3&B

PHILOSOPHICAL

TRANSACTIONS

OF THE

ROYAL SOCIETY OF LONDON

SERIES A.

CONTAINING PAPEKS OF A MATITKMATK 'AL <>K PHYSK'AL CHARACTER.

VOL. 20:i.

/T

LONDON:

PK1NTED BY HARRISON AMU SONS, ST. MAKTIN's LANE, W.C..
fjrinltrs in ©rbinarji to f is Slajt

SEPTEMBER, 1904.

a

v. 2 O

I iii J

CONTENTS.

(A)
VOL. 203.

List of Illustrations . p;icfe v

Advertisement

I. On the Propagation of Tremor* over the Surface of an Elastic Solid. By HORACE
LAMB, F.R.S. ..................

II. On the Structure of Gold-Leaf, and the Alsorptiori Spectrum of Gold. By

J. W. MALLET. F.R.S., Professor of Chemistry in the University of
Virginia ............ ......... 43

III. Mathematical Contributions to the Tlteory of Evolution. — XII. On a Generalised

Theory of Alternative Inheritance, trith special Reference to MENDEL'S Lmi:s.
By KARL PEARSON, F.R.S ................. 5;i

IV. On the Acoustic Shadow of a. Sphere. Si/ LORD RAYLEIGH, O.M., F.R.S.

With an Appendix, giving the Values of LEGEXDRE'S Functions from P0 to P20
at Intervals of 5 degrees. By Professor A. LODGE ........ 87

V. On the Integrals of the Squares of Ellipsoidal Surface Harmonic Functions. By

G. H. DARWIN, F.R.S., Plumian Professor and Fellow of Trinity College, in
the University of Cambridge ............... Ill

a 2

VI. The Specific Heats of Metals and the Relation of Specific Heat to Atomic

Weight.— Part III. By W. A. TILDEN, D.Sc., F.K.S., Professor of Chemistry
in the Royal College of Science, London page 139

VII. An Enquiry into the Nature of the Relationship between Sun-spot Frequency

and Terrestrial Magnetism. By C. CHKEE, Sc.D., LL.D., F.R.S. . . 151

VIII. On Some Physical Constants of Saturated Solutions. By the EARL OF
BERKELEY. Communicated by F. H. NEVILLE, F.R.S. 189

IX. The Third Elliptic Integral and the Ellipsotomic Problem. By A. G. GREEN -

HILL, F. if.S 217

X. On the Resistance and Electromotive Forces of the Electric Arc. By W. DUDDELL,

Wh.Sc. Communicated by Professor W. E. AYRTON, F.R.S. .... 305

XI. On the High-Temperature Standards of the National Physical Laboratory: an

Account of a Comparison of Platinum Thermometers and Thermojunctions
with tlie (rax Thermometer. By J. A. HARKKR, D.Sc., Fellow of Owens
College, Manchester, Assistant at the National Physical Laboratory.
Communicated by I!, T. GLAZEBROOK, F.R.S., from the National Physical
Laboratory 343

XII. Colours in Metal Classes and in Metallic Films. By J. C. MAXWELL GARNETT,

B.A., Trinity College, Cambridge. Communicated by Professor J. LARMOR,
Sec.R.S. . ..." ' 385

Index to Volume . 4i>

L v

LIST OF ILLUSTRATIONS.

Plate 1.- Professor J. W. MAM,KT on the Structure of Gold-Leaf, and the Absorption
Spectrum of Gold.

Plate 2. — Mr. W. DUDDELL OH the Resistance and Electromotive Forces of the
Electric Arc.

[ vii 1

ADVERTISEMENT,

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*

f

upon any subject, either of Nature or Art, that comes before them. And therefore the
thanks, which are frequently proposed from the Chair, to be given to the authors of
such papers as are read at their accustomed meetings, or to the persons through whose
hands they received them, are to be considered in no other light than as a matter of
civility, in return for the respect shown to the Society by those communications. The
like also is to be said with regard to the several projects, inventions, and curiosities of
various kinds, which are often exhibited to the Society ; the authors whereof, or those
who exhibit them, frequently take the liberty to report, and even to certify in the
public newspapers, that they have met with the highest applause and approbation.
And therefore it is hoped that no regard will hereafter be paid to such reports and
public notices; which in some instances have been too lightly credited, to the
dishonour of the Society,

DtTTT AO ADTtrn A

mri -r< i

TONS.

l!fj HoRAGS L\MR, F.Jt.S.

INDEX SLIP.

Elastic Solid.

8, 1903.

LAMB, Horace. — Propagation of Tremors over the Surface of an Elastic
• Holid.

Phil. Tram., A, vol. 208, 1904, pp. 1-42.

isntropio fUt>ti<

Kiirthquakes, Relations of Elastic Theory to the1 Phenomena of.

LAMB, Horace. Phil. Trans., A, vol. 203, 1904, pp. 1-42.

•, Surface, und Tremors, in un Elastic Solid.

LAMB, Horace. Phil. Trans., A, vol. 203, 1904, pp. 1 -42.

iti ti'.ost tiit!^

.urfi-CK ; but some
v.i in. on- bricHv) <--.- " >

s&n thougliL best i
.•Ives.at tiu? ii<-
w character »t' I)IH
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in relation to '
•

he surface of a " semi-

a plane. For purposes

the solid as lying below

of force at a point. In
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il source of disturbance,

of the problem, it has
ions as they manifest
> latter introduces into
e solid are accordingly

leoretical grounds, and
TS on seismology have
lomena, at all events in
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I medium, as developed

special type of surface-
irface in modifying the
ban had been suspected.
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Although the circum-
lighly idealized state of

p. 98 (1898)) that in such
t of view, unimportant.
!, p. 441.

6.1.04

[ viii ]

upon any subject, either of Nature or Art, that comes before them. And therefore the

thanks, which are freqi

such papers as are read

hands they received th<

civility, in return for th

like also is to be said w

various kinds, which an

who exhibit them, free

public newspapers, thai

And therefore it is hoj

public notices ; which

dishonour of the Societ}

.SKI

tu •.•)i))-iu8 -irl,1
M»n »iS (o* ,/. .

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.(iil't

PHILOSOPHICAL TRANSACTIONS.

I. On the Propagation of Tremors over the Surface of an Elastic Solid.

By HORACE LAMB, F.R.S.

Eeceived June 11,— Read June 11,— Revised October 28, 1903.

INTRODUCTION.

1. THIS paper treats of the propagation of vibrations over the surface of a " semi-
infinite " isotropic elastic solid, i.e., a solid bounded only by a plane. For purposes
of description this plane may be conceived as horizontal, and the solid as lying below
it, although gravity is not specially taken into account.*

The vibrations are supposed due to an arbitrary application of force at a point. In
the problem most fully discussed this force consists of an impulse applied vertically to
the surface ; but some other cases, including that of an internal source of disturbance,
are also (more briefly) considered. Owing to the complexity of the problem, it has
been thought best to concentrate attention on the vibrations as they manifest
themselves at the free surface. The modifications which the latter introduces into
the character of the waves propagated into the interior of the solid are accordingly
not examined minutely.

The investigation may perhaps claim some interest on theoretical grounds, and
also in relation to the phenomena of earthquakes. Writers on seismology have
naturally endeavoured from time to time to interpret the phenomena, at all events in
their broader features, by the light of elastic theory. Most of these attempts have
been based on the laws of wave-propagation in an unlimited medium, as developed
by GREEN and STOKES ; but Lord RAYLEIGH'S discovery t of a special type of surface-
waves has made it evident that the influence of the free surface in modifying the
character of the vibrations is more definite and more serious than had been suspected.
The present memoir seeks to take a further step in the adaptation of theory
to actual conditions, by investigating cases of forced waves, and by abandoning
(ultimately) the restriction to simple-harmonic vibrations. Although the circum-
stances of actual earthquakes must differ greatly from the highly idealized state of

* Professor BROMWICH has shown (' Proc. Lond. Math. Soc.,' vol. 30, p. 98 (1898)) that in such
problems as are here considered the effect of gravity is, from a practical point of view, unimportant,
t 'Proc. Lond. Math. Soc.,' vol. 17, p. 4 (1885) ; ' Scientific Papers,' vol. 2, p. 441.
VOL. CCIII. — A 359. B 6.1.04

2 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

things which we are obliged to assume as a basis of calculation, it is hoped that the
solution of the problems here considered may not be altogether irrelevant.

It is found that the surface disturbance prodxiced by a single impulse of short
duration may be analysed roughly into two parts, which we may distinguish as the
"minor tremor" and the "main shock," respectively. The minor tremor sets in
at any place, with some abruptness, after an interval equal to the time which
a wave of longitudinal displacement would take to traverse the distance from the
source. Except for certain marked features at the inception, and again (to a lesser
extent) at an epoch corresponding to that of direct arrival of transversal waves, it
may be described, in general terms, as consisting of a long undulation leading up to
the main shock, and dying out gradually after this has passed. Its time-scale is
more and more protracted, and its amplitude is more and more diminished, the
greater the distance from the source. The main shock, on the other hand, is pro-
pagated as a solitary wave (with one maximum and one minimum, in both the
horizontal and vertical displacements) ; its time-scale is constant ; and its amplitude
diminishes only in accordance with the usual law of annular divergence, so that its
total energy is maintained undiminished. Its velocity is that of free Rayleigh waves,
and is accordingly somewhat less than that of waves of transversal displacement in
an unlimited medium.*

The . paper includes a number of subsidiary results. It is convenient to begin by
attacking the problems in their two-dimensional form, calculating (for instance) the
effect of a pressure applied uniformly along a line of the surface. The interpretation
of the results is then comparatively simple, and it is found that a good deal of
the analysis can be utilized afterwards for the three-dimensional cases. Again, the
investigation of the effects of a simple-harmonic source of disturbance is a natural
preliminary to that of a source varying according to an arbitrary law.

Incidentally, new solutions are given of the well-known problems where a periodic
force acts transversely to a line, or at a point, in an unlimited solid. These serve, to
some extent, as tests of the analytical method, which presents some features of
intricacy.

2. A few preliminary formula? and conventions as to notation may be put in
evidence at the outset, for convenience of reference.

The usual notation of BESSEL'S Functions " of the first kind " is naturally adhered to ;
thus we write :

2 f^"
JG (£) = - cos (£ cos &>) dot (1).

77 JQ

* Compare the concluding passage of Lord RAYLEIGH'S paper :

" It is not improbable that the surface-waves here investigated play an important part in earthquakes,
and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great
distance from the source a continually increasing preponderance."

The calculations indicate that the preponderance is much greater than would be inferred from a mere
comparison of the ordinary laws of two-dimensional and three-dimensional divergence.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 3

By a known theorem we have also

2 f"
J0(£) = sin (£ cosh u) du (2),

TT JQ

provided £ be real and positive. For our present purpose it is convenient to follow
H. WEBER* in adopting as the standard function "of the second kind"

2 f"
KO(£) = — cos (£ cosh w) du (3).

It is further necessary to have a special symbol for that combination of the two
functions (2) and (3) which is appropriate to the representation of a diverging wave-
system ; we write, after Lord HAYIYEIGH,!

7T J()

so that

We shall also write, in accordance with the usual conventions,

For large values of £ we have the asymptotic expansion

In the two-dimensional problems of this paper we shall have to deal with a
number of solutions of the equation

ff + ft+/^ = 0 ......... (8),

ox- cy~

constructed from the type

'

where £ is real, and

a = /- or =

* 'Part. Diff.-Gleichungen d. math. Physik,' Brunswick, 1899-1901, vol. 1, p. 175. HEINE (' Kugel-
functionen,' Berlin, 1878-1881, vol. 1, p. 185) omits the factor 2/w. In terms of the more usual
notation,

T£0=2{- Y0 + (log2-y)J0},

7T

where y is EULER'S constant. The function J^Ko has been tabulated (see J. H. MICHELL, ' Phil. Mag.,'
Jan., 1898).

t 'Phil, Mag.,' vol. 43, p. 259 (1897); 'Scientific Papers,' vol. 4, p. 283. I have introduced the
factor 2/7T, and reversed the sign.

B 2

4 PROFESSOE HORACE LAMB ON THE PROPAGATION OF

according as £3 5 /r, the radicals being taken positively. In particular, we shall
meet with the solution

. _ 1 f" e-^d( _ 2 f"

TT J-» a TT Jo

and it is important to recognize that this is identical with D0 (hr), where
r = ^/(x? + i/3). To see this, we remark that <f>, as given by (11), is an even
function of x, and that for x = 0 it assumes the form

2re-"»d€ 2 F e-^

= s = -7775

TT Jo a TT Jo v/(/i-

\ ......

+ I?2)

by the method of contour-integration.* This is obviously equal to D0 (hy}. Again,
the mean value of any function <j> which satisfies (8). taken round the circumference
of a circle of radius r which does not enclose any singularities, is known to be equal
to J()(&r).<£0, where <^0 is the value at the centre. t We can therefore adapt an
argument of THOMSON and TAIT} to show that a solution of (8) which has no
singularities in the region y > 0, and is symmetrical with respect to the axis of y, is
determined by its values at points of this axis. We have, accordingly,

_! (•£=

7T J — on (Z

Again, in some three-dimensional problems where there is symmetry about the
axis of z, we have to do with solutions of

based on the type

......... (15),

where TS = \/(x~ -\-y~), and a is defined as in (10). In particular, we have the
solution

which (again) reduces to a known function. At points on the axis of symmetry
(CT = 0) it takes the form

* If we equate severally the real and imaginary parts in the second and third members of (12), we
reproduce known results.

t H. WEBER, ' Math. Ann.,' vol. 1 (1868).
J ' Natural Philosophy,' § 498.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 5

Since the mean value of a function <j> which satisfies (14), taken over the surface of
a sphere of radius r not enclosing any singularities, is equal to

sin hr ,

hr '

where <£0 is the value at the centre,* the argument already borrowed from THOMSON
and TAIT enables us to assert that

where-

Finally, we shall require FOURIER'S Theorem in the form

I

/(X) «*<-*> c?X ....... (19)4

— M _»

and the analogous formula

As particular cases, if in (19) we have f(x) = 1 for x* < a~, and = 0 for x- > a", then

/•/^.\ 1 I* sin ft* ,-j, 7> -2 f°sin£a ^ ,,>
/ (x) = g e^dfss- -£-i30B&dg .... (21);

7T J -» ff 77 J 0 g"

and, if in (20) /(CT) = 1 for CT < «, and = 0 for CT > o, then

Jo^Ji^df ........ (22).

These are of course well-known results. ||

* H. WEBER, 'Crelle,' vol. 69 (1868).

t If in (18) we put z = 0, and then equate separately the real and imaginary parts, we deduce

I J0 (£ cosh «) cosh « du = — ~ ,
Jo C

J0 (i sin w) sin it du = --,-- .

These are known results. Of. RAYLEIGH, ' Scientific Papers,' vol. 3, pp. 46, 98 (1888) ; HOBSON, ' Proc.
Lond. Math. Soc.,' vol. 25, p. 71 (1893) ; and SONINE, ' Math. Ann.,' vol. 16).

f H. WEBER, 'Part. Diff.-Gl. etc.,' vol. 2, p. 190. Since A. occurs here and in (20) only as an inter-
mediate variable, no confusion is 'likely to be caused by its subsequent use to denote an elastic constant.

§ H. WEBER, 'Part. Diff.-Gl. etc.,' vol. 1, p. 193.

|| It may be noticed that if in (20) we put / (^) = «-i/"r/tir, we reproduce formulae given in the foot-note t
above.

PROFESSOR HORACE LAMB ON THE PROPAGATION OF

PART I.

Two- DIMENSIONAL PROBLEMS.
3. The equations of motion of an isotropic elastic solid in two dimensions (x, y] are

where u, v are the component displacements, p is the density, X, /x, are the elastic
constants of LAME, and

These equations are satisfied by

provided

8</> 0\li 8</> 3»// /„.-•> *

u = -- + J , v = 5- — a ....... (25),*

3x fy cy ex

-ML^vty 3V ^/fv2^ ....... (26).

In the case of simple-harmonic motion, the time-factor being d?*, the latter
equations take the forms

(V* + A*) $ = 0, (V2 + F)^, = 0 ...... (27),

where

70 jJ fJ 22 7 ^ r r 2 JtS/

~ r+ 2lj.~p< p. ~ p '

the symbols «, 6 denoting (as generally in this paper) the views-slownesses,^ i.e., the

reciprocals of the wave-velocities, corresponding to the irrotational and equivoluminal
types of disturbance respectively.

The formulas (25) now give, for the component stresses,

-=—A-(-2 = — K~(f> — 2 •5—5- -p 2 -~ «—
/A p. ox By* cxdy

&* = ^ + ^ = 2 -u ^ - ^ - 2 "-» .... (29).

fi ox d?/ dx dy dx~

ii 11 ty a*2 axay-

* GREEN, 'Camb. Trans.,' vol. 6 (1838); ' Math. Papers,' p. 261.

t The introduction of special symbols for wave-slownesses rather than for wave-velocities is prompted by
analytical considerations. The term " wave-slowness " is accredited in Optics by Sir W. R. HAMILTON.

TREMORS OVER THE SURFACE OP AN ELASTIC SOLID. 7

In the applications which we have in view, the vibrations of the solid are supposed
due to prescribed forces acting at or near the plane y — 0. We therefore assume as
a typical solution of (27), applicable to the region y > 0,

(30),

where f is real, and a, /3 are the positive real, or positive imaginary,* quantities

determined by

«s = P_/,2, & = £*-& ..... (31).

For the region y < 0, the corresponding assumption would be

4> = A.'e«>e*c, i/» = B'eft"e* ........ (32).

The time-factor is here (and often in the sequel) temporarily omitted.

The expressions (30), when substituted in (25) and (29), give for the displacements
and stresses at the plane y = 0

«0 = (#A-£B)e**, i>0 = (-aA--ifB)c*' .... (33),
and

= t - 2»«A 2 - *»B

..... (34).
8 - F A

The forms corresponding to (32) would be obtained by affixing accents to A and B,
and reversing the signs of a. /3.

4. In order to illustrate, and at the same time test, our method, it is convenient to
begin with the solution of a known problem, viz., where a periodic force acts
transversally on a line of matter, in an unlimited elastic solid, f

Let us imagine, in the first instance, that an extraneous force of amount Ye'*'' per
unit area acts parallel to y on a thin stratum coincident with the plane y = 0. The
normal stress will then be discontinuous at this plane, viz.,

[PyJ^+o ~ l>.J, = -o = - Y«''fl (35)>

whilst the tangential stress is continuous. These conditions give, by (34),

- 2ifa (A + A') + (2f 2 - **) (B - B') = 0

Again, the continuity of u and v requires

=

* This convention should be carefully attended to ; it runs throughout the paper.
t RAYLEIGH, ' Theory of Sound,' 2nd ed., § 376.

8 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

Hence

A = -A' = ^- B = B' = f'4 ' ' ' ' ' (38>-

We have, then, for y > 0,

" -*

To pass to the case of an extraneous force Q concentrated on the line x = 0, y = 0,
we make use of (19). Assuming that the /(X) of this formula vanishes for all but
infinitesimal values of X, for which it becomes infinite in such a way that

we write, in (39), Y = Qf/£/2ir, and integrate with respect to f from — oo to + co.*
We thus obtain, for y > 0,

, Q r e~v#, + = -« f &:*£•* . . .

47T/C-/A J -x 47T&->t J -oc p

or, on reference to (13),

where r = \/(%2 + ?/2).

If we put x = r cos 6, y = r sin #, we find from (25), on inserting the time-factor,
that for large values of r the radial and transverse displacements are

a<£ ai// _._ Q v/JL e; ( /•'-*-- to s; nfl

r" -

cos ^

respectively.! Use has here been made of (7).

A simple expression can be obtained for the rate (W, say) at which the extraneous

* The indeterminateness of the formula (19) in this case may be evaded by supposing, in the first
instance, that the force Q, instead of being concentrated on the line .<• = 0, is uniformly distributed over
the portion of the plane ,'/ = 0 lying between x= ±a. It appears from (21) that we should then have

Y=Q «in&dc
2?r £a

If we finally make a = 0 we obtain the results (40).

t The second of these results is equivalent to that given by RAYLEIGH, loc. cit., for the case of
incompressibility (A. = oo ).

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. <)

force does work in generating the cylindrical waves which travel outwards from the
source of disturbance. The formula) (40) give, for the value of dv/dt at the origin,

This expression is really infinite, but we are only concerned with the part of it in the
same phase with the force,* which is finite. Taking this alone, we have

? • <">•

Discarding imaginary parts, we find that the mean rate, per unit length of the axis
of z, at which a force Q cos pi does work is

, .x

5. We may proceed to the case of a " semi-infinite " elastic solid, bounded (say)
by the plane y = 0, and lying on the positive side of this plane. We examine,, in
the first place, the effect of given periodic forces applied to the boundary.

As a typical distribution of normal force, we take

........ (4G),

the factor e'f* being as usual understood. The constants A, B in (30) are determined
by means of (34), viz. :

+ (2f 2 - F) B = 0,

Hence

2<r-F_Y B_. (4g)

~EX?T /*' "F(f) /*
where, for shortness,

F (f ) = (2^2 - A2)2 - 4^«/8 ....... (49).

We shall find it convenient, presently, to write also
, .. /(0 = (2^2-*8)s + 4fsa/8 ...... . . (50).

*i

* The awkwardness is evaded if (as in a previous instance) we distribute the force uniformly over a length
2a of the axis of x. This will introduce a factor I^-S^\ under the integral signs in the second
member of (44).

VOL. CCIII. — A. 0

10 PEOFESSOR HORACE LAMB ON THE PROPAGATION OF

The surface-values of the displacements are now given by (33), viz. :

P - 2a£ «** Y

_
:

Y

The effect of a concentrated force Q acting parallel to y at points of the line
x = 0, y = 0 is deduced, as before, by writing Y = — Qd£/2ir, and integrating
from — oo to oo ; thus

tQ f g_(2_e - V-Jtap)^
°~ ~2«iL. ~F(f) ~

f ..... (52).
Q f" PaeF^dg

~2vJ-. F"f

In a similar manner, corresponding to the tangential surface forces :

[f.ry]o = Xe'f*, \_pyJ\Q = 0 (53),

we should find

XO f-2 7,2 Y
T> ^JC /^ -^X

And, for the effect of a concentrated force P acting parallel to x at the origin,

P

"2ir^-.

h ..... (55).
iP p ^(2^-F-2«^)^c^ I

2»/tJ- ~'F"(f)"" J

The comparison of ??„ in (52) with -«„ in (55) gives an example of the general
principle of reciprocity.*

We may also consider the case of an internal source of disturbance, resident
(say) in the line x = 0, y=f, the boundary y — 0 being now entirely free. The
simplest type of source is one which would produce symmetrical radial motion (in
two dimensions) in an unlimited solid, say

<£ = D0 (hr), V = 0 . . ....... (56),

where r, = ^/{x° + (y — /)3j, denotes distance from the source. If we superpose on
this an equal source in the line x = 0, y = — f, we obtain

<£ = DO (/«•) + D0 (Ar'), ,/, = o ..... . . (57),

* RAYLEIGH, 'Theory of Sound,' vol. 1, § 108.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 11

where / = </{a? + (y +/)2}. It is evident, without calculation, that the condition
of zero tangential stress at the plane y = 0 is already satisfied ; the normal stress,
however, does not vanish. It appears from (13) that in the neighbourhood of the
plane ?/ = 0 "the preceding value of <f> is equivalent to

_ i p er<*-f> (ft* dg -If"

IT J— » « TT J_

TT _» a

2 f" cosh 0.1

f ,tr

e e

Substituting in (29) we find that this makes

a

Comparing with (46), we see that the desired condition of zero stress on the
boundary will be fulfilled, provided we superpose on (57) the solution obtained from
(30) and (48) by putting

TT a

and afterwards integrating with respect to f from -co to co . The surface-
displacements corresponding to this auxiliary solution are obtained from (51), and if
we incorporate the part of UQ due to (58), Ave find, after a slight reduction,

...... (GO).

These calculations might be greatly extended. For example, it would be easy,
with the help of Art. 4, to work out the case where a vertical or a horizontal periodic
force acts on an internal line parallel to z. And, by means of the reciprocal theorem
already adverted to, we could infer the horizontal or vertical displacement at an
internal point due to a given localized surface force.

6. It remains to interpret, as far as possible, the definite integrals which occur in
the expressions we have obtained.

It is to be remarked, in the first place, that the integrals, as they stand, are to a
certain extent indeterminate, owing to the vanishing of the function F (f ) for certain
real values of f. It is otherwise evident d priori that on a particular solution of any
of our problems we can superpose a system of free surface waves having the wave-
length proper to the imposed period '2-ir/p. The theory of such waves has been given

C 2

12 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

by Lord RAYLEIGH,* and is moreover necessarily contained implicitly in our
analysis.

Thus, if we put Y = 0 in (47), we find that the conditions of zero surface-stress are
satisfied, provided

A : B = 2/r — k~ : 2iKOil = — 2i.K/31 : 2*c — kz . . . . (61),

where K is a root of F (f ) = 0, and «1( /8ls denote the corresponding values of a, /3,
Now, in the notation of (49) and (50),

Equating this to zero, we have a cubic in f8/^, and since k~ > /r, it is plain that
there is a real root between 1 and oo . It may also be shown without much difficulty
that the remaining roots, when real, lie between 0 and h~/kz. The former root makes
a, ,5 real and positive, and therefore cannot make f(£) = 0. The latter roots make
a, /B positive imaginaries, and therefore cannot make F (f) = 0. This latter
equation has accordingly only two real roots £ — i K , where /c > I;.
Thus, in the case of incompressibility (X = oo , h= 0) it is found that

K/k= 1-04678

and that the remaining roots of (62) are complex, t On POISSON'S hypothesis as to
the relation between the elastic constants (X = JM, 7r = ^P), the roots of (62) are all
real, viz., they are

eiV = \, U3-V/3), i(3 + v/3),
so that

K/k = i v/(3 + v/3) = 1-087664 . . . ;

this will usually be taken as the standard case for purposes of numerical illustration.
In analogy with (28), it will be convenient to write

«=pc ........... (63),

where c denotes the wave-slowness of the Rayleigh waves. The corresponding
wave-velocity is

k 7 _-, k /u,

cl=.bl=. A / C .

K K V p

According as we suppose X = QO , or X = /x, this is '9553 times, or '9194 times, the
velocity of propagation of plane transverse waves in an unlimited solid.

The further properties of free Rayleigh waves are contained in the formulae (61)

* 'Proc. Lond. Math. Soc.,' vol. 17 (1885); 'Scientific Papers,' vol. 2, p. 441.

t Cf. RAYLEIGH (loc. tit.), where it is also shown (virtually) that they are roots of / (£), not of F (£), if
a, /3 be chosen so as to have their real parts positive.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 13

and (30). We merely note, for purposes of reference, that if in (33) we put £ = i K ,
and accordingly, from (61),

A = (2K" - F) C, B = ± 2ifc«1C ...... (64),

we obtain by superposition a system of standing waves in which

u0 = — 2* (2/r — k* — 2^) C sin KX . e*', v0 = 2&2a1C cos KX . e!f{ . (65).

The theory here recapitulated indicates the method to be pursued in treating the
definite integrals of Art. 5. We fix our attention, in the first instance, on their
" principal values," in CAUCHY'S sense, and afterwards superpose such a system of free
Rayleigh waves as will make the final result consist solely of waves travelling
outwards from the origin of disturbance.

It may be remarked that an alternative procedure is possible, in which even
temporary indeterminateness is avoided. This consists in inserting in the equations
of motion (23) frictional terms proportional to the velocities, and finally making the
coefficients of these terms vanish. This method has some advantages, especially as
regards the positions of the "singular points" to be referred to. The chief problem
of this paper was, in fact, first worked through in this manner ; but as the method
seemed rather troublesome to expound as regards some points of detail, it was
abandoned in favour of that explained above.

7. The most important case, and the one here chiefly considered, is that ot a
concentrated vertical force applied to the surface, to Avhich the formulae (52) relate.
The case of a horizontal force, expressed by the formulas (55), could be treated in an
exactly similar manner.

Since w0 is evidently an odd, and v() an even, function of x, it will be sufficient to
take the case of x positive.

As regards the horizontal* displacement it(), we consider the integral

L ({) dr =
J

3 - A8)

taken round a suitable contour in the plane of the complex variable £, = f + *V
If this contour does not include either "poles" (± *, 0), or "branch-points"
(± h> 0)) (± &» 0) of the function to be integrated, the result will be zero.

