)... 3.000 LiCl 20 5-079 5 x203
fluoride ..... mash eie Lik 12 9:207 53x 20°1
LOGS cu. Wen senna Lil 56 2°227 6 x20°8
TIAN BAC pts aetiess a5 ee LiNO, 34 2°821 44 X21°3
perchlorate............. Liclo, 52 1-742 +X 20°1
piliga te. (8)... cs.ine seen Li,Si0, 44 3°756 8 x20°8
sulphate acid ......... LiHso, 52 1-636 4 X21°3
SulpHAabe: cea (hae Li,SO, 54 2°680 7 X207

(2) Richards and Meldrum (1917) give the melting-point of pure
lithium chloride as 618° C. This makes for better agreement, giving
Nv x 10717 =5x 2056.

(b) The melting-point of lithium silicate is given as a standard tem-
perature (1201° ©.) in the Smithsonian Physical Tables.

* H.§. Allen, Phil. Mag. vol. xxxiv. p. 478 (1917).
+ H.S. Allen, Phil. Mag. vol. xxxv. p. 338 (1918),

Frequency and Molecular Number. AAT
Sodium Compounds (N=11).
Name. Formula. N. »xl07. Nyx fo-™.
Sopium
borate tetra- ......... Na,B,O, 98° 1:596 73 X20°9
BRGHIAUE! Ceccts sess: NaBrO, 70 1-801 6 X21:0
SMO MhO: 22) con dSe sce e ce NaBr 46 3037 63 x 20°7
GREROOALE | .--s 2. a 5.'. Na,CO, 52 2878 7 x214
CEREALS ici sacdesss sus NaCloO, 52 1°962 5 xX20°4
ehiloride (@)...:.....'... . NaCl 28 4-407 6 x206
MMGLIGE 5.05.62 0200 65% NaF 20 6-794 +x 209
hydroxide 30.3.5... NaOH 20 6°787 65 X 20°9
MILO Vc cicciecls oe sian ci Nal 64 2°268 tx 207
MPERGES ooo fh ews aden: NaNO, 42 2417 5 x20°3
MaMMIEG)(O). 2.0. eds NaNO, 34 2°724 4>xX206
phosphate meta...... Na,P,0O;,5 200 0°829 8 X20°7
MMI PHELO . 0 ..s0s.ee nc a,SO, 70 2337 8 x20-4
sulphate acid (¢).,... NaHSO, 60 1°859 52x 20°3

(a) Melting-point of sodium ehloride (801° C.) has been used as a
standard temperature.

(0) Melting-point of sodium nitrite, 271° C. (Divers, 1899),

(c) Melting-point of NaHSO, “ over 315° C.” (Gmelin- Kraut).

Potassium Compounds (N=19). .

Name. Formula. Ne 107-4, Nv pein; 14.
PorassIUM
arsenate acid ......... KH, AsO, 86 1-564 53 X21°3
EOMALE 252.0605 5.0.) KBrO, 78 1°708 63x 20°5
EOMIES) 555004000... KBr 54 2°575 62 x 20°6
carbonate ............ K,CO, 68 2-310 73x 21:0
EIOEBIS <. co ccsoe inet KCI1O, 60 1886 53x 20°6
BAMGRIGE 5.025 cies KCl 36 3°493 6 x20°9
Giromate <..:..:..... K,CrO, 94 1-882 83x 20°8
dichromate............ K,Cr,0, 142 0-974 65 x 21-4
MATING. cso 2: . 2 32+ KE 28 4°789 63 x 20°6
PONMUALE feos -2 oc 0s0e 00 KCHO, 42 1955 AX 205
hydroxide ............ KOH 28 3°431 4*-X 21:3
MEDD ape a vacev covets. KIO, 96 1-609 6 x204
STL ee KI 72 1:988 7 X20°4
AOORAGI EET oo ve tiece KI, 178 0-543 45X21-5
MEREMED SC. 522 d-iis 4000 KNO, 50 2:080 5 X20'8
perchlorate............ KCIO, 68 2-046 65 X 21:4
poriadate. 6.2 :.-cse2- KIO, 104 1-488 74X 206
hosphate 12 AL Ae ON:

ae } KH,PO, 68 1308 41x209
ROLDDALG (J). oc1k0 2500 K,SO, 86 2°121 9 X20'3

Be RILEY ag 5 coe ae: KHSO, 68 1:536 5 x20°9
sulphocyanate......... KCNS 48 1778 4 x213

(a) Melting-point of the anhydrous salt, 96° C.,’Tilden (1884).
(6) The melting-point of potassium sulphate (1070° C.) is used as a
standard temperature.

The results for sodium and potassium compounds are
specially interesting, and should be compared with one
another where that is possible. There are eleven cases in

212

448 Dr. H. 8. Allen on Molecular

which compounds of sodium are represented in the second
list by the corresponding potassium salts. For five of these
(the bromide, chloride, fluoride, iodide, nitrate) the frequency
number is the same for the sodium as for the potassium salt.
For the bromate, carbonate, chlorate, the frequency number
for the potassium compound is greater by 4 than that for
the sodium compound. for normal potassium sulphate
Nvx 10-”=9 x 20°3, as compared with 8x 20:4 for the
sodium salt. The results for the acid sulphates and the
hydroxides are less reliable.

The bromides of sodium and potassium require special
consideration. For NaBr the melting-points recorded fall.
between the early value of Carnelley (708° C.) and that
of Ruff and Plato (765° C.). Taking the latter value,
Nv = 139:'7 x 10%. For KBr Carnelley gave the value
699° ©., whilst Ruff and Plato in 1903 found 750° C.
Using the highest value according to the principle adopted
generally, it is found that Nv=139'0x10". Thus if the
melting-points of Ruff and Plato are correct, the values of
Nv for NaBr and KBr are in close agreement with one
another; but they cannot be represented by using frequency
numbers of the form n or n+ 4 unless high values for vp, are
employed, the product for NaBr being equal to 64 x 21°5 x 10
and that for KBr being equal to 64 x 21-4 x10”. A redeter-
mination of the density and melting-point for each of these
salts is to be desired. It may be noticed that both LiBr
and RbBr give normal results, the value of Nv for the
former salt being 6 x 20°5 x 107, for the latter 7 x 20°5 x 10!2;
the melting-point of CsBr is not recorded.

Assuming that the value of Nyx 10~” for NaBr is written
in the form 64x 21°5, calculation of the probability by the
formula of Laplace shows that the chance of an accidental
concordance in the values of v, for the sodium salts is about
1 in 36.

The agreement between the values of v, in the case of the
compounds of rubidium and cesium is only moderately
good. Perhaps it may be safe to predict from a comparison
of the results with those already given that the melting-
point of cesium iodide will prove to be higher than the
recorded value, 621° C., so as to give a frequency number 8
instead of 73. It will be noticed that in general the
frequency numbers for the compounds of the alkali
metals tend to increase as the atomic number of the metal
increases.

Frequency and Molecular Number. 449
Rubidium Compounds (N=37).

Name. Formula, N. vx10-!% Nvyx10—):
Rvusipium
PenmNGe 66. os eeccceess RbBr 72 1:990 7 x20°5
chloride (a) ......... RbOl 54- 2-310 6 x20°8
i = Se RbF 46 3021 62x 21-4
| PpRRPTIANG — oo occas. sc RbOH 46 2:297 5 x211
“LOTT aS A Rbl 90 1-616 7 x208
Gonde Ga. Sk ita Rb,O, 90 1-674 hi SAU
pentasulphide......... Rb,S, 154 0-685 5 x21

(@) Melting-point of rubidium chloride, 714° C. (Richards & Meldrum, 1917).

Cesium Compounds (N=55).

CzsIuM
chloride (a)............ CsCl 72 2:065 7 X212
7S) Dee eeenreere CsI 108 1-480 74X213
A eh ere CsNO, 86 1-542 63 x 20°4
baie tri-. ....:.....-- Cs,0, 134 1-076 7 X206
SEP ROLY = Leet ss. s suk Cs,0, 142 1-073 73 X 20°3
pentasulphide.. ...... Cs,S, 190 0°612 5321-1

(a) Melting-point of cesium chloride, 645° C. (Richards & Meldrum, 1917).

Only a few copper compounds can be included ; it is of
interest to note that the frequency number for cupric
chloride is 63 as compared with 6 for the cuprous salt.

Copper Compounds (N= 29).

Name. Formula. Wp K10T Ne 103:
Cuprovus
Bromide oo cc cce CuBr 64 2:267 1 ST
LE ee CuCl 46 2-726 6 x20°9
MMO bodes cas eue Cul 82 2048 8 x21°0
sulphide ........-.:.-. Cu,S 74 3002 11 x202
Curric
eaeterthe . 3. sos ean CuCl, 63 2°136 63 x 20-7
SLE RES ee eee CuO 37 5°458 10 x20:2
Silver Compounds (N=47).
SILVER
TE 2 ee ree ee AgBr 82 1-935 tae 2 2
chlorate ........ cere _ AgClO, 88 1-421 6 x20°8
BUREN Yoo oct owcivecue = Ag(Cl 64 2°343 (fees ie
20 TES CS et ra ele AgF 56 2-609 7 X21:0
MUA 5 3, -sonntscebane AgI 100 1657 8 xX20°7
MATOS (1) © 2-2 cons o 00 AgNO, 78 1-543 6 xX201
phosphate ortho...... Ag,PO, 188 1309 15 x206
J Pyro ..5..; Ag,P,0O, 274 0756 10 x20°7
BUI BAEG oo. svese<se-00- Ag,SOx 142 1°378 3X 20°8
2 ATT Aas Ag,S 110 2004 103x21-0
felitieide 2. 3.05<305- Ag,Te 146 1686 12 x205

(a) Silver nitrate changes from the rhombic to the hexagonal-rhombo-
hedric form at 159°°8 C. The density of the rhombic form has been
used in the calculation.

The salts of silver show fair agreement with the proposed
relation. It is doubtful whether silver nitrate should have

450 Dr. H. S. Allen on Molecular

been included, as the density corresponds to a crystalline
form different from that to which the melting-point applies.

No results are available for compounds of gold.

The melting-point and the density are known for the
greater number of the halogen derivatives of Group L, so
that it is possible to institute a comparison between the
frequency numbers for this group of compounds. ‘The
results are collected in the Table following, which contains
the atomic or molecular number in italics and the product
Nv x 107" in roman numerals. The values for the thallium
compounds have been added as they resemble closely the ©
corresponding rubidium compounds*. A close examination
of the table will reveal many interesting relationships ; the
close resemblance between the sodium and the potassium
salts is at once evident; the frequency numbers for these
compounds exceed the corresponding numbers for the
lithium salts by unity (the bromide excepted); the fre-
quency numbers for cuprous chloride, bromide, iodide form
the sequence 6, 7, 8; the frequency numbers for silver
chloride, bromide, iodide form the sequence 7, 74, 8.
Further, it will be noticed that silver chloride, cuprous
bromide, sodium iodide have the same frequency number, 7;
ceesium chloride, rubidium bromide, potassium iodide form a
similar chain with frequency number, 7.

Monohalides of Group I.

Element. Fluoride Chlorde Bromide Iodide
P93: Cl=77,. Br=35, 1=53,
li=3 12 20 38 56
54x 201 5x 203 6 x 20°5 6x 20°8
Naor 320 28 46 64
63x 20°9 6 x 20°6 62x 20°8 TX207
K=i9 28 36 54 72
63 x 20°6 6 x 20°9 62 x 20°6 7X 20°4
Cu=29 38 46 64 82
—_— 6 x 20°9 x 20% 8x 21:0
Rb=387 46 54 72 90
64x 21-4 6 x20°8 7x 20°5 7x20°8
Ag=47 56 64 82 100
7x 21:0 7 X21°4 72X21:2 8xX20°7
osligan Vey 2 60 108
—- rp ob | — 743X213
T1=81 90 98 116 134
LUNES 8 x 21-4 dl (iB heal. \ hens

* Tutton, Roy. Soc. Proc. vol. Ixxix. p. 351 (1907).

Frequency and Molecular Number. 451
Group II.

On the whole the results tor compounds containing
elements belonging to the second group are less reliable
than those already considered. This arises partly from the
fact that in many cases there is uncertainty as io the true
value of the melting-point, no values later than those of
Carnelley being recorded for several compounds*; further,
the product Nv is in general greater for these compounds
than it is for the corresponding compounds containing
elements of the first group, so that larger frequency numbers
are required. Consequently the frequency numbers given in
the following Tables are not to be regarded in all cases as
final; some of them—especially amongst the larger numbers
—may require revision when more exact data have been
obtained.

In spite of such uncertainties a comparison between the
compounds in these Tables, and also between these and the
corresponding compounds for the first group, will show
many suggestive relationships.

Grovp II.
Beryllium Compounds (N=4).
Name. Formula. Ne 107 7) No 107
BeEryLuium
That Cah See eee BeF, 22 5213 535 X 20°9
ERAS 2. eo Sac esa eke sae Bel, 110 1:339 oe xX21-0
Magnesium Compounds (N=12).
Macnesium
ehloride C .......:.... MgCl, 46 2°807 6x21°5
PUBIC io cosas MeF, 30 5-446 8 x 20°4
Calcium Compounds (N = 20).
CaLciuM
aluminate ........+++- CaAl0, 78 S011 112%20-4
BEBHUGOG! (202.8 ews th a CaBro 90 1:783 8 x20°1
PERI G) occ cations CaCl, 54 2°302 6 x20-7
MOEVGG ys tients teks ye CaF, 388° 4757 9 x2071
BARE a o's osicutva sais Calz 126 1464 9° <20°5
nitrate (2) Cece... Ca(NOs)2 82 1715 7 x20-1
rE rs es Ca3No 74 2°532 9 X20°8
EET eS a eee Pe CaO 28 7696. 10520°5
SELES ois oe weet twe CaSiO3 58 3°53 10 x20°5
PIEGUAGO 22s. ocieenes' CaSO, 68 2979) TOK 20'S

(a) Highest. density recorded, 2°472 (Landolt-Bornstein).

* These cases are indicated in the tables by the letter C following the
name of the compound.

452 Dr. H. 8. Allen on Molecular
Strontium Compounds (N=38).
Name. Formula. N. x10
STRONTIUM
bromiden€ ccs: nemees SrBr, 108 1512
chloride 22... 0 hncaetee SrCle 72 2:201
fuoridejG, 3:25 SrF, 56 3°037
hydroxide #3.:55.208 Sr(OH) 56 2:204.
1OUUEC,.2 sk ek cence aces Srl, 144 1197
NitKate Clg... .c cena Sr(NOs3)2 100 1549
ORIG “soc ee aso eee SrO 46 67195
silicate; 2... tees SrSi03 76 2°940
Bien. Barium Compounds (N=56).
EORTC ition ee cearne BaBr, 126 1°531
carbonate (@) ......... BaCO3 86 2°485
chloride @.ir/iaraees BaCl, 90 1-982
HUGE see dieccteue eee BaF, 74 2°768
TOUTS rose se de eae Bal, 162 1°153
narrate: OC [i.o. cues Ba(NOs), 118 1:298
StlVeaie! wartiesueeseeee Ba(Si0,) 94 2°418

Nyx107”,

8 x 20-4
74X21°1
8 X21°3
6 X20°6
82 Xx 20°3
73 X20°7
14 x20-4
11 x20°3

9. X2is
103 x 20°4
83x 21-0
9 x20°5
9 x20°7
74 X 20°4
Ll. 207

(a) ‘“Schmilzt in CO, bei 1380° noch nicht” (Boeke).

Comparing together the compounds of calcium, strontium,
and barium we find that, in general, the frequency number
corresponding to a particular radicle tends to increase with
an increase in the atomic number of the metal. Again,
comparing the chloride, bromide, and iodide for a particular
metal, itis noticed that the frequency number for the bromide
is greater than that for the chloride, and the frequency
number for the iodide is at least as great as that for the
bromide. In the case of the fluoride, however, the frequency
number is about the same as that for the iodide.

Zine Compounds (N=30).

Name. Formula. N. vx107!%. Nvyx10-”.
ZINC
SUCOUAEC,.cokan viene conte Zn(C,H30,), 92 11138 5 xX20°5
bromide CO nee Zn Bro 100 1:408 7 x204
chloride (a): cens che ZnCl, 64 1849 54x 21°5
fluorides Gz c.csnenee ZnF, 48 3410 8 x 20°5
POMUAG IO. sce mame Zul, 136 1143 73 X20°5
sulphide, ..c: Hc anate ZnsS 46 3°936 9 x20'1
‘‘All lower values are

(a) Melting-point of zine chloride, 365° C.
obtained from impure material.”

Cadmium Compounds (N=48).

CADMIUM
bromide cc. eoraee CdBr, 118 1°412 8 xX20°8
chloride sc. seccscac: CdCl, 82 1:846 75 X20°2
finomde.C “i kos) ee CdF, 66 2°500 8 x206
TOMI ota. cnts vehement Cal, 154 1:027 Gatst
eyo) oo: ie aR RCORE RED CdSO4 96 2°153 10 x207
Wpeutae Mercury Compounds (N=80).
romrdelie,c cscs ae HeBr, 150 0-927 65 X 21:4
chloride: (cise HegCl, 114 1:200 63 x 21°1
LOMO scat ececes nae Hgl, 186 0°798 7. x2t2
Mercurovus
LOGIC (@) ec vnesupenee Hel 133 1157 74X 21:2

(a) Melting-point of mercurous iodide, 290° C., Yvon (1878).

a

Frequency and Molecular Number. 453
Group III.