A convenient contour for our purpose is a rectangle, one side of which consists of
the axis of £ except for small semicircular indentations surrounding the singular
points specified, whilst the remaining sides are at an infinite distance on the side 77 >0.
It is easily seen that the parts of the integral due to these infinitely distant sides
will vanish of themselves. If we adopt for the radicals ^/(Z,* — h") and v/(£a — &2),

* The sense in which the terns "horizontal" and "vertical" arc used is indicated in the second

sentence of the Introduction.

14

PROFESSOR HORACE LAMB ON THE PROPAGATION OF

k K

e

Of, ft

Fig. 1.

at points of the axis of f, the consistent system of values indicated in fig. 1,* we
find, for the various parts of the first-mentioned side,t

F'(-K)

where the terms with F'(— K) and F'(/c) in the denominator are due to the small
semicircles about the points (i K, 0). Equating the sum of these expressions to
zero, we find, since F'(— K) = — F'(K),

= _ «. H cos

, ,x , ,

-

r

- - 2.VH cos KX -

* The function under the integral sign in (66) is uniquely determined (by continuity) within and on
the contour when once the values of the radicals v/(f2 - A2) and ^/(f 2 - k3) at some one point are
assigned. The convention implied in the text is that the radicals are both positive at the point ( + « , 0).

It will be noticed that over the portion of the axis of £ between - k and - h the function in (66)
differs from that involved in the value of w0 as given by (52). This is allowed for in the second member
of (67). Corrections, or rather adjustments, of this kind occur repeatedly in the transformations of this
paper.

t The symbol $ is used to distinguish the " principal value " of an integral (with respect to a real
variable) to which it is prefixed.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 15

where

F(K)

a numerical quantity depending only on the ratio X : ^.
To examine the value of v0 we take the integral

__ , /69x

* _ p)2 _ 4 ^(£2 _ AS) ^^ _ *

round the same contour. Integrating along the axis of g we find

_

T

and thence by addition, since the terms due to the infinitely distant parts of the
contour vanish as before,

it* fit

cos K — - ^

f =° IfZnf-it

2f — J-,,
J* * f

'(«

~r 2Ar ~T^//-\ r/TA \'W>

JA

where

Hence if to the principal values of the expressions in (52) we add the system of
free Rayleigh waves,

u —i^R sin KX, v0 = — i ^ K cos KX (72),

p. p

16 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

which is evidently of the type (65), we obtain, on inserting the time-factor,

^J- (73).

Q

~

f* ff^-

J ,( (2^ - F)*

•<-M

#

+

(74).

This is for x positive ; the corresponding results for x negative would be obtained by
changing the sign of x in the exponentials, and reversing the sign of u0.

The solution thus found is made up of waves travelling outwards, right and left,
from the origin, and so satisfies all the conditions of the question.

The first term in u0 gives, on each side, a train of waves travelling unchanged with
the velocity c~l. The second term gives an aggregate of waves travelling with
velocities ranging from lrl to a""1. As x is increased, this term diminishes indefinitely,
owing to the more and more rapid fluctuations in the value of e'H

On the other hand, the part of v0 which corresponds to the first term of w0 remains
embedded in the first definite integral in (74). To disentangle it we must have
recourse to another treatment of the integral |^ (£) c/£. One way of doing this is to take
the integral round the pair of contours shown in fig. 2, where a consistent scheme of

-K -h -h

f

a, -ft -0,,-p

a,

Fig. 2.

values to be attributed to the radicals ^(t* — A2) and v/(£2 — L-) is indicated. For
the only parts of the left-hand contour which need be taken into account we find

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 17

= f- — *^*

- Jo 2 F* - 4r2

+ F)* - 4r?2 + V)

Similarly, in the right-hand contour,

f ° * //) dr = f °

~ " - T)

We infer, by addition,

Pf k-ae^dg > v .

" TT7]tV = 2:rK sm ^ + 2

*—> J ?

(

If we multiply this hy — Q/27T/X, and add in the term due to the free Rayleigh
waves represented by (72), we obtain, as an equivalent form of (74),

_

4jQ f* F^(g2- /^) v/(As -J2) e^^ cf ^
" " "

, Q .w f"_

h Jo 28

_ _ _

(2V8 + F)* - 47,2 ^(/i* +.^) ,/(P + r/2) '

It is evident that all terms after the first diminish indefinitely as x is increased.

* From this we can deduce, by the Scime method as in Art. 4, an expression for the mean rate W at
which a vertical pressure Q cospt does work in generating waves, viz.,

w =

/i '2lTj

12 ft

+ ~J"t. I ^LS>_\*_

VOL. CCI1I. — A. D

18

PROFESSOR HORACE LAMB ON THE PROPAGATION OF

If in (73) and (76) we regard only the terms which are sensible at a great distance
from the origin, we have, for x positive,

V-

V-

and similarly for x negative we should find

un = —

(77);

(78).

These formulae represent a system of free Rayleigh waves, except for the
discontinuity at the origin, where the extraneous force is applied. The vibrations
are elliptic, with horizontal and vertical axes in the ratio of the two numbers
H and K, which are denned by (G8) and (71), respectively. To calculate these, we
have, since F (K) = 0,

and therefore

J^&J
= -KF(

where, by dift'ereutiation of (62),

2&X (2/c2 - #»)»

, .

. . (so).

In the case of incompressibility I find

H - '05921, K= -10890;
whilst on POISSON'S hypothesis

H = -12500, K = -18349,

so that the amplitudes are, for the same value of ju, and for the same applied force,
about double what they are in the case of incompressibility.

A similar treatment applies to the formulae (55), which represent the effect of a
concentrated horizontal force Peipt. Taking account only of the more important terms,
I find, for x positive,

' (81),

and, for x negative,
where

MO - _ !? HVp(|-">, v0 = — K'e**'-")

/A /I,

H,

'

K'= -

= — — H'e*(l+er), «„=:—? K'e
V- /*

, 2/1-'^, (2>c2 - P)2
"

(82),

. . . . (83).

TKEMORS OVER THE SUEFACE OF AN ELASTIC SOLID.

19

The ratio of H' to K' is, of course, equal to that of H to K ; K' is, moreover,
identical with H, in conformity with the principle of reciprocity already referred to.
It appears, therefore, from the numerical values of H, K above given, that for X = oo

and for X =

H' = -03219, K' = -05921 ;
H' = -08516, K'= -12500.

Again, in the case of the internal source (56) I find, for large positive values of x,

u0 = - 8/cH'e-a'/e'''>('-«>, t'0 = StVcK'e— 'e^'—' (84),

and, for large negative values,

u0 = 8K-H'e~ai/e1/>(' cx\ v0 = 8?'/cK'e~a'/e?p(<+") (85).

The factor e~aif indicates how the surface effect (at a sufficient distance) varies
with the depth of the source.

8. If in any of the preceding cases we wish to examine more closely the nature and
magnitude of the residual disturbance, so far as it is manifested at the surface, it is
more convenient to use the system of contours shown in fig. 3. With this system we

I i

-K -h

-h

-6-

Fig. 3.

k K

— £

can so adjust matters that the radicals ^/(^ — W] and ^/(£2 — k") shall assume in all
parts of- the axis of f exactly the values a, ft with which we are concerned in formulae
such as (52). It is convenient, for brevity, to denote by ± a.', fi' the values assumed
by the same radicals on the two sides of the lines £= — h, and by a", i/» their
values on the two sides of the line g = — k, these values being supposed determined

D 2

20 PEOFESSOE HOEACE LAMB ON THE PEOPAGATION OF

in accordance with the requirements of continuity. Thus, with the allocation shown
in the figure, we shall have, for small values of 17,

a' = - v/(2/ii?)e-1'>, & = i v/(&3 - /i2) "I

-.... (86),

H
approximately.

Taking the integral (66) round the several contours, in the directions shown by
the arrows, we find

r £(2£3 - k* - 2
\ ~F77

cos KX

rx [ 0/2 _ 7.2 _ 0«,"/3" 9/2 _ 7,2 I oV/C// 1

i (,-**[ J /4 -- /t - Z,ct ft /(,_•- Ic + Za. ft 1 r „, • /

Jo l(2£* - yt2)3 - 4^""^ (2^ - P)2 + 4£V'£" J 4e C ^

J_ --*. f J 2^^3_Z-_2a'/S/ 2^-P + 2a^ 1 . ... ,

Jo [(2? • - A;2)2 - 4CVyS/ "" (2^3 - £2)2 + 4^V/i/J 4<? ^ '

cos KX

_________

o (2^ - /r)1 + 16^ (C2 -

where, in the first integral, £= — k + iy, and, in the second, £ = — h + irj.
The integral (69), taken round the same contours, gives

tfa.

= 2,7 sn

, ,.crr _^<_ -p^ i

Jo l(2>-F)8-4Ca'/8' (2^-l2)2+4CV^J6 ^

+ Ste-*" f- -^^-C? ~ /i2) ^
Jo 22-F* 162-

* f __ V(2^J?£*r?ds
Jo(2^-F)*+16£*£«--A*F

on the same understandino-.

o

The definite integrals in these results can all be expanded in asymptotic forms by
means of the formula

and when kx, and therefore also kx, is sufficiently large, the first terms in the
expansions will give an adequate approximation.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 21

Thus, taking account of (86), the last members of (87) and (88) are equivalent to

cos KX + 2 v/(2ir) A/( 1 - ~] • ie '

V \ #7 (kx)*

and

27rK sin KO; - V(2,r) f 1 - ff) . *-e-~-!-(-

\ «*/ (to)1

7,27.2
//r> \ /* *v 6> IP

- ^(2iO -(TJ— g^i • -pjjr

respectively. Substituting in (52), and adding in the system (72) as before, we have,
for large positive values of x,

r> n /9 // j,~\ ^(ft-tx-ii,)

u, = - Q He'<*-"> + S ,Y/.- V i1 - P) ' "Trv-

/x, /u. V TT V \ A;-/ (^x)5

Q/ 2 // 3yL-'3 //" /- — // - ^ it'1 ( ;" -**-W
— A / • — VJ>^ /r/ . i .e,,,

^ V TT (^-2A8)» (Ax)»

9'

,'O 9O /o / 7(2\ ,V(i''-^-l")

t,0 = _ ?« Ke'^'—' + ^ A/ - • 1 - ';, - i —1^1

fJ. p V 7T \ A,-/ (/.'X)'

Q
"*-

The first terms in these expressions have already been interpreted. The residual
disturbance constitutes a sort of fringe to the cylindrical elastic waves which are
propagated into the interior of the solid, and consists of two parts. In one of these
the wa ve- velocity p/k, or b~l, is that of equivoluminal waves; the vibrations (at the
surface) are elliptic, the ratio of the vertical to the horizontal diameter of the orbit
being 2^(1 — A,2//c2), or 1'633 for \ = p.. The remaining part has the wave-velocity
p/h, or a"1, of irrotational waves ; the surface vibrations which it represents
are rectilinear, the ratio of the vertical to the horizontal amplitude being
(&3 — 2h2)/2h (kz — /i2)*, or "3535 for \ = p.. With increasing distance * the
amplitude of each part diminishes as x~'s, whereas in an unlimited solid the law
is jc~*, as appears from (42).

Similar results will obviously hold in the case of the other problems considered in
Art. 5.

9. It has been assumed, up to this stage, that the primary disturbance varies as a
simple-harmonic function of the time. It is proposed now to generalize the law
of variation, and in particular to examine the effect of a single impulse of short
duration. From this the general case can be inferred by superposition.

22 PEOFESSOR HOEACE LAMB ON THE PEOPAGATION OF

It is to be noticed, in all our formulae, that if we write

£ = p0, k = pa, k = pb, K = pc,

••J -- ' .

the symbol p which determines the frequency will disappear, except in the
exponentials; this greatly facilitates the desired generalization by means of
FOURIER'S theorem. Thus, in the case of a concentrated vertical pressure Q (t) acting
on the surface, the formulae (73) and (74) lead to

- 62)2 - 402 v/(02 - a2)

The definite integrals represent aggregates of waves, of the same general type,
travelling with slownesses ranging from a to b, and from b to oo , respectively.

If we suppose that Q (t) vanishes for all but small values of t, it appears from
(92) that the horizontal disturbance at a distance x begins (as we should expect)
after a time ax, which is the time a wave of expansion would take to travel the
distance ; it lasts till a time bx, which is the time distortional waves would take to
travel the distazice ; and then, for a while, ceases.* Finally, about the time ex, comes
a solitary wave of short duration (the same as that of the primary impulse) represented
by the first term of (92). This wave is of unchanging type, whereas the duration
of the preliminary disturbance varies directly as x, and its amplitude (as will be seen
immediately) varies inversely as x.

If we put

(«)cfe .......... (94),

the integration extending over the short range for which Q is sensible, the
preliminary horizontal disturbance will be given by

«0 —

provided

U (ff) = -

— r--- ......... (95),

irp.bx \x

(202 - b*Y + 160* (ffi - a2) (W - ffi)

where a < 0 < b. The following table gives the values of U (0) for a series of values
of 8/a, on the hypothesis of X = /A, or b/a = 17321.

•• This temporary cessation of the horizontal motion is special to the case of a normal impulse. If
the impulse be tangential, the contrast between the horizontal and vertical motions, in this respect, is
reversed.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID.

23

6/a.

U(0).

9/a.

U(0).

0/0.

U(0).

6/a.

U(0).

1-000

0

1-025

+ •62777

1-10

+ •22789

1-550

- -15122

1-001

+ •31247

1-030

+ -59351

1-15

+ •10295

1-600

- -15842

1-002

+ -42080

1-035

+ •55806

1-20

+ -02722

1-625

-•15927

1-003

+ -49148

1-040

+ -52308

1-25

-•02311

1-650

-•15681

1-004

+ •54191

1-050

+ • 45741

1-30

- -05905

1-675

- -14845

1-005

+ •57926

1-060

+ •39889

1-35

- -08622

1-700

-•12795

1-010

+ -66493

1-070

+ •34746

1-40

- -10771

1-725

- -07021

1-015

+ •67536

1-080

+ •30238

1-45

- -12527

6/a

0

1-020

+ •65744

1-090

+ •26279

1-50

- -13975

—

—

1

The function has a maximum value + '67643 when#/« = 1 '01368 ; it changes sign

when 0/a = T22474 ; and it has a minimum value — '159319 when 9ja = 1'62076.*

The graph of this function is shown in the upper part of fig. 4. If the scales be

Fig. 4.

properly chosen, the curve will represent the variation of MO with t, during the
" preliminary" disturbance, at any assigned point x. For this purpose the horizontal
scale must vary directly, and the vertical scale inversely, as x.

* The calculations were made almost entirely by Mr. H. J. WOODALL, to whom I am much indebted.

24

PROFESSOR HORACE LAMB ON THE PROPAGATION OF

The interpretation of the expression (93) for the vertical displacement r0 is not
quite so simple. For a given value of x, the most important part is that corre-
sponding to t = ex, or 0 = c, nearly, when the integrand in the second term changes
sign by passing through infinity. This is the epoch of the main shock ; the minor
disturbance which sets in when t = ax leads up continuously to this, and only dies
out gradually after it.

As a first step we may tabulate the function V (6} defined by

V (0) - - -

*

--

(20* - #>)* + 160* (0* - a2) (62 -02)

(20* - b-)3 -

for a < 0 < b

, for 0 > b .

(97).

i

9la.

V(0).

0/a,

V(0).

0/a.

V(0).

0/a.

V(0).

1-000

0

I • 025

- -39425

1-10

- -08981

1-550

- -22781

1-001

- -21995

1-030

- -36340

1-15

- -02454

1-600

- -31645

1-002

- -29488

1-035

- -33293

1-20

- -00218

1-625

- -37299

1-003

- -34284

1-040

- -30387

1-25

- -00193

1-650

- -44110

1-004

- -37630

1-050

- -25142

1-30

- -01508

1 • 675

- -52493

1-005

-• 40039 1-060

- -20681

1-35

- -03796

1-700

- -63087

i-oio-

- -44907

1-070 -• 16932

1-40

- -06941

1-725

- -76935

1-015

- - 44543

1-080

- -13795

1-45 -• 10989

b/a

- -81649

1-020

- -42324

1-090

- -11173

1-50

- -16137

— .

—

i

0/a.

V(0).

0/a.

V(0).

0/a.

V(0).

0/a.

V(0).

b/a

-0-81649

1-90

+ 20-38685

2-10

+ 1-99591

2-5

+ •91464

1-75

-1-39031

1-95

+ 5-42335

2-15

+ 1-69743

3-0

+ •60196

1-80

-2-98197

2-00

+ 3-31759

2-20

+ 1-48891

4-0

+ •38179

1-85

-8-65843

2-05

+ 2-46398

2-25

+ 1-33404

10-0

+ •13292

c/a

00

_

.

The function has a minimum value — '45120 when 0/a = 1 '01 170, and a zero
maximum when 0/a = 1 '22474 ; it changes from — oo to + oo when 0/b = 1 '08767,
or 0/a = 1-88389.* Its graph is shown in the lower part of fig. 4, and also (on a
smaller scale, so as to bring in a greater range of 0) in fig. 5.

It is postulated that the function Q (t) is sensible only for values of t lying within
a short range on each side of 0 ; the function Q (t - 0x) will therefore be sensible
only for values of 0 in the neighbourhood of t/x. We will suppose that for given
values of x and t its graph (as a function of 0) has some such form as that of the

* As in the case of U (6), the calculations are due chiefly to Mr. WOODALL.

TEEMORS OVER THE SURFACE OF AN ELASTIC SOLID.

25

dotted curve in fig. 5. If a; be constant, the effect of increasing t will be to cause
this graph to travel uniformly from left to right ; and if we imagine that in each of

Fig. 5.

its positions the integral of the product of the ordinates of the two curves is taken,
we get a mental picture of the variation of v{) as a function of t, on a certain scale.

For the greater part of the range of t, the integral will be approximately
proportional to the ordinates of the curve V(0), viz., we shall have

(98),

TT/Jibx ' \X/

in analogy with (95). But for a short range of t, in the neighbourhood or ex, the
statement must be modified, the dotted curve being then in the neighbourhood of the
vertical asymptote of the function V(#). Since the principal value of the integral is
to be taken, it is evident that as t approaches the critical epoch and passes it, v0 will
sink to a relatively low minimum, and then passing through zero will attain a
correspondingly high maximum, after which it will decrease asymptotically to zero,
the later stages coming again under the formula (98).

Although the above argument gives perhaps the best view of the whole course of
the disturbance, we are not dependent upon it for a knowledge of what takes place

VOL. CCIII. — A. E •

26 PEOFESSOE HOEACE LAMB ON THE PEOPAGATION OF

about the critical epoch ex. We may proceed, instead, by generalizing the expres-
sions (77). This introduces, in addition to the given function Q(t), whose Fourier
expression is

Q(\)cosp(t-\)dX ..... (99),

TT JO

the related function

(100);

TT Jo J -oo

viz., we have

u0 = - H Q (t - ex) + &c., v0 = — Qx (* - ex) + &c. . . (101).

It does not appear that the connection between the functions Q (t) and Qv (t) has
been specially studied, although it presents itself in more than one department of
mathematical physics. The following cases may be noted as of interest from our
present point of view :

(104).

It is evident, generally, that if Q be an odd function, Qv will be an even function,
and vice versd.

The values of UQ and v0, as given by (101), are represented graphically in fig. 6,
for the case where Q (t) and Qv (t) have the forms given in (102).* Moreover, writing

HQ/2T7/AT = /, KQ/27T/AT = g, t — ex •= T tan x,
we have

2x) •/. vo = sin 2X • 9 ..... (105) 5

the orbit of a surface-particle is therefore an ellipse with horizontal and vertical semi-
axes f and g. And if from the equilibrium position 0 we project any other position
P of the particle on to a vertical straight line, the law of Fs motion is that the
projection (E) describes this line with constant velocity. See fig. 7, where the
positive direction of y is supposed to be downwards.

' The relation between the scales of the ordinates in the graphs of w0 and v0 depends upon the ratio of
the elastic constants A, /*. The figures are constructed on the hypothesis of X = /*.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID.

27

Fig. 6.

A similar treatment would apply to the formulae (81), and (with some modification)
to (84).

It remains to justify these approximations by showing that the residual disturbance
tends with increasing x to the limit 0. For this purpose we have recourse to the
formulae of Art. 8. As a sufficient example, take the second term in the last
member of (88). If we multiply by e''11, take the real part, and substitute rj = p<l>,
k = 2)b, the corresponding term in the value of v0, as given by (52), assumes the form*

-Q cos/; (t - bx) f°F U) e-** d<f> + Q- sin p (t - bx) f f(<f>) e~^ d<j>,
fj. Jo U. Jo

where the functions F (<£) and /(<£), which do not involve p, are of the order </> l when
<j> is large. If we generalize this expression by FOURIER'S Theorem (see equation (99)),

we obtain, in the case of an impulse Q of short duration,

-S f F (<f>) d<j) \ e-*4p cos p (t — bx) dp + S f / (<£) d<j> [ e-*p sinp(t - bx) dp

7T/X ^0 Jo TTjLt Jo Jo

Q f rr / JL \ ^^ "<P i y I f/sL\ (^ to) Cc-«p -. ^/,\

^^ I H / fi\ \ _ —I— — r I fn I >- ~ . , 11UOJ.

TTfJ,

* The symbols <^>, F, / are here used temporarily in new senses,

E 2

28 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

For any particular phase of the motion, t varies as x, and the expression (10(5)
therefore varies inversely as x. This confirms, so far, our previous results (95) and
(98). Hence with increasing distance from the origin the disturbance tends to the
limiting form represented by (101).

Before leaving this part of the subject, it is to be remarked that the peculiar
protracted character of the minor tremor which we have found to precede and follow
the main shock is to some extent special to the two-dimensional form of the question.
It is connected with the fact, dwelt upon by the author in a recent paper,* that even
in an unlimited medium a solitary cylindrical wave, whether of the irrotational or
equivoluminal kind, is not sharply defined in the rear, as it is in front, but is prolonged
in the form of a "tail." In the three-dimeiisional problems, to which we are about
to proceed, this cause operates in another way. The internal waves are now
spherical instead of cylindrical, and so far there is no reason to expect a protraction
of a disturbance which in its origin was of finite duration. But at the surface they
manifest themselves as annular waves, and accordingly we shall find clear indications
of the peculiarity of two-dimensional propagation to which reference has been made.
On the whole, however, it appears that the epochs of arrival of irrotational and
equivoluminal waves are relatively more clearly marked and isolated than in the two-
dimensional cases.

PART II.

THREE-DIMENSION A L PKOBLEMS.
10. Assuming symmetry about the axis of z, we write

w = ^/(x' + ?/), u = _c v, v — y_q (107),

CT CT

so that q denotes displacement perpendicular to that axis.

A typical solution of the elastic equations, convenient for our purposes, is derived
at once from Art, 3, if we imagine an infinite number of two-dimensional vibration-
types of the kind specified by (25) and (30) to be arranged uniformly in all azimuths
about the axis of -., and take the mean. In this way we obtain from (33), with the
necessary change of notation,

A | V~ cos w d(a = _

,, = (- «A - #B) . -i I e'*"°" da = - («A + ifK) J

* Cited on p. 37 post,

.. (108).

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 29

Also, from (40), for the corresponding stresses at the plane 2 = 0, we have

(109).
J0 = p. {(2e - V) A + 2i

Although the above derivation is sufficient for our purpose, it may be worth while
to give the direct investigation,* starting from the equations

32?.< /, x 3 A „., 32y \ 3 A

' a? = (x + ^ te + ^ p w =(x + ^ a,,

" (HO),

U(/ UfJ

where

O.K Si/ 3;
In the case of simple-harmonic motion («'/'') these are satisfied by

U = •—£ -|- U1 , V = - - -f- i./, W = -» -- + w' . . . , (112)

ox ay oz

provided

(V- + /,-)c/> = 0 (113),

and

(V- + £3) «' = 0, (V- + /,'-) v' — 0, (V- + /••-) ('•' = o j

3 / ? •' ~> <•' L ... (1 14).

+ - + ('1 = "

where /r, P are defined as before by (28). A particular solution of (114) is

provided

(V3 + F)X = 0 (116).

On the hypothesis of symmetry about O: we have

__, 32 1 3 o" / , 1 7\

V- = -, + - - ~- + x -., ....... (l17).

CTS~ -us ons 04

and the formulae (112), (115) are equivalent to

* C). ' Proc. Loud, Math, Soc,,' vol. 34, p. 276, for the corresponding statical investigation.

30 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

If we take, as the typical solution of (113) and (116),

^ = A«-«J0(^r), X = Rr"J0(&r) ..... (119),

where a, ft have the same meanings and are subject to the same convention as in
Art. 3, we have, from (118),

q = (- f Ac- + fiSRT*) J, (frr) "I

12 .
w = (_ aAe-« + f3Be-*) J0 (frr) '

and tlience for the stresses in the plane z = 0

- (2f * - tf) f B} J, (frr) ]

f • ( /•

- F A - 2^/3B} J0 (&r)

J

The formula- differ from (108 i and (109) only in the substitution of igR for B. The
notation of (119) is adopted as the basis of the subsequent calculations.
If we are to assume, in place of (1 19),

...... (122),

the corresponding forms of (120) and (121) would be obtained by affixing accents
to A and B, and changing the signs of a and ft where they occur explicitly.

1 1. As in Art. 4, we begin by applying the preceding formulae to the solution of a
known problem, viz., where a given periodic force acts at a point in an unlimited solid.

Let us suppose, in the first place, that an extraneous force of amount Z . J0 (£CT) e'P',
per unit area, acts parallel to z on an infinitely thin stratum coincident with the
plane z — 0. The formulae (119) will then apply for : > 0, and (122) for z < 0. The
normal stress will be discontinuous, viz. :

[^],=+o-[p4=-o=-Z.J0(^) ...... (123),

whilst p:a. is continuous. Hence

(2f* - **) (A - A') - -zeft (B -f B') = - Z |

P > • - • (124).

2a (A + A') - (2f- - F) (B - B') = 0 J
Also, the continuity of q and CT requires

--(B-B') = o|
We infer

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID.
and therefore, for z > 0,

31

(127).

To pass to the case of a concentrated force Re1'', acting parallel to z at the origin,
we have recourse to the formula (20), where we suppose f(\) to vanish for all but
infinitesimal values of X, and to become infinite for these in such a way that

f"

J
Jo

2ir\d\ = R.

We therefore write Z = Rf d£/2Tr, and integrate with respect to f from 0 to QO.*
We thus find, for z > 0,

- ' °28)'

which are equivalent, by (18), to

R 3 e~*

-- .

' Bz r

Y —

R

.
4:Trp2p ' r

(129)

This will be found to agree with the known solution of the problem.! If we retain
only the terms which are most important at a great distance r, we find, from (1 18),

/7 —
" " "

R f 1

— - J
" "

4 77 [ X + 2u r3

,,.3

R J

IV = . 4:

4irU + 2/irs p.

Inserting the time-factor, the radial displacement is

(130).

ZW

R

and the transverse displacement in the meridian plane is

— zq _ R CT >(,_M

- o c/

Returning to the exact formulae (128), the expression for the velocity parallel to z
at the plane z = 0 is found to be

o (£,)£# ...... (133),

* A more rigorous procedure would be to suppose in the first instance that the force R is uniformly
distributed over a circular area of radius a, using the formula (22). If in the end we make a = 0, we
obtain the results in the text.

t STOKES, 'Camb. Trans.,' vol. 9 (1849); ' Mathematical and Physical Papers,' vol. 2, p. 278.

32 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

or, taking the real part,

47TP/3

+ terms in sin pt

The terms in cos pt remain finite when we put ts = 0 ;* and the mean rate W at
which a force R cos pt. does work in generating waves is thus found to be

W =

Jo

(a3 + 2fc3) . (135),

24ir/o V

a and ?> denoting as before the two elastic wave-slownesses. The result (135) can be
deduced, as a particular case, from formulae given by Lord KELVIN, t

12. Proceeding to the case of a semi-infinite solid occupying (say) the region z> 0,
we begin with the special distribution of surface-stress :

[>«](I = Z. J(1(&0, [>-„] = 0 ....... (136).

The coefficients A, B in (119) are now determined by

Z ]

** ^ ....... (137),

2«A - (2^2 - jfe9) B = 0 J

whence

A_2P-/^ Z B_ 2« Z n .

A~ FffT'7' *X£) 7 ...... (38)>

the function F (^) having the same meaning as in Art. 5. The corresponding
surface-displacements are

. Z

F (€) ?

}> (139).

Fa ,, x Z

'o = -p (£) ' ° (€™' ' —

This result might have been deduced immediately from (51) in the manner indicated
at the beginning of Art. 10.

t The terms in sin pt become infinite. If the force R be distributed over a circular area, the awkwardness
is avoided. A factor

iteOV
ifo J

is thus introduced under the integral signs in the first line of (135), where a denotes (for the moment) the
radius of the circle. Finally, we can make a infinitely small,
t 'Phil. Mag.,' Aug. 1899, pp. 234, 235.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 33

It we put Z = 0 in (137) we get a system of free annular surface- waves, in which

n — __ trl^ir- l-~ O« /9 \ T / .. \

*yO"~ A. y ^iv A/ _ '/. | ^ j . i . ti i i Kco /

. . . (140),

0 = /^ . J0

where K is the positive root of F(f) = 0, and a], & are the corresponding values of
«, y6. These are of the nature of " standing " waves.

To pass to the case of a concentrated vertical pressure Re'>' at 0,* we put in
accordance with (20), Z = — Rfr/£/27r.. and integrate from 0 to oo.f The formulae
(139) become

R r^(2p_F-2«/3) , ., w -]
)0 F /Ji ($»)<#

.... (Hi).
J

Again, the case of an internal source of the type

+ = -*-, X=0 ..-.:.... (142),

where r denotes distance from the point (0, 0, /), can be solved by a process similar
to that of Art. 5. First, superposing an equal source at (0, 0, — /), distance from
which is denoted by r , we have

- Mr g — ihr'

r r' '

and therefore, by (18), in the neighbourhood of the plane z = 0,

(•* „«(:-/)

Jo(6r)

JD a

.........

a
This makes

This may be regarded as the kinetic analogue of BOUSSINESQ'S well-known statical problem.

t It might appear at first sight that a simpler procedure would be possible, and that the effect of a
pressure concentrated at a point might be inferred by superposing lines of pressiire (through 0) uniformly
in all azimuths, and using the results of § 7. It is easily seen, however, that such a distribution of lines
of pressure is equivalent to a pressure-intensity varying inversely as the distance (w) from O. This is
not an adequate representation of a localized pressure, -since it makes the total pressure on a circular
area having its centre at 0 increase indefinitely with the radius of the circle.

VOL. CCIIT. — A. F

34
and

PROFESSOR HORACE LAMB ON THE PROPAGATION OF

. • • (H6).

The additions to (143) whicli are required in order to annul the stresses on the
plane z = 0 are accordingly found by writing

a

in (139), and then integrating with respect to
obtain, finally,

from 0 to oo . In this way we

......

In a similar manner, with the help of Art. 11, we might calculate the effect of a
periodic vertical forqe, acting at an internal point.

13. For the sake of comparison with our previous two-dimensional formulae, it is
convenient to write, from (2) and (6),

7T Jo

TT

cosh

. . . (148).

The formulae (141) are thus equivalent to

R r

<ln = — 0 jT cosl1

/!7r/Ll Jo
*T ) *x

^ _l\J ' 7

'"'n == ~o~ "'(

'27T/U. - l)

These results are closely comparable with (52), and our previous methods of treat-
ment will apply. It is, however, unnecessary to go through all the details of the
work, since the definite integrals with respect to £ which appear in (149) can be
derived from those in (52) by performing the operation — ?'3/3.x upon the latter, and
then replacing x by 57 cosh u.

Thus, from (67) and (70) we derive

~
(s)

= 27r/cH sin (m, cosh u)

. (150),

TREMOltS OVER THE SURFACE OF AN ELASTIC SOLID. 35

f |2fe ^' c°8h " d% - 2™K sil1 (* w cosh w) ~ 2^$ f " Jri* e~*w cos" " d£
J-» Jb (f) J*.b (f)

- 2P I ^^^ ~ ^')^t: a-^acothu Jf

L F £

where H aud K are the numerical quantities defined by (68) and (71). Substituting
in (149) we have

TT / \ i -7

» - - . H . K, (K.) + I)

,

where the notation ol the various BKSSKL'S Functions is as in Art. 2.
Superposing the system of free waves in which

f/0 = ^ . H . J\ (KCT), ?«;„ = — '' - . K . J0 (/era) .... (154),

Z^, ^/A

we obtain, finally, on inserting the time-factor,

* — -H-^M^ +

Since these expressions are made up entirely of diverging waves, they constitute
the complete solution of the problem where a periodic normal force He** is applied to
the surface at the origin.

• An alternative form of (156), which puts in evidence that part of the vertical
disturbance which is most important at a great distance from the origin, is obtained
from (75). Attending only to the "singular" term, we find

r* i'~ffiL

^ F (£\ ei£wcosl1 " d£ = — 2iV/eK . cos (/era cosh u) + &c. . . (157),
and therefore, from (149),

(158).

Adding in the system (154) we have altogether

g0 = - £~ . H . D! (/era) 0* + &c., v>0 = JR . K . D0 (/era) <£* + &c. ( 159).

F 2

36 PROFESSOR HORACE LAMB ON THE PROPAGATION OF

Hence, by (7), we have, at a great distance CT,

*-»-w< ^go).

. --., 0 .

2/A V TTKiy 2p. TTK7S

This may be compared with (77). The vibrations are elliptic, with the same ratio of
horizontal and vertical diameters as in the case of two dimensions ; but the ampli-
tude diminishes with increasing distance according to the usual law CT"* of annular
divergence.

In the same manner we obtain, in the case of an internal source of the type (142),

. . . . (161),

where the factor e~"'-' shows the effect of the depth of the source.

The expressions for the residual disturbance might be derived from the formulae
of Art. 8 by the same artifice. Without attempting to give the complete results,
which would be somewhat complicated, it may be sufficient to ascertain their general
form, and order of magnitude, when ACT and kvs are large. To take, for example, the
parts due to the distortioual waves, if we perform the operation — id/dx on the
second terms of the unnumbered expressions which occur between equations (89)
and (90), above, and then replace x by CT cosh u, the more important part of the
result in each case is

e-ika"x>i*tt/(kvr cosh u)3-2,

multiplied by a constant factor. This result is to be substituted for the definite
integrals with respect to f which occur in (149); the corresponding terms in
i/o and IVQ are therefore of the types

1 r e-ikacosh"du _md 1 rxe
o1 Jo cosh M* £CT' Jo

(cosh M)* (£CT)' o (cosh w)3

respectively. By the method by which the asymptotic expansion (7) of the
function D0 (£) is obtained, it may be shown, again, that these terms are ultimately
comparable with

where the time-factor has been restored. In the same way, the terms in <?0 and i00
which correspond to the expausional waves are ultimately comparable with

The attenuation with increasing distance is much more rapid than in the case of the

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 37

annular liayleigh waves, so that the latter ultimately predominate.* It is also
much more rapid than in the case of elastic waves diverging from a centre in an
unlimited medium, where the amplitude varies inversely as the distance.

14. The generalization of the preceding results, so as to apply to an arbitrary
time-variation of the source, follows much the same course as in Art. 9. The full
interpretation is however more difficult, so far at least as regards the minor tremors.

The main part of the disturbance, in the case of a local vertical pressure applied to
the surface, is obtained by generalizing the formulas (159). These may be written

o0 = H - K-- \*ei»(t-a>eo*">du + &c., w,= -lK n- J fV("-»OMh«>dw + &c. (162).

TT fJL Cm Jo TT fj. Ct Jo

Hence, corresponding to an arbitrary pressure R (t), we have

«/0 = — H— R(£ — cts cosh u) C/M+&C., iv(}=- — LT(£ — CCTCOsh «) c/«-|-&c. (163),

ff/A OwJo 7T/X Ot Jo

where, in analogy with (100),

K (t) = -1- f dp ( R (X) ship (t -X)dk ..... (1G4).

77 JO J -*

The character of the function oft represented by the first definite integral in (163)
has been examined by the author t for various simple forms of R (t), and a similar
treatment applies to the second integral. For example, if we take

it is found, on putting

t — COT = T tan x,§

that for values of ro large compared with r/c, and for moderate values of x,

Til (t - cv, cosh M) (^« = R A/( 2T ) cos O - k) v/(co« x) • ( ' ^)t»

Jo fiT " \CTSj

I ' K (< - rw cosh u) dn = - ^ \/( 2T ) sin (iff - -JX) v/'(cos x) • ( L67)>
Jo 2r v \CBT/

approximately. Substituting in (163), we have, ignoring the residual terms,

'/o = -/«i" (^ ~ tx) cos'
">u = .'/ cos (** — tx) cos' X

* Cf. the footnote on p. 2 ante.

t "On Wave-Propagation in Two Dimensions," ' Proc. Loud. Math. Soc.,' vol. 35, p. 141 (1902).
t cy. Equation (36) of the piper cited. It may Ije noticed that the functions on the right hand of (166)
and (167) are interchanged, with a change of sign, when we reverse the sign of x-
§ The symbol x is no longer required in the sense of equations (115), &c.

PROFESSOIJ HORACE LAMB ON THE PROPAGATION OF

Rt

38
where

The following numerical table is derived from one given on p. 155 of the paper-
referred to : —

2X/7T.

(t-em)lr.

W/-

ioaig.

- -9

-6-314

- -014

- -060

- -8

-3-078

- -078

- -153

- • 7

-1-963

- -199

- -233

- -6

-1-376

- -365

- -265*

- -5

-1-000

- • 549

- -228

- -4

•727

- -719

- -114

- -3

•510

- -838

+ •066

• 2

•325

- - 882*

+ •287

- • 1

•158

- -837

+ -513

0

0

- -707

+ •707

+ '1

+ -158

- -513

+ •837

+ '2

+ -325

- - 287

+ -882*

+ '3

+ -510

- -066

+ -838

+ -4

+ -727

+ -114

+ •719

+ '5

+ 1-000

+ -228

+ -549"

+ '6

+ 1-376

+ • 265*

+ • 365

+ • 7 + 1 • 963

+ -233

+ •199

+ •8 +3-078

+ -153

+ -078

+ •<) +6-314

+ -060

+ •014

* Extremes.

The graphs of (/,, and v^ as functions of t, in the neighbourhood of the critical
epoch era, are shown in fig. 8, which may be compared with fig. 6.t The corresponding
orbit of a surface particle is traced in fig. 9, where the positive direction of z is
down wards ; it may be derived by a homogeneous strain from a portion of the curve

whose polar equation is

,•* = «' cos 1 (6 - ITT).

The amplitude of this part of the disturbance diminishes, with increasing distance
from the source, according to the law TO-"*.

Complete expressions for the disturbance are obtained by generalizing (155) and
(156). They may be written

H £

or o

. ) du.d0 (169),

TTO//, J a ovs . o

I ^V(0)- f f R (« - #" cosh u) du.de ... (170),

Ja ot J()

where U (0) and V (0) are the functions defined and tabulated in Art. 9.

t See the footnote on p. 26 ante.

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID.

39

Fig. 9.

Fig. 8.

The method applied in that Article to obtain a general view of the whole progress
of the vertical displacement at any point might be employed again here, the upper and
lower curves in fig. 4 being combined with auxiliary movable graphs of

Or" 3 r™

— - R (t — #arcosh u) <ln and 6 R (t — 6ts cosh «) du,

CIS Jo ot . a

considered as functions of 6. In the case of a primary impulse of the type (105), both
graphs would have somewhat the form of the lower curve in fig. 8, the functions being
practically (except for a constant factor) of the type

n

v

- sin (477 — |x) cos!"x , wliere x — ^an

_, t — ^ra-

in the more important part of the range. Both graphs, if drawn to the scale of
fig. 4 or 5, would be excessively contracted horizontally when we are concerned with
values of is large compared with T/CO. Owing to the compensation between positive
and negative ordinates in the auxiliary graphs, it is plain that the disturbance
expressed by the ^-integrals in (109) and (170) will be relatively very small except
when t/zs has values 0 for which the gradient of U (0) or V (0) is considerable. As

40

PROFESSOR HORACE I, A Ml? OX THE PROPAGATION OF

regards the horizontal displacement </,„ the minor tremor will consist of a single
to-and-fro oscillation about the epoch anr, followed after an interval by a somewhat
similar oscillation about the epoch bis, with almost complete quiescence between. As
regards the vertical displacement, there will be a to-and-fro oscillation about the epoch
ats, then a period of comparative quiescence, and finally a gradually increasing
negative displacement (with a slight irregularity at the epoch bis) leading up to the
main shock, after which there is a gradually decreasing positive displacement.

The expression for the horizontal displacement </0 may be treated in a different
manner. Transforming (169) we have

q0 = ~ a^fo R (* - CBT cosh u) du - ~- ("' 0U (0). 3 f R (t - OTS cosh u) du.dd

H 3 f*

= — - — - R (t — cis cosh u) du

TT/JL OWJO

+ ,f f" W (0) + U (0)} . TR (t - 0zs cosh u) du.dO . (171).
iro/iwJa Jo

A rough sketch of the graph of 0U'(0) + U(6») is easily made, and the function

0

R (t — Q-ss cosh u) du

is, in such a case as (165), one-signed, but its integral with respect to 0 does not
converge when the lower limit is large and negative. The method therefore fails to

V

Fig. 10,

TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 41

give us a convenient view of the progress of q() as a function of t. The difficulty is
due to the peculiarities of annular propagation to which reference has already been
made.

In fig. 10 an attempt, based on the former method, is made to represent (very
roughly) the whole progress of the horizontal and vertical displacements due to a
single impulse of the type (165) at a distance large compared with T/C.

SUMMARY.

We may now briefly review the principal results of the foregoing investigation, so
far as they may be expected to throw light on the propagation of seismic tremors
over the surface of the earth.

It has been necessary to idealize this problem in various ways in order to render it
amenable to calculation. In the first place, the material is taken to be compact and
homogeneous, to have, in fact, the properties of the " isotropic elastic solid" of
theory. Moreover, the curvature of the surface is neglected. Again, instead of a
disturbance originating at an internal point, we study chiefly the case of an impulse
applied vertically to the surface. Under these conditions the disturbance spreads
over the surface in the form of a symmetrical annular wave-system. The initial form
of this system will depend on the history of the primitive impulse, but if this be of
limited duration, the system gradually develops a characteristic form, marked by
three salient features travelling with the velocities proper to irrotational, equi-
voluminal, and Rayleigh waves, respectively. As the wave-system, thus established,
passes any point of the surface, the Jwrizontal displacement shows first of all a single
well-marked oscillation followed by a period of comparative quiescence, and then
another oscillation corresponding to the epoch of arrival of equivoluminal waves.
The whole of this stage constitutes what we have called the "minor tremor" ; it is,
of course, more and more protracted the greater the distance from the source, and
its amplitude continually diminishes, not only absolutely but also relatively to that
of the " main shock," which we identify with the arrival of the Rayleigh wave. It may
be remarked that the history of the minor tremor depends chiefly on the time-
integral of the primitive impulse ; the main shock, on the other hand, follows the
time-scale of the primitive impulse, and is affected by every feature of the latter.*

Similar statements apply to the vertical displacement, except that the minor
tremor leads up more gradually to the main shock, being interrupted, however, by a
sort of jerk at the epoch of arrival of equivoluminal waves.

The history of the horizontal and vertical displacements, about the epoch of the
main shock, in the case of a typical impulse of the type (165), is shown in fig. 8 ;

* Observational evidence in favour of the existence of the three critical epochs in an earthquake
disturbance has been collected and discussed by R. D. OLDHAM, "On the Propagation of Earthquake
Motion to Great Distances," 'Phil. Trans.,' A, 1900, vol. 194, p. 135.

VOL. CCIII. — A. G

42 PROFESSOE HOKACE LAMB ON THE PROPAGATION OF TKEMOBS, ETC.

whilst fig. 9 shows the corresponding orbit of a surface-particle. In fig. 10 a sketch
is attempted of the whole progress of the disturbance.

These results are of a fairly definite character, but they are based, as has been said,
on purely ideal assumptions, and it remains to inquire how far they are likely to be
modified by the actual conditions of the earth. The substitution of an internal source
for a surface impulse will clearly not affect the general character of the results at a
distance great compared with the depth of the source, although differences of detail
in the wave-profile at the critical epochs will occur, and we can no longer assume that
the disturbance is the same in all vertical planes through the source. Again, the
chief qualitative difference introduced by the curvature, of the earth will be that
the minor tremor, whose main features are evidently associated with the outcrop of
spherical elastic waves at the surface, will be propagated directly through the earth,
so that the first two epochs Avill (at distances comparable with the radius) be
accelerated relatively to the main shock,* which being due to the Tlayleigh waves
will travel, with the velocity proper to these, over the surface.^

It is a more difficult matter to estimate the nature and extent of the modifications
produced by heterogeneity. It is, perhaps, possible to exaggerate these, for the
qualitative effect of a gradual charge of elastic properties would not be serious, and
even considerable discontinuities would have little influence if their scale were small
compared with the wave-length;}] of the primitive impulse. A covering of loose
material over the solid rock probably causes only local, though highly irregular,
modifications, with some dissipation of energy.

It must be acknowledged that our theoretical curves differ widely in two respects
from the records of seismographs. In the first place, they show nothing corresponding
to the long successions of to-and-fro vibrations which are characteristic of the latter.
It would appear that such indications, so far as they are real and not instrumental,
are to be ascribed to a succession of primitive shocks, in itself probable enough.
Again, the theory gives vertical and horizontal movements of the same order of
magnitude, and in the case of the Rayleigh waves, at all events, where a definite
comparison can be made, the vertical amplitude is distinctly the greater. The
observations, on the other hand, make out the vertical motion to be relatively small.
The difficulty must occur on almost any conceivable theory, and appears indeed to be
clearly recognised by seismologists, who are accordingly themselves disposed to
question the competence of their instruments in this respect,

* Cf. E. D. OLDHAM, he. cit.

t The theory of free Rayleigh waves on a spherical surface is known ; see Professor BROMWICH, loc. cit.

| This term is used in the same general sense in which in hydrodynamics we speak of the " length " of
a solitary wave travelling along a canal. There is no question, in the present connection, of anything
analogous to " oscillatory waves."

INDEX SLIP.

lectrum of Gold.

>e University

MALLET, J. W. — On the Structure of OJold-Leaf, and the Absorption
Spectrum of Gold. Phil. Trans., A, vol. 203, 1904, pp. 43-51.

Colloidal Gold, Absorption Spectrum of.

MALIET, J. W. Phil. Trans., A, vol. 203, 1904, pp. 43-51.

Gold-Leaf, Microscopic Structure of ; Mechanism of Gold-beating.

MALLBT, J. W. Phil. Trans., A, vol. 203, 1904, pp. 43-51.

us seen niri< . ; i... mi,
•' which .sij»-r-,.
I. The coins

>le proportif

4i-biue. Tin- «u

'ickness of the •_

well knowj i to i

.-. in addition, nu

ing irregularly

•lisiii, but I-') (1;,

' illustrates this

with MII avt<

M plate of ;r

VDAY'S liak-

•finienUt

Fo

(360.)

G 2

resents a remarkable
all or only slightly
en, unless silver in
n¥ case the colour is
>q>ected, not uniform,
of the surface. All
•f the leaf.

ulerate amplification,
;• some tendency to
most irregular way.
iry commercial gold-
rom the eye-piece to

311 February 5, 1857,
Light,"! there occur
e did not escape his
irts of the leaf were
he leaf appearing as
ross both the thicker
.o specimens of gold-
t is the thicker folds
est." And again he
irregular corrugated
on in one direction

.ted to the Eoyal Society,

29.1.04

42 PROFESSOR HOR

whilst fig. 9 shows the

is attempted of the wh

These results are of ;

on purely ideal assump

modified by the actual

for a surface impulse \v

distance great compare

in the wave-profile at '

the disturbance is the

chief qualitative cliffe

the minor tremor, who

spherical elastic wave?

so that the first twc

accelerated relatively

will travel, with the v

It is a more difficult

produced by heterogt

qualitative effect of ;

even considerable disc

compared with the

material over the sol

modifications, with so

It must be acknowl

from the records of se

to the long succession

It would appear tha

are to be ascribed t

Again, the theory g

magnitude, and in tl

comparison can be i

observations, on the <

The difficulty must o

clearly recognised 1

question the compete

* Of. R. D. OLDHAM, l<

t The theory of free R

| This term is used in

a solitary wave travelling

analogous to " oscillatory

A ... -,.„,,*,

i

.Ifl-Rt .qq ,i

6 •miJjifitfr

.lo«»iT
VI ,80fi .Io* ,A ..arunT .Ii/14 .'// I. mult

' /; ; I , •

€1 ,ROS,.!oT fA. f.HanT .liil^ .

^t^ir

II. On the Structure of Gold- Leaf, and the Absorption Spectrum of Gold.

By J. W. MALLET, F.R.S., Professor of Chemistry in the University

of Virginia.

Received May 22,— Read June 11, 1003.

[PLATE 1.]

GoLD-leaf, as seen under the microscope by transmitted light, presents a remarkable
appearance which seems to have been hitherto either not at all or only slightly
noticed. The colour of the transmitted light is bluish-green, unless silver in
considerable proportion be alloyed with the gold; in this latter case the colour is
purplish-blue. The amount of light transmitted is, as might he expected, not uniform,
the thickness of the gold film varying within very small areas of the surface. All
this is well known to anyone who has ever looked through a bit of the leaf.

But, in addition, numerous black lines are visible under verv moderate amplification,
ramifying irregularly over the surface, here and there showing some tendency to
parallelism, but for the most part running into each other in the most irregular way.
Fig. 1* illustrates this ; it is a microscopic photograph of ordinary commercial gold-
leaf, taken with an amplification of 75 diameters, and a distance from the eye-piece to
the camera plate of 378 millims.

In FARADAY'S Bakerian lecture, read before the Royal Society on February 5, 1857,
on the " Experimental Relations of Gold (and other metals) to Light," t there occur
two or three sentences which prove that this peciiliar appearance did not escape his
keen observation. For example, he says " when the thicker parts of the leaf were
examined they seemed to be accumulated plications of the gold, the leaf appearing as
a most irregular and crumpled object, with dark veins running across both the thicker
and thinner parts, and from one to the other." Again, referring to specimens of gold-
leaf which had been heated in oil, he says " it will be seen that it is the thicker folds
and parts of the mottled mass that retain the original state longest." And again he
remarks, " there is a little difficulty in admitting that such an irregular corrugated
film as gold-leaf appears to be, can possess any general compression in one direction

* All of the microscopic photographs referred to in this paper have been presented to the Royal Society,
but only Nos. 4, 6, and 8 have been reproduced for publication,
t 'Phil. Trans.,' 1857, pp. 145-181.
(360.) G 2 29.1.04

44 PKOFESSOE J, W. MALLET ON THE STEUCTUEE OF GOLD-LEAF,

only." But FARADAY does not seem to have specially investigated the peculiarity in
question, or its cause, and, in view of the process by which gold is extended into
these thin films, the terms " plications " and " folds " which he uses must be understood
as referring to the appearance only of the leaf and not to its actual structure.

The idea first suggested by the ramification and reticulation of black lines was that
they might depend in some way on the crystalline structure of the alloyed gold used
for making commercial gold-leaf, modified and distorted during the process of beating.
Hence specimens of gold-leaf variously alloyed were compared with each other. The
following samples were furnished me by the manufacturers, the W. H. Kemp Company,
of 165, Spring Street, New York, with a statement of their composition :—

A. Dark or red gold-leaf, made with an addition of 18 grains of copper to each

Troy ounce (480 grains) of pure gold, or, more strictly, gold assaying about
998-999 fine.

B. Gold-leaf of medium colour, made with an addition of 12 grains of copper and

1 2 grains of silver to each Troy ounce of fine gold.

c. Pale or light-coloured gold-leaf, made with an addition of G pennyweights
(144 grains) of silver to the Troy ounce of fine gold.

Figs. 1, 2, and :] show the appearance of these three samples respectively under
the amplification already mentioned for No. 1, which represents the gold alloyed with
copper only, No. 2 that containing both copper and silver, and No. 3 that containing
silver only. The three exhibit some differences, but not much greater than are
presented by different samples of leaf of the same composition, and the general
character is evidently the same. In consequence of the small amount of light
transmitted by the leaf, exposures of the photographic plates for two or three minutes
were necessary, and changes in the state of the sky and character of the light during
this time prevent the photographs giving quite a correct idea ot the different degrees
of translucency of the specimens. Owing probably to slight shaking of the floor
affecting the position of the camera, the ramified lines do not appear quite as sharp
and Avell defined as when viewed directly through the eye-piece of the microscope.

It was desirable to see whether the same appearance, if referable in any way to the
original molecular structure of the metal, would present itself in leaf beaten from
pure gold free from all alloy. On applying to two firms of gold-beaters — one in
New York and the other in Philadelphia — to make for me a small quantity of leaf
from fine gold, I was assured by both that it was impossible to beat pure gold thin
enough to be seen through. Dentists' gold foil could be had, but it is quite opaque.
The reasons assigned for the difficulty were the excessive tendency of the pure gold
to cohere, so that it could not be manipulated without different parts touching each
other and sticking together, and also the tendency of the pure metal to stick to the
" gold-beaters' skin " or animal membrane used to separate the leaves in beating.

AND THE ABSORPTION SPECTRUM OF GOLD. 45

After a good deal of persuasion I succeeded in inducing the manager of the
W. H. Kemp Company — Mr. W. II. HAN'NA — to try the experiment of beating into
leaf, as thin as could be had, a sample of fine gold which I sent him. This was
" proof gold" from the assay department of the United States Mint at Philadelphia,
and therefore of the highest attainable purity. The result was quite satisfactory for
the intended purpose, though it would not have been so in a commercial sense, there
being a good deal of waste, and many torn leaves and large holes. The microscopic
appearance of this pure gold-leaf is shown in fig. 4 (Plate 1). It is in general like
the commercial specimens, but the lines are bolder and more strongly defined — a
consequence, as I think will be shown, of the greater softness of the pure metal.

Study of these microscopic appearances, and comparison of them with each other
and with the micro-photographs of OSMOND, RoBEETS-AustEN, ARNOLD, ANDREWS
and others, did not seem to support the idea that the lines in question are due to
more or less distorted crystalline structure. In order to learn whether the lines are
to be referred to, and originate in, the process of gold-beating by which the leaves
have been produced, the attempt was made to obtain galvanically-deposited films of
something like the same thickness, so that these latter might be microscopically
examined by transmitted light.

Pieces of thin rolled silver foil, much larger than would be needed for microscopical
examination only, were varnished on one side and then electrolytically coated with

«/ ' ti \J

fine gold on the other, using a specially prepared pure cyanide solution and an anode
of fine gold. As there was no guide by which to determine in advance the thickness
of the gold film which would admit of being satisfactorily seen through, the current
was passed for various periods of time, producing films of several different thicknesses,
and, after the subsequent treatment, one or two were selected which gave the best
results. About a square centimetre cut from each piece of foil was well washed with
ether to free it from varnish, and was then cemented — the gilded face downwards —
upon a slide of .thin microscope cover glass by means of Canada balsam somewhat
diluted with ether. After time had been afforded for the balsam to harden, the silver
was dissolved oft' slowly by very dilute nitric acid, and the gold film was ready for
microscopic examination. Fig. 5 shows the appearance presented, the amplification
and distance from eye-piece to camera plate being the same as for fig. 1 and for all
the other microscopic illustrations of this paper. It is evident that the mottled
structure of this film, showing varying thickness, is unaccompanied by the ramifica-
tions of well defined black lines to be seen in beaten gold-leaf. No attention should
be given to the two large bars of shadow crossing each other at right angles in this
photograph ; they are due to the shadow of a part of the window sash having been
inadvertently allowed to fall on the illuminating mirror of the microscope.

To test whether the black lines are really due to minute threads or wires of gold
with diameters considerably greater than the thickness of those parts of the leaf
which can be seen through, it was proposed to protect a piece of gold-leaf by placing

4(5 PROFESSOR J. W. MALLET ON THE STRUCTURE OF GOLD-LEAF,

it between two sheets of silver foil, roll the whole down to a fraction of the original
thickness, remove the silver by means of nitric acid, and see whether the lines in the
gold had been broadened out by flattening of the wire-like threads if present. A
rectangular piece of fine silver foil, '019 millim. thick, was folded in two across the
middle of its length, a piece of the " fine" gold-leaf which had been specially beaten
for me by the W. H. Kemp Company was spread out flat between the two folds of
silver, and then by the same firm the whole rolled down until the double thickness,
•088 millim., had been reduced to 'OOG millim. Care was taken to introduce the
folded edge first between the rolls, so as to prevent as far as possible slipping of one
surface of foil upon the other. Examination with nitric acid of different parts of the
rolled-down foil showed that, although there had been no small tearing of the gold
and many holes had been produced in it, there were quite sufficient areas of it left in
a practically continuous state. Assuming that the gold had been rolled out pari
passu with the silver, each had been reduced to something like one-sixth or one-
seventh of the original thickness.