The compounds met with in connexion with elements of
the third group do not require much discussion. The
melting-points of boron oxide and boric acid are determi-
nations of Carnelley in 1878, whilst those for aluminium
bromide and aluminium iodide are of an even earlier date,
for these latter compounds the degree of molecular asso-
ciation is uncertain also.

The results for the compounds of thallium form a fairly
consistent series, and it is of interest to compare the frequency
numbers with those previously given for the compounds of
the alkali metals. The mean value of v, deduced from the
thallium salts is 21:05 x 10'sec.~1, and the chance that the
agreement should be purely accidental is about 1 in 39.

Grove ITI.
Boron Compounds (N=5).
Name. Formula. NS oo M105 1, Nom
Boron
oxide 0 eens ts B,O3 34 3°185 5 X2E7
sulphide tri ......... B.S, 58 1613 45 X20°8
Bi) cjeeuta:..... B.S. 90 1271 528x208
Boric
Pe HBO, 32 2387 = 32 x 218
Aluminium Compounds (N=13).
ALUMINIUM
MGC. o-tona use: Al; Br; 236 0°858 5 X20°3
dee ee ALI, 344 0682 58x 21°3
TLS ee Al,O, 50 4949 12 x206
Berne. oa... woe. sess A1,S3 76 2°212 8 x20°5
Yttrium Compound (N =339).
YrrriuM
PRMET EAR Foo 2 28 sce YtCl 90 1114 5X 20-1
Lanthanum Compound (N=57).
LANTHANUM
paberide) 5.092 25a2 Lets LaCl 3 108 1-693 9X 20°3
Cerium Compound (N=58).
Crrovus
CLC 2 [a eee CeCl3 109 1°645 9x 20:0
Thallium Compounds (N=81).
THALLIUM
UD) ae TlBr, 116 1-466 8 xX21°3
CATDOWALS. ....2-0---- T1,CO3 192 0°825 73 X 211
e.g (Cee ee T1Cl 98 1-623 6 X21'1
On Ee NA a Ti 134 1:247 8 X<20'9
TRAE in is as o's TINO; 112 1-137 6 X21:2
Guide Cmy eo... T1,0, 186 1067 923x209
perchlorate ........... TIC1O, 130 1-310 8 X21°3

sulphate ............... TLSO, 210 0981 10 x206

454 Dr. H. S. Allen on Molecular

Group LY.

The compounds of carbon are so numerous as to require
separate consideration. It is only necessary to mention for
the sake of comparison with compounds containing other
elements of the same group, that for carbon tetrabromide
(N=146) the value of Nv is 5x 20°5x 10”, and for carbon
trichloride (C,Cl,) the value is 44 x 20°6x 10". For silicon,
titanium, and zirconium only a few results can be given.

Grove LV.
Silicon Compound (N=14).
Name. Formula. N. »xl107?. Ny xloo:
SILICON
tetrabromide ......... SiBr, 154 0°553 4x21:3
Titanium Compounds (N=22).
TiraNium
tetrafluoride ......... a, 58 1848 5X 21-4
tetrabromide (a)...... TiBr, 162 0°545 4x 22:0
OWES: Wr RIN TO, 38 5°536 10x 21:0
(a) Melting-point, 39° C., Duppa (1856).
Zirconium Compound (N=40).
ZIRCONIUM
CSREES + ah seca nee ZrO, 56 5°283 14x211

Disregarding the high value for v, in the case of titanium
bromide, for which no modern determination of the melting-
point is recorded, the values of vs for the compounds of this
group are generally higher than those for the earlier groups.
In the case of the lead salts, however, a lower value is found
for the chloride, bromide, and iodide; but the close agreement
between the values of the product Nv for these three salts
deserves special notice, as also the agreement for the oxide

and the sulphide.

Tin Compounds (N=50).

Name. Formula. NN. yxlo-!. Nex 10s
STANNIC
Mupride b..sc/e.veee ee SnF, 86 2°049 85 X20°7
MOCECE sik. sce Sucugeenmaees SnI, 262 0-491 6 x21°4
15-6116 | Ree Ae p55 SnO, 66 oo0e 105x211
STannous
DrOnaTde YU. bak eens SnBr, 120 1:075 G xis

sulphide: <:./ecveueedene SnS ~ 66 2-780 9 x20°4

Frequency and Molecular Number. 455
Lead Compounds (N =82).
Name. Formula N. y»xlo7 Nvxio7™.
LEap
UCCTICTS (SSR ae a ee PbBr, 152 1:067 8 x203
ING Fence Ocenia sal as PbCl, 116 1-414 8 x20°5
CE 0 LeU ee me dado is 188 0-865 8 x203
ch eae PbO 909 2448 103x21-0
PleMUMOWice a iccsesaee ste: PbSe 116 2-029 ll x21-4
SDIBRE GS deca Cate wic ees PbSO, 130 L795 Ih <2Te2
BIPPHUES, foi. 2 kolo nns ye PbS 98 2°274 103 X 21:2
Thorium Compound (N =90).
THORIUM
EIOEIME och eno secees ThCl, 158 1-214 9x 21°3

Group V.

Almost all the ccmpounds so far considered have been
metallic salts. In the fifth group a number of non-metallic
compounds are met with; and. although many of these
conform to the proposed relation with frequency numbers n
or n+4, several require the use of values n+2 orn+?. It
is at present impossible to decide whether this means that the
relation fails to hold in these two cases, or whether the
apparent failure is to be attributed to our ignorance of
the degree of molecular association in such compounds. It
may be pointed out that the compounds in question belong
to the ‘‘ non-polar” type*. According to Langmuir solid
polar compounds are built up of atoms bound together by
secondary valencies, whilst solid non-polar compounds consist
of “ group molecules” in which the atoms are usually held
together by primary valencies; these group molecules in turn
are bound together by secondary valencies to form a large
“‘erystal molecule.” As data for determining the molecular
frequency are available for only a small number of inorganic
non-polar compounds, further discussion may be deferred till
organic compounds are considered. |

Group V.
Nitrogen Compounds (N=7).
Name. Formula. Ni vxdo—? Ny x 10n:
NirroGen
chlorophosphide ... N,P3Cl, 168 0'580 45X21°6
pentoxide ............ NO, 54 1-278 4X 21°2
sulphide 3...032c.00-.5% N,S, 92 1117 5 X20°5

* See Langmuir, Amer. Chem. Soc. Journ. vol. xxxvili. p. 2221 (1916).

456 Dr. H. S. Allen on Molecular
Phosphorus Compounds (N=15).
Name. Formula N. vx107 Nyx107™.
Puospuoric ;
ACIAUORTMO=~. sesctoee H3P0, 50 1-470 35 X 21-0
PuosPHorous
acid hypo-............ H3P0, 34 1855 3 X21:0
pio HOLUNO=\5.4 de, Sener HsPQ3 42 1°794 33 X21°5
Puospnorus
Oxige trina a aus P40, 108 0-761 4 x20°5
oxybromide ......... POBr; 128 0°706 +201
oxybromdichloride.. | POBrCl, 92 0°807 32 X 21'2
oxychloride............ POCI, 74 0-920 34 X20°9
sulphobromide ...... PSBr3 136 0659 444X211
sesquisulphide ...... P4S3 108 0-913 43 x 207
pentasulphide ...... P.S, 110 1024 523x205

Vanadium Compound (N=23).

VANADIUM
pentoxide ...... ..... v0; 86 1-841 73 X21:1
Columbium Compound (N =41).
CoLUMBIUM
pentachloride ........ CbCl, 126 0-876 55 X20°1
Arsenic Compounds (N=383).
ARSENIC
disulphide ........-. maa. 98 1299 6 x212
ARSENOUS
DY OMMOC Giada odhene AsBr3 138 0-688 42x 21:1
TORUS | erty: cee eetmeeeie AsI3 192 0:629 6 X20°]
OXIGS Aus. secon eee As,O, 180 0712 6 X21°3
Selenide: t.csccsae<n seme As,Se3 168 0-910 75 X 20°4
Bulpliide’ if. .j.asceweees As,83 114 1144 6 x21-'7
Antimony Compounds (N=51).
ANTIMONY
bromide tri- .......es SbBr3 156 0°702 5 x21°9?
chloride tri=..tssas SbCl 102 0:907 42x 20°6
fluoride tri-............ SbF3 78 1-602 6 x20°8
Lodide viri= toe ae SbI3 210 0°615 6 x21°5
Bismuth Compounds (N=83).
BismuTHu
bromide: cee. ceee BiBrs 188 0-749 1. x20Ts
chloride di-.......-.... Bicl, 117. «0999 s«BS x 21-2
chloride tri- ......... BiCls3 134 0 943 6 K2Ich
Tantalum Compounds (N=73).
TANTALUM

chloride ...... Ak Nee TaCl, 158 OFT 6 x20°5
FIMOKIIS 1 check ees TaF, 118 0-930 5 X2ZL9%

Frequency and Molecular Number. 457
Group VI.

In the sixth group of the Periodic Table only a dozen
compounds need be considered. Several of these are oxides,
and it is noticeable, both here and elsewhere, that these in
general necessitate large values for va.

It is interesting to find that salts of the heaviest metal,
uranium, appear to conform to the same rule as salts of the
lightest metal, lithium. Thus lithium fluoride, with molecular
number 12, has for Nv the value 54x 21:0x10!", uranium
hexafluoride, with molecular number 146, has the value
5x 21:0x 10”; lithium iodide (N=56) gives Nv=6 x 20°8
x 10”, whilst uranium tetra-iodide, with the large molecular
number of 304, gives Nv=9 x 20°7 x 10”.

Grovur VI.
Sulphur Compounds (N=16).
Name. Formula, N. vx10—!2. Nyx10—!”.

SuLPHUR

trioxide 6 ...........- (SO3), 80 0-816 3 X21°7
SULPHURIC

ML YEO «2.04. - Ree H,8,0, 90 0-890 4 x20°0

Selenium Compounds (N=34).

SELENIUM

dioxide ....... Rae ee SeO, 50 2-473 6 x20°6

oxychloride........... SeOCl, 76 0:984 33 xX 21°4
SELENIC

BRIM eae ect ans, as-i0- H,SeO, 68 1-269 4 x216

Tellurium Compound (N=52),

TELLURIUM

tetrabromide ......... TeBr, 192 0-792 75 X 20°3

Chromium Compounds (N= 24).

Curoaium

EEIOMIONG ) S.ceesentlt. . CrO, 48 2:030 42 21°7
CHROMIC

LE a eee Cr,0, 72 3°876 13 X 215
Curomous

ct) gl Ea ie CrF, 42 4°300 83 X 21-2

Molybdenum Compound (N= 42).

Mo.tyspEnuM

PEABRAMG yo 5 foc ds tense MoOz3 66 2576 8 x<21:2

Uranium Compounds (N=92).

URANIUM

fluoride hexa-......... UF, 146 0°719 5 x21°0

iodide tetra- ........ (ag 304 0°614 9 x20°7

RING Cie 5. ares: <ycaws vO, 108 3°182 16 X21°5

458 Dr. H. S. Allen on Molecular

Group VII.

In the seventh group the only metallic compounds repre-
sented are those of manganese. ‘The four sults in question
give concordant values for y,. The chance of an accidental
agreement is about 1 in 180.

Group VII.
Manganese Compounds (N=25).
Name. Formulaa N. vx107-'. Nvx10-2,
MANGANESE
ehiloride i pesa. scapes MnCl, 59 2°252 63 X 20-4
MOPIDE oh ccesacenseeer MnF, 43 3°756 8 x20°2
CUBE TCE RO Ey any ERE esc MnSiO; 63 3°057 93 x 20°3
SUL pH ebe ). J, <eseasen one MnSO, 73 2°107 73 X 20°5

In the same group are a number of compounds containing
chlorine or iodine. Most of these require the provisional use
of frequency numbers of the form n+ 4 or n+32.

Chlorine and Iodine Compounds.

Chlorosulphonicacid... C]SO,.OH 58 1:335 53 X 20°6
Iodine monochloride &, ICl 70 1°125 3¢ xX 21-0
Iodine tetrachloride ... ICls 104 0°836 41x 20°5
lodine fluoride ......... TF. 98 0-869 4 X213
Lodic anid’ csectieacee HIOs 78 1:342 5 x20°9

Grovur VIII.

In the eighth group the data are scanty, only six com-
pounds of iron being represented, two of cobalt, and one
each of nickel, ruthenium, and osmium. Rhodium, palladium,
platinum, and iridium are not represented atall. The results,
as far as they go, tend to show that the compounds of this
group fall into line with the metallic salts of the earlier groups,
the only serious discrepancy being the case of cobalt carbony]
where it is possible that molecular association occurs.

Frequency and Molecular Number. 459

Group VIII.
Iron Compounds (N= 26).

Name. Formula. N. vxl0-!*, Nvyx10-.

Iron

disulphide ............ FeS2 58 3°748 103 x 2U'7

phosphide ............ Fe,P 67 3°653 12 x204
Ferric

Tt a ri FeCl 83 1-498 6 x20°7

Sammapeey eco O88 Fe,03 76 3°330 12 x21-1
Ferrous

Nelphide ......2..... we FeS 42 4°856 10 x20°4
Ferroso-Ferric

TT Ess 2 Os Eee Fe304 110 2-762 15 x20°3

Cobalt Compounds (N = 27).

CoBaLT

EERE. £2, 2 bores: Co(CO)s 83 0:927 3% x 20°5
CoBALTOUS

AE MAEG 201. <2. 202005 CoSO, 75 2477 9 x20°6

Nickel Compound (N= 28).

NIcKEL

sulphide mono-...... NiS 44 3914 83 x 20°3

Ruthenium Compound (N=44).

RoUTHENIUM

2 RuO, 76 1-399 5 x213

Osmium Compound (N=76).

OsMIUM

tetroxide ............06- OsO, 108 1:079 52x 21°2

In the foregoing pages a number of cross-correspondences
between the values of the frequency numbers for related
compounds have been noticed. But instances might also be
cited in which the frequency number has not the value which
might have been predicted for it byanalogy. For example,
the chlorides of potassium and rubidium have the same
frequency number 6, but though the frequency number for
calcium chloride is 6 that for strontium chloride is 734;
again the iodides of calcium, strontium, barium give the
unexpected sequence of numbers 9, 83, 9. In view of the
hypothesis that has been put forward relating the frequency
number to the number of valency electrons, it may be sug-
gested that the frequency number determined from the
melting-point may not have, in all cases, a unique value,

460 On Molecular Frequency and Molecular Number.

In consequence of the presence of traces of impurity, or
through imperfect crystallization, it is at least possible that
fusion may sometimes occur when the number of valency
electrons is smaller than the normal value.

It is noticeable that for a large number of compounds the
frequency number is one of the series — — 44, 6, 74, 9,-—;
suggesting that the factor 3 plays an important part in the
determination of its value. Itmay not be entirely accidental
that a number belonging to this series frequently occurs when
the compound contains an element of valency 3 (e. g. amongst
the compounds of As, Sb, Bi, and Fe). It is not unlikely
that the factor 3 should occur in such cases if it is supposed
that the chemical valency is associated with a certain number
of valency electrons.

The facts recorded in this paper are regarded as proving
that the characteristic frequency calculated for a metallic
compound by means of Lindemann’s formula can, in general,
be expressed in the form ny, or in the form (n+ 4)y,, where n
is an integer and vy, is approximately 21x10! sec.-1. The
evidence for this may not be considered conclusive when com-
pounds of one single metal are examined, but the evidence in
the present case is cumulative. The degree of probability
of this result may be small when compounds of a single
element are found to conform to the rule; but it is larger
when the compounds of all the elements in a group are found
to give concordant values for v4, and becomes very great
when the number of compounds is extended to include all the
groups in Mendeléeff’s classification. Further, these results
for compounds and those recorded previously for the elements
mutually support one another, and lead to the conclusion that
we have to deal with a property that is fundamental and
characteristic of the solid state of matter. It is to be borne
in mind that the formula of Lindemann is looked upon as
giving only an approximate value for the characteristic
frequency, and the final justification for the proposed
relation must come from observations on the specific heat
of compounds at low temperatures.

[ 461 J

LIV. On the Operator V in Combination with Homogeneous
Functions. Second Paper. By Frank L. Hircucoc.,
Massachusetts Institute of Technology, Cambridge, Mass.*

1. JNTRODUCTION.—In a former paper (Phil. Mag.

vol. xxix. May 1915, p. 700), two theorems were

developed, which yield properties of any homogeneous vector

in connexion with the operators SV and VY. I now pro-

pose to study in a somewhat similar manner the Laplacean
operator A, or — \/”, defined by

OL een. O-
Aa Se a eae (1)

in three dimensions (with analogous definition for two di-
mensions). The results obtained are of two sorts. First, as
a more elementary matter, I shall show how a large number
of reduction formulas may be written down, usefu! in the
evaluation of multiple integrals. Later I shall prove a
theorem similar to those of the first paper, giving the form
of the function by which various homogeneous sealars or
vectors differ from harmonic functions of the same degree.