A small piece of the foil in this condition was varnished on one side, and the other
side stripped of silver by very dilute nitric acid. A number of specimens were spoiled
at this stage, since the acid getting through any holes would attack the silver on the
other side and eat its way between the varnish and the gold film, which was so
exceedingly thin as not to bear any manipulation when unsupported. A few good
specimens, however, were secured. These were cleared of varnish by soaking in
ether, cemented by the gold face with diluted Canada balsam to slips of thin
microscope cover glass, and, after hardening of the balsam, the second film of silver
was gradually removed by very dilute nitric acid. Fig. 0 (Plate 1), representing,
under the same amplification as in the other figures, the microscopic appearance of
one of these specimens of rolled-down pure gold-leaf, exhibits very distinctly the
flattening out of the minute metallic threads, favoured by the greater softness of the
gold than of the silver which enclosed it.

As a further test of the black lines being due to minute wires or threads of gold,
specimens of the fine gold-leaf were thinned down by partial solution, in order to see
whether the lines would remain visible longer than the general surface of the leaf,
and the thicker lines longer than the more delicate. The solvent used was a |- per
cent, aqueous solution of potassium cyanide, to which had been added a little
hydrogen dioxide. The result is shown in fig. 7, and in fig. 8 (Plate 1), the former
of these representing a less, and the latter a more, advanced stage of the solvent
attack upon the leaf. The more gradual obliteration of the black lines than of the
rest of the surface is quite apparent.

As it seemed to be established that the black lines under examination represent
microscopic threads or wires, and that these are developed in the gold during the
process of beating, it was natural to look for their possible origin in some correspond-
ing peculiarity of structure in the " gold-beaters' skin" or animal membrane between

AND THE ABSORPTION SPECTRUM OF GOLD. 47

sheets of which the leaves of gold are extended. But this idea is not borne out by
microscopic study of that material. The thin gold foil with which the process is
begun is first beaten for about twenty minutes only between surfaces of " cutch "
paper, which has simply the structure of a felted mass of vegetable fibres. The
principal extension of the gold is brought about by beating for about four hours in a
" shoder" or packet of leaves of old or previously often used gold-beaters' skin, the
packet, containing a thousand leaves, being from time to time bent between the
fingers to loosen the gold films and prevent their sticking to the membrane, and
finally by beating for another four hours in a " mould" or similarly made up packet
of leaves of new or much less used gold-beaters' skin, repeating the bending of the
packet to maintain the looseness of the gold films. The cutch is beaten with
hammers of about sixteen pounds in weight, striking about sixty blows a minute, the
shoder with hammers of about ten pounds and at the rate of about seventy-five blows
per minute, and the mould with six-pound hammers and at the rate of about ninety
blows per minute. Figs. 9, 10, and 1 1 represent respectively the cutch paper, the
already much used gold-beaters' skin of the shoder, and the new, or nearly new, skin
of the mould. There is nothing in any of these to account for the black lines seen in
the gold-leaf. As far as any distant resemblance to these is suggested by some of
the vegetable fibres in fig. (J, it is to be remembered that fibres in relief would
produce in the gold corresponding furrows, appearing as lines of greater, not less,
translucency than that of the rest of the surface. The animal membrane or gold-
beaters' skin in which by far the greater part of the beating is done, including all the
later part of the work, exhibits in figs. 10 and 11 the simple and nearly uniform
structure of the serous coat of the intestine— said to be the c;t-cum — of the ox which
is used for the purpose.

A careful personal inspection of the process of gold-beating at the establishment of
the W. H. Kemp Company in New York, has led me to the belief that the production
of the ramified lines of microscopic wires or threads in the gold-leaf is due to the
following cause. The face of the hammer used is slightly convex, and hence a blow
struck with it tends to stretch each sheet of gold, and the animal membrane enclosing
it, outwards in all directions from the centre of impact. The membrane is elastic and
not absolutely uniform in thickness or tensile strength. Hence it tends to form,
along lines of weakness, wrinkles running irregularly outwards, such as may be
produced in any stretched piece of cloth by a push of the finger in any given
direction. These wrinkles constitute microscopic troughs or furrows into which
the soft gold is driven, forming corresponding rods or wires of extremely minute
size. The elasticity of the membrane leads to the momentarily developed wrinkles
being almost instantly obliterated, while the plasticity of the gold admits of no
corresponding disappearance of the wire-like threads produced. The complicated
ramification of the lines is no doubt due in part to the irregular distribution of lines
of weakness, and therefore of easy stretching, in the membrane, partly to the blows

48 PROFESSOR J. W. MALLET ON THE STRUCTURE OF GOLD-LEAF,

of the hammer falling in rapid succession upon different adjacent parts of the surface,
and partly to the lack of uniformity of support given by the other leaves above and
below in the packet. The view now stated receives confirmation from a point
strongly insisted upon by Mr. HANNA — the very intelligent superintendent of the
W. H. Kemp Company's workshops — namely, that for success in the gold-beating
process much depends on the condition of the animal membrane as to moisture or
dryness. If it be very dry the gold-leaf cracks or breaks, while if the membrane be
too moist the leaf sticks to it. The membrane requires to be dried or dampened to
correct the opposite effects of change in the atmosphere. This accords with the idea
that a certain amount of elastic stretching of the membrane, from which this
recovers, is necessary for the permanent or inelastic extension of the gold. In fact,
as the area of the gold-leaf is permanently extended by the beating, while that of
the membrane is not, the one film manifestly must slide over the other. It is
scarcely conceivable that this sliding shall occur at the moment at which a blow
falls, when friction between the surfaces is at a maximum. If not, it must occur
just afterwards, as a result of the elastic resilience of the membrane, Avhich leaves
behind it the plastic gold.

It is evident that the statements to be found in the books as to the actual thickness
of gold-leaf — based as they are upon weighing of measured areas— represent only
averfif/e thickness, and that, in view of the decidedly greater thickness of these
microscopic threads of gold running through the mass than of the intervening parts,
the thickness of these latter parts must be notably less than the average. The
following determinations were carefully made with several square decimetres of leaf
in each case, accurately measured as to area, and weighed on a delicate assay balance.
The results are stated in " microns" (thousandths of a millimetre).

Average thickness.

Commercial gold-leaf, alloyed with copper only, represented by Fig. 1

•0797/1

,, „ .. copper and silver ., ,, 2

•0822/1

,, „ silver only ,, .,3

•0937/1

Gold-leaf specially beaten from " fine" gold „ ,, 4

•1082/1

A galvanically deposited film of " fine " gold ,, ,,5

•1203/1

Maximum thickness of " fine " gold film which can be seen

through — about .

•2000u

Dentists' " fine " gold foil '9228/1

In connection -with the microscopic examination of gold films by transmitted light,
it seemed to be interesting to make some observations on the absorption spectrum of
the metal, especially as there have been recently published the results of spectroscopic
study of the light which the metal reflects.

It was proposed to examine for this purpose metallic gold in the following forms : —
1. Pure or " fine" gold-leaf.

AND THE ABSORPTION SPECTRUM OF GOLD. 49

2. Gold chemically reduced in a dilute aqueous solution of its chloride — so-called

colloidal gold — the metal being in sufficient quantity and state of aggrega-
tion to transmit greenish-blue light.

3. Ditto, a less amount of more finely distributed gold transmitting ruby-red

light.

4. Glass coloured by gold so as to transmit greenish-blue light — the so-called

saphirine glass.

5. Glass coloured ruby-red by very finely divided gold.

It was found to be impracticable to secure any result for the gold-leaf, on account
of the very small amount of light transmitted. For the colloidal gold in water, and
the gold-coloured glass, the following results were obtained.

Visible Spectrum.

The source of light was a strong electric arc between closely placed carbon poles,
with a slit of about ^ millim. in width. Dispersion was obtained by a Rowland
concave grating of 21'5 feet focal length, railed with about 15,000 lines to the inch,
using the spectrum of the first order. The photographs were taken on M. A. Seed
dry plates (" orthochrornatic, L"), specially sensitized for the region from green to
red inclusive. The original photographs were laid side by side, so that the positions
of like wave-lengths were the same for all, and then re -photographed together on a
reduced scale. The results are shown in tig. 12, with a few positions indicated in
Angstrom units.

Taking the strips in order from the top downwards, the first (uppermost) strip
represents the light transmitted simply through a sheet, about 2 rnillims. thick, of
colourless glass of the same kind as that on which the gold ruby-red is " flashed," and
which also formed the end plates of cells containing the colloidal gold in aqueous
suspension — time of exposure about 2 minutes — the darkness at the less refrangible
end is due, not to absorption, but to the insensitiveness of the photographic film for
rays in this region. The second strip shows the effect of transmission through a
column of water, 2 '2 5 centims. long, containing 75 milligs. of metallic gold to the
litre, reduced from the chloride by potassium acid carbonate and formic aldehyde,
and exhibiting dark greenish-blue colour — time of exposure 30 minutes. The third
represents also colloidal gold in watery suspension, but in a column of 9 '2 5 centims.
long, with 50 milligs. of gold per litre, and showing a blue or slightly violet-blue
colour — time of exposure 20 minutes. The fourth represents the same, in a column
of same length as the last, but with only 20 milligs. of gold per litre, and showing a
clear ruby-red colour — time of exposure 10 minutes. The fifth, shifted over to the
right to secure correspondence of position for equal wave-lengths, is the same as the
first, but with shorter exposure; the right-hand end is in the region of slight

VOL. ccm. — A. H

50 PROFESSOR J. W. MALLET ON THE STRUCTURE OF GOLD-LEAF,

sensitiveness of the film. The sixth (lowermost) strip shows the result of trans-
mission through the "flashed" ruby-red glass, with very long exposure— 1 hour and
10 minutes.

In these photographs there is no indication of well defined absorption bands. The
general absorption belongs mainly to the middle portion of the spectrum, and is, on
the whole, more marked at the less refrangible end, with notable increase of
absorption in this region as the amount of gold present is increased. The position of
maximum absorption is nearer to the long-wave end for the glass than for the
colloidal gold in water. It is interesting to note that, while no photographed results
could be obtained from the saphirine glass, the absorption being too far in the red for
the sensitiveness of the film, eye observation of this glass, using sunlight and a glass-
prism spectroscope, showed a distinct belt of absorption extending from about 5700
to 6250, beside the general absorption of rays of shorter wave-length. Allowance
has to be made in the photographs for insensitiveness of the film at the red end of
the spectrum.

Ultra-violet Spectrum.

Tins was examined with a quartz prism, and for the liquids a tube closed at the
ends by plates of quartz. The source of light was electric sparks between cadmium
poles placed pretty near each other. The results are shown in figs. 13, 14 and 15, a
few of the positions being indicated by the wave-lengths of the cadmium lines, as
before. No results could be obtained for the saphirine or the ruby glass, the glass
alone absorbing all rays in the ultra violet. Fig. 13 represents the water with
colloidal gold in suspension, 75 milligs. to the litre, in a column of 2 '25 centims. long.
Fig. 14 represents a like liquid, with 50 milligs. per litre, and in a column 9'25
centims. long. Fig. 15 is the same, with 20 milligs. per litre, and in a column
also (J'25 centims. long.

In each of these three figures the three uppermost strips represent exposures for
3, 5 and 10 minute* respectively (counting from above downwards), the light passing
through the colloidal gold liquid, while the four lower strips exhibit the results from
sparks through air (no gold liquid interposed) for 1, 2, 5 and 10 seconds respectively.
The general absorption, without indication of dark bands, begins to be well marked
at about 3500, and increases toward the more refrangible portion of the spectrum, the
effect increasing also with the amount of gold present.

Infra-red Spectrum.

This was examined, by the obliging permission of Professor S. P. LANGLEY,
Secretary of the Smithsonian Institution, Washington, D.C., in the astrophysical
laboratory of that institution, using sunlight, a rock-salt prism, and Professor

AND THE ABSORPTION SPECTRUM OF GOLD. 51

LANGLEY'S bolometer with photographic auto-registration of the results. These
results were in general as follows : —

The specimen of ruby-red gold glass almost totally absorbs the light of the short-
wave-length side of D, rapidly increases to the full transparency of the ordinary
colourless (unflashed) glass at about A, and continues as transparent as this ordinary
glass to about 2' 5 p. The saphirine or blue glass coloured by gold, cuts off the light
to near C, then rises very rapidly to great transparency at and beyond A.

The red colloidal gold liquid No. 1, 20 milligs. metallic gold per litre, contained in
a cell with end plates of thin microscope glass, 4 "5 centims. apart, produces great
general absorption in the visible spectrum, though not reaching to the point of
completely, or almost completely, extinguishing any rays included within the region
of spectrum studied, as was the case with the glass specimens. The absorption of
this liquid becomes practically identical with that of distilled water at and beyond A.

The violet-blue colloidal gold liquid No. 2, 50 milligs. gold per litre, and the
greenish-blue liquid No. 3, 75 milligs. gold per litre, behave on the whole like liquid
No. 1, except that they diminish the radiations throughout the spectrum to a very
great extent, as if by the interposition of opaque obstacles to the rays. Liquid No. 3
appears relatively less transparent in the visible spectrum, besides being generally
less transparent throughout the spectrum.

I regret that the blue-print tracings of the bolometer curves are so faint as not to
allow of photographic reproduction on a reduced scale.

For the microscopic and spectroscopic photographs I have to thank the kind
assistance of Professor A. H. TUTTLE and Dr. W. J. HUMPHREYS of this University.

H 2

J. W. Mallet.

Phil. Trans., A, vol. 203, Plate 1.

INDEX SLIP.

neojy n

solution. — XII. On a
h special Reference to

PEAKSON, Karl. — Mathematical Contributions to the Theory of Evolution.
— XII. On a Generalised Theory of Alternative Inheritance, with
Special Reference to Mendel's Laws.

Phil. Trans., A, vol. 203, 1904, pp. 53-86.

Correlation, Parental, Ancestral, Fraternal, and Midparental on Mendeliuu
Theory.

PEARSON, Karl. Phil. Trans., A, TO!. 208, 1904, pp. 53-86.

Inheritance, Mendel's Principles lead to Ancestral Law.

PEARSON, Karl. Phil. Trans., A, vol. 203, 1904 pp. 53-86.

Mendel, Relation of his Laws to Biometric Theory of Inheritance.

PEARSON, Karl. Phil. Trans., A, vol. 203, 1904, pp. 53-86.

Regression, flows from General Theory of Pure Gamete.

PEARSON, Karl. Phil. Trans., A, vol. 203, 1904, pp. 53-86.

theory a jn-niTafiv

theory of lien-

•/

d<v.relope< I —

the light of rr

country and .V

termed " Men

principles prn|

gamete formul

uula is the

at fill wit) i

is quiff

30.3.

(361.)

nhentancc, with special

iider rather at length a

not venture to call this

'cause it is not even the

3ause MENDEL'S original

O

d is not the theory here
have been discarded in
Mendelian, both in this
in replaced by what are
Cation the fundamental
i individual case a pure
ibing the facts, t This
as it fits well, badly, or

s is going on to term
he present investigation
: we speak of it merely
based on the conception
groups, while they may
dy fuse and lose their

had already worked at some

vvesen,' Jahrg. IV. (" Ueber
esellsch.,' vol. xviii. (1900),
pril and May, 1903.

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TTI. Mathematical Contributions to the Theory of Evolution. — XII. On a
Generalised Theory of Alternative Inheritance, ivith special Reference to
MENDEL'S Laws.

By KAIIL PEARSON, F.R.S.

Received September 11, — Read November 26, 1903.

(1.) Introductory. On a Generalised Theory of Alternative Inheritance, with special

Reference, to MENDEL'S Laws*

IT seems likely to be of interest at the present time to consider rather at length a
fairly full mathematical theory of the pure gamete. We do not venture to call this
theory a generalised Mendelian theory of inheritance, partly because it is not even the
most general theory of the pure gamete conceivable, partly because MENDEL'S original
theory of heredity was perfectly clear and perfectly simple, and is not the theory here
developed. The pure and simple Mendelian theory seems to have been discarded in
the light of recent experimental results by more than one Mendelian, both in this
country and abroad. The original Mendelian theory has been replaced by what are
termed " Mendelian Principles." In this aspect of investigation the fundamental
principles propounded by MENDEL are given up, and for each individual case a pure
gamete formula of one kind or another is suggested as describing the facts, t This
formula is then emphasised, modified or discarded, according as it fits well, badly, or
not at all with the growing mass of experimental data.

It is quite clear that it is impossible while this process is going on to term
anything whatever Mendelian as far as theory is concerned. The present investigation
is therefore not a generalised Mendelian theory of heredity : we speak of it merely
as a generalised theory of alternative inheritance, and it is based on the conception
that the gamete remains pure, and that the gametes of two groups, while they may
link up to form a complete zygote, do not thereby absolutely fuse and lose their

* I owe the incentive to this memoir to Professor W. F. R. WELDON, who had already worked at some
of the simpler special cases and who placed his results entirely at my disposal.

t See especially TSCHERMAK, ' Zeitschrift f. d. landwirthsch. Versuchswesen,' Jahrg. IV. (" Ueber
Ziichtung neuer Getreiderassen ") ; DE VEIES, ' Ber. d. deutsch. botan. Gesellsch.,' vol. xviii. (1900),
pp. 435-443 ; BATESON, ' Proc. Camb. Phil. Soc.,' vol. 12, p. 53 ; ' Nature,' April and May, 1903.

(361.) 4.3.04

54 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

identity. The analytical expression of this is represented by the fundamental
formula :

(AA') x (««') } = {

' Aa'

,, OO.

A'a

j L AV

where (AA') and (««') are the parental zygotes, and the right-hand side of the
equation represents the four possible constitutions of the offspring. Such a formula
as the above may be accepted without any hypothesis as to dominant and recessive
characters, but these terms were certainly essential to Mendelian theory as propounded
by MENDEL himself, and it becomes very doubtful whether we ought to attach his
name to any theory which discards these " recognition marks." It is very convenient,
however, to have names for the alternative elements expressed by capital and small
letters respectively. I propose for the purpose of this paper to term an A-element a
protogene, and an a-element an allogene. Two protogenic elements will give rise to a
protogenic zygote AA, two allogenic elements to an allogenic zygote aa, and a
protogenic and allogenic element to what Mr. BATESON has termed a heterozygote
Aa. We may thus class his homozygotes into protozygotes and allozygotes. We
reach pure Mendelianism by making our protozygotes " dominants," our allozygotes,
" recessives," and our heterozygotes " hybrids of dominant character." In so far as
our theory of pure gametes replaces protozygote, allozygote, and heterozygote by
" dominant," " recessive," and " hybrid with dominant character," it becomes a
generalised Mendelian theory, but only in this case. Otherwise we must look upon
it as an attempt — in one direction only of course — to give a consistent mathematical
basis to the various formula? which have been propounded for describing statistical
data classed under Mendelian categories ; shortly we shall endeavour to develop a
general pure gamete theory.

The results were worked out in a purely impartial frame of mind ; indeed, once
state the hypotheses, and the analysis is far too complex to allow us to predict rf
priori what can possibly result from it, nor does the investigation admit of any but
one solution. If the hypotheses are admissible, then any narrower pure gamete
formula must lead to results embraced under our general conclusions.

What we have to admit at the present time are the following conditions : —

(i.) The existence of a vast bulk of evidence that heredity, as far as measurable
characters are concerned, follows within a population perfectly definite laws.

(ii.) The existence of another mass of experiments, in which simple and pure
Mendelianism is certainly inadmissible, but in which certain ratios undoubtedly
approach the values they would have on such a simple and pure Mendelian theory.

It is possible, therefore, that a generalised theory of the pure gamete would account

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 55

for (ii.) ; it can only do so satisfactorily, however, if it does not contradict the results
of (i.). Hence arises the present attempt to develop in one direction a generalised
theory of the action of the germ-cell.

As we have frequently had to assert, the laws investigated under (i.) have nothing
whatever to do with any physiological hypothesis. That a physiological hypothesis
leads to them is not much test of its validity — it is a necessary, but riot sufficient,
criterion of its correctness. If, however, it contradicts them, we are bound to
discard it, and seek for its modification or replacement. The present study is an
attempt to see how far one generalised pure gamete theory leads to results in
accordance with the law of regression and the known nature of the distributions of
offspring in populations.

('2.) Nature of Hypothesis adopted.

'We start with a zygote consisting not of a single protogenic pair AA, but built
up of n such pairs,

AjAj -|- A.iA.i -|- A3A3 -f- . . . -j- A,,A,,.

We suppose this to produce gametes which unite with those of a similar allogenic
zygote

%«! + 03a2 + a3«3 + . . . + «„«„.

Any element of the protogenic gamete must unite with the corresponding element
of the allogenic gamete, i.e., A,, with ar, and by the fundamental principle (i.) above,
this gives rise to the four possibilities

A,.ar ,

A,a, ,
arA,. ,

which are all of the same constitution. The result is the hybrid group, symbolised by

a^ + «j,A3 + «3A3 -(-...+ «,,A«,

the perfect multiple heterozygote.

The population will now be supposed to consist of any number of such perfect
heterozygotes, which we shall suppose to again cross. We shall now have

arA,. X a,Ar = r arar
arA.r

lA,Ar,

56 PKOFESSOK K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

or, each couplet will give rise to four possibilities, representing, however, only three
constitutional differences, expressed by

arar + 2arA.r + ArA,.

Since these four possibilities may occur with each of the n couplets, we shall have
when two perfect heterozygotes cross, 4" resulting possibilities. These form the
resulting population of the second generation. Our first problem will be to find the
distribution of this population. This, according to MENDEL, is the segregating
generation. We must inquire into the frequency of each constitutional difference in
this segregating generation.

We now reach our second limiting hypothesis, which is needful if we are to apply
our theory to sexual reproduction. We suppose :—

There to be an absence of homogamy (including self- fertilisation), and the members
of the second generation to cross absolutely at random and with equal fertility.*
We have then to ask what is the distribution of constitutional differences in the
third generation. Does the process of segregation begun in the second generation
continue to the third, or does the population now remain stable ? Is the continual
segregation into pure protogenic and allogenic individuals a necessary result of any
pure gamete theory, or does the belief in such necessity depend upon the first
Mendelian experimenters working only with self-fertilising individuals ?

(3.) PROBLEM I. — To find the Distribution of the Offspring of the Perfect

Heterozygotes.

We shall here use a symbolic form of analysis. Let u stand for act, v for «A, and
\v for AA ; then any corresponding couplets will give rise to

u

and any one of these constitutions may be associated with one of the similar
constitutions in any of the remaining n — 1 couplets. Hence the general distribution
of the population will be given by the terms of the multinomial

(u + 2v + to)".
This equals t

«'< 4- 7m"-1 (2r + w) + ^L"-1) u"~~ (2v + w)- + ...

1 . _

* If one is to study heredity in populations with a view to the problem of evolution, the conditions as
to fertilisation should approach as far as possible the conditions we suppose them to be under in a
natural state; we must fix our attention on the mass relations between successive generations of the
population.

t Throughout this memoir the symbol cn, v, ,, is used for the expression \n / { \n-p-q [g \q).

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 57

Thus, for example, there would be out of the total population of possibilities 4" :
1 purely allogenic individual, n X 3 individuals with n — 1 allogenic couplets ; 2n of
these would have one heterogenic couplet, and n would have one protogenic
couplet.

Generally there would be 3* cH>tt0 individuals with n — s allogenic couplets, and
these individuals would be distributed according to the terms of the binomial
(2v + iv)'.

We are thus able to write down at once the number of any class of individual that
can appear in the segregating generation. For example, how often do individuals
like uH~P~ivf-itfi appear, i.e., individuals with n — p — q allogenic, p heterogenic, and
<2 protogenic couplets ?

To answer this problem all we have to do is to pick out the coefficient of iiu~i
in the above multinomial, and the result is

We are thus fully able to predict how many individuals of each kind ought to
occur when a population of perfect n-eouplet heterozygotes are crossed.

Corollary (i.). — Let us consider only the number of allogenic couplets in the
distribution of the segregating generation. If we were " pure Mendelians " we should
for the purpose of character classification make v = w, as the heterogenic couplet
would then give the dominant character. But without doing this we can assume
v and w to be non-it's.

Hence the distribution of allogenic characters in the population follows the simple
binomial

4" (± + f )"•

Thus we see that the distribution would be a skew binomial closely approximating'
to my skew curve of Type 111.,* and becoming indefinitely close to a normal
distribution of the form

y « y<fr*"t

when the number of couplets, n, is indefinitely increased.

For any Value of n the mean of the skew binomial as measured by the formula of
the memoir on " Skew Variation,"!

= n + 1 - (1 + f») = in,

i

and the Standard deviation = \/T$n.

Thus the mean number of allogenic Couplets in the members of the segregating
generation is J of the total number of couplets.

* 'Phil. Trans./ A, vol. ISO, p. 373.
t Ibid., p. 346;

VOL. com. — A. i

58 PROFESSOK K. PEAESON ON A GENERALISED THEORY OF ALTERNATIVE

Corollary (ii.). — The distribution of heterogenic couplets in the segregating
generation is given by the symmetrical binomial

The mean number is therefore \n, and the standard deviation = v/i^.

This is a symmetrical binomial, and approaches extremely closely, even for a fairly
small value of n, to the normal curve. We see that if any character depends solely
upon heterogenic couplets, the distribution will be nearly normal, and the variability
slightly greater than one depending on the allogenic (or, of course, the protogenic)
couplets only.

To sum up, then, so far as the distribution of characters depending upon allogenic
or heterogenic couplets goes, we may say that a generalised theory of the pure
gamete leads us to those normal and skew distributions of frequency with which
biometric studies of variability have made us already familiar. It would not be
possible to base a crucial experiment on the existence or non-existence of such
frequency distributions. The generalised pure gamete theory would, however,
account for their appearance, which, of course, a purely descriptive statistical theory
cannot do. On the other hand, distributions diverging much beyond the errors of
random sampling from binomials of the above types would tell pro tatno against the
pure' gamete theory in its above form. The presence of binomials of two types only,
(I + ?)" :lll(i (i + £)"> ought to be capable of detection, even if it would not already
have been discovered, had it been the rule.

(4.) PROPOSITION II. — To determine the Distribution of the Offering of the
Segregating Generation, supposing them to Mate at Random and 'without
Differential Fertility.

The solution of this problem may be reached as follows :—

Suppose P any male, and Q any female, say each of n — 1 couplets, producing an
array of offspring, which we will denote by R ; now suppose an additional couplet, the
/ith, added to both male and female zygote. The male may be now :

P + a,^,,, or P + aflAH, or P + A»a,, or P + AHAU ;
and the female may be

Q + a,a«, or Q + a,,A,,, or Q + A,,a,,, or Q + A, A, ; '

that is, we get 4x4 new mating individuals, with 4x4x4 new offspring
possibilities.

Now consider the first father P + aKan ; the possibilities which arise from mating
him with the four mothers are the array K of offspring combined with any one of the

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 59

•
16 possibilities 8anan + 8a,,A,,, or this is the same thing as multiplying the R array

by the symbolic factor 8 (u + v) = 1GU, say.

The next pair of fathers 2 (P + auAn) with the four mothers reproduce the array R
of offspring combined with 8 (a,,an + 2a,,A« + A,,AM), or 32 possibilities. But this is
the same as multiplying the R-array by the symbolic factor 8 (u -\- 2v + w) = 32V,
say. Lastly, the P + A,,AM father with the four mothers gives 16 possibilities of the
form 8a,,A,, + 8A,,AB to be combined with the R-array of offspring, which is the same
thing as multiplying the R-array by the symbolic factor 3 (v + w) = 1GW, say,

We have at once the symbolic relation among the operators ;

U + 2V + W = u + 2v + w ;

and, further, the important result that the array of offspring due to any pair P and Q
of n — 1-couplet parents can be converted into the arrays of offspring due to the 16
pairs of parents formed by adding an additional couplet to P and Q, by multiplying
that array by the symbolic factor

16U + 32V + 16W = 16 (u + 2r + w).

We have thus by induction a means of finding the array of offspring due to a
population of parents of n couplets from the series of arrays due to a population of
n — 1 couplets. Since all the arrays are to be multiplied by the same symbolic
factor, we can multiply their total by this factor. Or the distribution of offspring of
(n — l)-couplet parents being J, that of H-couplet parents

= 16 (u + 2v + 10) J = 4 X 4 X 4 . (£M + |v f » J.

Now consider parents of one couplet, their distribution is given by aa + 2«A + AA,
and they are to mate with the same series, an -\- 2aA + AA.
But

art X aa = 4na,

2 (aa X 2aA) = 2 (4aa + 4aA),
2 (aa X AA) = 2 (4«A),
2aA X 2aA = 4aa -f SrtA + 4AA,
2 (2aA X AA) = 2 (4aA + 4AA),
AA x AA = 4AA,

Total = IGaa + 32aA + 16AA.

= 16 (u + 2v + w) = 4 X 4 X 4 (> + f v + £w)
symbolically.

I 2

60 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

* Hence, by the above proposition, the distribution of offspring- of parents of two
couplets is

4 X 4 X 4 . (\u + \v + » X 4 X 4 X 4 . (±11 + f y + »

and, by induction, the distribution of offspring for the random mating of parents of

n couplets is

4" X 4" X 4" . (\u + f v + \w}"-

This, except for the constant factor 4" X 4", is absolutely identical with the
distribution of the parental population, and accordingly if the next generation also
mates at random, the mixed race will continue to reproduce itself without change.
We therefore reach the following result :—

However many couplets we suppose the character under investigation to depend
upon, the offspring of the hybrids — or the segregating generation — if they breed at
random inter se, will not segregate further, but continue to reproduce, themselves in
the same proportions as a stable population.

It is thus clear that the apparent want of stability in a Mendelian population, the
continued segregation and ultimate disappearance of the heterozygotes, is solely a
result of self-fertilisation ; with random cross fertilisation there is no disappearance
of any class whatever in the offspring of the hybrids, but each class continues to be
reproduced in the same proportions. Thus our generalised theory lends no countenance
to the appearance of any "mutations" within a hybrid population under random
mating; the only appearance of new constitutions is in the segregating generation, or
the first generation of hybrid offspring. Except at this stage, the appearance of the
unfamiliar is only the chance occurrence of a very rare normal variation. When we
recollect that a purely allogenic individual is only to be expected once in a population
of 4" individuals, or if there be ten couplets, once in more than a million individuals,
it will be clearly seen that the rarity of some of the more exceptional normal
constitutions may easily lead to their being looked upon as " mutations," even if they
appear in the offspring of a population many generations removed from hybridisation.

(5.) PROPOSITION III. — To find the Array of Offspring due to a Parent of given
Gametic Constitution mating at Random.

This can be again deduced by the method of induction adopted in the last
proposition.

Supposing a male P of n — 1 couplets to mate with all possible females, and R,,_1
to be the array of offspring, then we have seen in the last proposition that if we add
an nih couplet aaau-io P, the array of offspring due to P + aHaJt will be 16UKB_j ; if
we add a couplet cr,.A.H, the array of offspring due to fathers of type P + a«A,, will be

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 01

10VRa_1; and if we add a couplet of form A,,A,,, the array will he of the form
Now start with a father of one couplet ; this must lie alnl, or a, A,, or A,A], or in our
symholic notation u, r, or w ; the offspring array arc respectively 8«,a, + S^A] or
4«1a1 + S^A! + ^A^ or Sa^ + SAjAj, i.e., 1GU, 16V, or 1GW. These, therefore,
are the possihle values of P^. Hence, hy the principle just developed above, the
array of offspring due to a father of type

un~f~i
is

or remembering that such fathers occur with a frequency of 2>'cMt!, ?, we have for the
total distribution of offspring of all fathers of type

u»-p-i ,•/> ic'i,
the symbolic result

4" X 4". c,,,,,, /[?'-/'-? (2V)'' W?.

Substituting, the following expression would give all offspring of fathers of the
type u"~P~ivr>w'!, i.e., with n — p — q allogenic, p heterogenic, and q protogenic
couplets

4" X 4" . ca,M ($u + -\vy-f-i ('-« + v + ^''Y Q-'.' + -I-"1)7-

Therefore, given n and given p and <j, it is merely a matter of expansion to find the
array of offspring due to any special class of father.

Corollary (i.). — So far we have supposed our special class of father to lie defined by
the exact couplet distribution constitutional to him. But it is of interest to consider
the array of offspring we get supposing only the allogenic couplets fixed in number,
for example, in a generalised Mendelian theory if the number of recessive couplets be
fixed, but the heterogenic and dominant, as both exhibiting dominant characters, be
considered as indifferent. Let s = number of allogenic couplets, then we have to

sum all arrays like

4" X 4". c,]]iV/U'(2V)nV?,

subject to the condition that p + q = n — .s.
The result is clearly

4" X 4».c^.ie2{c,+,,Pie(2V)>W'}
= 4" X 4" . cBi.i0U' (2V + W)"-'
= 4» X 4" . CB>1, , (§u + i-r)' (i« + ft' + w)—
= 4" X 4" . c,, .,.(£« + Jv)' {($u + %v) + (v + u<)}"-'.
This, we note, is not a pure binomial., or the arrays of offspring of a father with a

(}2 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

given allogenic constitution are not either symmetrical or skew binomials, but of a
much more complex character. The only exception is the array of offspring of pure
allogenic fathers,* which is given by

4" X 4" X (%u + %»)*•

This is a symmetrical binomial. This result is, of course, of special interest, for it
gives us the distribution of offspring if the hybrid offspring were at any time crossed
with the pure allogenic race, which was one of the original factors of the hybridisation.
The deviation from binomial distribution in the above arrays ought to be further
considered, for if this deviation should turn out to be very significant, it would form a
convenient test for any generalised theory of pure gametes.

Corollary (ii.). — If we sum the above expressions for the array of offspring of all
fathers of p allogenic couplets for values of s from o to n, we have the total offspring
population

= 4" X 4" . Sc,, ,, 0U< (2V +

= 4" X 4" X (U + 2V + W)"

— 4" x 4" X (u + 2v + «')",

a result we have already found in Proposition II. as giving the distribution of the

total offspring population.

(G.) PROPOSITION IV. — -To find the Mean Number of Allogenic Couplets in the
Offspring of all Fathers having in their Constitution s-allogenic Couplets.

By the first corollary to the last proposition the distribution of such offspring is
given by

4" X 4" . c, , 0 i + Iv

where 17 is written for \ (v -f w), a quantity which is unity so long as we consider not
the distribution, but the total number of the non-allogenic couplets. Now this is
clearly the sum of a number of symmetrical binomials in ^u + ^v, and may be put

= 4" X 4" . S c,,, v- (%u + ly)"-'' (217)''.

i = 0

Now the means of each of these binomials can be found from the general theory of

the binomial, t If we take our origin at n~+T allogenic couplets, with a frequency
zero, the mean of the first binomial, or

* Or, of course, the array of sons from pure protogenic fathers,
t 'Phil. Trans.,' A, vol. 186, p. 373.

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 63

(iu + i^)" is at 1 + |-n, and its total frequency J\ = 4" X 4" X cn> ,t 0 ;
the mean of the second binomial, or

the meaii of the third biuomial, or

the mean of the (i + 1 )"' binomial

The total frequency is accordingly

= 4" X 4"c;v,,03"-.

Hence if ?>i, be the distance from the same origin of the mean of the above
system of binomials

; X 3

4. 4. 2- -"^

1.2.8...4

i 4- i 4. ~ 4.

2 / "J"

Now

"-' = 1 1 + 2 (n - s)x + 22 (!LrL«H*L

L » Zi

Multiply by ar, differentiate both sides and divide by 2, finally putting x — I, and
we find

3-* + (n - s) 3"-'-1 SB 1 + 2(n - s) f

1 . — •

| 4-...

,

n - A- n - s

1.2.8...*

or

Hence we deduce

/! X 3"-'wf =/! (3"-' + (n - s) 8s-'-1 + 4»i3"-'},

«

W, = 1 -f |»

64 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

But the mean of the whole population of offspring is at 1 + fw from our origin.
Thus we have the final results :

Mean number of allogenic couplets in offspring of fathers with s allogenic couplets

= LH _ J (/,. _ s) allogeuic couplets.
Deviation from mean of general population of this array of offspring

Deviation of fathers from mean of population

— s — -L-/6 = i'(4s — n).
Thus

Deviation of offspring from mean of jxtpulation j_
Deviation of fathers from mean of population

We have then the following results, which could certainly not have been fore-
seen : —

(<(.) The regression is constant for all arrays, or the regression curve is a straight

line.

(b.) The slope of this straight line is 3, or, since we have seen that the population
is stable, the parental correlation is -$ also.

Now these results are of very singular importance. A very general theory of the
pure gamete type leads to linearity of the regression curve, a result amply verified
by observations t>n inheritance in populations ;* and this result is quite independent
of the number of couplets supposed to form the total character of the parent, or of
the fact that in this case the arrays of offspring are skew and do not obey the normal
law.t Further, the value of the correlation reached is numerically identical with the
value obtained by FBANCIS GALTON in his original investigations on the inheritance
of stature ! The generalised theory of the pure gamete is thus shown, whatever the
number of couplets taken, to lead to precisely the chief results already obtained by those
who have studied heredity statistically. So far then it might appear that a
generalised theory of the pure gamete was capable of being brought into accordance
with the chief results of biometric experience in heredity. This would undoubtedly
be a great step forward, as linking up perfectly definite inheritance results with a
physiological theory of heredity. Unfortunately the whole drift of recent biometrie
observations on heredity emphasises three points ;

First. — That the parental correlation appears to be markedly greater than -j, nearef
to "45 to '5,

* GALTON, ' Natural Inheritance,' p. 96 ; ' Biometrika,' w>l. 2, pp. 216 and 362-3.
t This is further demonstration that linearity of regression has nothing whatever to do with the Gauss-
Lapkcian law of errors, i.e., normal curves or surfaces,

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 65

Secondly.— That this correlation appears to vary slightly from character to

character.
Thirdly. — That it does not appear to be absolutely the same for all species.

It is most unfortunate for this general theory of the pure gamete, that it throws
the Mendelian back into the position of the biometrician of 1885.* One might have
hoped that the generality involved in n couplets would have led to the requisite
elasticity, or, failing this, to a numerical value of parental correlation nearer the
cluster point of existing measurements than ^. We can only say, at present, that a
generalised theory of the pure gamete leads to precisely the same general features of
regression as have been observed by the biometricians, but it appears numerically too
narrow to describe the observed facts.

(7.) PROPOSITION V. — To find the Standard Deviation of the Array of Offspring
due to Fathers with s-allogenic Collets.

We have to find the standard deviation crs of the combination of binomials dealt
with in the previous proposition. Each component standard deviation must, of
course, be weighted with the total frequency of the component, and there must be
the proper reduction to the mean of the array as a whole.

The (i + l)th binomial (%u -f £?')"-; has V\n — i)% X £t for its standard deviation,
and the distance of its mean from the mean of the array

= [n +"l - (i + 1 + 1 (n - i))} - [$n - J- (n - *)}

~— ~ ~~ s) ~~ 2^'

~S

Further, the frequency of this component is

We thus see that it contributes

— i ,

-+

n — s 2

3 2

to the total second moment about the mean of the array. This gives us
f * T'-' v ,r 2 -

/! x 6 x <rs =

* " GALTON'S law makes the amount of inheritance an absolute constant for each pair of relatives. It
would thus appear not to be a character of race or species, or one capable of modification by natural
selection." More ample statistical experience of populations since 1885 shows that absolute constancy of
the heredity coefficients is not consonant with actual measurements. — ' Roy. Soc. Proc.,' vol. 62, p. 411,

t 'Phil. Trans.,' A, vol. 185, p. 373.

VOL. CCIII. — A, K

66 PEOFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

or

Now

and by differentiating

2 (7i _

Repeating the process

4 (» - s) (n - * - 1) (1 + 2ic)"-'~2 = 2cB_fi ,., 0 2;i (i - 1) a1'-2.

Hence, putting a; = 1, we have the required expressions on the right of the above
result, or

0.-.-S - Q,-, J" i (n ~ *)2 _i_ (Ji_- *) (n -.Sj-J) _. 2_(?1
U 9 9 9

Therefore

a-,2 = ^»i — i (n — s) = 31,; (5w, + 4s).

Now the standard deviation of the whole population, as far as allogenic units are
concerned, is

a- = v/ni f = X/I-G".
Thus

n

This result is of singular interest. The variability of an array of offspring
corresponding to a father of given allogenic constitution is not independent of the
father, but increases steadily from a minimum of \/jfant when there are no allogenic
couplets in the father, to a maximum of \/\n when there are only allogenic couplets.
In other words, fixing our attention on the same character, let the offspring of the
hybrids inter se be crossed first with one pure race, and then with the second pure
race, i.e., first with pure allogenic and then with pure protogenic individuals, there
ought to be a marked difference in the variability of the resulting offspring in the
two cases.

Corollary (i.). — In the theory of linear regression as apart from the theory of
normal correlation on the basis of the Gauss-Laplacian distribution,* if <r be the
standard deviation of any character, and r its correlation with a second character,
then

o- v/1 _ r2

* YULE, 'Journal of Royal Statistical Society,' vol. 60, December, 1897,

INHERITANCE, WITH SPECIAL KEFEKENCE TO MENDEL'S LAWS. 67

is the mean standard deviation of all the arrays of the first character for a given value
of the second. This expression is no longer the actual standard deviation of each
array.

It is of interest to see that this general law of linear regression is verified in the
present case. We have cr = x/fV1 and r = i- Hence if 2m be the mean standard
deviation of the arrays, we should expect

V 2 — _3.n /I 1\ _. !„

-IB — i6n(L 97 — ¥n-

Remembering the weight of each array,* we have

4» X 4" X 4" X 2,«3 = 'I" j 4" X 4»cB , 03"-»

» = o L

36

= 4" X 4" (1 + 3)" 3%n + 4" X 4"n4"-1 -3*6-
= 4" X 4" X 4" X -J-n,

whence S,,,2 = -^n, as we anticipated.

Corollary (ii.). — It is clear that some arrays of offspring will be more, others less
variable than the general population. The standard deviation of an array will be
equal to that of the general population when s is found from

-re ($n + 4s) = -?6-n, or s = T70-n.

Now the mean number of allogeriic couplets in the general population is \-n. Thus
the offspring array equally variable with the general population is at distance -fan
from the mean. But cr for the general population = \/-fan. Hence, if we take
fathers deviating from the population mean by \/-fan X cr, we should expect their
offspring to be equally variable with the general population. Supposing, therefore,
the theory under discussion were true, we should have a means of finding, at least
approximately, the number n of couplets corresponding to the character under
consideration. All we should have to do would be to find the standard deviation of
each array of offspring corresponding to a given father ; these standard deviations
ought to increase or decrease steadily across the table, their squares giving a straight
line when plotted. Smooth the results, interpolate a value equal to cr, and we shall
have the character of the father whose offspring are equally variable with the general
population ; but the deviation of this father from the mean ought to be <*/-$$n X cr.
Hence, since cr can be found, we have at once an approximate value of n. Approxi-
mate only, of course, because our arrays are classed by units of measurement, inches,

* The number of offspring in the array due to the fathers with s-allogenic couplets is found at once
by putting u=o-w in the formula of Corollary (i.) on p. 61, and equals 4"x 4™ < <•„, „, 03n~'.

K 2

68 PROFESSOR K. PEARSON ON A GENERALISED THEORY OP ALTERNATIVE

centimetres, &c., and each such unit will not, as a rule, represent one allogenic
couplet ; but interpolation ought to give a result not widely divergent from the

truth.

The method would of course fail practically if n were very large. For example, if
n were 48, the deviation of the required group of fathers would be 3(r, and hence
such a father would only occur once in 1000 individuals. In a manageable population,
therefore, we are very unlikely to have enough such fathers to form any reliable
measure of the variability of their offspring. At the same time, the squares of the
variabilities of the arrays of sons due to quite frequent fathers ought to give a
straight line, and if this line be determined properly, there should be no difficulty in
finding the theoretical position of the above father, and so finding n.

Many other physiological theories besides the present might give this peculiarity
of the diminishing variability of the arrays of offspring as we pass from one side of
the correlation table to the other. Such changes in variability are familiar to those
who have had to deal with skew correlation. But, as far as we are aware, they have
Hot hitherto been noticed in inheritance tables. The existence of this changing
variability would not affect in any way the general theory of linear regression applied
to heredity in populations. It would, however, lead to an immediate extension of
that theory consisting in the tabling of the standard deviations of the arrays.
Should the standard deviations of these arrays show no bias towards a linear
distribution, but only the fluctuation to be expected from random sampling about
the mean value cr */ (l — r-), we should have a strong argument against the present
general theory of alternative inheritance. We seem here, therefore, to have a crucial
test of the validity of the theory, which may be quite as easy to apply as the, previous
test of the numerical value of the parental correlation.

Of course the results now reached are not consistent theoretically with normal
correlation surfaces with their elliptic contour lines. The fact that Mr. GALTON came
to his elliptic contours in the first instance on the basis of his observations, and not
from any theory,* shows that they must in the case he was dealing with be approxi-
mately correct. Further, there is no doubt that in other statistics for characters in
man there is within the limits of random sampling a close approximation to normal
distribution. It might be hard to consider that such a deviation as would arise with
a continuously increasing variability of the arrays from one side to the other of the
table could exist and escape notice, had we not in physics had evidence that theory
has often led to the discovery of an obvious relation which time after time must have
been overlooked by previous observers unprovided with the theoretical hint of what
to seek for. Hence while we may say that the parental correlation given by the
theory is too rigid for the facts, we must leave this second test until more careful
examination ad hoc has been made of the ample existing data.

* 'Natural Inheritance,' p. 101.

INHERITANCE, WITH SPECIAL ItEFEKENCE TO MENDEL'S LAWS. M

(8.) PROPOSITION VI. — To find the Array of Offspring due to a Grandfather of
s-allogenic Couplets, supposing Complete Random Mating in the Population.

The general distribution of the population is

The fathers with s-allogenic units in their correlation are given by

f,,i.so?/,*(2r + w)"~\

The array of offspring due to these fathers is simply obtained by writing U ior u,
V for r, and W for iv, and multiplying by 4" X 4". This is a general rule for getting
the offspring from any father if lie mates at random. It gives us, as on p. 61, for the
offspring distribution

4" X 4" X f,, ; ,,,_,!? (2V + W)"-
- 4" X 4" X «„,,,„ (±». + £»')' {(£» + ir) + (r -f w)}— .

To get the offspring of this array treated as fathers and mating at random, we
have only to repeat the process, and we find

Offspring of grandfather of s-allogeuic couplets

= 4" X 4" X 4" X 4"e,M,u(iU + ±\)> {^U + 2V + W}-

= 44"c.,,.,.(*« + |c)'(s" +^+ H''+ ^))-J X (*)»-,
where

e^Sr + lu',

and is equal to unity if we identify -o and w as something not allogenic. This can be
dealt with exactly as in Proposition IV. we dealt with the array of offspring due to
a father of s-allogenic couplets, i.e., by analysing the array into the sum of a number
of weighted binomials ; in this case all. skew.

Writing as before, i? = £ (r + iv), we have to expand

The general term is

This has a total frequency OB_,, ,, „ (!)' X /]s and its mean is at a distance i+ 1 -H("~ *')
from the origin which is taken at (n + 1) allogeuic couplets.

70 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

The total frequency of the array is (1 + £)"-/,. Hence, if m1, be the mean of the
grandchildren measured from the same origin, we have

/! X (I)"" X m', =/ (1 + fn + i-(n - s) {2 + %(n -- 1)}

+ / & \ o ( ^ & ) ( n s •"•" i ) ( o i 5 / ^ o \ /

it) ^ — > & ~r~ 8 ' (n "It • • •

\bl 10 I I o \ /'

or

Thus

Mean of grandchildren = f n — ^ (n — s).

Deviation from general population mean = \n — $ (n — s) = -^ (4s — n).
Deviation of grandparent from general population mean = s — \n — \ (4s — n). *

Hence

Deviation of offspring _ ,
Deviation of grandparent

This ratio is the same whatever he the allogenic constitution of the grandparent.

(9.) PROPOSITION VII. — Tu find the Array of Offspring due to an mth Great-grand-
father of s-allogenic Couplets, supposing Complete Random Mating in each
Generation.

The array due to a father of s-allogenic couplets is

4" X 4" X cv,6 {%(u + v)}> II (u + v) + (v + w)}"-',

and, as we have already seen, we must multiply by 4" X 4" and put | (u -j- v) for u,
£ (u + 2v -\- w) for v, and \ (c + w) for w to get the array due to the grandparent of
s allogenic units. This process must be repeated m times if we wish to obtain the
array due to the -)«"' great-grandparent.

We must first investigate what happens to | (u + v) if this interchange be made m
times. Suppose that it has been done * times, and let the answer be

Kepeat the operation, and the expression becomes

(f M,- + iM',-) 4 (u + v) + (iM; + f M',-) | (v + w),

INHERIT ANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 71

or

M,-+1 = f M, + |M', ,

M'<+i = iM, + fM',-.

Therefore

M,-+1 + M',-+1 = M; + M', = M0 + M'0 = 1 + 0 = 1.
Hence -

M/+1 = 'M; + }, Wi+l = pi', + £,

M,+1 - i = HM; - i) = A (Mo - £) = A- ,

1 — 1 fAP. . . J \ — 1 /"VF . . l\ —

. M . -- m. n — f ; = - .+

Hence, finally,

Thus the result of ??i changes on ^(n 4- v) is known.
Similarly the result of TO changes on ^(v + w) is

We can now write down the array of offspring due to an ?/ith great-grandparent of
s-allogenic couplets. It is

/ ,„ ,.\ » -Kf <-<•' ~T~ V I 117 <• ~

(4« x 4°)'Vn,,,0 M,,,- ^ + M'ffl _}

\ A -J

X |(MW + 2M'W) n -+ - + (M'w + 2MW) " +

We must now find the mean of this array. For brevity let us write
M«J(M + v) + M'W-J('' + »') = PI + Xe

(M, + 2M'.) «± 2 + (M'. + 2M.) ^ = M" + 2M - (^ + Xc

where

1 M'»

' l+M',

, _ r _i f i » w

£— i~i Tl» fc ^ 1 I AT/ W'

2_(M,,,2 - M'^) 2 (M. - M/M)
MM + 2M'M 1 + M'»

72 PEOFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE
Hence we have to find the mean of the system

(pu + Wdiu + Xc + vriY-:

The ith component binomial of this sum of binomials is

e,,_I,,>i»yOnw + Xe)— '.
It therefore has its mean at a distance

•i + 1 + X (n — i)

from (n + 1) allogenic couplets, and a frequency given by

fi = cn _,,,-„ i/ft.

The total frequency of the whole array = (1 -j- v}"~'f\.

Hence, taking moments round the origin at n + 1 allogenic co\iplets, we have, if
m', be the mean of the array,

„)-•/; x mf, =

_ ,s) (-2 + X (« - 1))

>(8 + X(n-2)) + ...

+ ^H-^^-s-2^;>-^-{+1)(i + i + x(,-^) + ...}.

Summing and dividing by (1 -f- v)"~! we find

m', = 1 + Xn + (" " *^ ~ X) .

Hence the mean number of allogenic couplets in the members of the array
= n + 1 - m', = n (1 - X) - (n - s) - v - (l - X).

Deviation of offspring from mean of general population

v(l — \)

= s - - ' — n

We now substitute for v and X in terms of M,,, and Wm, and find

= -1- - - 1 n 4- M' } - l ~ 2M'" -
~ ell M- - -

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 73

A Q ^^ 1)

Hence : Deviation of offspring = -^ ^ • but the deviation of m"1 great-grand-
parent = s — £n.

Thus we have

Deviation of offspring l 1

Deviation of mttt great-grandparent ~" ^ 'lm

This result is independent of \ and of n.

Thus we conclude :

(i.) The regression of offspring on any individual ancestor is linear;
(ii.) The correlation coefficient is halved at each stage in ancestry ;
(iii.) The result is perfectly independent of the number of couplets introduced into
the formula.

The first two results are very familiar to biometric workers in heredity.

The actual numerical values of the grandparental, great -grandparental, great-great-
grandparental correlations are y, -j^-, -/j, &c.

These are distinctly less than the values so far readied fur ancestral correlation,
the grandparental correlations, for instance, Iving between "2 and "•'>.

The results show, however, that a general theory of the pure gamete, embracing
the simpler forms of the Mendelian principle, leads us directly to a series of ancestral
correlations decreasing in a geometrical progression. Thus, when we suppose a
population arising from hybridisation to cross at random, we find that it obeys the
second fundamental assumption of the biometric theory of heredity.* In other words,
ancestry is of the utmost importance, and the population follows laws identical in
form with those propounded in the biometrical theory on the basis of a linear
regression multiple correlation. Only the values of the constants deduced for the
law of ancestral heredity from the present theory of the pure gamete (which appears
to cover the bulk of Mendelian formulae hitherto propounded) are sensibly too small
to satisfy the best recent observations on inheritance.

It is of interest to find "Mendelian Principles" when given a wide analytical
expression leading up to the very laws of linear regression, of distribution of
frequency, and of ancestral inheritance in populations, which have been called into
question as exhibiting only a blurred and confused picture of what actually takes place.

It would be an immense advantage if we could accept such a theory of the pure
gamete as has been here analysed as a physiological basis for the theory of heredity.
We should then have a physiological origin for the ideas of regression and of ancestral
inheritance which statistics of heredity in populations have made familiar to
biometric workers. Unfortunately, even such a general pure gamete theory as we
have here dealt with, while leading to results which form a special case of the law of
ancestral heredity, is not sufficiently elastic to cover the observed facts. The lesson

* ' Biometrika,' vol. 2, p. 220.
VOL. COIII. — A. L

74 PROFESSOE K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

to be learnt from the present investigation is, however, that there is no essential
repugnance between any of the main results of the biometric school and a theory of
the pure gamete, but, on the contrary, it is perfectly possible to test such theories by
biometric methods. We may fairly ask anyone who propounds in future a Mendelian
or pure gamete formula as a general theory of heredity, to remember that it involves
in itself definite laws regulating the reproduction of a population mating at random,
and that it is incumbent on the propounder to test whether or not such laws are
consistent with what we already know of the inheritance statistics of such popula-
tions. When we remember that deducing all the effects of such a formula within the
whole field of inheritance will almost always form a very laborious piece of
mathematical analysis, there seems a touch of scientific irresponsibility in propounding
an immense variety of formulae to suit one or other special case, and the modifying or
withdrawing them when they are found to fail in another.

(10.) PROPOSITION VIII. — To find tlic Per/region and Correlation of Brethren
on the Theory developed in thix Paper.

We shall suppose the group of brethren to consist of 4^ members, or any pair of
parents to have a family of 4^.

Consider first parents of one couplet only, the offspring of the 16 possible pairs are
• nveii in the table below :—

Father.

<i 4- a.

. + A.

4X»

A + a.

A + A.

Now let us take every pair of brothers out of eacli of these 1G families and form a
correlation table of brothers. The following table gives the distribution of the
various types :—

First brother.

'/.

r.

«•.

il /:
iii.-

4x(9X-4)
24X2
4x2

24X"
80X2-32X
24X2

4X2
24X2
4x(9X-4)

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS.

75

This gives a total of 25Gx° — (34^= 16 X 4^ (4^ — 1) pairs of brothers, as it should,
every brother in 16 families having 4^ — 1 brethren, and there being 4^ in each
family. Clearly we can divide by 4y, and we have the simplified form :—

"I-

''I-

Wi.

/I.,

9X - 4

6x

X

l'-2

6X

20x-8

6x

W-,

X

fix

9x-4

where the subscripts 1 and 2 refer to the first and second brothers.

Let us simplify this by considering only the allogenic elements, 17 denoting either a

Then we have : —

heterogenic or a protogenic element.

"l-

9x-4

'H-

•X
41X-12

Thus the distribution of pairs of brothers in the case of the character being fixed
by a single couplet is

- 4)

(41 x - '-)

This is to be read as follows : there are (J^ — 4 cases of both brothers being
allogenic, to 41^ — 12 cases of neither brother allogenic, to ?x + 7x cases of one onlv
of the two brothers being allogenic.

Now when we pass from a character fixed by one couplet to a character fixed by
two, the above distribution can occur in either couplet, and every possible pair of
brothers will be got by squaring the above expression. Proceeding in this way to
ft couplet characters, we have the following symbolic expression for the distribution
of brothers

(4X)" X {9X - 4)

(41X -

the omitted factor 4X being restored.

This, I think, represents the distribution of a character measured by allogenic
couplets in a population of pairs of brothers, i.e., is the correlation table for brothers
in the population — any term involving «/«/ representing when we put 77! and 77.,
equal to unity the number of pairs in which the first brother has p and the second «/
allogenic couplets in their constitutions.

Hence if we find the coefficient of uf in terms of u.it we shall have the array or

L 2

76 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

brothers to be found associated with a brother of p allogenic couplets. The above
expression may be written

(4X)" [{(9X — 4) u* + 7X?7, } MJ + (7XM2 + (41X — 12) y^ 773]".

The term involving uf is

(4X)H ((9x - 4)«. + rxntf {7Xw2 + (4ix - 12) 7,!}"-*%-^,,,..