2. The fundamental reduction formula.—Let us first take,
as in the former paper, F(p) a function of the point-vector p,
homogeneous of degree m. In this case, however, I restrict
m to be positive, and shall at first suppose F'(p) to be a scalar
function. Write r=Tp, and let f(r) be a scalar function
of r. Consider the problem of integrating the product
I(r). F(p; over the volume of a sphere of radius a with
centre at the origin of coordinates: I shall prove the
following reduction formula,—

Formula (A)

INFO) : ye (dV = pa Dat

imran \((AFm(p)aV, (A)
where the triple integrations are to be taken over the volume
of the sphere. We begin with Gauss’s theorem,

\\(UX+4 mY +nZ)dS= (\((S == SH ag a) ie ie 0)

where X, Y, and Zare scalars, J, m, and n are the direction
cosines of the oulward normal to a closed surface, and the
integrations are carried over the surface and throughout its

* Communicated by the Author.
Phil. Mag. 8. 6, Vol. 35. No. 210. June 1918, 2K

462 Dr. F. L. Hitchcock on the Operator V in
volume, respectively. On the surface of the sphere l, m,

and n become ~, - and =. Now let X= — oe . Yee a and
lie o Substituting i in (2) we have

siie (0S tyS' +287) as =(((AFdv.. (3)

The left side simplifies by Euler’s theorem giving

—(jF.dS=(((ArdY, <7.

where F’,, is any homogeneous scalar function of positive
degree m and the integrations are with reference to the
sphere of radius a with centre at the origin.

Next start afresh with the left member of (A). By the
form of the definition of a homogeneous function which was
laid down in the first paper, F,,(p)=r"F(u), where F(w)
denotes a function of Up, that is, a function of a point on
a unit sphere. Therefore by performing the integration with
respect to r (writing dS =7? dS, and dV=r?d8odr), we shall
obtain

SSS7@)-En(pdV=[2r"+2f(r)dr . [FECu)dS, 5)

where the double integral is carried over the surface of a
sphere of unit radius. “But on the surface of the sphere of
radius a we have r constant and equal to a, hence

(fE(u)dSy= 5 I, (p) a5, -. op (6)

the integral on the right being taken over the sphere of
radius a. ‘The formula (A) now follows at once by elimi-
nation of the double integrals between (4), (5), and (6).

By (A) we may solve a variety of simple problems. As an
example, if the density at a point in the sphere is equal to
the distance from the centre, the moment of inertia about
OZ is found by writing f(r)=", and F(p)=2?-+y?, when (A)

becomes

3
== 5 ae LOT pune
(yi r(a?+y")d\ =i dr. iV AdV = ra = g 70.
3. Extension to vector functions—In proving (A) by the
aid of Gauss’s theorem, (Fp) was a scalar function of p.
There is nothing in the result itself a require this restriction.

Combination with Homogeneous Functions. 463

In fact, since A is a scalar operator, the formula can be
applied in turn to the scalar components of a vector F(p),
hence is true for the vector. That is, F(p) in formula (A)
may be either vector, scalar, or quaternion, /(7) remaining a
scalar.

4. Laxtension by geometrical transformation.—Any problem
which can be worked out by (A) for the sphere may be made
to yield an indefinite number of other examples by performing
geometrical transformations upon both members of (A). Let
y(p) be a veetor function of (p). If we replace p by ¥(p)
and if dy(p)=¢dp, the Jacobian of the transformation is
Hamilton’s third invariant for the linear vector function
which in this case I shall call J. The Laplacean operator A,
which is the same as —V7”, is transformed into —(g!-1V/)?,
that is to say the operator V is transformed by the reciprocal
conjugate of the operator ¢. And r is transformed into
T'y(p), the tensor of the transformed point-vector. Formula
(A) now becomes, supposing that we began with a sphere of
radius unity, |

S\\A2yp) . F..(yp) I (p)aV
== famtagtr) de [MSV Crp) T(erdV,

where both triple integrals are taken throughout the trans-
formed surface. On the right we note that V, as indicated
by the stop, acts on F(yp) but not on J(p). In general V
acts on the constituents of ¢. The propriety of the inte-
erations is assumed. In case yp is a homogeneous function
of p of degree n, Fn(yp) is homogeneous of degree mn,
and may therefore be taken as any homogeneous function
which gives the problem a meaning.

As perhaps the simplest illustration, let yp=¢p, a self-
conjugate linear vector function. The unit sphere is carried
into an ellipsoid. The scalar f(r) becomes a function whose
level surfaces are similar to and concentric with the bounding
ellipsoid. Since n=1, F,.(yp) may be any homogeneous
function of p under the same restrictions of continuity as in
Gauss’s theorem. The constant Jacobian cancels out. Hence

[SS AC$p) -F,(—aV= =F em2finddr SOV YE, (p)aV,

where ¢@ is any self-conjugate linear vector function with
constant constituents, and the triple integrals imply inte-

gration over the ellipsoid ’¢p=1.
22

464 Dr. F. L. Hitchcock on the Operator VY in

To take a specific example, let us extend to the volume of the
ellipsoid the problem worked at the close of Art. 2, the equation

peg:

2
of the surface being = 73 + ~ =1, the density equal to

the square root of the left side of this equation, and the moment
of inertia about OZ required. The transformation is given

by bo=i+j% -- k-, The Laplacean operator A trans-
)
forms into —(f-!\7)?, that is into
0 Ou wee

a? A hee

Ox? | Oy Uy ee

and the transformed equation reads
2 2 pee
BRC Sars: Se mates Be)
WW atty ee a eee

: 0 0 fe :
= an pdr . ese + 6° ae +053) (2? + y?)dV,

where the voiume integrals are taken over the volume of
the ellipsoid. The right side by inspection is Lf (\(et b*)dV,
or 27(a?+0* )abe.
5. Inverse method of transformation.— When a surface is
given, and when we can transform it into a sphere, we can
apply (A) directly if the integrand is homogeneous after the
transformation. The transformation is then the inverse of
yp in the last article. This method, while resting on the
same principles as the foregoing, differs in using an untrans-
formed A. To illustrate, let p=A(p')=y~*(p') be the trans-
formation which carries a given surface into a sphere. Let
the Jacobian of the transformation % be J’, and let 2x be
homogeneous of degree n. We then obtain by (A)

(SSAC yp) - F,(p)aV
=\fA(r') - FinQye) J'(o)dV, by the trl. p=Ap’,

u . ig ! / I I
= Pe ret? t(r)dr \(VATF,OA,p') -J'(p!)]aV",
where p=mn+3(n—1), the degree of the integrand after
the transformation. The integration on the left is carried
over the volume of the original surface, the others are over
a sphere of radius a and centre at the origin of coordinates

Combination with Homogeneous Functions. A465

into which the given surface is carried by the transformation
p=A,(p'), or simply Ap’. We note that A, as indicated by
the brackets, operates on J’ as well as F. To take an
example, the problem worked at the close of Art. 4 would
read by the present method

{ile + Treas Hawes ty AatoaV’
where the right-hand integral is over a unit sphere, leading
to the same answer as before. Here Ap’ =iaw' + jby' + kez’,
so that the transformation is equivalent to putting aa’
for x, &e.

6. Analogue in two dimensions.—By parallel reasoning we
may work problems of the same character ina plane. Thusif
F be homogeneous in a plane point-vector p, that is in wand y,

and if A= 2, + se

Formula (B)

- I Ca ee
Wie). B(p)dA= —_lPertf(r)dr. NAB, (dA, - (B)
where the double integrals are to be taken over the area of a
circle of radius a with centre at the origin of p.
As an example easy to verify by other methods, let it be

gree a
required to carry the integral | ease dA over the area

included within the circle a?+y?=2ax. The transfor-
mation #=2'?, y=a'y’, carries the area within the circle into
the area included between the axis of X’ and the new
circle z'?+y'?=2a, on either side of OX’. The Jacobian

19
. . . fe % . .
is 2x”. The transformed integral is (\> wd A’, which if
xe

taken over the whole of the new circle gives twice the
original quantity. We therefore have, dropping accents am:
applying (B), (remembering that the new radius is VW 2a),

1 1) ee A
ig Ge ( dr | AvdA=2 1a’,
mim « 0

the double integral being over the whole circle. _

7. Application to a hemisphere.—By the combined use of
(A) and (B) we may obtain a reduction process for inte-
grating a homogeneous function over a hemispherical volume.
The relation (5) is unchanged in form, but we now carry the

A466 _ Dr. F. I. Hitchcock on the Operator V in

triple and the double integrals over the volume and curved sur-
face of the hemisphere, respectively. In employing Gauss’s
theorem, however, we must add a term to the left member of (2)
corresponding to integration over the base of the hemisphere.
If the base is taken in the wy plane J and m vanish and n=1.

it Zi e as before, the term to be added to the left of the

equations (2)—(4) is | a dA, where the integration is over

a circle of radius a in the xy plane with centre at the
origin. Eliminating the integration over the curved surface
as before we now have

Formula (C)

(Wf) .F(p)dV = = ily tflr)de | ( onda + SISAF av |
| (C)

where the triple integrals are carried over the volume of
_ the hemisphere, the double integral over its base, the latter
to be further reduced by (B) if necessary.

Asan elementary example, let us find the gravitational
moment of the hemisphere with respect to its base, the
density being any function of the radius. We have

(Sif) -2dV = 508407) dr. [fad +0] =m fers) dr,

Such formulas and examples can be multiplied in great
number, Thus by quite similar reasoning we have the two-
dimensional analogue of (C),

Formula (D)

a ta, EK
400) +n (p) Ca =~ pot lt(n) dr| Ce aes \(AF aA |

-a OY
(D)
where the notation is as in (B), and the double integrals are
over a semicircle of radius a and centre at the origin, having
its diameter on the w axis. The single integral is taken
along this diameter, that is, y is set equal to zero before
integrating.

These formulas may be extended to cover cones, cylinders,
segments of parabolas, and even triangles. As a final
exumple of the sort let us use (D) to find the area between
the parabola #?—162+4y=0 and the axis of w. By writing
47? for y and leaving « unchanged we carry the segment of
the parabola intoa semicircle. We need not move the origin,

Combination with Homogeneous Functions. A467

since the new integrand does not contain x. The-area is
therefore given by integration over a semicircle of radius 8
standing on the 2 axis, by (D),

| 08 16 S° D912

§ fay ia a) 7 eT, » de—0]= a 3° 16s ae

It may fairly be said, however, that we are here intruding
on ground properly belonging to ordinary rectangular coor-
dinates. Yet the method may with equal ease be applied to
problems of much greater complexity; and has some theoretic
interest in that we use a process of differentiation to arrive
at the definite integral.

8. A more general problem.— We may now travel a little
further afield and bring the foregoing integrals in touch with
the theory of the potential function. As a first step in this
direction let it be required to extend formula (A) so as to

evaluate
{{ [S10 Fear. pense teisi at:

where 7” is the distance from the variable point p to some
fixed point other than the origin. Thisis the same as finding
the potential, at the fixed point, due to the attraction of the
sphere if its density at p is f(r).F(p). The homogeneous
function F'(p) is, however, assumed at present to be a
polynomial.

9, Theorem on the expansion of a polynomial in terms of
harmonics.—The solution of the above problem depends on
the following theorem:—

Any homogeneous polynomial F,,(p) differs from an harmonic
function by a sum of terms, each of which consists of an even
power of the radius vector multiplied by an harmonic function of
lower degree.

In symbols this Seen may be stated as

Formula (£)
A a eee ie Pe. Eg oe at CD)

if m is even; whileif m is odd the last term is 7”~1H,. Here
the H’s are harmonic polynomials of degrees indicated by
their subscripts.

To prove this result we have first to find the effect of V
on any term of the form r’t, where ¢ is any homogeneous
scalar function of degree n inp. By direct expansion, (re-
membering Vr=u),

V (rt) = hr tut + Vt kr" pt +rFVt, . . (8)

468 Dr. F. L. Hitchcock on the Operator V in
and by operating a second time with V7
V2 (rt) =k(k— 2 re 4 pt —Bhr*-*t + Dkrk-*SpV7 t + Vt
=V7t—khan+k+ jr’ 2. |. sr

because p?=—?”, and SpVt=—nt by EHuler’s theorem. If
we prefer to put V*=— A we may state this identity as

Formula (F)
A(Mt) =r Ai+k(2n+k+1)r).,” (ae

where ¢ is homogeneous of degree n.
If ¢ is harmonic we may conveniently write

Formula (G)
A(T) =cr' "He, ; . .

where c denotes the constant k(2n+4+1).

Weare now able to prove the existence of the expansion (E)
inductively, by showing that if it exists for all polynomials
Fy of degree p it exists for all polynomials of degree p+2.
Let m=p+2. Then AF, is of degree p. If the theorem

is true for degree p we may write
A¥a=Hy477b,2 tb at...y .
but by operating with A a both sides of (K)
AF a=cH, 2-er,_,+¢ rb, +...

where the c’s are positive constants by formula (G). Since
p=m—2 we may take as a possible set of values

1 1 1
1 baat ee é H 3 H,,-4= rips 3 HH, -6= a fa Ve &e.
By substituting these values in (H), since Fis known, H,, is
known. It is therefore evident that the expansion (11) is
known by comparison with (10). Hence (H) is known.
Now a constant and a linear expression are always harmonic.
That is, (E) exists when m=0 and when m=1. Hence it
exists when m=2 and when m=3, and so universally.

The same inductive argument shows that the expansion
(EH) is unique; for if expansion in the form (10) is uniquely
possible (11) is uniquely possible. But the expansion is
unique for polynomials of degree 0 and 1, hence for all
polynomials.

Combination with Homogeneous Functions. 469

Asa simple example let it be required to expand a* in the
form

f=, 2 Hs rH,

where the subscripts denote the degrees of the harmonics,
Operating with A and determining the numerical coefficients

on the right by (F),
he ee A 207 ee Ala = 2212005,
whence

H,=}, H,=1(62?~2r?), Hy=at—1r?(62? —2r’) —} aaa

Tt appears that, in general, each of the H’s will be expressed
in terms of all the H’s of lower subscript.

10. Term by term evaluation of the integral.—Let us now
use the elementary theory of the potential function to evaluate

the integral
{ (\ 7) Hav

over a sphere of radius a with centre at the origin. ‘This is
the same as finding the potential at a point O' due to a
volume-density of f(7).H, throughout the sphere. Suppose
first that O’ is outside the sphere. Following Maxwell’s
notation*, we may write H,=r"Y,, where Y, is a surface
harmonic. The potential at an external point due toa
surface-distribution Y, over a sphere of radius a with

om . . . Aa Ne s
centre at the origin is known to be = —» that is
Amant? (2n +1)r”

it af nes Nips + ° °
n+ 1)rt and, since H,=Y,a" at the surface, a distri-

bution /(a)H,, over the surface will give an external potential
Anat?(q) Ee

(2n+ | Vieigli :
sphere we therefore have

Formula (HH)

+ 13 | Uae oe Aq H.,, Cd onto US
(\[Fs0) Baa = Fy eet y ede Sa)

the accents being dropped after the integration, i. e. r is put
for T(00’). We can now evaluate the integral propounded

For the external potential due to the solid

* Elect. and Mag. 3rd Ed. vol. i. Art. 131 a.

470 Operator V in Combination with Homogeneous Functions.

in Art. 8 when the point O’ is outside the sphere by applying
(E) and (H) in succession, viz.

ArH,
\\\; nite F ma V = me Lyeto Tate (r r)dr

4a, a, 2m
(2m— ye pny) det ie.
To illustrate, let us find the external potential when the
density of the solid sphere varies as the fourth power of the
distance from a diametral plane. By the expansion of a4
already obtained

(( Sav 4a Hy a Agen ae AnH a!
f Ve A es)

where the values of the H’s are those calculated in Art. 9
for the expansion of 2*, and ris written for T(OO’) on the
right.
11. The potential inside the sphere.—It is known thata
surface-distribution Y, over a sphere of radius a and centre

at the origin causes a potential inside the sphere a
4H, aati! Genin:
or Qn+ 1a" The potential at a point inside the soli
sphere may be regarded as due to two additive causes, first a
solid sphere of radius r on whose surface the point lies,
second a shell of thickness a —r fitting outside the first. The
first part of the resulting potential is, by (H),

apie pen F rar;

and the second part, due to a surface-density /(7)H,dr on
each infinitesimal shell of radius greater than 7, is by the
formula quoted at the beginning of “this ar ticle,

AqrH,,-

(2n + 1) Sr eh

Hence the potential inside the solid sphere due to volume-
distribution /(7rH),, is given by
Formula (A’)

ort AdrHn Laie tee Y .
ih od (”) HadV= (Qn ap 1) | ei gent vie, dr + iN rf (7) dr \ ;
(H’)

whence by means of the expansion (H) we can find the value
of (7) when the point is inside the sphere.

Graphical Methods of correcting Telescopic Objectives. 471

Completing the application to a distribution 2, we have
the internal potential

Lo 4nH, ¢ 7? oe Appa Rye. hen
\\{eav= Cie she ae Bi Wa

7
4m Ho (7° eit
paresis a

* 1 7? 6

As a yerification, the internal and external expansions
coincide if we write r=a. The external potential satisfies
Laplace’s equation while the other yields 47a* when operated
on with A, agreeing with Peisson’s equation*.

LY. On Graphical Methods of correcting Telescopic Objectives.
By A. O. Auten, Lecturer in Optics, The University of
Leeds t.

AVING had occasion recently to use the N.P.L. tables
relating to small objectives, it occurred to me that the
information there furnished, as well as much more of equal
or even greater importance, could be given in a very small
compass by means of a few formule, in combination with
graphical methods. It is true that this means substituting
calculation for a direct extraction of values from tables, but
a number of considerations may be set vffagainst this. First,
the calculations I propose are quitesimple; most of them are
also fairly short. Such as they are, they are not likely to act
as a deterrent ; for it must be remembered that both the
tables and the equivalent calculations lead to figures such as
no manufacturer with a reputation to keep up would employ.
It may safely be assumed that in future all lens-makers will
use the services of an expert computer, and the labour
of computing is so great in any case that a little more
at the outset will not be objected to, especially if that
little extra work saves a great deal of labour further on.
Again, the tables are only for a few selected glasses; no
tables of reasonable bulk could include all available glasses,
and even with these few it is necessary to apply sundry
corrections for variations of refractive index and dispersive
* Tn general the polynomial
r?Hm mH yo il ey

D(2m+3) t d2m+1) + 62m—l) 1”

yields the right member of (E) when operated on by 4, (proof by (F)).
+ Communicated by the Author.