Neglecting the constant factor, the distribution in u.z is given by
{(9X - 4)7*. + 7^}" [7xu, + (41X -

We require to find the mean of this array.
Put

X=9*-4 „- 7X_ , =

16X-4' 16X~4' 7X

Then again, but for a factor independent of the power of u.2, the array may be
read

The eneral term is therefore

and we must sum from s = o, to .s- = n — p.

The general term has therefore its mean at the distance 1 + /* Q> + s) from
p + * + 1 allogenic couplets, or its mean

= (1 — p.) (/> + -s')5 allogenic couplets,

and its total frequency = vn~p~'cn_j>iai0.

This gives a total frequency of the array proportional to (1 + v)n~p.
Hence, taking moments, we have for the mean m of the array given by

1 hereiore

m = l-^n + v-^-rtp.
1 + v I +v *

Hence we see that

(i.) The regression between brothers is linear.

(ii.) The fraternal correlation which is equal to the regression

_
3(4X-1)'

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS.

77

and is quite independent of the number of couplets. It is, however, a function of ^,
the size of the family used in forming the table. We have the following values :—

Size of family.

Value of x-

Value of fraternal
correlation.

8

X = 2

• 3333

1L>

3

•3636

16

4

•3778

L'O

5

•3860

24

6

•3913

32

8

•3978

40

10

•4017

~£>

GO

•4007

The value of fraternal correlation thus varies with the size of the family dealt with
from '3 to '4. Probably the more correct way of looking at any fraternal correlation
table would be to suppose it a random sample of all the pairs of brothers which would
be obtained by giving a large, or even indefinitely large, fertility to each pair, for
what we actually do is to take families of varying size and take as many pairs of
brethren as they provide. In this case we ought to reach a fraternal correlation of '4.
precisely the value reached by the ancestral law when we take FRANCIS GALTON'S
original series of ancestral correlations.*

Thus we conclude that on the general theory of the pure gamete here dealt with,
the fraternal correlation is slightly larger than the parental. This is in accordance
with the general result of biometric investigations on populations. But the value, as
in the case of the parental correlation, is very sensibly lower than the value-- --about
'5 — found from recent investigations on man. It is further very inelastic even if we
allowr for some variation in the size of families dealt with. There can be little doubt
that fraternal correlation varies from character to character and species to species in
a manner sensibly beyond what can be accounted for by differences in the size of the
family dealt with.f

Corollary. — We can exhibit the regression in the form :

Mean of array — mean of general population

_ v\ - p.) Deviation Of brother from mean of general population},
1 + v

by observing that 1 — /x = |-(l + "/*)> whence

* ' Roy. Soc. Proc.,' vol. 62, p. 410.

t There is sensible variation even for different characters, when we take the same series of pairs of
brothers, and only one pair from each family.

78 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

or

m — 4-H = V -—(f> — T>0.

1 + v v

(11.) PROPOSITION IX. — To find the General Formula for Biparental Regression on
the Theory of the Pure Gamete, and the Value to be c/iren to the " Midparent."

If we applied without further consideration the general formula for biparental
regression to this case, we should have, if m-f,,/ be the mean of the offspring due to
fathers of /^-allogenic couplets, mated with mothers of '/-allogenic couplets,

JL I i _ / \ . \ i i / i \

P'l 4 I" 3V / 4" / "I 3 W 4 /'

This follows at once, since the mean of the general population = %n, the regression
coefficient for either parent = ^, and there is no assortative mating.
Flence we should have

Now suppose both parents of pure allogenic race, then /> = <y = //, and all the
offspring will be of pure allogenic race, or we must have m^ — n.
But the above formula gives

»»/, == i»,

which is not correct.

In other words, while the above formula gives the best plane to fit the array of
points determined by the parental constitutions, that array of points does not truly
lie in a plane. Or, although the simple regressions are linear, the compound
regression is not truly planar. We have therefore to find its true form, and measure
the amount of deviation from the truth involved in using a biparental formula of the
type indicated.

Given a character resulting from n couplets, we require 4" X 4" individuals, 4" male
and 4" female, to form the whole possible system of random matings. In such a
population there would be

<'//. /:,<y~'' fathers of /^-allogenic couplets,
and

</«../.«:5"~'/ mothers of ry-allogenic couplets.
The chance therefore of a mating of a ^-allogenic father and a cy-allogenic mother is

''//,/)," ''«,'/,« 3~"~y'~'//4:J" ',

and when n be even moderately large, this gets very small if p and q at all nearly
approach n. For example, if n were 5, and father and mother were both pure
allogenes, the chance of such a pure allogenic mating would be only

1/1,048,576,
or m a population of a million would hardly occur once.

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS.

79

Still keeping n = 5, take p = (/ = \n = 4. Then we have the chance of such a

mating

= 1/4215
still extremely improbable.

Thus when n is even moderately large, pure allogeriic matings are so rare that they
have vanishingly small influence in the population at large. Even if n were 2, the
chance of a pure allogenic mating is only -j-^. These points must be borne in mind
in what follows.

Consider first a father and a mother of one couplet each, their zygotes are either u,
i' or w, involving a gametic constitution «, -f- «• «> + A or A -j- A. We have the
following scheme :—

Zygotc of fathe

II
IV

II-

V

Zygotu of mother. Number of mating.

2 (!'•+'•)
2 ('• + /')

Hence it there be

1 allogenic couplet in lather and I in mother, offspring

4",

— 4(»f + 2")
= 4(2» + u)
= 4n + 4r

1 .......... i)

0 .......... I

0 .......... i)

Let us write

l(i/0 = 1G ($n + o + /r),

Then consider tlie relation

(Moeu + n/e.) x (,,/y + ,,/IT?I) = 4o (yo£iy +ya ^i^, + ^^^ +|/.,ey),

where e and r; are mere symbols, and 0, 1, etc., denote their powers. », u' refer
respectively to father and mother, and their powers denote the number of allogenic
couplets in the zygotes of father and mother. Then the above is a symbolical relation
which gives, by equating any power or product of e and r) on either side, the offspring
of a pair of parents of definite constitution.

Now suppose the parents not to consist of a single omplet, but of n couplets, then
the total distribution of offspring that we have given above for any couplet may occur

80 PROFESSOR K. PEAESON ON A GENERALISED THEORY OF ALTERNATIVE

in each couplet, and each such distribution must be combined with every other couplet
distribution. We then reach, dropping unnecessary indices, the general symbolic
relation,

X (Jt'° + ur) + ' + • • • «

= 4" X 4" X ( j(> +ji (e + 17) + />??)"•

Tims the array of offspring due to parents of zygotes with p and q allogeuic
couplets respectively— i.e., 'to w''X ?'/''— is the coefficient of e'ty' on tne right-hand side,

or in the expansion of

4" X 4" X (./„ +Ji (c + i?) +>»«?)"•
This may be written

4" X 4" X (./, +jfl)e+jo+jflY-

Thus the coefficient of e'' is

4" X 4" X (/, + jsr})"(Ju +jtfT~Pcn,p,0.

We require to pick the coefficient of T?" out of this in order to get the array ot
offspring due to fathers of j>, and to mothers of q, allogeuic couplets. But this is
clearly

4" X 4" X r,.f,.{J**Jif~9J<ri'ct,*,«

.>•:'"' JT '''in'" •''f',,-rtr...t>1,-r..

or, more briefly,

A" V -I" V /•

X 4 X ',,,j,,

We shall first find the mean and frequency corresponding to the /•"' term as given
above of this series. What we have to deal with is

We may write this

[ X 'it
4"

X (

where x = '' ail(l x' == ir ~l~ tw> an<^ l')Oth may be put unity when we are merely
rinding the distribution of allogeuic couplets.

Now the general term in the above expression is

INHERITANCE, WITH SPECIAL REFERENCE TO MENDEL'S LAWS. 81

and s must be taken from o to n — p — r. The frequency of this term is

Q»-S+r

_ _ nn-i>-r-» „

4« ^»-p-r,t,<it

and its mean is 7 — r + 3 (p — </ + 2r + .s) allogenic couplets.
The total frequency of the ?•"' term is therefore

Q»-3 + i- 30.1 -j.-/

/i _i_ 2V'~''-'' —
4" 4» '

which of course must ultimately be multiplied by the factorials in r omitted above.
If mr be the mean number of allogenic couplets in the r"' term, we have

— 7 + )'

- 9''-i'-r-v 'a — r 4- ' / /) — o 4- •>;• 4- «V-

An " c»-J-r,»,oia ' I 3 VJP 71^"' T */t

4t

= <«-«- + i(p-Sf+ 2r)} 3"^'" + * (n -jp - r) ?^-) -

Thus :

ni.(. = gr — r + 1 ( i> — </ + 2r) + -,1, (« - p — r),

= 9^ + 37 + I? — »>'•

This is the mean of the /•"' term, and its total frequency is

:T-" -*-'/

4. C-.ftrXCA,

Hence, if

F = 4" x.33"~^~» S'

>'=

_ 4»3a,-,-, x^;
we shall have

/X «V/ = S l(ayi + ii'/ + f.^) - u'1} ('^,r.. X

where j/;yjf is the mean of the arrav of offspring due to fathers of p and n. others of </
allogenic couplets.

The only difficulty here is summing the series

But this may be written

r-I=«-l

S (n X c, _,,„,,_,

1—1=0

or multij)led by 4"~132("~])~/-"'^~1) /)? it represents the total offspring of fathers of p and
VOL. com. — A M

82 PROFESSOR K. PEARSON OX A GENERALISED THEORY OF ALTERNATIVE

mothers of q — 1 allogenic units in a population with (n — 1) couplets in their
constitution

_ _!«-! v ,, v r Q-2(»-l)-/>-7-l

X C»-i,j>, o A O,,_ 1,2-1,0 °

Hence

,•-1=1-1

S {« X G.-1,!,,,-.! X CR ,_!_,+!,.} = «• X CM_lij))0 X CH_i, ,_,,..

Now

4« x 32«-p-j xf= -i" X f.,,,, . X (•„,,,„ 32"
Thus

f—Cn,r,o X 0,,,,.,

or, if'/' denote the above series, we have

v _ // (n - />) ,
^ ii.

This lends us to

4 f/ ('»• — 1>)

-

0,0

This is a most remarkable result, for it shows that the regression surface is not a
plane but a hyperboloid. Let us measure all the quantities in deviations from the
mean of the general population, i.e., put m/lt/ — m'fr/ -f i", 1> = p' + i», 5 = </ + i"-'
We find

This foi-mula reconciles at once the Mendelian and C4altonian positions. When the
number of couplets is large, parents having a number of allogenic couplets comparable
with n are vanishingly small in number. The standard deviation a- of the population

is \/'.\nj 4- (see p. 57).
Hence we may write

But // can only once per thousand cases be as big as 3cr, and accordingly when n is
large, both terms of the product will be small. In this case the surface becomes
practically planar, and we have the Galtonian result of 1885,* that the offspring from
the Galtonian "midparent" are one-third nearer the general mean of the population.
On the other hand, when n is small we see that for midparents not differing too
widely from the population mean, Galtonian regression of the value ^ holds, but that

* ' Natural Inheritance,' p. 97.

INHKKITANCK, WITH SPECIAL KEFEKENCE TO .MENDEL'S LAWS.

8:5

as we pass away from the mean of the population the regression gets less and less,
becoming absolutely zero when we take two pure allogenic parents. Just as the
regression is reduced when we emphasise the allogenic constitution of the parents, it
is increased when we emphasise the non -allogenic elements.

To illustrate these points 1 have drawn a diagram for n = 9 of the contour lines of

\ \

\ v y

\N

Mdndelian

Gcltonian

Str

Hyperbolas

ight. Lir

ies

\\
\

\\

\
\ \

\

A9

Is

\ \
\

\

*,'

£

o o
y% a

I*

T3 "
I,

V

\ X

\

\

<£

1,

A

\

M

' ..... 2. 3 5

7 8 9

•^ Number of Allogenic Couplets in Father.

Midparental contours.

the regression surface as a plane and a hyperboloid. The area ABCD contains all
possible parents; 999 out of 1000 random matings fall in the loop abed; 99 out of
100 random mating within the loop a'b'c'd', which has been carried right round to
mark that it excludes matings in which both parents have no allogenic couplets. The

M 2

84 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE

means of the array of offspring due to any pair of parents are marked in slim figures
along the Mendelian hyperbolic contours, and in heavy figures along the Galtonian
straight lines. We see comparatively small differences as long as we deal with
matings within the 99 per cent, loop, somewhat greater differences as we approach
the 9 9 '9 per cent, loop, and very marked differences as we go beyond the latter
boundary towards pure allogenic parentage. A study of the diagram illustrates at
once how the Mendelian theory exhibits for the bulk of the population Galtonian
regression, such regression becoming, however, less and less as we proceed to
individuals, the frequency of whose matings may be less than one in a million.

It will be seen again that in this proposition we have no fundamental antagonism
between the Mendelian and biometric standpoints. We reach a single formula whicli
approaches more and more closely to the biometric standpoint when we deal with
characters depending on many allogenic couplets.* On the other hand, it gives the
absence of regression which is obtained when pure allogenic parents are mated. We
see, however, quite clearly that it is totally erroneous to argue from this single case
against regression in general. Such regression actually exists on " Mendelian
Principles" when any population breeding at random is taken, and involves in itself
the whole conception of ancestral correlation and the influence of ancestry.

Of the formula for the midparent now reached, however, we can only say that on
the basis of our experience in populations the factor § seems too inelastic to work.
Here again the data must be especially investigated from the standpoint of a
midparent given by

before judgment can be final on this test.

Theoretically, by assuming the midparent to be

we should have a means of finding ^ by averages, and therefore n, the number of
couplets involved.

(12.) General Conclusions.

In this paper we have dealt with a general theory of the pure gamete— possibly
not the widest that could be conceived— but sufficiently wide to indicate the real
bearing of Mendelian formulas when applied to a population mating at random. We
see that under such circumstances :

(i.) The^ population which results from the offspring of hybrids remains stable;
every variation which appears, appears with a certain definite and predicable

* For n = 50 or 100 the population within the 1 in 1000 line sensibly obeys ordinary linear midparental
regression.

INHERITANCE, WITH SPECIAL INFERENCE TO MENDEL'S LAWS. 85

frequency ; there is no room for the appearance of " mutations," although certain
variations with very small frequency would he extremely rare in a limited population.
A mutation — a variation not hitherto observed — would only appear in the offspring
of the hybrids between two pure races ; after this with random mating the mixed
race woiild remain perfectly stable until disturbed by sexual or natural selection.
These are the only mutations which arise on the generalised theory of the pure
gamete, i.e., two pure races form one mixed race, breeding true to itself; it is difficult
under these circumstances to account for the origin of the two pure races by a
mutation-theory of the differentiation of species !

(ii.) Between any two relations— ^if we measure the character by the number of
allogenic or protogenic couplets in the zygote of the individual — we have a linear
regression. The frequency distribution of any character is skew, approaching closely
the normal distribution as the number of couplets which determines the constitution
of the zygote is increased.

(iii.) The correlations between pairs of blood relations take definite numerical
values absolutely independent of the number of couplets, and the same for all
characters and races.

(iv.) The ancestral correlations form a geometrical series of common ratio one-half.

(v.) Fraternal correlation is fixed between narrow limits depending on the number
of brothers per familv dealt with, and is verv slightly larger than parental
correlation.

(vi.) The theory of the midparent for a considerable number of couplets approaches
closely that originally given by FRANCIS GALTOX, except for extreme values of the
character, when the regression becomes rapidly smaller and ultimately vanishes.

We thus see that a generalised theory of the pure gamete would be of very great
advantage if it could be accepted. It would lead to a system of inheritance' in
randomly mating populations with non-differential fertility, which in its broad
features would be essentially the same as that which has been biometrically
developed not from theoretical hypotheses, but from the statistical description of
observed facts in populations.

Unfortunately, however, when we come to the actual numerical values for the
coefficients of heredity deducible from such a theory of the pure gamete, they do not
accord with observation. They diverge in two ways : First, they give a rigid value
for these, coefficients for all races and characters — a result not in reasonable
accordance with observation. Secondly, they give values distinctly too small, as
compared with the average values, or with the modal values of large series of
population observations.

We thus reach the point we have so often had to insist upon : that the biometric
or statistical theory of heredity does not involve a denial of any physiological theory
of heredity, but it serves in itself to confirm or refute such a theory. Mendelian
formulae analytically developed for randomly mating populations are either consistent

«(•> OX A GENERALISED THEORY OF ALTERNATIVE INHERITANCE, ETC.

or not with the biometric observations on such populations. If they are consistent, it
shows their possibility, but does not prove their necessity. If they are not, it shows
they are inadequate. The present investigation shows that in the theory of the pure
gamete there is nothing in essential opposition to the broad features of linear
regression, skew distribution, the geometric law of ancestral correlation, etc., of the
biometric description of inheritance in populations. But it does show that the
generalised theory here dealt with is not elastic enough to account for the numerical
values of the constants of heredity hitherto observed.

It will be time enough to consider other more or less general Mendelian formulae
when there is far better evidence than exists at present that they cover a real range
of observation, and have not been solely invented to describe isolated experiences, the
numerical results of which are not in complete accordance with simple Mendelianism.
Given such neo-Mendelian formula?, there is a perfectly straightforward mathematical
method of applying them to randomly mating populations, but that method is
excessively laborious, and the biometrician may well hesitate to undertake the task
of their investigation. A few minutes suffice to invent a Mendelian formula, but
weeks of labour may be involved in testing whether it leads to legitimate results
when applied to sexually crossing races. Let us therefore have a few simple general
principles stated which embrace rill the i'acts deducible from the hybridisation
experiments of the Mendelians; these can form the basis of a new mathematical
investigation, but it is idle to undertake such an investigation so long as Mendelian
Principles remain in a state of flux.

Any combination of the theory of pure gametes here discussed with homogamy, or
with fertility correlated with homogamy, or again with prepotency of individual or of
type, would emphasise the correlations which we have found above to be too low;
but such hypotheses would involve a fundamental alteration in the formula

(ft + a') (A + A') = «A + «A' + «' A' + AA'.

Such a formula would then give the j>oxsil>ilitiex of the cross, but the proportions of
these possibilities actually occurring would be quite different*

Such loading of the possibilities — not only of the individual couplet — but very
probably of associated couplets in the constitution — might conceivably enable us to
deduce better values for the ancestral and collateral correlations. But it would
abolish not only the simplicity of the fundamental Mendelian formula, it would also
involve lengthy preliminary studies on homogamy, fertility, and prepotency before
any effective formula could be propounded.

> Toss two pennies, and the result of 4n tossings will closely approximate to the distribution
n (HH + 2HT + TT). Load one or both coins, and the possible variations will still be HH, HT or TT, but
their proportions will be far from n : 2ft : n.

INDEX SLIP *i>here.

fit/ Low? RAYLEIOH, O.M., F.K.S.

II "itli an Appendix, yii'iny f/i*'

ir.t »f .,»:«

KA YLKIOH, (Lord). — On the Aroustio Shadow of a Sphere, with au Appendix
giving the Values of Legendre's Functions from P,, to P.,, at Intervals
of 5 Degrees, by Prof. A. Lodge.

Phil. Tran«., A, vol. 203, 1904, pp. 87-110.

Leeendre's Functions — Approximate form when n is large.

RATLEIGH, (Lord). Phil. Trans., A, vol. 208, 1904, pp. 87-110.

Hhado vr, Acoustic, of a S phew.

RATLBIGH, (Lord), Phil. Trans., A, vol. 203, 1904, pp. 87-110.

source of sound of a rigid sphejv

The question turns upon tliM .-Lttivc inn-iii1
radius (c) of tlie sphei-i-. 1C >
lui« but littlt; ett""CT lit

The following bthli-
.somewhat larger sjihc

•motions from P0 to P00
LODGE.

904.

Lissed the effect upon a
source.

'ave-length (X) and the
e presence of the sphere
nee.
> principal directions of

-f- G' represents the intensitx of s«

M'OHS such that ft is the cosine n1

'Ugh the source. Up<

<J ' — 11 values of /A, when i

The

more than doubled when kc = '2.
(362.)

In looking al

stance from the sphere
?n them and that radius
rneasureineJit adopted,
hen the propagation is
H = 1 show that the
direction being already
ires, the first point which
0.3.04

86

ON A GENERAL!

or not with the biometric
shows their possibility, bi
they are inadequate. Th
gamete there is nothing
regression, skew distribm
biometric description of
generalised theory here (
values of the constants of

It will }>e time enough
when there is far better e
of observation, and have
numerical results of whic
Given such neo-Mendeliai
method of applying the1
excessively laborious, and
of their investigation. 1
weeks of labour may be
when applied to sexually
principles stated which
experiments of the Men<
investigation, but it is idl
Principles remain in a sta

Any combination of th
with fertility correlated A\

J

type, would emphasise tl
but such hypotheses woul

(ft + o

Such a formula would t
these possibilities actually

Such loading of the p
probably of associated coi
deduce better values for
abolish not only the simp]
involve lengthy prelimim
any effective formula coul

* Toss two pennies, and the
n (HH + 2HT + TT). Load one or
their proportions will be far from n

•

••n to the bnnul i-

<>f ancesfcftklB X3QHI
ti:H!fi. But it doex the

enough to acctoini flu

iln-r i'inr«j «.r MVSH s^ciici-:il

xiboaqqA miifoii? ,wi»dq8 n io -wolmilS oiJeiiojA'dll nO — .Omul) .

\K ,K1 <>•> ift rao-rt haolJaituH t'-nbnyjipj lo nuit-'f a/11 Sniyiji

.•)Mb«pJ .A .loll 7ff ,88-)T4'(fT O lo

Ml

,K08 .\<rt .A ..

T ,BOS IOT .A .

ill.

jitoau1 i'
AWJrtHi

method i

.OII-T8 .qq ;K*f ,R<)8 .fo» ,A ,.

formula, but

t} mate results

imple general

ie iivliridisation

mathematical

u MS Mendelian

!>i.iuogamy, or
individual or of

. U- too low ;

l.'it i 'a- proportions of

• MI i;!et -but very
..!-i\ Pliable us to

.-id collateral correlations. But it would
amenta! Mendelian formula, it we..
unogamv, fertility, and inv

be propounded.

the

s will closely «})proximato to tJ

••

IV. On the Acoustic Shadow of a Sphere.
By Lord KAYLEIGH, O.M., F.fi.8.

With an Appendix, (jiving the }ralnes of LEG END UK'S Functions from P0 to P20
at Intervals of 5 degree*. By Professor A. LODGE.

Received December 28, 1903,— Head January 21, 1904.

IN my book on the ' Theory of Sound,' § 328, I luive discussed the effect upon a
source of sound of a rigid sphere whose surface is close to the source.

The question turns upon the relative magnitudes of the wave-length (X) and the
radius (c) of the sphere. If kc he small, where k — 'liri X, the presence of the sphere
has but little effect upon the sound to be perceived at a distance.

The following table was given, showing the effect in three principal directions of
somewhat larger spheres : —

ke.

//..

P + G»

1

•294291

i

-1

•259729

0

•2:51999

1

•502961

1

- 1

• 285220

0

•236828

1

•6898

2

-1

•3182

0

•3562

Here FJ + Gr': represents the intensity of sound at a great distance from the sphere
in directions such that //, is the cosine of the angle between them and that radhis
which passes through the source. Upon the scale of measurement adopted,
F2 -+- G'' = ^ for all values of fj., when kc = 0, that is, when the propagation is
undisturbed by any obstacle. The increased values under /JL = 1 show that the
sphere is beginning to act as a reflector, the intensity in this direction being already
more than doubled when kc = '2. " In looking at these figures, the first point which
(362.) 9-3.04

88 LORD BAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

attracts attention is the comparatively slight deviation from uniformity in the
intensities in different directions. Even when the circumference of the sphere
amounts to twice the wave-length, there is scarcely anything to be called a sound
shadow. But what is, perhaps, still more unexpected is that in the first two cases
the intensity behind the sphere [/*= — !] exceeds that in a transverse direction
[ju, = Oj. This result depends mainly on the preponderance of the term of the first
order, which vanishes with /JL. The order of the more important terms increases with
kc ; when kc is 2, the principal term is of the second order.

" Up to a certain point the augmentation of the sphere will increase the total energy
emitted, because a simple source emits twice as much energy when close to a rigid
plane as when entirely in the open. Within the limits of the table this effect masks
the obstruction due to an increasing sphere, so that when /j. = — 1, the intensity is
greater when the circumference is twice the wave-length than when it is half the
wave-length, the source itself remaining constant.''

The solution of the problem when kc is very great cannot be obtained by this
method, but it is. to be expected that when (U, = 1 the intensity will be quadrupled,
as when the sphere becomes a plane, and that when p, is negative the intensity will
tend to vanish. It is of interest. to trace somewhat more closely the approach to this
state of things — to treat, for example, the case of kc = 10.* In every case where it
can l)e carried out the solution has a double interest, since in virtue of the law of
reciprocity it applies when the source and point of observation are interchanged, thus
giving the intensity at a point on the sphere due to a source situated at a great
distance.

But before proceeding to consider a higher value of kc, it will be well to
supplement the information already given under the head of kc = 2. The original
calculation was limited to the principal values of /A, corresponding to the poles and
the equator, under the impression that results for other values of p would show
nothing distinctive. The first suggestion to the contrary was from experiment. In
observing the shadow of a sphere, by listening through a tube whose open end was
presented to the sphere, it was found that the somewhat distant soui-ce was more
loudly heard at the anti-pole (//,= — ]) than at points 40° or 50° therefrom. This
is analogous to POISSON'S experiment, where a bright point is seen in the centre of the
shadow of a circular disc — an experiment easily imitated acoustically! — and it may
be generally explained in the same manner. This led to further calculations for
values of //, between 0 and — 1, giving numbers in harmony with observation. The
complete results for this case (kc = 2) are recorded in the annexed table. In
obtaining them, terms of LEGENDBE'S series, up to and including Pfi, were retained.
The angles 6 are those whose cosine is /z.

* See RAYLEIGH, 'Proc. Roy. Soc.,' vol. 72, p. 40; also MACDONALD, vol. 71, p. 251 ; vol. 72, p. 59 ;
PoiNCAiiri, vol. 72, p. 42.

t 'Phil. Mag.,' vol. 9, p. 278, 1880; 'Scientific Papers,' vol. 1, p, 472.

WITH AN APPENDIX BY PROFESSOR A. LODGE.

89

kc = 2.

ft

F + iG.

F2 + G2.'

4(F* + G2).

0

+ -7968+ -2342z

•6898

2-759

15

+ • 8021 +• 1775 i

•6749

2-700

30

+ -7922+ -0147*

•6278

2-511

45

+ -7139- -2287 i

•5619

2-248

60

+ -5H4_ -4793i

•4912

1-965

75

+ -1898- -6247?

•4263

1-705

90

- -1538- -5766;

•3562

1-425

105

- -3790- -3413«

•2601

1-040

120

- -3992- -0243 i

•1600

0-640

135

- -2401 + -2489*

•1196

0-478

150

- -0088+ -4157 i

•1729

0-692

165

+ -1781 + -4883«

•2701

1 • 080

180

+ -2495+ -5059e

•3182

1-273

A plot of 4 (F~ -4- G2) against 6 is given in fig. 1, curve A.

The investigation for kc = 10 could probably be undertaken with success upon the
lines explained in ' Theory of Sound ;' but as it is necessary to include some 20 terms

\

Curves

of Intensity

A, Ac

B, Ac

45

60

75

90
Fig. 1.

105

120

135

150

165

180

of the expansion in LEGENDRE'S series, I considered that it would be advantageous to
use certain formulae of reduction by which the functions of various orders can be
deduced from their predecessors, and this involves a change of notation. Formula
convenient for the purpose have been set out by Professor LAMB.* The velocity -

' Hydrodynamics,' § 267 ; ' Camb. Phil. Trans.,' vol. 18, p. 350 1900.

VOL. CCIII. — A.

N

90 * LORD EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

potential <// is supposed to be proportional throughout to elhlt, but this time-factor is
usually omitted. The general differential equation satisfied by \jj is

of which the solution in polar co-ordinates applicable to a divergent wave of the
ntu order in LAPLACE'S series may be written

^l = S,tr»Xn(kr) ......... (2).

•

For the present purpose we may suppose without loss of generality that k = 1.
The differential equation satisfied by x« (r) is

and of this the solution which corresponds to a divergent wave is

Putting n = 0 and n =• 1, we have

e~ir (1 + ir)e~ir

Xo(r)-~> XiO') = L

It is ea,sy to verify that (4) satisfies (3). For if x« satisfies (3), ?'-1x'« satisfies the
corresponding equation for x«+i- And r~}e~ir satisfies (3) when n = 0.
From (3) and (4) the following formulae of reduction may be verified :

X'»(>-)=-rXn+l(r) ......... (6),

rX'»(r) + (2n + I)Xn(r) = Xa_l(r) ....... (7),

By means of the last, ^2, ^3, &c., may be built up in succession from XQ and Xi-
From (2)

d^/dr = S,,(nr"-\n + r"x'n),
or, with use of (7),

Thus, if UM be the wth component of the normal velocity at the •surface of the
sphere (r = c),

T7. = <f-ift,{Xi.1(c)-(ii + l)x,(c)J ...... (10).