472 Mr. A. O. Allen on Graphical Methods

power. All the lenses are achromatic doublets for C and F,
whereas in practice the objective will often be required to
have some chromatic error. All the lenses are cemented, so
that they are corrected either for sphericity or for coma, but
not for both; no tables of moderate bulk could include the
possibility of air-gaps.

Again, all the lenses are computed for an object at infinity;
whereas reading-telescopes should be computed for a com-
paratively near object. Finally, the tables refer only to
doublets, whereas the methods given below can be applied
also to triplets; or, for that matter, to systems with any
number of thin components, but combinations of four or five
lenses as telescopic objectives are to be regarded as mere
scientific bizarreries. Now all these variables (refractive
index, dispersive power, position of object, air-gaps, number
of components) are taken account of below, and without any
serious addition to the labour involved. But it must always
be remembered that the results arrived at in this way (or by
the tables) ought never to be seen by the lens-grinder; they
are simply intended to give the computer a favourable start.

The assumptions made are: (1) that the thickness of each
lens or gap is negligible ; (2) thatall the angles in the caleu-
lation are so small that the excess of any angle above its
sine is exactly equal to a sixth of the cube of the angle. In
other words, the rays could all travel within a capillary tube
lying along the axis of the lens.

The symbols employed below are chosen to suit the present
problem, and would not necessarily fit into a more general
scheme. Focal lengths, radii, and intersection-distances are
avoided; it is the reciprocals of these quantities which are
more important. The four curvatures are ¢, Co, ¢3, ¢4, from
left to right; the powers of the two lenses are p;, po, and it
is assumed that the power of the combination is chosen as a
unit, so that p,+p,=1. So far as this paper is concerned
there is no condition whatever connecting p,; and po; they
may be quite independent, or may be chosen to give achro-
matism between any two colours, or to give a desired
chromatic error, or to satisfy some other condition not
stated ; they may be of like or unlike signs. Ifa ray incident
on the system is converging toward a point beyond the
system, the reciprocal of its intersection-distance will give
the initial convergence, uw.

After the ray has passed through the first surface the quantity
u, becomes u,’; and as the thickness of the lens is neglected,
u;'=Uz,and soon. All these c’s, u’s and p’s are to be thought
of as “angles per unit height of incidence.” ‘The excess of

of correcting Telescopic Olyectives. 473

¢3 above ¢., called g, is the air-gap between the two lenses ;

when it is ee } the gap is age aay The

negative aseial ss
refractive index of the first lens is N; of the second n;
these may be taken to refer, for instance, to the brightest
part of the spectrum. Clearly

Cg = Cy P1 =, ey ee qe OR Oia Ol = GASES ——— -+ Is
N-1 N-1 N-1 n—1l

so that any one of the c’s determines the rest; c, will be
used as the independent one.

The problem is to express (1) the spherical aberration,
(2) the offence against the sine-condition, as functions of
N, ”, ¢, 4, 9g. The only rational way to express the first is
by means of the angle between the emergent ray (in air)
and the line joining the point of emergence to the ideal
image-point. The other aberration can also be expressed as
an angle. If the sine-condition were fulfilled the locus of
the intersections of corresponding incident and emergent
rays would be a circle of curvature u,+w,' (see Steinheil &
Voit, p. 66; Southall, p. 409; Cheshire, J. R. Mier. Soc.).
The excess of the actual curvature above u,+ uw,’ is the an gle
required. The expressions I obtain are these :—
_ (1) The spherical aberration is

Ac? + Buy? + Cg? + Dewy + Heyg + Pug + Gey + Huy + Ky + L;
(1)

(2) The sine-error is
Deeg FS, see, Wena aa

where |
9
A=4p,(1+ nyt Spo( 1+

4
n
2 2 2
C=}p,(1 ae ) D=—m(2+ =) —p(2+ a ;

ATA Mr. A. O. Allen on Graphical Methods

(3+ “| (3+ *)

N n 2n+2
ae eR) 1 2 PiP2 .
Hpi (ih ae ty 5 (an+2+ Wor)

N ( n
if
me ae (2+ =) Pip n+2
eat Ppa TRS '8 = Rl Ae 9 pedtiienee *.
2 P2 et o (2n +24 wo}
ae
y ae
1 1 (34 =—x)
L=$p,’. if 2 taPY That pps, ee
rn) le (1-5)
N

These expressions have perhaps a forbidding aspect; but
it is to be noticed that owing to the constant occurrence of

: and * along with integers, the computation is not nearly
as toilsome as it looks; even L and 8 yield quite readily.
The expression (1) is merely an expansion of Seidel’s first

term, $8;. Hissecond term, 48,, expands into a form which, —

if S; is zero, contains expression (2) as a factor; so that the
expression is only to be regarded as measuring the departure
from the sine-condition, and the amount of coma, provided
there is no spherical aberration; and the meaning of Cheshire’s
‘‘intersection-surface” is subject to the same limitation. More
generally, the connexion with the Seidel terms may be stated
thus:—If the object-point is at a finite distance, and ata height
y; above the axis, then the spherical aberration of a ray from

S

LS

it, striking the objective at height h, is (yi—h)?(p1 + pro)” - 7

ee ee a a ee ee ee

of correcting Telescopic Objectives. 475
and the comatic error is —3y;(y,—h)?(p, +p). Se. and ex-
pression (2) is (S.—S8 )u,; butif the object is at infinity, and
the ray from it strikes at height h and slope @, the spherical

aberration is then {9 —A(p, +p.) }*. a the comatic error is
30{6—h(pi+pz2)}?. iL and expression (2) is (S,—S2)(p1 + p2)3

and in either case expression (1) is $8). Of course only
rays in a meridian plane through the object are here
considered.

For the purpose of merely calculating the aberrations,
without discussion, I find it very convenient to follow
Professor Conrady’s plan of introducing a fictitious air-gap
between all cemented surfaces, and then computing (by slide-
rule) the aberrations for each lens, rather than for each sur-
face. The ray is first taken through the system by the schedule
1;=Cy—y, Nay’ =1, wy'=c,—%', ug=u,', and so on. Then

the form taken by 8; is (i- x) Stiri —uy') + (%9')?(tg—1') F,

while the expression ts x) D{i,(4;— uy’) + to! (ug—t,')} is

S.—S, multiplied by uw, or by —P as the case may be;
P denotes the power of the whole combination, p,+ pe.

There is no difficulty about modifying the coefficients
A, B, &c. to allow for a third component, especially as most
examples of this class would be cemented, so that all terms
in g, or g2 would disappear.

I now proceed to deduce some consequences of the above
formule. As (2) and (1) are respectively of the same shape
as the equations of a plane and a conicoid, it is natural that
a number of familiar expressions should occur in the
deductions from them.

We notice first that (2) is linear. Therefore when two
of the variables (say c,; and u,) have been fixed to remove
spherical aberration, there is only one value of the third which
will remove coma, and there always is one; 7.¢., with the
form of the leading lens fixed, as well as the position of the
object, there is only one air-gap, and therefere only one form
appropriate for the second lens. And a cemented lens cannot
be free from coma (except by good luck) if ¢, and wu, have
been fixed by other considerations. The graph of the comatic
error is, for spherically corrected combinations, a straight line,
whether it is plotted against g, or uw, or any of the c’s. For
all other combinations, as soon as two variables are selected,

476 Mr. A. O. Allen on Graphicat Methods

the graph of the comatic error against the third is a
parabola.

Expression (L) is quadratic in c,, u,, or g; when any two
of the variables have been fixed, there are two values of the
third which will remove spherical aberration (though these
two may coincide, or be imaginary). With g and uw, fixed,
the graph of the aberration against any one of the c’sisa
parabola with its axis vertical ; its vertex will be downward
if p, and p, are both positive, and also if p, is positive and p,
negative, provided the combination is a converging one.
The matter upon which above all I would lay stress is that
the latus rectum of this parabola, being the reciprocal of A,
is entirely independent of ¢,, w,, and g; in other words, so
long as we keep to the same two glasses, and to the same two
powers, we may vary the curves and the air-gap and the
position of the object as much as we please, but we shall
always get the same parabola; all that will change will be
the position of its vertex. Therefore we need only calculate
the latus rectum (1/A), plot the parabola once for all (either
by the focus-directrix property with compusses, or from the
equation y* =cw with logarithms or a slide-rule), and then cut
out a templet of this parabolic form. Thereafter, in any
problem on lenses of these two glasses, and of powers pro-
portionate to p, and pg all we need do is to calculate the
position of the vertex, lay the templet in position and draw
the parabola (1), and immediately we see all the possibilities
within reach by variations of the four curves. We can see
what curves (if any) will remove spherical error (or introduce
a desired amount), and whether these curves will also correct
for coma, and if not, how bad the coma will be. The vertex
is to be found thus:—Knowing w, and g, calculate c, from
the equation

2Ac,+ Du, + Eg+G=0. e ° ° ° (3)

This gives the abscissa; then calculate the ordinate (the
aberration) by substituting for c,, uw, and g in (1).

But it may be asked, how can the method be made to
include all values of u; and g, when the above graph is for a
particular u, and g? Suppose in the first place that we do
not object to having a fixed value for w, but wish to exhibit
all the possibilities which follow from varying g as well as ¢.
We begin by finding the locus of the vertices of all such
parabolas as the above, as g is varied with uw, fixed. It is
easy to see what this locus will be; it means plotting the
aberration (1) with the restriction (3) imposed upon it ; and
as (3) is linear, the graph will still be a parabola, although of

a eee oe ee

oj correcting Telescopic Objectives. 477

a new shape (II). Itis easy to show tliat the reciprocal

of the new latus rectum is A(4ho —1). At the vertex of
this parabola we not only satisfy equation (3), but also

He, +Fujt2Cg+K=0. . . . . (4)

We therefore find the vertex by solving (3) and (4) together
(uw; is given), which gives the abscissa ¢,, while the aberration
is calculated from (1). Having the vertex and the latus rectum
the locus is quickly drawn. It would be used as follows.
The position of the object is given by uw, and we wish to know
what can be done with a series of values for g. Begin with one
of them, insert it and w, in (3), and find c,. Pick out the point
on parabola II which has this ¢,. Lay the templet for
parabola I with its vertex at this point, and immediately we
see all the possible aberrations with that air-gap. Now
change g, make the necessary change in ¢, (by simple pro-
portion, as we see from (3)), shift the templet to this new
vertex, and so on.

Had it been desired to vary wu, while g was fixed, the only
difference would be that we should get parabola III; its

jatus rectum is the reciprocal of A(4 hs = 1), and its vertex
is given by using

DC,+2Bu,4+Fg+H=0, ... . (3)
instead of (4).

Finally, suppose we wish to generalize parabola II so as to
include all values of u,; we must first find the locus of the
vertices of all such parabolas as IT while u, varies. It will
again be a parabola (LV); its latus rectum is the reciprocal

4AC— TE? pene .
of Geb =F} CABC + DEF—AF*—BE*— CD"), and its
vertex is found by solving (3), (4), and (5) for c,, and then
finding the ordinate from (1).

If LIT is generalized in this way, we get parabola V, with
the same vertex as IV, but its latus rectum is the reciprocal
of CAB— ©) (4ABC +DEF—AF?—BE?—CD")

gen—pry OO 3

To use V, weshould first assign a value tog; solve (3) and
(5) for c, and pick out the corresponding point on V; fit
to that point as vertex the templet of III, and draw III.
Assigna value to 1, solve (3) for ¢, pick out the point on III,
and use it as a vertex for the templet of I. All variations in

Phil. Mag. 8. 6. Vol. 35. No. 210. June 1918. 21

478 Mr. A. O. Allen on Graphical Methods

aberration as the curvatures are altered are now shown for
the selected g and 2}.

I propose next to illustrate the use of these methods by
solving a few numerical problems, but instead of employing
vraphs (which would involve expensive plates) the solutions
will be entirely algebraical. In a discussion of principles it
it desirable to use 5-figure accuracy, although in practice
4 figures would be ample. I take the case of two glasses
for which 1/N=0°658805, 1/n=°61721; also p,;=2°508,
pz —1:508. These two powers were, as a matter of fact,
selected for C—F achromatism, but we have no concern
with that. The combination is ene which was studied
in some detail in Professor Conrady’s autumn class last
year. I take first a few simple questions on the front
lens alone; », must then be put equal to zero in the coeffi-
cients; A =2°9063, B=5°4143, D= —8°3206, G= —24:5088,
B=63' (267, L= bite:

Problem (1). If the object is at infinity, what curves give
least spherical aberration? Solve equation (3) with w, and
G=9;3 ¢=—G/2A=4:2165; c, is less than c, by 4°8426 in
each example, and is here —0°6261; the so-called “ crossed
lens.” The aberration, according to (1), is 16°089.

Problem (2). If the Jens is given, for what point will
its aberration be least? For example, suppose ¢,=38°'0,
(y= —1°8426. Solve (5) with cq =3 and g=0; y4=—(B8D
+ H)/2B= —0°80942, 2. e. the object is about 12 focal
lengths in front of the lens, and the aberration is 16°843.

Problem (3). Is it possible for a single thin lens to be
free from aberration? For correction, parabola I must cross
the horizontal axis, so that its vertex must not be above that
line. The critical positions are where parabola III crosses
the line. Its vertex is given by solving (3) and (5), with

g=O0; these give w= Ors | 254, ¢,= = 2°4213,

RI NON —=
2 2(N —1)

as common-sense shows. For these values the aberration
is 16°9400. Parabola III is therefore given by: “ excess
aberration above 16°9400=square of excess curvature above
24213.” Where it crosses the line, aberration =0, there-
fore cy=10°4309 or —5°5883, and, by (3), u,=4°3396 or
—6°8494, So that aberration cannot be corrected except
for object-points within about + of the focal length on one
side (real) and } on the other (virtual), and even then the
curves have to be very strong. ‘There is no useful case. As
a mere arithmetical exercise, we may take u,= —8; then c

has to be —8'5342, and c, is —13°377.

ai

of correcting Telescopte Objectives. 479

Problem (4). To correct simultaneously for spherical aber-
ration and for sine-error; in other words, to tind the two
‘“aplanatic points.” The position of these is well-known, so
that the exercise affords a useful check on the foregoing
formule. quate the expressions (1) and (2) to zero,
and solve simultaneously, with P=4°1603, Q=—6°6683,
S=—18-4354, and g=0; the solutions are u,=4°8426,
18-1922, c,—7-3506, or) = —7°3506, c= —7°3506,
€9=—12°1932. These tally with the familiar solutions
Us! =Co=p,/(N —1); w=4=—p,/(N—1). In one case the
rays enter the first face normally, in the other they leave
the second face normally. The cases are both useless, so far
as telescopic work is concerned.

I take next a few examples onthe doublet. The coefficients
are. now :-—A = 1°22153, B= 2°-22153, C = —1°6847a,
D=—3-44306, E=—3°36951, F=4-87751, G= —3:'73215,
ae3o46, Kk = 20° T1990. = 181770, P= L72158,

= —2°72153, R= —2°43875, 8S = —2-66771.

The Jatus rectum of all parabolas of class I is 1/(1°22153);
TI, —1/(2°10720); Ill, —1/(0-103042); LV, —1/(0-099481) ;
V, —1/(1°19840).

Problem (5). With parallel light and a cemented lens,
what curves give (algebraically) least aberration? Solve
Poway 2,0, g—0 .c =F a2ii, co=t3=—3' 3149, t= 63
+ 2°4315=0°8834 ; the aberration is —1°0330.

Problem (6). Given u,=0 and g=0, find the aberration

for ¢-=—4'5. Parabola (1) gives: excess aberration above
—1:0330 = 1°22153 x (square of excess curvature above
—3°3149), so that the aberration =— 1:0330 + 1°7155

Problem (7). Given u,=0 and g=0, what curves will
remove spherical aberration? We must have 1°0330 =
1°22153(e,—1°5277)?, or c,=0°60806 or 2°4473. Then, as
usual, deduct 4°8426 for c,, and add 2°4315 to ¢3 for cy.

Problem (8). Repeat (5), with g=0°1. The vertex of the
parabola is now given by ¢;=1'6656 instead of 1°5277;
indeed, every 0°1 in g means 0°1379 extra in ¢,; so, an avial
gap (g= —0°1) would mean c,=1:3898. With u=0, g=0'1,
and e,=1°'6656, the aberration is 0°4897. Asthis is positive,
no choice of curves with air-gap 0°1 (and parallel rays) can
give freedom from aberration. Yet for a suitable value of
uy this could be done.

Problem (9). Over what range of values of u; could it be
done? To answer this, we require parabola III. Its latus
rectum has been given above; its vertex is found by solving (3)

and(5) with g=0°1; namely, ¢, =0°24933, aberration =0°6965.
2L2

480 Graphical Methods of correcting Telescopic Objectives.

It crosses the axis, where — 0°6965 = — "103042 (¢,— 0°24933)?,
2. @. where c,=2°8492 and —2°3505, and (3) gives for the
corresponding values u,=0°8398 to —2°8497. Provided
that w, does not lie within this range, an air-gap 0:1 is not
incompatible with spherical correction.