When n = 0,

....... (11).

WITH AN APPENDIX BY PEOFESSOR A. LODGE. 91

The introduction of S,, from (10), (11) into (2) gives \f>H in terms of U« supposed
known.

When r is very great in comparison with the wave-length, we get from (4)

v»(r) = l"e ' (12)

K»\ / r»+i V1^/'

so that

^ = S^"e~ (13).

M

In order to find the effect at a great distance of a source of sound localised on the
surface of the sphere at the point //,-=!, we have only to expand the complete value
of U in LEGENDRE'S functions. Thus

U,, = i (2n + 1) PH (M) J *' UPH(/i) dp

" JrfS . . (14),

in which JJUc^S denotes the magnitude of the source, i.e., the integrated value of U
over the small area where it is sensible. The complete value of \\i may now be
written

._ . ,t

"^-«

When n = 0, x«-i (c) ~ C/l + !) X« (c) ^s t° be replaced by — c-^i ('')•
If we compare (15) with the corresponding expression in " Theory of Sound," (3),
§ 238, we get

c"+1 (x«-i (c) ~ (» + 0 X« (c)l = " ^-"F. (/c) . . . (1(5).

Another particular case of interest arises when the point of observation, as well as
the source, is on the sphere, so that, instead of r = oo , we have r = c. Equation (15)
is then replaced by

4™

It may be remarked that, since i/> in (17) is infinite when ^ = + 1 and accordingly
PM = 1, the convergence at other points can only be attained in virtue of the factors
Pfl. The difficulties in the way of a practical calculation from (17) may be expected
to be greater than in the case of (15).

We will now proceed to the actual calculation for the case of c — 10, or kc — 10.
The first step is the formation of the values of the various functions x«(10)> starting
from xo(10), ^ (10). For these we have from (5)

N 2

92

and

LOED KAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,
10Xo(10) = cos 10 — ism 10,

102Xl (10) = -A, cos 10 + sin 10 + * (cos 10 — -& sin 10).
The angle (10 radians) = 540° + 32° 57''4G8 ; thus

sin 10 = - -5440210, cos 10 = — '8390716,

10Xo = - -8390716 + -5440210 i,
102Xl = — -6279282 - 78466957:.

From these, X.,, Xs> • • • are to ^e comPutecl 'm succession from (8), which may be
put into the form

io«+~x,,+1 = '2n^~ 10"+1x« - io-x-1-

For example,

103X2 = -3(lo2Xi) " 10Xo = + '6506931 -- 7794218 i.

When the various functions 10"+1X// have been computed, the next step is the
computation of the denominators in (15). We write

D,= 10"+1{X;,_1-(», + l)xj

= 10 X I0"x»-i -(» + 1)10"+1X» • (18),

and the values of D,, are given along with 10ii+1X,, in the annexed table.

M.

10»+1x»(10).

IV

0

0 -83907 + 0- 54402 i

+ G-2793 + 7-8467z

1

0-62793-0-78467 /

7 -1349 + 7 -0095 i

2

+ 0-6.r,OG9- 0-77942 i

8-2314-5-5084J

3

+ 0 -95327 + 0- 39496 i

+ 2-6938- 9-3741 i

4

+ 0-01660 + 1 -05589*

+ 9 • 4498 - 1 • 3299 i

5

0-93834 + 0-55534 i

+ 5-7960 + 7-2269 i

6

1-04877-0-44501 i

2 -0420 + 8 -6685 i

7

0-42506-1-13386 i

7-0872 + 4-6208z

8

+ 0-41117-1 -25578 i

7-9512-0-0366*

9

+ 1 -12406-1 -00096 /

7 -1288 -2 -5482 i

10

+ 1-72454- 0-64605 i

1 -7293 -2- 9031 i

11

+ 2 -49747-0- 35574 i

12-7243-2-1916?:

12

+ 4-01964-0-17216?:

27 • 2807 -1- 3194 »

13

+ 7-55164-0-07465 i

65-5265-0-6764*

14

+ 16-36978-0-02941 i

170-030 -0-3054J

15

+ 39-92071-0-01062 i

475-033 -0-124i

16

+ 107-3844 - 0-00353 i

1426-33 -0-047 i

17

+ 314-45 -0-OOlOi

4586-2 -0-017*

18

+ 993-19 -0-000 i

15725-0 -0-010*

19

+ 3360-3

- 57274

20

+ 12112

- 220750

21

+ 46299

- 897460

22

+ 186974

- 38374 xlO2

WITH AN APPENDIX BY PKOFESSOR A. LODGE. 93

It will be seen that the imaginary part of 10"+1x/, (10) tends to zero, as n increases.
It is true that if we continue the calculation, having used throughout, say, 5 figures,
we find that the terms begin to increase again. This, however, is but an imperfection
of calculation, due to the increasing value of y^ (2u +1) in the formula and
consequent loss of accuracy, as each term is deduced from the preceding pair. Any
doubt that may linger will be removed by reference to (4), according to which the
imaginary term in question has the expression

• ,1+1 ( d \" sm r
\ r drj r

Now, if we expand r-1sinr and perform the differentiations, the various terms
disappear in order. For example, after the 25th operation we have

25 sin r _ 5_0^_48 . . . 4 . 2 __ 52 . 50 . . ._ 6 . 4 ^ 54^ . .6 )A &c
r 51 ! 53 ! 55 !

the first term being in every case positive and the subsequent terms alternately
negative and positive. The series is convergent, since the numerical values of the
terms continually diminish, the ratio of consecutive terms being (when r — 10)

100 100 100

- ) ) , <XC.

2. 53 4. 55 I! . 57

Accordingly the first term gives a limit to the sum of the series. On introduction of
the factor 10''+1, this becomes

102fi

1.3.5 ... 49. 51 '

i.e., approximately 10~8 X 3'0. A fortiori, when n is greater than 25, the imaginary
part of 10"+1x,, (10) is wholly negligible.

We can now form the coefficients of P« under the sign of summation in (15), i.e.,
the values of

iH(2n+ I)!),"1 (19)-

For a reason that will presently appear, it is convenient to separate the odd and
even values of n.

94

LORD EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

It.

»» (2»+ 1) D^-1.

n.

in(2n+l)Dn-\

0

+ 0-06217-0-07769;

I

+ 0-21020-0-21396J

2

+ 0-41955-0- 28076 i

3

+ 0-68978-0- 19822 i

4

+ 0-93391+0- 13143 i

5

+ 0-92629 + 0-74289J

6

+ 0-33469 + 1 -42083 i

7

-0-96831 + 1 -48517 i

8

-2-13800 + 0-00984 i

9

-0-84474 -2 -36328 i

10

+ 2 -38104-0- 89430 i

11

+ 0-30236 + 1 -75549 i

12

-0-91426 + 0-04422 i

13

-0-00425-0 -41200 i

14

+ 0-17056- 0-00031 i

15

+ 0-00002 + 0 -06526 i

16

- 0-02314 +0-00000 i

17

-0-00000 -0-00762*

18

+ 0-00235

19

+ 0-00068i

20

-0-00018

21

- 0-00005 i

22

+ 0-00001

In the case of 0 = 0, or \j, = + 1, the P's are all equal to + 1, and we have
nothing more to do than to add together all the terms in the above table. When
Q = 180°, or p. = — 1, the even P's assume (as before) the value + 1, but now the
odd P's have a reversed sign and are equal to — 1. If we add together separately
the even and odd terms, and so obtain the two partial sums S: and S2, then 2^ + S3
will be the value of S for 0 = 0, and 2j — S3 will be the value of 2 for 0 = 180.
And this simplification applies not merely to the special values 0 and 180, but to all
intermediate pairs of angles. If Sj -f- S3 corresponds to 0, Sj — S2 will correspond to
180 - 6.

For 0 and 180 we find

2t = + 1-22870 + -35326 i,

S3 = + 0-31135 -f -85436 i;
whence for 0 = 0

S = 2 (F + iG) = + 1-54005 + 1 '20762 i,
and for 0 = 180

S = 2(F + iG) = + 0-91735 - 0-50110?.

When 0 = 00°, the odd P's vanish, and the even ones have the values

— . . i

"

1.3

2. 4

P« = -2:4T6> fe-

For other values of 0 we require tables of Pn (0) up to about n = 20. That given
by Professor PERRY* is limited to n less than 7, and the results are expressed only to
4 places of decimals. I have been fortunate enough to interest Professor A. LODGE
in this subject, and the Appendix to this paper gives a table calculated by him
containing the P's up to n = 20 inclusive, and for angles from 0° to 90n at intervals
of 5°. As has already been suggested, the range from 0° to 90° practically covers
that from 90° to 180°, inasmuch as

P2/, (90 + 0)= P2S (90 - 0), P,u+l (90 + 0) = - P2,,+I (90 - 0).

* 'Phil. Mag.,' vol. 32, p. 516, 1891 :. see also FARR, vol. 49, p. 572, 1900.

WITH AN APPENDIX BY PROFESSOR A. LODGE.

95

In the table of coefficients it will be observed that the highest entry occurs at
n = 10, in accordance with an anticipation expressed in a former paper.

As will readily be understood, the multiplication by P,, and the summations involve
a good deal of arithmetical labour. These operations, as well as most of the
preliminary ones, have been carried out in duplicate with the assistance of
Mr. C. BOUTFLOWER, of Trinity College, Cambridge.

kc = 10.

9.

2(F+*G).

4(F2 + G2).

6

+ 1-54005 + 1-207 62 1

3-8300

5

+ 1-58407 + 1-14959 z

3-8309

10

+ 1-701 86 + 0- 96603 i

3-8295

15

+ 1 -84773 + 0-63523 t

3-8176

30

+ 1-52622-1 -17708 i

3-7148

45

- 1-13754-1 -48453 i

3-4978

GO

- 0-74695 + 1 -59745 i

3-1098

75

+ 1-45160-0 -62553 i

2-4984

90

- 1-31 954-0 -09924 i

1-75104

105

+ 0-94204 + 0-41681?

1-06117

120

-0-57769-0-48417*

0-56815

135

+ 0-29444 + 0-43841 i

0-27890

150

-0-08146- 0-35600 i

0-13338

165

-0-12081+0-28341 i

0 • 09492

170

+ 0-35454 + 0-01457 i

0-12591

175

+ 0-76023-0- 34059 i

0-69395

180

+ 0-91735-0-50110J

1-09263

The results are recorded in the annexed table and in curve B, fig. 1. The
intention had been to limit the calculations to intervals of 1 5°, but the rapid increase
in (F~ + G2) between 165° and 180° seemed to call for the interpolation of two
additional angles. This increase, corresponding to the bright point in POIWSON'S
experiment of the shadow of a circular disc, is probably the most interesting feature
of the results. A plot is given in fig. 1, showing the relation between the angle ft,
measured from the pole, and the intensity, proportional to F- + G2. It should,
perhaps, be emphasised that the effect here dealt with is the intensity of the pressure
variation, to which some percipients of sound, e.y., sensitive flames, are obtuse.
Thus at the antipole a sensitive flame close to the surface would not respond to a
distant source, since there is at that place no periodic motion, as is evident from the
symmetry.

I now proceed to consider the case where the source, as well as the place of
observation, are situated upon the sphere ; but as this is more difficult than the
preceding, I shall not attempt so complete a treatment. It will be supposed still
that kc =10.

96

LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

The analytical solution is expressed in (17), which we may compare with (15).
Restricting ourselves for the present to the factors under the sign of summation, we
see that the coefficient of P« in (17) is

__ (2«_+ l)c'^x,, (c) = (2n_+ 1) c

while the corresponding coefficient in (15) is

(2n + l)i"
"D,,

If these coefficients be called C,,, C',, respectively, we have

C. = i--o"+1x.(o).d' (20),

in which the complex factors ti»+\u(c), C',,, for c — 10, have already been tabulated.
We find

n.

(•2ft + l)10»+V

D,;

it.

(2n, + l)10»«X»
Dn

0

-0-0099 + 0 -0990 i

1

-0-0306 + 0-2999z

2

-0-0542 + 0-5097 i

3

-0-0835+0-7358*

4

-0-1233 + 0-9883;

5

-0-1827 + l-28l7z

6

-0-2813 + 1 -6390 i

7

-0-4666 + 2-0956«

8

- 0-8667 +2-6889z

9

-1 •8111+3'3152t

10

- 3 -5284 + 3 -0805 i

11

-4-2766 + 1-3796*

12

-3 -6673 + 0-3351 /

13

-3-1110 + 0-0629z

14

- 2-7920 + 0-0100 i

15

-2-6051+0-0014i

1C

-2-4844 + 0 '0001 i

17

-2-3998

18

- 2 • 3369

19

-2-2881

20

-2-2496

21

-2-2183

22

-2-1925

23

-2-1711

24

-2-1528

25

-2-1374

2G

-2-1240

The product above tabulated shows marked signs of approaching the limit — 2, as
n increases; so that the series (17) is divergent when P« = 1, i.e., when 6 = 0, as
was of course to be expected. The interpretation may be followed further. By the
definition of P,,, we have

f -t fy ft i^ O

(. * * I

so that, if we put a = 1,

1 + P! + P2 + P3 + . . . =

= 1 + P, . a + P.. a~ + . . . + P, . a" + . . .

2 sin

(21);

(22).

WITH AN APPENDIX BY PROFESSOR A. LODGK. 97

Thus, when 9 is small, and the series tends to be divergent, we get from (17)

= ._

" 27T . 2c sin

(23);

and this is the correct value, seeing that 2c sin (^6} represents the distance between
the source and the point of observation, and that on account of the sphere the value
of \jf is twice as great in the neighbourhood of the source as it would be were the
source situated in the open.

When 0 = 180°, i.e., at the point on the sphere immediately opposite to the source,
the series converges, since P» takes alternately the values + 1 and — 1. It will be
convenient to re-tabulate continuously these values from 7; = 18 onwards without
regard to sign and to exhibit the differences.

71.

Function.

First difference.

Second difference.

Third difference.

18

2-3369

19

2-2881

- -0488

—

. —

20

2-2496

- -0385

+ •0103

—

21

2-2183

- -0313

+ -0072

- -0031

22

2-1925

- -0258

+ -0055

- -0017

23

2-1711

- -0214

+ •0044

- -0011

24

2-1528

- -0183

+ •0031

- -0013

25

2-1374

- -0154

+ -0029

- -0002

26

2-1240

- -0134

+ •0020

- -0009

In summing the infinite series, we have to add together the terms as they actually
occur up to a certain point and then estimate the value of the remainder. The
simple addition is carried as far as n = 21 inclusive, and the result is for the even
values of n

— 18-3939 + 9-3506 i,

and for the odd values

- 19-4734 + 9-1721 i,

or, with signs reversed to correspond with PoB+i (180) = — 1,

+ 19-4734 - 9-1721 i.

The complete sum up to n = 21 inclusive is thus

+ 1-0795 + -1785t .

(24).

The remainder is to be found by the methods of Finite Differences. The formula
applicable to series of this kind may be written
VOL. com. — A. o

98

LOED RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

in which we may put

= 2-1925.,

= 2-1711, &c.

Thus

$ (o) _

... = + 1-0962 + -0054 -f- "0004 = 1-1020,

and for the actual remainder this is to be taken negatively. The sum of the infinite
series for 0 = 180° is accordingly

- -0225 + -1785 t

(25),

from which the intensity, represented by ('0225)- -f ('1785)2, is proportional to
•03237. Referring to (17), we see that the amplitude of i/» is in this case

(26).

X v/('03237)

4TTC

We may compare this with the amplitude of the vibration which would occur at
the same place if the sphere were removed. Here

(27),

477?'

47TC

since r = 2c. The effect of the sphere is therefore to reduce the intensity in the
ratio of -25 to "03237.

In like manner we may treat the case of 0 = 90°, i.e., when the point of
observation is on the equator. The odd P's now vanish and the even P's take signs
alternately opposite. The following table gives the values required for the direct
summation, i.e., up to n = 21 inclusive : —

(2n + l)10»+1x».Pn(90)

(2»+l)10n+1x«.P,,(90)

n.

Dn

n.

D.

0
4
8
12
16
20

- -0099+ -0990i
- -0462+ -3706J
. - -2370+ -7353 i
- -8273+ -0756i
•4879+ -OOOOi
- -3964
'

2
6
10
14

18

+ -0271- -2548 i
+ -0879- -5122 i
+ -8683- -758H
+ -5848- -002H
+ -4334

-2-0047 + 1 -2805 i

+ 2-0015-1-5272J

WITH AN APPENDIX BY PROFESSOR A. LODGE.

99

The next three terms, written without regard to sign, and their di (Terences are as
follows : —

22

•3688

24

•3470

- -0218

26

•3292

- -0178

+ •0040

The remainder is found, as before, to be

+ £(-3688) + i('0218) + -J (-0040) = + "1903.

The sum of the infinite series from the beginning is accordingly

+ -1871 - '2467 i (28),

(•1871)8 + ('2467)2 = -09588.

in which

The distance between the source and the point of observation is now
2csin45° = c.v/2.

The intensity in the actual case is thus '09588 as compared with '5 if the sphere
were away.

For other angular positions than those already discussed, not only would the
arithmetical work be heavier on account of the factors P,,, but the remainder would
demand a more elaborated treatment.

O 2

100

LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

APPENDIX.
By Professor A. LODGE.

TABLE of Zonal Harmonics ; i.e., of the Coefficients of the Powers of x as far as P20 in
the Expansion of (1 - 2x cos 6 + x~)-* in the form l+P,a;+P3a;3 + . . . + P,X'+ • •
for 5° Intervals in the Values of 0 from 0° to 90°. The Table is calculated to
7 decimal places, and the last figure is approximate.

0
5

10
15

20
25
30

35

40

45

50
55
GO

65
70

75

80
85
90

0
5

10
15

20
25
30

35

40
45

50
55
60

65
70
75

80
85
90

PI (= costf).

1-0000000
•9961947
•9848078
•9659258

•9396926
•9063078
•8660254

•8191520
•7660444
•7071068

•6427876
•5735764
•5000000

•4226183
•3420201
•2588190

•1736482

Po.

1-0000000
•9215975
•7044712
•3983060

+ -0719030

- -2039822

- -3740235

- -4114480
•3235708

- -1484376

+ -0563782
•2297230
•3232421

•3138270

•2088770

+ -0431002

- -1321214

- -2637801

- -3125000

1-0000000
•9886059
•9547695
•8995191

•8245333
•7320907
•6250000

•5065151
•3802362
•2500000

•1197638
•0065151
•1250000

- -2320907

- -3245333

- -3995191

- -4547695

PS-

1 • 0000000
•9772766
•9105688
•8041639

•6648847
•5016273
•3247595

+ -1454201

- -0252333

- -1767767

- -3002205

- -3886125

- - 4375000

- -4452218

- -4130083
•3448846

- -2473819

1-0000000
•9622718
•8532094
•6846954

•4749778

•2465322

+ -0234375

- -1714242

- -3190044

- -4062500

- -4275344
•3851868

- -2890625

- -1552100
•0038000

+ -1434296

+ -2659016

1-0000000

•8961595

•6164362

+ -2455411

- -1072262

- -3440850

- -4101780

- -3095600

- -1006016
+ -1270581

•2854345
•3190966
•2231445

+ -0422192

- -1485259

- -2730500

- -2834799

- -1778359

Nil

1-0000000

•8675072

•5218462

+ -0961844

- -2518395
•4062285

- -3387755

- -1154393
+ -1386270

•2983398

•2946824
+ -1421667

- -0736389

- -2411439

- -2780153

- -1702200

+ -0233080
•2017462
•2734376

1-0000000

•8358030

+ -4227908

- -0427679

- -3516966

- -3895753

- -1895752

+ -0965467
-2900130
•2855358

+ -1040702

- -1296151

- -2678985

- -2300283

- -0475854
+ -1594939

•2596272

•1912893

Nil

1-0000000
•9436768
•7839902
•5471259

•2714918
+ -0008795

- -2232722

- -3690967

- -4196822

- -3756505

- -2544885

- -0867913
+ -0898437

•2381072
•3280672
•3427278

•2810175

ue< LOO/

Nil

- -5000000

— 11VUI OU

Nil

O-tUfU/U 1U/ UUGI

• 3750000 Nil

Po.

PT.

PS-

P P

± p. X

10-

.

1-0000000
•8012263
+ -3214371

- -1650562

- -4012692

- -3052371

- -0070382

+ -2541595

•2973452

+ -1151123

- -1381136

- -2692039

- -1882286

+ -0323225

+ -2192910

•2316302

+ -0646821

- -1498947

- -2460938

WITH AN APPENDIX BY PEOFESSOR A. LODGE.

101

Table of Zonal Harmonics ; i.e., of the Coefficients of the Powers of x as far as P2r, in

the Expansion of (1 — 2zcos0-|-a;3)-i in the form l+P^+PjX* + . .. + Pnx" + .. .
for 5° Intervals in the Values of 6 from 0° to 90°. The Table is calculated to
7 decimal places, and the last figure is approximate — continued.

9.

PH.

Pi*

P«.

PH.

P».

0
5
10
15

1-0000000
•7639723
+ -2199746
- -2654901

1-0000000
•7242508
+ -1205620
- -3402156

1-0000000
•6822849
+ -0252742
- -3868998

1-0000000
+ -6383094
- -0639478
- -4048245

1-0000000
+ -5925694
- -1453436
- -3948856

20
25
30

- -4001361
•1739692
+ -1607048

- -3528461
- -0223995
+ -2732027

- -2682722
+ -1215469
+ -3066580

- -1585374
+ -2332489

+ -2584895

- -0376336
+ -2952537
+ -1465789

35
40
45

•3096940
+ -1712040
- -1041843

+ -2532528
- -0211959
- -2467193

+ -1130760
- -1892595
- -2393239

- -0565267
- -2599246
- -0972709

- -1950586
- -2083112
+ -0903925

50
55
60

- -2640939
- -1769491
+ -0638713

- -1987621
+ -0522404
•2337529

- -0019170
+ -2209602
+ -1658041

+ -1821884
+ -1959135
- -0571737

+ -2281988
+ -0110216
- -2100185

65
70

75

+ -2351950
+ -1864450
- -0305439

+ -1608831
•0787947
- -2274796

- -0863490
- -2239288
- -0850288

- -2197701
•0745390

+ -1687887

- -0989734
+ -1597121
+ -1638193

80
85
90

0
5
10
15

- -2145820
- -1988401
Nil

- -1307104
+ -1041876
+ -2255858

+ -1544264
+ -2010073
Nil

+ -1730902

- -0629592
•2094726

- -0860215
- -1982155
Nil

Pic.

PIT.

Pis-

£]<!.

P-,o.

1-0000000
+ -5453192
- -2173739
- -3594981

1-0000000
+ -4968206
- -2787566
- -3024136

1-0000000
+ -4473403
•3284945
- -2284640

1-0000000
+ -3971492
•3658960
- -1432466

1-0000000
+ -3465207
- -3905880
- -0527721

20
25
30

+ -0801110
•2997862
+ -0036143

+ -1815511

+ -2495290
- -1318805

+ -2560661
+ -1566049
- -2254922

+ -2965867
+ -0399984
- -2553464

+ -3002029
•0780855
- -2169986

35
40
45

- -2565851
- -0654984
+ -2150310

- -2244163
+ -0986597
+ -2100803

- -1151189

+ -2088162
+ -0857609

+ -0289684
+ -2180388
- -0809310

+ • 1556356
+ -1273281
- -1930653

50
55
60

+ -1133974
- -1714205
- -1498551

- -0732822
- -2012353
+ -0522168

- -1986904
- -0625381
+ -1922962

- -1792842
+ -1207910
+ -1377671

- -0359655
+ -1945128
- -0483584

65
70
75

+ -1249926
+ -1757158
- -0760903

+ -1956924
- -0336558
- -1924117

+ -0427633
- -1883363
- -0249700

- -1501989
- -0935549
+ -1696995

- -1644051

+ -1165241
+ -1093683

80
85
90

- -1912133

+ -0255526
+ -1963808

+ -0165069
+ -1908789
Nil

+ -1861639
+ -0082151
- -1854706

+ -0473145
- -1794383
Nil

- -1608344
- -0383005
+ -1761970

102 LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

EXPLANATION OF THE METHODS OF COMPILING AND CHECKING THE ABOVE

TABLES.

Calculation of the Even Orders.

The Zonal Harmonics of even order in the foregoing tables were calculated from
the formula obtained by expanding (1 — 2x cos 6 + x")'* by means of the form

(«„ + ape* + ...+ <*&#* + ...+ o^V" + . . .)
where

i q F. /o™ _. i \

J.*O«t/*»*ljU/JLf -i

2 . 4 . 6 . . . 2r

whence

P2j! ($) = a,~ + 2an_lan + l cos 20 + . . . + 2«0a2n cos 2«0.

To calculate the coefficients a,,", 2a,,_]r/,/, + 1, . . . 2a,,r«2,1, an auxiliary table of values
of log]0 «, was formed from r = 1 to ?• = 20, to 8 decimal places ; and a similar table
of log]0 2«, from r = 0 to ?• = 9 ; so as readily to combine them to form (to 7 decimal
places) the logarithms of the required coefficients for different values of r>.

The coefficients were then calculated to 7 decimal places from their logarithms, and
checked for each value of n by seeing that they added up to unity in each case.

Next, a table of values of log cos 26, log cos 46, ... to 7 decimals, was formed for
all values of 6, at 5° intervals, from 5° to 90°. The addition of these to the
logarithms of the corresponding coefficients gave the logarithms of the various terms
(except as regards sign) in the above expansion of P2ll (ff). From these logarithms
the terms themselves were calculated to 7 decimals and tabulated, the positive terms
in black, and the negative terms in red ink. The accuracy of these terms was
checked by making use of the identities

(1.) 2 cos 60° = 1,
(2.) cos 50° + cos 70° = cos 10°,
(3.) cos 40° + cos 80° = cos 20°.
This, in addition to the primary identity

checked all the terms effectually except those which were multiples of cos 30°.,
These were checked by adding a number of them together and comparing their sum
with the sum of the coefficients multiplied in a lump by cos 30°.

WITH AN APPENDIX BY PROFESSOR A. LODGE. 103

In these ways all the separate terms were ensured to be free from errors due to
carelessness in taking proportional parts, or any other incidental errors.

Then the terms were added together for each value of 0 in P2/1 (6} for a given value
of n, so obtaining the values required for the actual table. By adding I mean to
include also subtracting, the artifice of putting positive terms in black and negative
terms in red being a great help in this part of the work. Errors in this work were
corrected by adding all the values of P2l, (6} from 6 = 5° to 6 = 90° for a given value
of «, and comparing the result with the sum obtained in a different way (see note at
the end of the second auxiliary table appended). Up to P1;J the additions and
subtractions and checkings were all done without mechanical aid, but for the later
values of i), from Pu to P2u, I made use of an EDMONDSON'S calculating machine which
was very kindly lent to me by Professor McLEOD.

In this way all the even harmonics were calculated and were ensured to be free
from errors, except those incidental to the last figure, which is, of. course, only
approximate, as the terms used in the calculation were evaluated to 7 decimals only.
I am confident, however, that the last figure is never far from the real value, and
that it would be more accurate in every case to retain it in numerical work with the
tables than to omit it. The error is not usually more than i '2 in the 7th place,
and I am confident that it never exceeds ^b 3, whereas omitting it would lead to a
possibility of ^ 5 m Addition to its actual error, i.e., to a maximum error of i 8.
I have assumed that there are very few numerical calculations requiring an accuracy
greater than an approximate 7th decimal place, and that, therefore, the vastly increased
difficulty which would have been caused by working throughout witli 8 decimals
would have been wasted labour.

Calculation of the Odd Orders, and Final Checking.

When the even orders were calculated, the question arose as to the be.st way of
calculating the odd orders. P,, of course, gave no difficulty, being merely cos 6. P3,
also, was quite easy to calculate directly from its value -g- (3 cos 6 -f- 5 cos 3$),
EDMONDSON'S machine being used for the purpose. The remaining odd functions
were calculated from the even ones by means of the identity

(In - 1) ?,?„_, = nPa + (n - 1) P....

The accuracy of the results was checked by recalculating the even P's from the odd
ones by means of the same formula. This clinched everything.