Problem (10). What is the largest air-gap which could be
substituted for 0°1 in problem (8) so as to make spherical
correction possible? This time parabola IJ is needed. For
the vertex, solve (3) and (4) with w,=0; namely, cy =4°2164.
aberration =14°201. It crosses the axis where —14:201=
— 2°1072(¢,—4°2164)*, 2. e. where c,=1°6204 or 6°5124, and
(3) then gives for g the values 0°06726 and 3°8317. So:that
for spherical correction for a distant object the air-gap must
not exceed 0°06726 (unless it has an absurdly high value).

Problem (11). Jf uw, lies outsidea certain range, all values
of the air-gap are compatible with spherical correction ; what
is this range? Parabola IV supplies the answer. Its vertex
is found by solving (3), (4), and (5) simultaneously; namely,
¢y=3°6073, w= —0°42546, and g=19426; aberration=
14-2374. Itsequation is: (aberration —14°2374) = — 0:099481
(c,—3 6073). It crosses the axis at c;=15°570 or —8°356,.
which means, by equations (3) and (4), that uy=7-931 or
—8:783. For stronger convergence or divergence than
this, correction is possible with any air-gap, provided the
curvatures are chosen properly.

Problem (12), Returning to problem (7), let us find the
comatic error for the two spherically corrected lenses there
given. Hxpression (2) gives (for c,=0°60806) the value
—1°62092; and for c,=2°4473, the error is 15454. Hquating
(2) to zero, with u,; =0 and g=0, we have ¢; = 1:5496 to comply
with the sine-condition. What the manufacturer would pro-
bably do, if he were tied down to these two glasses and a
cemented doublet, would be to adopt a compromise; and this
again shows the difficulty of dealing with lenses by tables,
for no tables can be so voluminous as to provide for
compromises.

Problem (13). Giving up the idea of a cemented doublet,
let us find what air-gap will remove both spherical aberration
and coma for a distant object. quate (1) and (2) to zero, and
solve with w;=0. The result is: y='06721 (or 3°8339, which
does not matter), c,=1°6448, c.=—3°1978, c,;= —3°1306,
Cy= —0°6991. These curves, then, forma favourable starting
point for calculating an achromatic aplanat. They would be
improved by making small allowances for the thickness of
the lenses, but this matter must be postponed. It also
remains to show what actual residue of error remains in all

Electrical Theories of Matter. A481

three respects; this involves trigonometric computation.
Most important of all is the question how best to utilise the
above methods in order to make a much better attempt for
the next trigonometric computation, instead of striking out
blindly.

Two other similarly computed objectives are :—

(1) For u.=—0°5 (object equal to image): ¢,=0°83d1,
€9= —4:0075, c3= —3°9539, c,= —1°5224.

(2) For ujy=—1:0 (object at focus): ¢,=0°0414, =
—4°8012, c3= —4:7498, cy= —2°3183: this may be regarded
as corrected for infinity, but with flint leading.

LVI. Electrical Theories of Matter and their Astronomical
Consequences with special reference to the Principle of
Relativity. By A 8S. Npvineron, I.A., P.RS., Plumian
Professor of Astronomy in the University of Cambridge™.

M* WALKER’S paper in the Philosophical Magazine
for April has given a new turn to the discussion
arising out of the motions of the perihelia of the planets,
and perhaps some remarks on the points raised may be
useful. I think that he greatly overestimates the differences
of opinion between us; our views seem to coincide on what
he calls “‘the main point at issue,” and itis difficult to believe
that relativists in general hold any other view.

Walker describes my method of dealing with the problem
as depending on an unsatisfactory assumption, and concludes
therefore that my treatment is invalid; but I am afraid he
gives a wrong impression by not mentioning that my argu-
ment tended to disprove the assumption in question. 1 did
not advocate the assumption leading to the equations of
motion used in my paper; on the contrary I showed that
the results disagreed with observation. ‘The suggestion had
been made that the famous discordance of Mercury could
be accounted for by the variation of mass with velocity,
according to the well-known hypothesis m=m(1—w?/c?)~?,
if account be taken of its interaction with the sun’s motion
through the «ther. ‘his hypothesis was examined, and the
conclusion was unfavourable, the detailed results being irre-
concilable with astronomical observation. That is to say,
I attempted to disprove the hypothesis m=mo(1—w?/c?)~*
which Walker rejects more summarily. Sir Oliver Lodge
also concludes that ‘‘ If therefore the theory fails to give all
the known perturbations correctly, something must be wrong;

* Communicated by Sir Oliver Lodge.

A482 Prof. A. 8. Eddington on Electrical Theories

and by finding out what is wrong, we may perhaps discover
something instructive”*. We may therefore start from this
point of general agreement.

The foregoing law of inertia corresponds to the Lorentz
electron in steady motion, but the method applies equally
to any other law of variation of inertia with velocity; the
only difference is a numerical factor which is of no con-
sequence to the argument. A fundamental revision of the
theory is therefore necessary. Walker has stepped in with
the suggestion that a trial should be made with the more
general (and, I consider, more plausible) assumption that
the inertia involves the acceleration as well as the velocity.
I cannot predict whether his mode of developing this view
will lead to an accordance with observation ; but I certainly
do not undertake to prove a general negative.

Walker’s thesis that the Lagrangian function depends on
the acceleration as well as on the velocity cannot be a point
of cleavage between relativity dynamics and non-relativity
dynamics—if I have rightly grasped the meaning of the
statement. In Newtonian particle dynamics, and also in the
quasi-stationary treatment of these problems, the Lagrangian
function is supposed to consist of two parts, (1) the kinetic
energy, involving the velocity only, and (2) the force-
function involving position in the field of force. The con-
tention is that this separation is inadmissible, and that there
is a cross-term involving both the velocity and the force (or
acceleration). Walker’s standpoint tends to associate this
cross-term with the kinetic energy, so that the kinetic
energy differs from the value calculated without regard to
the acceleration. Sir Oliver Lodgef and the writer find it
more natural to group the term with the force-function, and
say that the force of gravitation involves a term depending
on the velocity. The distinction appears to be purely verbal.

It is essential to an out-and-out relativity theory that this
cross-term should exist, and it is surprising to find relativists
represented as opposed to it.

Nor can the quasi-stationary assumption be regarded as a
fundamental point of difference. It would, I think, be
absurd either to affirm or deny the quasi-stationary principle
irrespective of the particular application proposed. The
question is whether it is a legitimate approximation in a
definite problem. I sympathize with Walker in demanding
a justification of this approximation in the cases where it has

been used—whether by relativists or others. The problem of

* Phil. Mag. February 1918, p. 143.
tT Loc. cit. pp. 155-156.

Se

of Matter and their Astronomical Consequences. 483.

the motion of the planets seems to afford a good illustration
of its fallibility. But it seems rather unfair to blame
relativists for a method which was introduced by Abraham
in a non-relativity theory. I am not sufficiently versed in
the history of the subject to know how extensively relativists
have followed his example; but I should have regarded it as.
one of the sins of our youth—due to the influence of evil
associates—and long since repented.

I understand that the true relativity theory of Kaufmann’s.
experiment (which seems to be the point in dispute) runs
something like this:—Consider an electron momentarily at
rest, but continually accelerated, in an electric field of force
Hf and a magnetic field H; then the acceleration « is given
by the equation

Ke = ma,

where e/m is a certain universal constant fer the negative
electron ; we need not inquire into its nature. This equation
expresses the ordinary definition of H. Now choose new
axes of coordinates with respect to which the electron bas.
an instantaneous velocity w. Referred to these axes the
electric and magnetic forces take known values EK’, H’, and
the new value of the acceleration @’ is obtained by making
Lorentz’s transformation of the coordinates and the time.
Accordingly the relativity-theory predicts that an electron
moving with velocity w in an electromagnetic field HE’, H'
will experience this acceleration «’. Keeping Hi’ and H’
constant, we find how a’ depends on w. Kaufmann’s expe-
riment—or rather the recent repetitions of it—confirm the
predicted relation with considerable accuracy. Not only
does this give the prestige of successful prediction to the
relativity theory, but it confirms that part of the hypothesis
most in doubt. It is generally admitted that the Lorentz
transformation holds for the differential equations of the
field; the question is, Does it hold for the boundary con-
ditions (whatever they may be) at the surface of an electron ?
The Kaufmann experiment, -dealing with a single isolated
electron, answers this in the affirmative. If then the
differential equations and the boundary conditions satisfy
the transformation, nothing more is needed to establish its
validity*. It should be noted that the experiment does not

* Experiments are, however, still needed to test whether the Lorentz
transformation covers the phenomena of quanta, which appear to involve
something outside the ordinary electromagnetic theory. The exception
is of special importance because it includes the vibration of an atom, which

is the simplest form of a natural clock that could be used for measuring
the time in the two systems.

484 Prof, A. 8. Eddington on Electrical Theories

tell us what are the boundary conditions at the electron, but
only how they are transformed by uniform motion.

There is no reference to the quasi-stationary principle in
this theory. It is true also that there is no reference to the
mass, energy or momentum of the electron ; the motion is
treated geometrically. It is, I think, inappropriate to speak
of the energy or momentum of an electron in accelerated
motion: these quantities are being radiated, and it is im-
possible to define the precise moment at which an element
of energy or momentum ceases to be attached to the electron
and passes into the general field. For uniform motion,
however, the values can be clearly defined. We do not
determine them directly from Kaufmann’s experiment ; but
we arrive at them indirectly because the relativity trans-
formation is verified. These expressions for the momentum
and energy of a uniformly moving electron are of limited
utility; as Walker rightly points out, it is not permissible to
differentiate them.

I gather from Walker’s remarks on p. 329 that he has
doubts whether the Fitzgerald-Lorentz contraction should
theoretically take place under circumstances such as those of
the Michelson-Morley experiment; that is to say, the cor-
telation found by Lorentz and Larmor is a possible one, but
it need not necessarily be the correlation occurring in Nature.
But a proof based on statistical mechanics has been put
forward, which seems to be sound*. The arrangement of
the particles constituting a solid is one of an infinite number
of possible states, and the form taken up by the solid is that
which is statistically most probable ; since the possible states
of the stationary and moving solid are correlated one to one,
the most, probable states (and therefore the actual states)
satisfy the same correlation. In other words entropy is
invariant for the Lorentz transformation.

The relativity principle has the great advantage that it
leads directly to the law m=m,(1—w?/c?)~? for uniform
motion of matter in bulk, and it is unnecessary to consider
the behaviour of an electron, or indeed to adopt an electrical
theory of matter. I seeno way of deducing from the various
electrons treated by Walker the corresponding laws of mass
for matter in bulk, so that the discussion of these does not
seem to advance the astronomical problem except by sug-
gesting possible analogies.

* E. Cunningham, ‘The Principle of Relativity,’ p. 206. Elsewhere
this book warns the reader against assuming that the correlation holds
for a non-uniform translation; and indeed those who accept Einstein’s
latest theory assert definitely that it does not hold.

of Matter and their Astronomical Consequences. 485

Turning now to Walker’s astronomical calculations, it may
be pointed out that his detailed calculations deal with a solar
system at rest inthe ether. The discussion therefore does
not relate to Sir Oliver Lodge’s suggestion as to the effects
of a solar motion, nor does it throw light on the point sub-
sequently brought out—that the motion of the solar system
(if any) bas no . observable effect on the motions of Venus
and the Earth. At the end of his investigation Walker hints
that a satisfactory theory similar to Lodge’s might be con-
structed by using the more general type of Lagrangian
function. He has, however, already three unknown con-
stants, kj, ko, k3; the components of the unknown solar
motion will give him three more; with six constants at
disposal, he can scarcely fail to secure a forced agreement
of the perihelia and eccentricities of the four inner planets,
and it is difficult to find any observational test for sucha
theory.

On p. 337 it is stated that the observed motion of peri-
helion of Mercury is satisfied by supposing that the attraction
depends on the velocity (2. e. relative velocity) according to
the law

= plo(1+3 w/c’),

and that Einstein implicitly introduces this comparatively
large dependence on speed. WHinstein’s law may be trans-_
formed in a great many ways; but | do not think any
possible interpretation of it reduces to this. If it is desired
to put the new wine into old bottles, | think we must say
that the theory involves different effects of the radial and
transverse components of velocity in modifying gravitation, or
to quote Walker’s earlier remark “ the modified Lagrangian
function depends on the acceleration as well as on the speed
of the system and involves also the relative direction of these”’*.
The point is perhaps not of great importance; because in any
case a theory which deduces the exact motion of Mercury
from a general principle stands on a different footing from
theories which merely use the motion of Mercury to obtain
an empirical determination of their arbitrary constants.

As a closely connected subject, the question of the alleged
discordance of the node of Venus deserves some remarks.

Dr. Jeffreys (Nature, April 11, p. 103) has commented on

* Walker's method of taking this into account is to give k, and k,
appropriate values, but in calculating the number % he has used the
quasi-stationary values, presumably as a concession to relativists. Iam
afraid I must reject the concession, and insist on agreeing with his true
opinion on this point.

486 Electrical Theories of Matter.

the scant attention paid to this as compared with the peri-
helion of Mercury. It may be well to explain why the
former discordance has been considered unimportant. The
residual of the node of Venus is 43 times its probable error”,
and the theoretical chances against such an error are about
400 to 1. But it must not be forgotten that this element has
been deliberately selected out of 16 elements as showing the
greatest discordance. To apply the test of probable error we
must select fairly and not pick out the worst cases. Let « be
the probability of an error less than a, then the probability
that all sixteen residuals are less than zis a!®, For a limit
of 44 times the probable error this gives a probability
(9976)!®=-962, so that the chance of the largest residual
being as much as 42 times the probable error is *038, or
about 1 in 26—an adverse probability, but not very emphatic.
To put the matter another way, we find (by solving a®=3
that the largest discordance of the 16 elements should just
exceed 3 times the probable error. We may therefore ask,
What is the probability that Newcomb underestimated his
errors in the ratio 2 owing to unsuspected sources of error ?
The evidence for a genuine discordance seems very flimsy.
To the astronomer, no doubt, it is an indication well worth
looking into; but it would be extremely rash to build a theory
on so slight a foundation.

The present state of the problem of the elements of the
four inner planets appears to be as follows:—The theory
given in Sir Oliver Lodge’s and my own papers leads to
secular perturbations of the Earth and Venus, which ought
to be perceptible to observation if the sun’s motion is greater
than about 10 km. per sec. Since these are not observed,
we conclude either that the sun’s motion happens to be very
small, or that there must be compensating terms in the more
complete theory. Following out the second alternative, there
are again two possibilities. Hither the compensation is an
accident due to the particular elements of the orbits and their
relation to the direction of the sun’s motion, or it is a general
compensation. <A theory can no doubt be constructed which
gives an accidental cancelling for Venus and the Harth, pro-
vided it contains a sufficient number of disposable constants
not otherwise determined ; but such a theory cannot carry
much conviction. If we suppose that the compensation is
general, then we are adopting effectively a relativity. theory
of gravitation—that uniform motion of a gravitating system
produces no observable effects. This involves a dependence

* The discordance of the perihelion of Mercury from tle Newtonian
theory is 30 times its probable error.

Audibility Factor of a Shunted Telephone. 487

of gravitation on the velocity of the planets; or, if preferred,
the same thing may be expressed in Walker’s phraseology.

In this last case the motion of the sun can have no influence
on the perihelion of Mercury, and the observed excess must
be ascribed either to outside causes, such as the mass or
resistance* of the Zodiacal Light, or to the laws of relative
motion. In my paper only 4 of the observed excess} is
given by the relative motion; and the difficulty is to bring
it up to the required amount without extravagant or ad hoc
assumptions. Walker’s investigation appears rather to em-
phasize this difficulty. Nevertheless Hinstein’s generalized
relativity theory gives the precise value required without
any arbitrary constants. It may perhaps be said that the
factor 6 (or 3) must be implicitly contained in the assumptions
of his theory. I suppose that the results of any theory are
implicitly contained in its postulates; so I cannot deny that
the factor 6 is concealed in Einstein’s Principle of Equi-
valence—‘“‘ that it is impossible by any experiment to dis-
eriminate between a gravitational field and a field of force
(such as the centrifugal force) arising from a transformation
of the coordinates of reference.” But at least it is cleverly
camouflaged ! Many pages of analysis are required to
obtain the result for Mercury, and | do not think any simple
interpretation of the occurrence of the factor can be given
at present.

LVII. On the Relation of the Audibility Factor of a Shunted
Telephone to the Antenna Current as used in the recepticn
of Wireless Signals.

To the Editors of the Philosophical Magazine.
GENTLEMEN ,— |
i your January number you published a paper with

the above title in which I criticised a previous
communication from Mr. van der Pol; my paper was
followed by an explanatory note in which Mr. van der Pol
attempted to justify his methods and conclusions. He
suggests that I had rather lost sight of the motives of his
paper and of certain experimental difficulties; I wish to
show, however, that, through depending too much on
Prof. Love’s account of Austin’s work and apparently

* Sir Oliver Lodge, ‘ Nature,’ March 21, p. 44.
+ Or 3, if the added inertia is subject to gravitation,

488 Prof. G. W. O. Howe on the

failing to consult Austin’s original paper, he has lost sight
of the facts and has attempted to prove something which
arose solely owing to an unfortunate misunderstanding on
the part of Prof. Love.

Mr. van der Pol says, “In discussing some experiments
by Dr. Austin, Prof. Love makes use of the audibility
factor defined as (R+8)/S, where S is the resistance of the
shunt and R the telephone resistance.” This is true, but
unfortunately Prof. Love overlooked the fact that, although
Austin gives the resistance of the telephone receiver as
600 ohms, he adds a footnote to the effect that “the in-
ductive resistance of each telephone used in calculating the
shunt ratio was 2000 ohms.’ No one can doubt that by
inductive resistance Austin means impedance, so that both
Austin and Hogan used the impedance in calculating the
audibility factors which constitute the ordinates of the points
in figs. 3 & 4 which Prof. Love reproduces from Hogan’s
paper. Prof. Love's Table IV. based on the simple re-
sistance of the telephone receiver gives results which are
quite incomparable with the plotted results of Hogan’s
observations. It should also be pointed out that Hogan
definitely states that in bis experiments R is the zmpedance
of the telephone, as correctly quoted by Prot. Love (p. 126).
For this reason his adverse criticism of the conclusions
reached by Austin and Hogan needs revision.

If Table IV.is recalculated using the value of R employed
by Austin, viz. 2000 ohms, the following results are obtained:

Shunt 8. Current I. LP S
Ohms. 10-6 amp. Rae
0:5 672 113
1 474 112
10 150 112
50 68 1125
100 49 114
400 26 112°5
3000 is Lis

The last column shows that there is not the slightest
foundation for Prof. Love’s statement that ‘the results of
these experiments are recorded by him | Austin] in a table
which does not support the conclusion that the current is
proportional to the square root, of the audibility factor.” In
view of the experimental difficulties the preportionality, as
indicated by the constancy of the values given in the last
column, is wonderfully exact and speaks well for the expe-
rimental skill of those who carried out the measurements.

It is seen therefore that the peculiar relationship which

Audibility Factor of a Shunted Telephone. 489

Prof. Love found between the received current and the
audibility factor was due entirely to this oversight; his
results involve the relationship between the impedance and
the resistance of the telephone and would vary with different
receivers. There can obviously be no simple relation between
the antenna current and the audibility factor, unless the latter
be calculated in such away as to give correctly the ratio between
the total sound-producing current and that fraction of it which
passes through the telephone receiver. It is obvious, moreover,
that for large values of (R+S8)/S, the wrongly calculated
audibility factor will also vary as the square of the antenna
current, but that for small values of (R+S8)/S, that is, for
weak signals, the wrongly calculated audibility factor will
vary less rapidly, as is clearly shown in the table given in
my paper (Phil. Mag. Jan. p. 133).

' This then is the simple explanation of Prof. Love’s sug-
gestions which the experiments of Mr. van der Pol were
intended to test.

Mr. van der Pol’s experiments have simply confirmed the
fact that if, when using such detectors, one miscalculates the
audibility factor by taking the resistance instead of the im-
pedance of the telephone receiver, the result will have the
peculiarity found by Prof. Love. This was obvious, however,
without any experimental confirmation.

Mr. van der Pol’s remark that “it is by no means clear
whether Austin or Hogan employed the true impedance of
their telephones, as in their papers no references at all are
given how they determined these impedances ” is obviously
unjust in view of the work of these two experimenters.
The table given above indicates, if it does not prove, that
the impedance of the receiver has been fairly accurately
determined.

In reply to Mr. van der Pol’s statement that “‘ no assump-
tions as that made by Prof. Howe that the true impedance of
the telephone under actual working conditions is equal to four
times | I gave 3 and 4 as alternatives] the steady resistance
has been justified by any experiments,” I wish to say that the
figures given were not assumptions but measured values at
the frequencies quoted. It will be noticed that Austin’s
receiver had an inductive resistance 34 times the steady
current resistance.

With respect to the closing paragraph of the note I think
that “ the uncertainty attending the constants employed by
Austin and Hogan and the difficulty of determining exact
values”? are by no means so great as Mr. van der Pol
imagines; but even were it otherwise, I cannot agree that it

490 Mr. R. F. Gwyther: A Doctrine

is “‘ better to base the reduction of the observations on known
measurements rather than on assumptions as to the ratio of
impedance to resistance,” unless the known measurements
are of some magnitude on which the observed phenomena
depend. In the case in point, no uncertainty as to the exact
wave-form can be regarded as a legitimate excuse for
neglecting the fact that one is dealing with an alternating
or pulsating current.
Yours truly,

31st January, 1918.
South Kensington. G. W. O. Hown.

LVI. A Doctrine on Material Stresses.
By R. F. Gwytuer*.

T is intended to provide a reasoned basis for a theory of
stresses, applicable in the first case to a body which
satisfies the geometrical conditions for acting as a Rigid
Body (not being in a state of constaint), but which shall be
capable of extension to Hlastic Bodies or to other approxi-
mations to Natural Bodies, without introducing the fiction of
an ‘undisturbed ” condition in which the body is assumed
to be free from stress. The doctrine, for the development
of which the several stages are indicated below, is the outeome
of a series of papers on the ‘“‘Specification of Stress,” pub-
lished by the Manchester Literary and Philosophical Societyf,
but it cannot be described as the motive of the papers, since
it has formulated itself during the progress of the series.

The body contemplated is not supposed to be crystalline,
fibrous, or annealed, and not to be subject to any special
conditions either locally or at the surface. The body may,
in the first instance, be regarded as either at rest or in
motion under the geometrical conditions which define rigidity,
and any modification of those geometrical conditions, such as
elastic modifications, are to be deduced from the stresses in
the initial instance, and from such definitions as may prove
necessary in the sequel.

The stages by which the theory is developed are stated
below. No analytical expressions are used, but reference is
made to such analytical expressions at all the stages. This is
inevitable, since the question is, at this stage, an analytical

question.

* Communicated by the Author.
+ Manchester Memoirs, No. 10, vol. 1vi. (1912), No. 5, vol. lvii. (1913),
No.5, vol. lviii. (1914), No. 14, vol. lx. (1916), No. 1, vol Lxii. (1917).

on Material Stresses. 491

(1) The nine elements of mechanical stress obey well-
known laws of resolution. By introducing the notion of infi-
nitesimal rotations of the coordinate axes about their own
positions, we can, for our present purposes, replace these
relations by the three differential-operators*, acting upon
the elements of stress, the determination of which follows
from the laws of their resolution ; that is, these operators
are based on mechanical considerations.

2) If we now consider any arbitrary vector (which we
shall speak of conveniently as a virtual or ee dis-
placement), we may look upon the nine first differential
cvefficients of its components as being replaced by the nine
elements of virtual or potential strain (including among them
the three rotations).

These nine virtual strains may now be affected by the
same infinitesimal rotation of the axes as is employed
in (1), and the three consequent difterential-operators *
acting upon the elements of strain may be deduced; in this
case on geometrical grounds. It will be noticed that the
forms of the two sets of operators are similar, and may easily
be made identical.

(3) I shall now introduce the fundamental assumption on
which the theory is based ; that the elements of a material
stress are functions of the first differential coefficients of the
components of some vector quantity ; in other words, functions
of some set of nine virtual strains.

(4) If we now turn to the two sets of differential-operators
concerning elements of stress and of virtual strain, we are
able, in consequence of assumption (3), to express each
element of stress in terms of the nine elements of virtual
strain; or, conversely, each element of strain in terms of the
nine elements of stress, by means of sets of simple partial
differential equations.

In obtaining these solutions constants will be regarded as
uniform and isotropic, and, consequently, we shall exclude,
among other things, the possibility of a crystalline structure.
We shall also suppose the body not to be in a state of con-
straint, or otherwise that that state has been eased. JI shall,
further, limit the solutions to relations of a linear form.
From this it will follow that the relations between the
elements of stress and of virtual strain agree, in form, with
the corresponding relations familiar to us in the Theory of
Hiasticity.

(5) Supposing the elements of virtual strain to be expressed

* Manchester Memoirs, No. 3, vol. ix. (1895), No. 1, vol. xii. (1917).

492 A Doctrine on Material Stresses.

in terms of the elements of stress, we may eliminate the former
by differentiation, and obtain a set of relations involving only
the elements of stress (stress-relations).

(6) I shall now suppose the set of three mechanical stress-
equations to be introduced which correspond to the state of
rest or motion of the body as a rigid body. With these
equations I shall suppose the stress-relations of the preceding
paragraph to be combined. From this combination we
may obtain Stress-Hquations, by which we may regard the
elements of stress to be defined : as far as they are capable of
being defined so long as the surface-traction conditions have
not been considered.

(7) Up to this point I have been contemplating such
cases as a cube of metal on a rough inclined plane, or a
connecting-rod moving in a prescribed manner, the geo-
metrical conditions for rigidity being preserved. The stresses
are not to be regarded as undefined, but as being determinate.

(8) The Theory of Elasticity is now introduced by the
definition of an Hlastic Body as a body such that the strain,
hitherto considered as a virtual or potential strain, is the
actual strain experienced by the parts of the body. The
stresses being definite, the strains become definite, ‘and the
displacement may be deduced.

The acceptance of the principles involved in this doctrine
would have the effect of removing the subject of stresses
from its accepted place in the Theory of Elasticity, and of
making it an integral part of the Statics and Dynamics of a
Rigid Body. Apart from the introduction of the cases of
motion of the body, this would appear, at first sight, to be
merely a matter of exposition of the Elastic Theory. Buta
further consequence would be that the determination of the
elastic strains from the Rigid Body Stresses would be only
the first stage in the Theory of Elasticity. A closer approxi-
mation to the values of the stresses would follow from the
estimated alteration of the surface-traction conditions con-
sequent on the displaced condition of the surface of the body,
and the subject would become one of continued approxi-
mations on the lines of certain other subjects of Mathematical
Physics, and an opportunity would be provided for a theory
of permanent set and of rupture.

Lymm, Cheshire.

fr e990

LIX. On the Wolf-note in Bowed Stringed Instruments. By
C. V. Raman, W.A., Sir Taraknath Palit Professor of
Physics in the Calcutta University”.

-: a the Phil. Mag. for June 1917 (page 536), Mr. J. W.

Giltay has questioned the correctness of the remark.
made by me in the Phil. Mag. for Oct. 1916 (page 394), that
the explanation of the effect of a“ mute” on the tone of
bowed stringed instruments is chiefly to be sought for in the
lowering of the frequencies of resonance of the instrument
produced by the loading of the bridge.

2. Before replying to the specific issues raised by
Mr. Giltay, I may be permitted to point out that the view
of the action of the mute suggested by me rests upon the
secure foundation of mathematical analysis. The effect of
adding inertia to any part of a dynamical system has been
considered by Lord Rayleigh, Routh and others, and it hag
been shown that the natural frequencies as altered by the
addition of the load are given by the roots of the equation
(see Routh’s ‘Advanced Rigid Dynamies,’ Section 76)

(N?P—n?)(N,2—17) &e., —an?(n? —n?)(n3?—n?) &e.=0,

In the above, N,, Nz, &e. are the frequencies before the
addition of load, 7, m2, n3, &c. are the limiting values of the
frequencies attained when the load becomes infinitely large,
and zis a positive quantity proportionate to the added inertia.

[n,=0, no >N,, n3>No, &e., according to the theorem
due to Routh].

The forced vibration due toa periodic force of frequency n
(assumed to act on the system at the point at which the
load is fixed) also depends on the magnitude of the ex-
pression on the left-hand side of the preceding equation,
being in fact inversely proportional to it except in the
immediate neighbourhood of the frequencies of resonance.
The expression may, for convenience, be written in the form
(p—aq). Assuming that the frequency n of the impressed
force lies between two of the natural frequencies, say N,
and No, of the system without any load, the effect of the load
on the forced vibration evidently depends on whether p and q
are of the same or of opposite sign. If n be less than n,,
they are of opposite signs, while if n be greater than ng, they
are of the same sign. In the former case, the load decreases
the amplitude of the forced vibration throughout. In the

* Communicated by the Author.
Phil, Mag. 8, 6. Vol. 35. No, 210, June 1918, 2M

494 Prof. C. V. Raman on the Wolf-note in
latter case, the vibration is increased by the addition of load
till the stage is reached at which p=ag, the amplitude then
becoming very large. Subsequent additions of load decrease
the forced vibration till it finally vanishes in the limit.

3. If, however, the point at which the load is fixed is not
the same as that at which the impressed force acts on the

system, the treatment is not equally simple. The expression

_ for the forced vibration then obtained from the Lagrangian

equations has the determinant for the free periods as its
denominator; but the numerator contains some additional
terms, the magnitude of which is proportional to the applied
Joad. If these terms are ignored, the sequence of changes
with increasing load would be exactly the same as that
stated in the preceding paragraph.

4. There is no difficulty in verifying the foregoing indi-
cations of theory experimentally. In the case of the violin
or ’cello, at least the first three of the natural modes of
vibration of the instrument have to be taken into account to
explain the phenomena produced by the mute within the
ordinary range of tone of the instrument. The two first
resonance-frequencies are those mentioned by Helmholtz in
his work. The pitch of the first is only slightly lowered by
the mute. The second is the well-known “ wolf-note,” and
the pitch of this is depressed by about 450 cents by the mute.
The pitch of the third resonance is about an octave higher
than that of the second, and this also gives a marked
‘“ wolf-note.” The mute lowers the pitch of this by about
700 cents. The mass of an ordinary brass mute is sufficient
to make the second, third, and higher resonance-frequencies
approximate to their limiting values. The effect of the mute
should accordingly be to increase the intensity of the graver
tones and harmonics of the instrument, and to decrease those
of high pitch. This is exactly what has been found experi-
mentally by Edwards (Physical Review, Jan. 1911).

5. Mr. Giltay’s criticisms may now be easily disposed of.
Experiment shows that he is incorrect in saying “ I suppose
that the change of pitch of the note of maximum resonance
of bridge, belly, &e. will practically be the same whether the
bridge be loaded at its highest point or as low as possible
and near to its left foot.”” Asa matter of fact, trial shows
that the lowering of the pitch of either of the two “ wolf-
notes ” is three to five times as much in the former case as
in the latter. As the observed mute-effect is less when the
load is placed at the foot of the bridge, the experiment
actually furnishes a strong confirmation of the correctness of
my views, and shows also that the interpretation given by

Bowed Stringed Instruments. 495

Giltay and De Haas to their observations (Proc. Roy. Soe.
Amsterdam, January 1910) requires revision. As a matter
of fact, it appears from my detailed observations that Giltay
and De Haas were in error in assum: ing that the motion of
the bridge in its own plane is practically that of a rigid body.
Owing to the form of the bridge, the cuts in it, &., this is
very far indeed from being the case, the elastic distortions
being very large. for instance, it makes all the difference
in the pitch of the wolfifa load be fixed immediately above
instead of immediately below the cut on the G-string side of
the bridge. This fact is inconsistent with the supposition
made by Giltay and De Haas that the motion of the bridge
in its own plane is one of simple rotation about an axis, and
proves that the theory of the action of the mute put forward
by these writers is untenable.

6. In view of what has been said in para. 4, the observed
muting of the high notes of the instrument which Mr. Giltay
suggests asa difficulty, is easily seen to be exactly what is to
be expected according to the view of the action of the mute
put forward by me. ‘In the absence of a mute, the resonance
of the violin is by far the strongest at the pitch of the
two wolf-notes. Theory thus indicates that the quality of
violin-tone and the effect of a mute upon it may be cha-
racterized as follows: the gravest tones have a weak funda-
mental with strong second and third harmonics, muting
increasing the fundamental at the expense of the harmonics :
in the middle of the scale the tones should have strong
fundamental and second harmonic with relatively weak
higher harmonics, all except the fundamental being decreased
by muting; the highest tones should have strong fundamental
and weak upper partials, all the components being decreased
by muting. The observations of Hewlett (Physical Review,
Noy. 1912) and those of Edwards already quoted are in
aan agreement with the above.

. Another interesting question which arises regarding
the Beton of ple imu a ss effect on the minimum “bowing
pressure necessary in order to elicit a steady vibration of the
usual type. I have investigated this question theoretically
by considering the effect of the mute on the motion of the
bridge and consequently on the minimum frictional force
which should be exerted by the bow on the string in order
that a steady vibration should be possible. The question has
also been studied experimentally using a mechanical player
in which an erdinary violin- bow excites the shine of a
violin under strictly controlled pressure and velocity of
movement, The quantitative data obtained clearly show the

Ia 2

496 Mr. Nalinimohan Basu on

great increase in the bowing pressure which becomes
necessary at the wolf-note pitch, and prove that the effect
of the mute is to increase the bowing pressure necessary at
low frequencies and to decrease itat high frequencies. With
this mechanical player, the “ cyclical” or “beating” tones
obtained in certain cases (Phil. Mag. Oct. 1916 and Feb. 1917)
may be steadily maintained and controlled by suitable adjust-
ment of the bowing pressure.

Indian Association for the
Cultivation of Science,
Calcutta.

LX. On a New Type of Rough Surface the Motion of a
Heavy Particle on which is determinable by Quadratures.
By Nauiimowan Basu, M.Sc., University Lecturer in’
Applied Mathematics, Calcutta *.

1. PT is well known that the motion of a heavy particle on
a rough surface is determinable by quadratures when

the surface is an inclined plane, a circular cylinder, a circular
cone, or a vertical cylinder standing on a logarithmic spiral
as the basef. |

The object of the present paper is to make known another
surface which has, in a certain sense, the same property as
the four surfaces mentioned above and which, it is believed,
has not been considered by any previous writer.

2. Let us consider the surface whose equation is

VvY=c—tana. tan-*2 —(x#?+y?—1)P(z)=0, . (1)
where P(z) is given by the relation

6
1 1+ tan; 1
2u cot a cosa a /1—p? cot? a
2
eee meee aE Ny
V1—pcotat V1+pcota.tan 5

x log a= =c—h, . (2)
V1—pcota— 71+pcote.tans

bo

tan 9 standing for 2P cos # and the awis of z being drawn
vertically upwards.

* Communicated hy Prof. G. Prasad.

+ For the first three cases, see any well known text-book on the
‘‘Dynamics of a Particle,” e.g. Routh’s book; for the last case, see
A. Razzaboni’s paper, ‘‘ Sul movimento d’un punto materiale sopra una
superficie non levigata.” (Giornale di matematiche, vol, xxxiv.)

a New Type of Rough Surface. 497
Then we proceed to investigate the path of a heavy particle
moving on this surface, the coefficient of friction being pz.

3. The equations of motion are:

oe eS :
g=mR—pRY,
oie pro 9,

where R is the normal reaction per unit mass and l, m, n are
the direction-cosines of the normal to that side of the surface

on which the particle lies.
Eliminating R we have
adil 2
ae nk: :)

v on y igvpset J
WeeAh dy & dz —po
ae i Py et

Thus we have

ly iL
vii = (us — 1) “(5 oe +42),

> dy d/(l1.,
BUY = bes va m) i ii 9° +92).

Writing

=4(,42), pad (ett), dood
AO HN dar Go meek ds Aaa:
and simplifying, we get
5 dv dz\ dz
po? a ee (1p «9 qe
oly dy de
v7 +m ws - + a+ (m—n ba. = A ==|/
Hence we obtain
v? yt? dz
Be | eS ES ae ds ih, Saleen,
(; dw -1") =" (i da d?y ( _ dy\ Px mot ey
EN ds ds Pe i a ey
nt 1 dy
dz "ds ds
UP da Bae. ay’ (3)
mds ds?

498 Mr. Nalinimohan Basu on

and
eee
de _ dz JE \ds det de dst) a

is en ae re \ g

Me 7-5 Tee
ds?) as

The path of the particle must therefore satisfy the
equation

moe a) dy
d dz ds ds
ds ds’ d?x d*y

m 732 les
ce ax da -

im
‘ dz ds’ ds? a °
=e ds* dx | a (4)
mM 73 = 173

The problem before us is therefore to find a curve
ste equations will satisfy

(i.) The initial conditions,
(ii.) The equation (1),
and Giii.) The equation (4).
Now from (1) we have

Ox ogy tan a MEE ey

or w+ y?
OY _ wtane ue
OY a+ y? gh)
Ox dP
—# =1—(2?+y7’?—1) —
OX <1 (t+ yl) F
fe i ie tan @ _9 P(e)|
Te a? aaa ee
2 ine
es 2,
m=— 9/3 zi +2 yb) |,

pe gatiane sl
das aL Bet oie ae?

|

a New Type of Rough Surface. 499

tan? a

: y dP
eu Dp 2\ Ogee pee
OF=1 + ae + 4 P(e? + y*) — 2(e* +y?—1) a
F 2
+(e +y—1) (S=) :

Assuming the equations of the path to be given by

w=cos(zcota), y=sin(zcote), . . . (5)
we have, for points on the curve,
OF = A4P?+sec? a,
QO.t = ytana—22P,
O .m=—(@ tan «+2yP),

and Oni ae

Also we find, for downward motion,
dz :
— =—sine
ds ‘
da °
Ge COS # SIN (z cot a),
(
d
= = — C08 a cos (z cot a),
d'z
aaa — cos” a cos (z cot «),
d*1
a = — cos” a sin (2 cot #).

Substituting these in the equation (4) and putting for w
and y their values given by (5), we see that the equation

saa

dz

must be true in order that the curve described by the
particle may be represented by the equations (5). But
differentiating (2) we find that the above eyuation is true.
The equations (5) also satisl'y the equation (1).
Therefore it 1s proved that, if a particle be placed on the
surface (1) and projected with « suitable velocity along the
helix (5), tt will continue to describe that curve.

—4{1—pcoota.cosaV4P?+ sec? a}

500 Mr. W. G. Bickley on Two-Dimensional

5. The velocity at any point of the curve is found from

(3) to be given by
v’?=29P(z).

We see from (2) that z=h makes P(z)=0. Thus the
level of no velocity is given by <=h.

6. To study the nature of the surface we observe at the
outset that, when w=0, 7. e. when the surface is smooth,
the limiting form of (2) becomes

P(z)=h—z,
and taking h=0, we obtain *

”
2(2?+y7)=tan a. tan“,
x

Now consider the general equation (1). Then we find
that (i.) the section by the cylinder #?+y?=1 is the helix
x =cos (z cota), y=sin (zcot«) ; (i1.) the section by the hori-
zontal plane z=/ is the straight line y=2 tan (hcota),z=h;
(iii.) the other horizontal sections are spirals of the form
r=ap+b; (iv.) generally, sections by vertical circular
cylinders are different from helices; (v.) the surface is not
a minimal one like the helicoid. Hence we conclude that,
although for small values of w the surface has nearly the
same shape as the surface discussed by Catalan, for fairly
large values of yw it differs essentially from Catalan’s surface
as well as from the helicoid.

I wish to express my thanks to Prof. Prasad for his kind
interest in the paper.

LXI. Two-Dimensional Motion of an Infinite Liquid.
By W. G. Bicxuiry, B.Se.f

§1. FN a recent paper (Phil. Mag. (6) xxxv. no. 205,
p. 119, Jan. 1918) Dr. J. G. Leathem has shown

how to determine the motion in two dimensions of an infinite
liquid occupying the space outside a solid body bounded by
a closed curve or polygon, due to prescribed motion of the
boundary. ‘The method used depends on the use of periodic
conformal transformations whereby the doubly connected
space outside the boundary is transformed into a semi-infinite
rectangle. The solution for the case of translatory motion
is neat and immediate, but this can hardly be said of the
solution in the case of rotation, although it is perfectly

x This is the surface discussed by Catalan (Journal de mathématiques,

ser. 1, tome xi.).
+ Communicated by the Author.

Motion of an Infinite Liquid. 501

complete and general. Whenthe paper appeared, the author
of the present paper was engaged in an attempt, since com-
pleted, to solve a particular case of the above problem, and
was led toa general method of attack which seems to give
results more immediately, and with a less complicated pro-
cedure, and it is thought that an outline of the method may
be of interest.

§ 2. Needless to mention, conformal representation plays a
large part. Instead of the periodic transformation advocated
by Dr. Leathem, it was found more convenient to use one
whereby the doubly connected region of the z-plane becomes
the upper half of the &-plane, the boundary in the <-plane
becoming the real axis in the ¢-plane. If the periodic
transformation is known, this may be effected by taking as

the auxiliary variable tan = in the notation of Dr. Leathem.

This will for the present be denoted by £(=&+ im). The

transformation may be written

a ea

In particular for the ellipse, of semiaxes ccosh a, csinh a,

e=c{2 cosh a+c(1—€?) sinh «}/(14 €?).

§ 3. The method now depends on the fact that, except.as
to a constant, the value of the stream function is known on
the boundary, and therefore on the axis of &. In particular,
if the motion is one of uniform translation with velocity U
in the direction inclined at an angle 8 to the z-axis, we
have on the boundary

yi =tU x ieee } + consis cuieeee | he)

For the case of rotation about the point 2) with angular

velocity w, on the boundary
2=7o|z—z|?+ const. . . - . (3)

On the boundary z=/(&) since »=0, therefore we have

as the values of y on the real axis in the &-plane,

“rian lifer. } + const...) 2. C2)
iota {(©)—2z9) + const: ©.) 2. (84)
Hence, as the corresponding values cf w (=¢+ uh),
st: dé

Revi oet ey ee
se ee oh ee.

PW ae il
W2= So M20)? pee

provided these integrals converge. This may always be
secured, since the boundary in the <-plane is by hypothesis

502 Geological Society: —

finite, and so /(£) tends to a finite limit as £+co. Thus we
obtain finally

mS Wyte ye) Sk ee

jk. OM (£)—fla » 1E
=e) Woes... @

These integrals, which express was a function of ¢, may
be evaluated by the method of residues, and the elimination
of ¢ between (1) and the results give w as a function of 2, as
1s required.

Loughborough,

March 5, 1918.

LXII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
(Continued from p. 292.]

January 9th, 1918.—Dr. Alfred Harker, F.R.S., President,
in the Chair.
HE following communication was read :—

‘The Highest Silurian Rocks of the Clun Forest District
(Shropshire).’ By Laurence Dudley Stamp, B.Sc., A.K.C.L.

Clun Forest is a large district—extending on both sides of the
Welsh Border—in which Upper Silurian rocks crop out over a wide
area, interrupted by outliers of Old Red Sandstone. The district
is separated from the typical Silurian area of Ludlow, which lies
some 15 miles away to the east, by the great line of disturbance
that passes through Church Stretton and Old Radnor.

The classification adopted for the highest Silurian strata is as

follows oT Thickness
an feet.
OLD RED) SANDSTONE! Jc ieee... .. cuca Purplish-red sandstones.
(Temeside Shales ......... 350 Olive-green shales with bands
of micaceous green grit; a
fragment-bed, with Eury-
TEMESIDE | pterid- and plant-remains,
GROUP. } forms the upper limit.
Downton Castle Sandstone 110 Yellow sandstones and tile-
Series. stones, with shales and
L Platyschisma Limestones.
C Upper... 50 Green laminated flags and blue
ee flagstones.
betas ‘ | ees Bes Lower ... 300 Ivregularly-bedded calcareous
Lana NS flagstones
GROUP. | 4 . .
‘| Rhynchonella Beds ...... 300 Grey calcareous flags with
massive blue flagstones.
Arumustry { Dayia Shales............... ?300 Striped laminated shales and
GROUP. { mudstones.
Lower Ludlow Shales............. Dark-grey shales and indu-

rated mudstones.
Towel) co. 1410

On a Flaked Flint from the Red Crag. 503

The distribution and characters of the beds are described. The
succession compares very closely with that in the Ludlow district
itself. The main differences are: (1) that the Aymestry Lime-
stone is represented by mudstones west of the great fault-line, and
(2) that all other divisions show greatly increased thicknesses.

There is no evidence of any stratigraphical break. On the.con-
trary, the sequence is complete from the Lower Ludlow rocks up
into the Old Red Sandstone, and the changes in lithology are usually
quite gradual. The oncoming of the Old “Red Sandstone conditions
is discussed, with regard to their effect on the lithological and
paleontological characters of the strata.

The extent of Old Red Sandstone, as indicated on present maps,
must be greatly restricted, since most of the supposed Old Red
Sandstone has been found to belong to the Temeside Group, which
in this district attains a great development. The Silurian age of
the beds in question is shown by the occurrence in them of “Lin-
gula minima, and of characteristic lamellibranchs, etc., also by
comparison with similar strata in the Ludlow area.

A comparison with other districts in which Upper Silurian rocks
are developed shows ‘that deposition attained its maximum along
the Welsh Border, the thickness of the formations decreasing
rapidly southwards and eastwards.

On the east of the district—in the neighbourhood of the great
fault-line—the strata are considerably folded along axes ranging
north-north-eastwards parallel to the main fault, with minor faults
following the same direction. Away from the major faults the
foiding is gentler in character, and a series of folds ranging nearly
due east and west make their appearance. Farther w est the north-
north-eastward folding and fracturing reappear.

January 23rd.—Dr. Alfred Harker, F.R.S., President,
in the Chair.
The following communication was read :—

‘On a Flaked Flint from the Red Crag.’ By Professor William
Johnson Sollas, M.A., Se.D., LL.D., F.R.S., V.P.G.S.

The remarkable specimen forming the subject of the paper
was obtained by Mr. Reid Moir from the base of the Red Crag
exposed in the brick-pit worked by Messrs. Bolton & Company
near Ipswich.

It is a fragment of a nodule of chalk-flint, irregularly rhombic in
outline, with a nearly flat base and a rounded upper surface which
retains the whitish weathered crust of the original nodule.

The base was formed by a natural fracture which exposes the
fresh flint bordered by its weathered crust.

Both upper and under surfaces of the specimen are scored with
scratches which are mainly straight, but in some cases curvilinear.

Two adjacent sides have been flaked bya force acting from below
upwards, in a manner that recalls Aurignacian or Neolithic work-
manship. The two edges in which the flaked faces meet the base
are marked by irregular minute and secondary chipping, such as
might be produced by use. On the hypothesis that the flint has
been flaked by design, these edges will correspond to the ‘surface

504 Geological Society :—

‘dutilisation’ of M. Rutt, and we should expect to find on the

opposite edges of the flint the ‘surface d’accommodation,’ as in
fact we do.

A singular feature, which seems difficult to reconcile with its use
as an implement, is the restriction of the flaking on one edge to the
weathered crust.

The origin of the flaking is discussed, and the author, while
admitting that the fashioning of the flint is not inconsistent with
intelligent design, concludes that the evidence is not sufficient to
establish this beyond dispute. It is eminently a case of,‘ not proven.’

February 6th.—Dr. Alfred Harker, F.R.S., President,
in the Chair.

The following communication was read :—

‘Some Considerations arising from the Frequency of Earth-
quakes.’ By Richard Dixon Oldham, F.R.S., F.G.S.

The publication* of an abstract of twenty years’ record of earth-
quakes in Italy gives an opportunity for studying the effect of the
gravitational attraction-of the sun; the period is so nearly coin-
cident with the lunar cycles of 19 and 18°6 years that the effect of
the moon may be regarded as eliminated, the record is of excep-
tional continuity and completeness, and the number of observations
is large enough to allow of the extraction of groups sufficiently
numerous to give good averages.

The distribution of the stresses is dealt with in text-books; there
is a Maximum upward stress, in diminution of the earth’s attraction
at its surface, at the two points where the sun is in the zenith or
nadir, and a maximum downward stress along the great circle
where it is on the horizon; but as, for the purpose of this investi-
gation, a decrease of downward pressure is equivalent to an increase
of upward, I shall take the line along which the downward stress is
greatest as the zero-line, and express the amount of stress at any
other time or place as a fraction of the difference between the net
force of gravity along this line and at the point where the sun is
in the zenith. The fraction, at any given time and place, depends
solely on the zenith distance of the sun, which is continually varying
with the revolution of the earth. At the equinox, when the sun is
on the equator, the curve of variation between 6 a.m. and 6 P.M.
is the same as in the other half of the day; at any other part of
the year it is not symmetrical in the two halves of the day, but
is the same during the day in the summer half of the year as during
the night in the corresponding part of the winter half, when the
declination of the sun is equal in amount, though opposite in
direction.

This gave the first suggestion for grouping the records. The
year was divided into two halves by the equinoxes, and the day
into two halves at six hours before or after noon, called day and
night for convenience, irrespective of the time of sunrise or sunset.
The result is given in the tabular statement below, the frequency

* Boll. Soc. Sismol. Italiana, vol. xx. (1916) p. 30.

On the Frequency of Earthquakes. 505

being expressed as a ratio to the mean, of each group, taken
as 100 :-—

DISTRIBUTION OF SHOCKS BY DAY AND NIGHT.

Italy, 1891-1910. Day. Night.
sleerte liege es. oes cs ae Se oe os os 50 § Ato
SANs og 1 Aree... 88 : 112
WWNIOR SYGATE ro sa. 4k STREP <3 oe 84 : 116
LUO ie: en... a SL dg
Peecember—January ....1.. sss +--+ (¢ 2. 123

Japan, 1885-1892. Day. Night.
Siren ese CY |e eee). 102; > ~_ 98
(CUS (ee) eres.” 5. ee Di er 403
VCE eld (i nce... 93e LOF

Assam Aftershocks.

SMAIEMBE TRIE 7. 255.5. ss eae os 0s LIB a 87
VAS 2 Ge er 10hiee 293
Misuse Halt. | Boo. 3. eee... TON Oo

From this statement it will be seen that the mean ratio of day
to night shocks over the whole period is represented by the figures
84:116; for the summer half of the year they become 88 : 112,
and for the winter half 81: 119, showing that during the day the
shocks are somewhat less frequent than the average in summer
and somewhat more frequent in the winter, with an opposite varia-
tion during the night. Taken by itself this difference might be
merely fortuitous, and further confirmation is required: this can
be got in two ways. In the first place by comparison with other
records, two of which, Milne’s catalogue of Japanese earthquakes
from 1885 to 1892*, and the aftershocks of the Indian earthquake
of 1897+ stood ready for use. They show a variation identical in
character with that of the Italian record. A second test depends
on the argument that, if the variation is in any way seasonal, the
divergence should be increased at the height of each season; the
figures for the months of January-February and of June—July
were taken out, as representing midwinter and midsummer respec-
tively, and found to show a divergence in each case greater than,
and in the same direction as, the respective half-years.

Taken by itself the variation, as between any pair of ratios, is as
likely to be in one direction as in the other, but the odds against
a complete concordance throughout the whole series is 31 to 1;
there is, therefore, a strong presumption that the variations are
not fortuitous, but due to some common cause which tends to
increase the frequency during the day and decrease it during the
night in summer, with the opposite in winter.

The variation in the frequency of earthquakes may, or may not,
be connected with the variation in the gravitational stresses due to
the sun ; but there is another line of investigation by which a con-
nexion may be better traced, dependent on the fact that the
prevailing effect of the vertical stress is in the direction of
lightening the load, and the prevailing direction of the horizontal
stress between east and south, during the six hours before the
meridian passages at noon and midnight, and of an increase in the

* Seismol. Journ. Japan, vol. iv. (1895).
+t Mem, Geol, Surv. India, vol. xxxv. pt. 2 (1903).

506 Geological Soctety :—

downward pressure and a horizontal stress between south and west
during the next six hours. The record was accordingly grouped by
the successive two-hour periods from XIT to XII o’clock, and the
mean amount of variation in the stresses was calculated for the
same periods. The result is set forth in the appended tabular
statement :—

DISTRIBUTION OF STRESSES AND SHOCKS IN Two-HouR PzRIoDs,
BEFORE AND AFTER MIDDAY AND MIDNIGHT.

MET Gap oretan cis 5 ke eae XE II Til Vi. VRE x XII

]
STRESSES.

Mean range of stress in each
two-hours, in Italy.

Wotal etresit s.) a6. .euedee —10 | —-27 | —-23 | 4-23 | 487 | 20
Horizontal component .. ... +°07 |---11 | —°20 | +:20 | +:11 | —"OF
Vertical component ......... —"14 | —-27 | —13 | 4°18) 22
SHOCKS.

Ratio of actual to mean fre-

quency of each two-hour |
)
|

period. |
TrAny, £69h-Lol0 cee £06 | eee ot 90 |
Japan, Aftershocks of Mino- | .
Owari, Oct. 28th, 1891 ...} 1°01 "95 96 ‘97 | 108 | ime
JAPAN, 1885-1890............ 00 |e ‘89 | ‘98 |) 106 “99
|

From these figures it is seen that, while there is no apparent
relation between the frequency and the total, or the horizontal,
stress, there is a close one with the variation of the vertical stress ;
the greatest number of earthquakes being in the period in which
there is the greatest increase of downward pressure; as the rate of
increase diminishes the number of shocks is less, suffering a further
diminution as the pressure begins to decrease, and reaching its mini-
mum in the period where the decrease in pressure is greatest,
increasing again in the same way to the maximum.

An attempt to apply the same method to the Japan record gave
a result which was, at first sight, contradictory and also inconsistent
in itself, for it gave an absolute maximum at the time when the
Italian gave a minimum, with another maximum, almost as great,
in coincidence with the Italian; but, in any comparison, it is neces-
sary to allow for the contrast in the character of the two records.
The Italian does not contain more than two, or at most three, great
earthquakes of the type that gives rise to long-distance records
(bathyseisms), and the aftershocks account for no more than a
quarter of the whole record; the Japanese record, on the other
hand, is dominated by bathyseisms and aftershocks. Not only
does the region give origin to an unusually large number of tele-
seisms, or bathyseisms, but aftershocks form fully three-quarters of
the record, and nearly a half consists of aftershocks of the Mino-
Owari earthquake of October 28th, 1891. Taking these separately,

On the Frequency of Earthquakes. 507

we get a curve of frequency similar to the Italian, except that the
maximum and minimum are reversed, the greatest number of
shocks corresponding to the period when the load is being lightened
most rapidly, indicating that these shocks are due to a general
movement of elevation rather than depression, a conclusion in accord
with field observations of other great earthquakes. In addition,
the shocks which occurred during the period 1885-90 were taken
out, as representing a more normal activity, though still one in
which aftershocks form fully half of the record, and the curve
was found, as might have been expected from the character of the
record, to combine the features of the Mino-Owari aftershocks with
those of the Italian curve of frequency, of earthquakes prevailingly
of the so-called ‘ tectonic ’ type.

These results are of twofold geological interest. In the first
place they confirm the conclusion drawn from a study of the Cali-
fornian earthquake of 1906*, that the great earthquakes differ from
the ordinary, not merely in degree but in kind. They indicate that
in the latter the main stress is compressive, probably due to settle-
ment, and in the former to elevation or tension, a conclusion which
is in accord with the fact that, in those cases in which it has been
possible to compare accurate measurements made before and after
the earthquake, the comparison has indicated an expansion, eleva-
tion, or both, of the area affected by the disturbance.

The second point of interest is that the figures give a means of
estimating the rate of growth of the strain which produces earth-
quakes. If we accept the hypothesis that earthquakes, in the limited
sense of their orchesis, are due to the relief by fracture of a growing
strain when this has reached the breaking point, it can be easily
shown that a variable strain, acting in alternate periods in increase
or decrease of the general growth of strain, while leaving the average
rate unaltered, will give rise to a corresponding variation in the
frequency of shocks in each period; and, besides that, there is a
simple relation between the magnitudes of the two stresses, to
which the strains are due, and the variations from the mean fre-
quency of earthquakes. A calculation on these lines shows that
the growth of strain, for Italy, is such that, accepting the pub-
lished estimates that an area of the earth’s crust of the magnitude
of Italy would crush under its own weight if left unsupported to
the extent of 1/400 of the force of gravity, the breaking strain
would be reached in about 33 years, starting from a condition of
no strain. The aftershocks of the Mino-Owari earthquake give a
little less than half this figure, which is again reduced to from five
to six months if account is taken of the difference between the
resistance of rock to tension and to compression. These figures are
given for what they are worth; at the least, they are of interest as
being the first authentic estimate which it has been possible to
make of the time required to prepare for, and, thence, of the rate

of growth of the particular tectonic process involved in the
production of earthquakes.

* Q. J. G. S, vol. Ixv. (1909) p. 14.

[ 508 ]

LXIII. Intelligence and Miscellaneous Articles.

On Relativity and Electrodynamics.

To the Editors of the Philosophical Magazine.
Dear Sirs, 27th April, 1918.

S the result of some correspondence, Dr. G. A. Schott has
detected an error of sign in my calculation of the trans-
verse inertia of a ‘“‘ contracted conducting electron.” In con-
sequence of this the following numerical correction should be
inserted in my recent paper on “ Relativity and Electrodynamics.”
For a contracted conducting electron

m,=m, (i- .)
in place of 41 v?
p Mm, =M, (1+ 75)
leading t 2
eading to pee ie _
in pl f if
in place o h—k,=— =

There is a corresponding correction for a Bucherer electron,
viz.

21 2

N,=M, (1 a 50 =.)
7 ‘ 2
in place of em, (1+ ae
giving ee ae
in place of , ae it

PLS 2°~ 90)"

I am,

Yours faithfully,
Grorce W. WALKER.

soo! 7

INDEX to VOL. XXX¥M

oh _- 7

J‘ q,
ra is di A \
———_<f>——_- - a?)
“Oo CA 4
F Cf ¥

AERIAL waves generated by im-
pact, on, 97.

Allen (A. O.) on graphical methods
of correcting telescopic objectives,
471.

Allen (Dr. H. S.) on molecular fre-
quency and molecular number,
338, 404, 445.

Anderson (Prof. A.) on the co-
efficients of potential of two con-
ducting spheres, 286; on the
problem of two and that of three
electrified spherical conductors,
388.

Arc, on some potential problems
connected with the circular, 396.

Archean rocks, on the radioactivity
of, 2085.

Astronomical bearings of the elec-
trical theory of matter, on the,
141, 327, 481.

Atom, on X-ray spectra and the
constitution of the, 293.

Audibility factor of shunted tele-
phones, on the, 131, 487.

Banerji (S.) on aerial waves gene-

© rated by impact, 97.

Barton (Prof. E. H.) on variably-
coupled vibrations, 62; on coupled
circuits and mechanical analogies,
203.

Basu (N.) on the diffraction of light
by cylinders of large radius, 79;
on a new type of rough surface,
496.

Bessel functions of equal order and
argument, on, 364.

Phil. Mag. 8. 6. Vol. 35. No. 210. June 1918.

AAS by Sa "ho “4

Neti

Bickley (W. G.) oi-sofie pwo-dimen-
sional problems “eonnected “with
the circular are, 306}>0n two-
dimensional motion of an infinite
liquid, 500.

Bocks, new :—Researches of the De-
partmentof Terrestrial Magnetism,
220; Centennial Celebratiou of
the United States Coast and Geo-
detic Survey, 291; Millikan’s
The Electron, 370; Napier Ter-
centenary Volume, 371; Hors-
burgh’s Modern Instruments and
Methods of Calculation, 372;
Hancock’s Elliptie Integrals, 372;
Annuaire du Bureau des Longi-
tudes, 444.

Browning (Miss H. M.) on variably-
coupled vibrations, 62 ; on coupled
circuits and mechanical analogies,
203.

Cloud, on the scattering of light by
a, 373.

Coloured flames, note on Fox Tal-
bot’s method of obtaining, 382.
Colours, on the resolution of mixed,
by differential visual diffusivity,

413,

Conductors, on the coefficient of
potential of electrified spherical,
286, 388.

Corona discharge, on the pressure
effect in, 261.

Coupled circuits and mechanical
analogies, on, 140, 203.

Currents, on the maximum force
between two coaxial circular, 13.

2N

510 INDEX.

Cyclones, on rain, wind, and, 221.

Cylinders, on the diffraction of light
by, 79.

Dee (R. M.) on rain, wind, and
cyclones, 221.

Diffraction, on the asymmetry of the
illumination - curves in oblique,
112.

Distribution law, on Wien’s, 190,
410.

Eddington (Prof. A. 8.) on electrical
theories of matter and their astro-
nomical consequences with special
reference to the principle of rela-
tivity, 481.

Edgeworth (Prof. F. Y.) on the law
of error, 422.

Electrical circuits, on coupled, and
mechanical analogies, 140, 203.
currents, on the maximum
force between two coaxial circular,

13.

discharge, on the pressure

effect in, 261.

potential of two conducting

spheres, on the coefficients of,

286.

theory of matter, astronomical
and gravitational bearings of the,
141, 481.

Electrified spherical conductors, on
the problem of two and that of
three, 388.

Electrodynamics, on relativity and,
327, 508.

Electrolytes, on the effect of inter-
ionic force in, 214, 352.

Error, on the law of, 422.

Evaporation, on some problems of,
270, 350, 481.

Films, on the lubricating and other
properties of thin oily, 157

Flames, note on Fox Talbot's method
of obtaining coloured, 382.

Focometry, on the nodal - slide
method of, 21.

Focus, on light distribution round
the, of a lens, 380.

Gas reactions, on velocity-curves
for, 281, 485.

Gaseous diffusion, on, 270, 350, 431.
Geological Society, proceedings of
the, 185, 292, 502.
Graphical methods of correcting

telescopic objectives, on, 471.

Gravitational bearings of the elec-

trical theory of matter, on the, 141.

Gray (Prof. A.) on the hodographic
treatment and the energetics of
a kia planetary motion,
181.

Greenhill (Sir G.) on coupled circuits
and mechanical analogies, 140.
Gwyther (R. F.) on a theory of

material stresses, 490.

Heat, on the value of the mechani-
cal equivalent of, 27.

Hemsalech (G. A.) on Fox Talbot's
method of obtaining coloured
flames, 382.

Hitchcock (Dr. F. L.) on the operator
V in combination with homo-
ceneous functions, 461.

Hodographic treatment of undis-
turbed planetary motion, on the,
181.

Holmes (A.) on the Pre-Cambrian
rocks of Mozambique, 135.

Howe (Prof. G. W. O.) on the rela-
tion of the audibility factor of a
shunted telephone to the antenna
current, 131, 487.

Dluminatiou-curves in oblique dif-
fraction, on the asymmetry of
the, 112. .

Interferometers for the study of
optical systems, on, 49.

Interionic force, on the effect of,
in electrolytes, 214, 352.

Todine, on the resonance spectra of,
236.

Ionization pressure in corona dis-
charge, on the, 261.

Ives (Dr. H. E.) on the resolution
of mixed colours by differential
visual diffusivity, 413.

Jeffreys (Dr. H.) on some problems
of evaporation, 270; on Wien’s
distribution law, 410; on trans-
piration from leat-stomata, 431.

Kimura (M.) on the series law of
resonance spectra, 252.

Laplacean operator, note on the, 461.
Larmor (Sir J.) on transpiration
through leaf-stomata, 350, 433.
Leaf - stomata, on _ transpiration

through, 275, 350, 481.

Leathem (Dr. J. G.) on the two-
dimensional motion of infinite
liquid produced by the translation
or rotation of a contained solid,
119.

Lens, on light distribution round
the focus of a, 30.

INDEX.

Light, on the distribution of, round
the focus of a lens, 30; on the
diffraction of, by cylinders of large
radius, 79; on the constancy of
velocity of, reflected from a moy-
ing mirror, 163; on the scattering
of, by a cloud of small particles,
3738.

Liquid, on the two-dimensional
motion of infinite, 119, 500.

Lodge (Sir O.) on the astronomical
and gravitational bearings of the
electrical theory of matter, 141.

Lubrication, notes on the theory of,
Faye

Majorana (Prof. Q.) on an experi-
mental demonstration of the con-
stancy of velocity of light reflected
from a moving mirror, 163.

Mechanical equivalent of heat, on
the value ot the, 27.

Mercury, on the motion of the peri-
helion of, 141, 327, 481.

Milner (Dr. S. R.) on the effect of
interionic force in electrolytes,
214, 352.

Mitra (S. K.) on the asymmetry of
the illumination-curves in oblique
diffraction, 112.

Molecular frequency and molecular
number, on, 338, 404, 445,

Morrison (J.) on the Shap minor in-
trusions, 292.

Motion, on the two-dimensional, of
an infinite liquid produced by the
motion of a contained solid, 119,
500; on the, of a heavy particle
on a new type of rough surface,
496.

Moving mirror, on the constancy of
velocity of light reflected from a,
163.

Mute, on the action of the, in bowed
stringed instruments, 493.

Nabla, on the operator, in combina-
tion with homogeneous functions,
461.

Nagaoka (Prof. H.) on the calcula-
tion of the maximum force be-
tween two coaxial circular cur-
rents, 13.

Nodal-slide method of focometry,
on the, 21.

Objectives, on graphical methods of
correcting, 471.

Oil films, on the lubricating and
other properties of, 1, 157.

dll

Oldham (R. D.) on the frequency of
earthquakes, 504.

Oswald (Dr. F.) on the Nimrud

- volcano, 138.

Pendulums, on the vibrations of
coupled, 62.

Perihelia of certain planets, on the
motion of the, 141, 327, 481.

Planetary motion, on the hodo-
graphic treatment and the ener-
getics of undisturbed, 181.

van der Pol (B., jr.) on the relation
of the audibility factor of a
shunted telephone to the antenna
current, 154.

Positive rays, on a new secondary
radiation of, 59. ;

Potential of conducting spheres, on
the coefficients of, 286, 388.

Potential problems connected with
the circular arc, on some, 396.

Radiation, on a new secondary, of
positive rays, 59 ; on the visibility
of, 174.

Radicactivity of Archean rocks, on
the, 206.

Radio-telegraphy, on the audibility
factor in, 1381.

Rain, wind, and cyclones, on, 221.

Raman (Prot. C. V.) on the wolf-note
in bowed string instruments, 493.

Rayleigh (Lord) on the theory of
lubrication, 1; on the lubricating
and other properties of thin oily
films, 157; on the scattering of
light by a cloud of small particles,
373.

Reactions, on general curves for the
velocity of complete homogeneous,
281, 435.

Reeves (P.) on the visibility of
radiation, 174.

Relativity, on the second postulate
of the theory of, 163; on, and
electrodynamics, 327, 508.

Resonance spectra of iodine, on the,
256 ; on the series law of, 252.

Rocks, on the radioactivity of
Archean, 206.

Searle {Miss N.S.) on the pressure
effect in corona discharge, 26].
Silberstein (Dr. L.) on light distri-

bution round the focus of a lens,

Smeeth (Dr. W. F.) on the radio-
activity of Archean rocks from
the Mysore State, 206.

ESS SELES Te

512 INDEX.

Sollas (Prof. W. J.) on a flaked flint
from the Red Crag, 503.

Sound, on the, produced by the
impact of two spheres, 97.

Spectra, on the resonance, of iodine,
236; on the series law of reson-
ance, 252; on X-ray, and the
constitution of the atom, 298.

Spheres, on the coefficients of poten-
tial of conducting, 286, 388.

Stamp (L. D.) on the Silurian rocks
of Clun Forest, 502.

Stresses, on a doctrine of material,
490.

Surface, on a new type of rough, 496.

Sutton (T. C.) on the value of the
mechanical equivalent of heat, 27.

Talbot’s method of obtaining co-
loured flames, note on, 382.

Telephone, on the audibility factor
of a shunted, 131, 487.

Telescopic objectives, on graphical
methods of correcting, 471.

Todd (Dr. G. W.) on general curves
for the velocity of complete homo-
geneous reactions, 281, 435,

Tomkins (J. A.) on the nodal-slide
method of focometry, 21.

Twyman (F.) on interferometers for
the study of optical systems, 49.

Tyndall (Dr. A. M.) on the pressure
effect in corona discharge, 261.

Vegard (Dr. L.) on the X-ray
spectra and the constitution of
the atom, 293.

Vibrations, on variably-coupled, 62,
203.

Visibility of radiation, on the, 174.

Visual diffusivity, on the resolution
of mixed colours by differential,
413. :

Walker (G. W.) on relativity and
electrodynamics, 227,508.

Watson (Prof. G. N.) on Bessel
functions of equal order and argu-
ment, 364,

Watson. (Dr. H. E.) on the radio-
activity of Archean rocks from
the Mysore State, 206.

Waves, on aerial, generated by im-

act, 97.

Wien’s distribution law, on, 190,
410.

Wind, rain, and cyclones, on, 221.

Wireless telegraphy, on the audi-
bility factor of a shunted telephone
in, 131, 487.

Wolf-note, on the, in bowed string
instruments, 4938.

Wolfke (Dr. M.) on a new secon-
dary radiation of positive rays,
59.

Wood (F. E.) on Wien’s distri-
bution law, 190.

Wood (Prof. R. W.) on the reson-
auce spectra of iodine, 236: on
the series law of resonance spectra,
252.

X-ray spectra and the constitution
of the atom, on, 295.

END OF THE THIRTY-FIFTH VOLUME.

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