The mode of using this formula which I adopted, between 6 = 5° and 6 = 60°
inclusive, was different from that adopted between 6 = 65° and 6 = 85° inclusive, so
as to minimize the effect of 7th-figure inaccuracies as much as possible.

104 LORD EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

Up to 0 = 60° I used it in the form

P - (_n + DP. + i + ttP— i
(2n+l)P,

where PI varied from 1 to |- ; each P being thus dependent on its immediate
predecessor and successor.

Beyond 60° I thought it better to use it in the progressive form

p . _ (2n - llP.P.-.-fr-lVP.-g

n

each P being thus calculated from the two preceding orders.

1 believe that in this way the maximum risk of a 7th-figure error occurs at 60°,
when Pj = |-, and is not very great even there, whereas the exclusive use of either
method would have greatly magnified the error at one end or other of the table.

Auxiliary Tables.
TABLE of Values of log ar and log 2ar.

r.

log «,..

log 2ar.

0

Nil

0-30103000

1

1-69897000

Nil

2

1-57403127

1-87506126

3

1-49485002

1-79588002

4

1-43685807

1-73788807

5

1-39110058

1-69213058

6

1-35331202

1-65434202

7

1-32112734

1-62215734

8

1-29309862

1-59412861

9

1-26827503

1-56930503

10

1-24599864

11

1-22579525

12

1-20731185

13

1-19027851

14

1-17448424

15

1-15976098

16

1-14597270

17

1-13300772

18

1-12077326

19

1-10919139

20

1-09819601

WITH AN APPENDIX BY PROFESSOR A LODGE.

105

TABLE showing the Acute Angles (in degrees) whose Cosines were required in
Forming the Terms belonging to the Harmonics of Even Order. The Signs pre-
fixed to the Angles Indicate whether their Cosines had to be Added or Subtracted.

9.

26.

461.

8ft

86>.

106*.

12ft

14-9.

166*.

189.

200.

5

+ 10

+ 20

+ 30

+ 40

+ 50

+ 60

+ 70

+ 80

+ 90

-80

10

+ 20

+ 40

+ 60

+ 80

-80

-60

-40

-20

- 0

-20

15

+ 30

+ 60

+ 90

-60

-30

- 0

-30

-60

+ 90

+ 60

20

+ 40

+ 80

-60

-20

-20

-60

+ 80

+ 40

+ 0

+ 40

25

+ 50

-80

-30

-20

-70

+ 60

+ 10

+ 40

±90

-40

30

+ 60

-60

- 0

-60

+ 60

+ 0

+ 60

-60

- 0

-60

35

+ 70

-40

-30

+ 80

+ 10

+ 00

-50

-20

+ 90

+ 20

40

+ 80

-20

-60

+ 40

+ 40

-60

-20

+ 80 + 0

+ 80

45

+ 90

- 0

+ 90

+ 0

+ 90

- 0

+ 90

+ 0

+ 90

- 0

50

-80

-20

+ 60

+ 40

-40

-60

+ 20

+ 80

- 0

+ 80

55

-70

-40

+ 30

+ 80

-10

+ 60

+ 50

-20

+ 90

+ 20

60

-60

-60

+ 0

-60

-60

+ 0

-60

-GO

+ 0

-GO

65

-50

-80

+ 30

-20

+ 70 +60

- 10

+ 40

±90

-40

70

-40

+ 80

+ 60

-20

+ 20 - 60 - 80

+ 40

- 0

+ 40

75

-30

+ 60

+ 90

-60

+ 30

- 0 +30

-60

+ 90

+ 60

80

-20

+ 40

-60

+ 80

+ 80 -60 +40

-20 +0

-20

85

-10

+ 20

-30

+ 40

- 50 +60 - 70

+ SO + 90

-80

90

- 0

+ 0

- 0

+ 0

- 0 +0

- 0

+ 0

- 0

+ 0

1

i

Note. — The terms in each of the columns headed 40, 80, 120, ... all balance, their
sum being zero.

The terms in each of the columns headed 20, G0, 100, ... balance except the
last term.

Hence the sum of all the 18 values of P« (0), from 5° to 90°,

This equivalence was made use of in finally checking the values of the harmonics of
each even order.

Some Cliaracteristies of the Functions as shown by the Tables.

The functions become more and more undulating as n increases, P«(0) having n
zeroes between 0=0° and 180°, similarly spaced on either side of 90°. The most
remarkable peculiarity noticeable in drawing their graphs is that the intervals
between the successive zeroes from the first to the «"' are almost exactly equal.

The graph of P20 is reproduced in fig. 2, to emphasize this peculiarity of equal
intervals.

VOL. com. — A. r

106

LORD KAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

oo

-p

O

WITH AN APPENDIX BY PROFESSOR A. LODGE. 107

In this respect LAPLACE'S approximate formula for high values of «, viz. : —

P,, (ff) =

-
y (mr sm 6)

cos nd

4

shows a wonderful resemblance to the actiial functions even for quite low values of n.
The numerical values of this function are, indeed, not very near the true values
even when n = 20, as will be seen by the following short table :—

Approximation.

True value.

Pro (15°)

30°

- -04577
- -21851

- -05277
- -21700

45°

- -19602

- -19307

60°

- -04962

- -04836

75°

+ -11051

+ -10937

90°

+ -17841

+ -17620

But, though its numerical values are not very close, the positions of most of its
zeroes are remarkably near the correct places. It can, of course, only be considered
between 0° and 180°, since sin 6 becomes negative beyond these limits. But
between these limits it has n zeroes, with n — 1 equal intervals between them, the
first zero being at 9 = 270° -=- (2n + 1), and the interval between successive
zeroes being 360° -f- (2n + 1), the formula for the required values of 9 being
9 — (4r + 3) 90° -f- (2n + 1), for integer values of r from 0 to n -- 1 ,

Taking n = 20, this would make the first zero approximately at G° 35', and the
constant interval 8° 47', very nearly; the roots given by the formula being, roughly,

6° 35', 15° 22', 24° 9', 32° 55^', 41° 42-£', 50° 29', 59° 16', . . .

The actual value of the first root of P.-,0(#) = 0 is slightly over f>° 43', and intervals
between successive roots are very nearly equal, varying between 8° 43' and 8° 47'.

The first ten roots are, to something like the nearest minute, 6° 43', 15° 20',
24° 11', 32° 57', 41° 44', 50° 30', 59° 10', 08° 3', 70° 50', and 85° 37'.

Professor PERRY has brought out a table of Zonal Harmonics to 4 decimals, for
every degree, as far as P7 (' Phil. Mag.,' December, 1891), and by help of this table
I have calculated the first root, and the intervals between successive roots, for P3 to
P7, to something like 1 minute accuracy. Their values, and the corresponding
approximations obtained from LAPLACE'S formula above, are given in the following
table, showing how far they differ for these low values of n : —

P 2

108 LOED EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE,

— ; —

First root, LAPLACE'S
and successive approximation.
intervals.

First root, LAPLACE>S
and successive
intervals. approximation.

P, 3°9 14 38 34

P,,

21 11

20 46

50 46 1 51 9(5
50 46 f

27 26]
27 35

27 36

27 41 £

27 35

P4 30 33 30 0

27 26

39 341-1

39 45 ). 40 0

39 34^J PT

18 24

18 0

23 45

P-, 25 1 24 33
32 24 1

23 54
23 57
23 57

-

24 0

32 35 1 „., 10

23 54

32 35 f

23 45 J

32 24 J

I

These examples indicate that the first root is always greater than the value given
by the approximate formula, and the successive intervals are slightly less.
The actual roots of P7 are, approximately,

18° 24', 42° 9', 66° fr 90°, &c.,
and those given by the Laplace formula are

18°, 42°, 66°, 90°,

so that the true roots are all a little ahead of those given by the Laplace formula,
so long as 6 is less than 90°. Beyond 90° the roots are, of course, similarly spaced in
everse order.

NOTE BY LORD RAYLEIGH.

Professor LODGE'S comparison of P20 with LAPLACE'S approximate value suggests
the question whether it is possible to effect an improvement in the approximate
expression without entailing too great a complication. The following, on the lines of
the investigation in TODHUNTER'S ' Functions of LAPLACE, &c.,'* § 89, shows, I think,
that this can be done.

MACMILLAN and Co., London, 1875.

WITH AN APPENDIX BY PROFESSOR A. LODGE. 109

We have

P,, = -,-~- -r {sin (n+l)0
irk(2n + 1) I

• - («),

with

When n is great, approximate values may be used for the coefficients of the sines
in (a). To obtain LAPLACE'S expression it suffices to take

1 1_._3 J_. 3 . 5

2' 2.4' 2 .~4 . 6'

but now we require a closer approximation. Thus

1 . (2n + 3) 2 \ 2n + 2

1.2. (27* + 3) . (2» + 5) == 2 . 4 \ ~ 2n + 2 " 2n + 4 >

and so on. If we write

r — l

2n'

the coefficients are approximately

1 1_._3 0 T_^3 . 5 o

2 lT 2 .1 X 274 76 *

and the series takes actually the form assumed by TODHUXTER for analyticnl
convenience. In his notation

1 3

C = t COS 0 -\- ifi COS ?>0 -j- £5 COS 50 + . . .,

2 . 4

and

S = t sin 6> + l <3 sin 30 + * ' 3 *6 sin 5(9 + . . .,

P, = , /04 lS (0 sin n0 + S cos n6\,
TTK (2n + 1)

where ultimately £ is to be made equal to unity.
By summation of the series (t < 1),

t t

C = —r- cos (0 + j<^)) S = —j~ sin (0 +
v/jo vp

110 LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE.

where

i2 sin 20 /<>•>

32 = 1 - 2£2 cos 20 + t\ tan 0 = — -^ rZa • • • - (<>)•

For our purpose it is only necessary to write C/t and S/< for C and S respectively,
and to identify t~ with x in (y). Thus

sn

(e),

and <f> being given by (3). We find, with t — 1 — ^

1

I

so that

and

whence

32 = 4sin20[l -

= 2 sin

1
8»

(0;

sin 20
tan^==2si,r0 + l/2»

cot <

4?;.

Using (£),

in (e) we get

^ Jl-ll.oosj

n sin 0)1 4«J

TT COt 0

, - Q

on

}...

which is the expression required.

By this extension, not only is a closer approximation obtained, but the logic of the
process is improved.

A comparison of values according to (6) with the true values may be given in the
case of n equal to 20.

VALUES of P20.

e.

True value.

According to (6).

15

- -05277

- -05320

30

- -21700

- -21712

45

- -19307

- -19306

60

- -04836

- -04834

75

+ -10937

+ • 10937

90

+ •17620

+ -17618

[ "1 ]

/ Tarmonic Functions.

INDEX SLIP.

/ Trinity College,

l)ju»wijj, (>. II. — On the IntegriiU of tlie Squares of Kllipsoiilal Surface
Harmonic Functions.

Phil. Trans., A, vol. 203, 19O*, pp. 111-137.

Kllipsoiilnl Harmonics- -Integrals of Squares of Surface Functions over
Surface of Ellipsoid.

lUiiwnr, (>. H. Phil. Trans., A, vol. 2(>3, 1904. pp. Ill -137.

• usactions,' namely, "' < >n 1 ,ilV ,-.••
"On the Pear-shaped Figure

pp. 301-.". __

vol. -JOO. I

'Pour-shaped Figure." uml

In - llnnumiM-.s; tl,.- tm,.-:.

LI tl.p

is therefon
Tin- an;

In ': HarHKiuics ' ti

it* solid harmonic fui

>proximate forni.s, tn

"tory results \\

ing it (litiiciili t

the harmonic '

.}fin \»- M|

in th»

of the ' Philosophical
-ol. l'J7, pp. 4(31-557,
-; of Licpiid," vol. 1 98,

of Equilibrium, &c.,"
, " Harmonics," " The

imately, approximate
isoid of the squares ot
piired whenever it is
he evaluation of them
s.

; integrals was very
;i improvement might
1 do not care to spend
sis.

en the three factors of
obtaining convenient
ossihle to obtain such
vd the disadvantage of
the surface harmonics,
ive are susceptible of
jme but not of all the
DUS expressions for the
it paper is to complete

ation. That used in
mmetrical expressions

25.3.04

112

PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

involved, but the notation used in the two later papers seems preferable where the
formulae are rigorous and symmetrical.

In " Harmonics " the squares of the semi-axes of the ellipsoid were

o 7 o i a 1 ~T~ /J \ 7 2 70, / ^ i\ 2 722

a" = «"( i/" ---- f>}> ° = * (" ~ 1)> c = k " •

The rectangular coordinates were connected with ellipsoidal coordinates v, p,, <f> by

x~

2

The three roots of the cubic

=1

were

u

/ -•> 2 7,2 2 I." 1 — ft COS

- K V , Mo — AT/A", M3 — ft" ~y_ o

Lastly v ranges from co to 0, /n between i 1, 0 from 0 to ^TT.
In the two later papers 1 put

,
1 + /8

K sin y

= sin

and for convenience I introduced an auxiliary constant /3 (easily distinguishable from
the ft of the previous notation) defined by sin ft = K sin y.
The squares of the semi-axes of the ellipsoid were then

.: _ k~ cos3 y ,.2 _ AT cos2 ft o _ k'
sin2 ft ~sm3"/8" ~ sin2 /3 '

The rectangular coordinates became

The roots of the cubic were

7. '2

This is the notation which will be used in the present paper.

F

K*

cos

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 113

If da- be ,in element of surface of the ellipsoid, and p the central perpendicular ou
to the tangent plane, it appears from the formula at the foot of p. 257 of " Stability "
that

P dv . - ^'3 cos ft cos y *2 cos2
'

AF

where A2 = 1 - /c2 sin2 0, F3 = 1 - K'- cos2 tf>.

In the previous papers I have expressed the two factors of which a surface
harmonic consists by $,*(/*) or P,-'(/x), and <£f ($), ds (<f>), S/ (<£) or S,'(<£), one of the
two P-functions being multiplied by one of the four cosine or sine functions.

Taking a pair of typical cases, the integrals to be evaluated are

> &f* do- and < &' da:

As it will be convenient to use an abridged notation, I will write these integrals
//(cos) and //(sin), according to an easily intelligible notation.

These functions involve integrals of even functions, and therefore we may integrate
through one octant of space, the limits of 6 and <j) being ^TT to 0, and multiply the-
result by 8.

It is clear then that

j,/cos) __ 8 cos cosy
sin:!/3

r

Similar expressions are applicable to all the other forms of function, but we may
proceed with this form as a type of all the others.

This formula shows that the variables are separable, and since we might substitute
5-77- — x// for (f> without changing the result, the (f) integrals are of the same type as the
6 integrals.

It^ias been stated above that two of the roots of the cubic equation are proportional
to K- sin2 6 and (1 — /c'2 cos2 </>). By the nature of the harmonic functions it follows
that if [^;s(/x)]~ is proportional to a certain function of /rshrfl, [(£/*(</>)]' is proportional
to the same function of (1 — /c'2 cos'3^)).

It follows that if ($,*)- = F(K~ - «- sin2 0) = F (K° cos2 6),

where a is a constant, which for the present we may regard as being unity. If then

we must have

[<£,•' («£)]2 = An - AlK'* sin2 j> + /V sin4 $ - A^K'* sin6
VOL. CCIII.— A. 9

114

PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

Accordingly if there is a term An [/r" + 2 cos2" + * 0 ( ' in j K~ cos2 B ($,*)2 ^ and a term

fc'J*sin2*^ -^ in ((C')2^> then there must be a term

cos

8"

in

[(•JjJ/)8 , and a term ( — }" A ,,U'2" + - sin2" "^> -2. It follows that the coefficient of

8/r5 cos B cosy •
s)-f. s.nif Ms

|K-"' ' = COS'" - - 0 '™ |K"-'"' Hill*" ,/> '^ - (-)— + ' f *5" COS5" 0 ^ f «'2" + 2 sin2" + * <f> <** .

For the snko of hrcvity I call this function [2?t + %2, 2m], and we may state that one
term in the required expression is ( — )'"AaA,a [2» + 2, 2m], where [ ] indicates the

T , i x 8&?1 COS B COSy

above [unction 01 the tour inteffrals. It follows that 1* (cos) -=- — .- ., n

sin' B

A-

.!,,/„ 4,0]- A? 4, 2]-f .1,J.,[4, 4

. i

0. o — ,-1,/J, [<;, 2 + A." [r,, 4 ] — . . .

.... (1).

Since

[•2n, 2m] = [*-" co8~"0(W jV-"' sin2'" f/> f/f - (-)"-'" f^'" sin2'" 6> fW f/c'2" sin2"</> ^,

it is clear that

[2>,, 2m] = -(-)"-'" [2,n, 2ft].

Hence if n and ??i differ by an odd number [2n, 2m] = [2m, 2ft], and if they differ
by an even number [2n, 2m] = — [2 in, 2n\. Also \2n, 2n~] = 0.

c1." lift riir r!A\

T J_ " 1 f n ^ i On /I tvv i' r\ ~) / /-^ji " OK I tt'li) i |

Let us write (2»ij = /c" cos2" ^ . !2»; = K -" sin2" </> --L, so th;

.'o A Jo r

[2n, 2m] = {2n} {2m}' — ( — )"-'" {2n}' {2m}.

We must now evaluate these functions.

Since A2 = 1 — K- sin2 0, we have bv differentiation

at

{ (-2n - 1 ) K2 cos-" 0 - (-2n - 2) (/c2 - *'2) cos2—2 0 - (2ft - 3) *'2 cos2" -* 0}.
Integrating between \-n and 0 and multiplying by /c2""2 we have

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 115

and by symmetry

{2n}' = - f- " 2 (K- - x'~) {2» - 2]' + *"_ ' 3 K~K'~ \2n - 4}'.

Multiplying the first of these by {2m'/ and the second by —( — )»-'" (2m| and
adding together we have

^, ,,)

'" -» ir-i T sttv Ufa /o\ PA i \ ~\ i -j/fc ^~ •> o /.i t-,^ . ~. -\

I 2n, 2m] = (K" — /c -) [2n — -2, 2m] + K-K - \_2it — 4, 2m].

By successive applications of this formula we may reduce any function [2«, 2m]
until it depends on [2, 0], but the result becomes very complicated after a few
successive reductions.

Now

*£-*. w'-ff

. o A . o I

{2j - =

cos-

sn

Then

o A Jo r

= KV + F'F - FF'.

But it is well known tliat this combination of the complete elliptic integrals with
moduli K and K is ^TT.*

Hence [2, 0] = \TT.

It seems unnecessary to reproduce the simple algebra involved in the successive
reductions, and I therefore merely give the results, as follows:—

[0, 2] = [2, 0]=^
-[0, 4] = [4, 0] = |(^-/<'-)47r

2, 4 | = | 4, 2]=4/<V~.^7
-[2, GJ= 6, 2]=.', (ic2-^)^3. I-TT

[G, 8] = [8,Gj = -l/cV(147r

These are the only functions of this kind which are needed for the evaluations of
integrals in this paper.

* See for example DUIIEGE'S ' Theorie der Elliptischen Functionen,' p. 293.

116 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

When these functions are introduced into (1) and the terms re-arranged, I find :—

T i v 7J- cos /3 cos y
If (cos) 4- - - T-£B~ ~ =
sm3/3

A* - £ M*A* + i
+ 2 (,c2 - /c'2) [-A. ^ _ ^ KWA,AZ + -6-?7. KVM

+ linr CH2-4) - 3W»;M043 - srb ti (4-6) -

+ 57779 [* (6- 8) - 7W*] *V*,M, - -7-e«n-[-£ (H. 10) - 92*V2J «VM A + . . .

' + 525*'^) A'J.i. - (2).

Li this result a good'many terms are added which are not deducible from the table
of functions given above, but every term as stated here has actually been computed.
The laws governing the succession of terms in the first six lines seem clear, but I do
not claim that the proof of the laws is rigorous. I do not perceive how each series
is derived from those preceding it, and I have no idea how the series beginning with
J,,,!.!. would go on. With sufficient patience it would no doubt be possible to
determine the general law of the series, but I do not propose to make the attempt at
present, since we have more than enough for the immediate object in view.

This result (2) is, of course, equally applicable to the integrals of the type //* (sin).

In order to effect the required integrations we must define the functions, and I
take the definitions (with a few very slight changes) from § '2 of " the Pear-shaped
Figure." In order to use the preceding analysis it is necessary that the square of the
P- function and the square of the cosine or sine function should be the same functions
of K- cos2 9 and of — K'~ sin2 (j>. But as in the definitions to be used this symmetry
does not hold good, a difficulty arises, which may, however, be easily overcome. If
the P-function be multiplied by any factor /, and the cosine or sine function by any
factor g, the integral will be multiplied by f'2g~. I therefore introduce such factors
j and (j as will render the residual factors of the squares of the P and cosine or sine-
functions symmetrical in the proper manner.

It seems desirable to show how the results found here accord with the approximate
integrals as found on pp. 548-9, § 22, of " Harmonics." In this connection I

remark that ~~'9a~~ f when written in the notation of "Harmonics," is

£V (z/3 — I)1 ( v1 — -• LLE) a factor which I denoted in that paper by M.
\ 1 — pi

It does not seem necessary to give full details of the analysis in the several cases,

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 11

since it is sometimes tedious, and it merely involves the substitution in the formula
of the values of AQ, A^ An, &c.

We will now take the several harmonics successively.

HARMONIC OF THE ORDELI ZERO.

This harmonic is simply unity, so that Al} = 1 and all other .1's vanish. The
formula is

This is obviously right since the integral is |^"/T, of which this is the known value.

HARMONICS OF TL-IK FIRST OKUEK.

Here we have all the .1's zero excepting .1,, ;md .1,. and \vlien the functions have
the proper symmetrical forms, we have from (i!),

r,/cos\ _ 47rA,") Cos B cos y ,- , ., i e « ..» 2 _L 9 / 2 >:\ i i -\ / . oh

' Isin/ " "sin;J {3~~

(I) Tlie Zonal Harmonic.
I define this thus,

1 . . . ,,.

= ( I - /c': cos- r/.)-' = // . (K- + «'- sin3 </;)', J

1

where J = , y = I.

K

On squaring Jj^i (p), it is clear that

Whence I find

T / \ 4?r/;3 cos B cos y /.2

Since with definition (-l)f~g-K° = 1,

A __ -l^3 cos )8 cos y

In " Harmonics " this harmonic is defined by

-^cos2^) . . . .'(6).

118

PROFESSOlt G. H. DAKWIN OX THE INTEGRALS Of THE

Now we must take for f and g values such as to bring the two definitions into
accord. This is the case if

and /?</V = 1 + A
Hence

/ x 47rM ,
(cos) - (1 +

agreeing with the result on p. 549 of "Harmonics" for the case i
type OEU.

(2) The Scctorial Cosine Harmonic.
I define this thus.

P/ (M) = (1 - K2 sin2 0)' = f. (K'- + K- cos2 0)*,

• • (7),
1, *• = 0,

<£,
where f =],;/= , .

s f/>

= / . K~ — K- sir

By symmetry with the last result

r , , x 47r/':i cos /3 cos y ..., :, ,., 4-rrk"' cos fi cos y
J11(cos)= , . 3 /./-ryV-= . 7 ^

J 3sms/3 3sm3p

In " Harmonics" 1 defined tlie functions thus,

• (9)-

(i^1 (^)) = cos (j)

If we take f = (] +/?f, </K' =

\\ — /8/

//(cos) = >AI

. (10).

J

I -/3

definitious agree, and we have

|ffM (1 + 2)8 + 2/3-) .... (11).

This agrees with the result on p. 54'J of " Harmonics' with / — !,•'>•= 1, type OOC.

I define this thus,

(3) The Sectanal Sine Harmonic.

,l(/jl) = Cos 0 - f, K cos 0, "1

' (<^) = sin rji = >/. i/ — I . K' sin (/>, f

(12),

where /'= , '/ —

K K\/ — 1

SQUAUKS OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. lit)

On squaring p^ we find A0 = 0, A\ = 1, and

In " Harmonics" the definitions were the same, and therefore

This agrees with the result on ]>. 548 of" Harmonics" with i = 1 , ,s- = 1 , type OOS.

HARMONICS OF TUK SKCOND ORDI.;U.
[n these the only coefficients are Alt, A}, A*, and (2) hecomes,

f , /COS \ _ 47I-F COS /3 COS 7 r , , __ ! 3 ,., .0,1 4 /| 4 3

-() * " '

with .v = 0, 1, 2.

(1) «W (4) The Zonal and Scctorial Cosine Harmonics.
These are defined thus,

$,*(/x) = /c--sm-0-7-, 1

^ ...... (15),

(•T/ (^) = 5'3 - K'* cos3 </,, (,• = 0, 2) J

where (/ = £[! + /c2 ^ (1 — K'V~)*], with upper sign for .s = 0 and lower for s = 2 ;
and (/'- = 1 — */2.
Writing

• OOO /.T /.T 1 I- O /.) - / , 0 /.T\l-l

ty = K~ — <f = q~ — K~ = jj IK" — K ~ ± (1 — K~K ~)1J,

..... (15),
= (J . t K~ sin2 </,), (,s = 0, 2) J

where f=l,g=l. It may he noted that t- is a symmetrical function in K" and
Squaring ^2* we find

120 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

After reduction I find, for s = 0, 2,
/,' (cos) = 4rf cos /3 cos y _

o SHI

^ (4 _

Now

9«* = 2 - 5/c'V3 ± 2 (/c - *'2) (1 - *V2)!,
27ifi = (4 - 7 *V2) (/c- - K"~) ± (4 - 13/cV2) (1 - /c/c'2)*,
81 1* = 8 - 40/<V2 + 4 1 /cV1 ± 4 (2 - 5/cV2) (/c3 - /c/2) (1 - ^/c'2)1.

Whence on substitution, with t/"2,(/3 = 1 ,

47Tn" COS p COS "V Li i

rt \

/g'(cos)=

_

O L I L

i i-/ . o /I\-T

. 4[(L - K-K~)~

' J

The upper sign being taken for the xoiml (x = 0), the lower for the sectorial
harmonic (x = 2).

If these expressions be developed in powers of K as far as three terms of the
series, I find, on re-introducing the factor f'~y~,

(17),

t

. . (18).

In " Harmonics" [ made the followin definitions

In order to make the two forms of definition agree we must take
Thus

. . (19).

Now on development

*8 = (DM1 - 5** + W*) =

- 10/3 + i

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 121

Whence

Introducing this I find

7 cos = 45

72 (cos) = 4?rM(l + 3/3 + 3/3*) ....... (20),

agreeing with the result on p. 549 of " Harmonics" with i = 2, a = 0, type EEC.
Again in " Harmonics "

3~ M = 3 ftP, (p.) + 7V (/.) = 3/8 + 3(1- f/3) cos- 0, ]

f . . (21).
&8 + cos 2(£ = 1 + l/S - 2 sin3 0 J

In order to make the two definitions agree we must take

or

So that /V: = 2' . 3* (1 + 8 - IF).

Introducing this in (18) we have

/3a (cos) = ^M. 12(1 -£ + /32) ....... (22),

agreeing with the result on p. 548 of " Harmonics " witli i = 2, *• = 2, type EEC.

(2) The Cosine Tesseral Harmonic.
These are defined thus, —

P.,1 (p.) = sin 0(l-K* sin- ff)* = f. (/ce - /ce cos- 6>)4 (K'- + /c2 cos3 0)4,

(23)

C3l (<£) = COS <^ ( I - K* cos--' ^)1 = g (K3 + K'2 SiU- «^)i (K'- - K'" sill3 ^)»,

, 1 1

where / = - , «/ = ^ •

Squaring P.21 we find

A0 = JK"\ A, = (K~ - /c'2), A, = - 1.

On substituting in the formula, I find, on putting f-g'icK'1 = 1 and reducing,

^ ....... (24).

5 SIB3
VOL. CCIII. — A. R

122 PROFESSOE G. H. DAE WIN ON THE INTEGRALS OF THE

In " Harmonics " the definitions were

P,1 M =

o1^) = (l - /3 cos 2^)* cos <f> =

cos

o

In order to make the two definitions agree we must take

= 3

so that /3</W'3 = 33 ^ + ^ = 3~ (1 + 3/3 + 4,8-). On multiplying (24) by this

1 — p

factor, we have

(26)

8i (cos) = *" M . 3 (1 + 3/3 + 4/3-).

O

agreeing with the result on p. 549 of " Harmonics " with i = 2, s = 1, type EOC.

(3) The Tesseral Sine Harmonic.
This is defined thus,—

P2i (p) = sin e cos e=f.K cos 0 (K- - /c2 cos- 6)*,

So1 ((^>) = sin (f>([ — K'~ cos" <£)4 = r/ . v/ - 1 K' sin ^ (/r

(27)

:- sn-

, 1 1

where / = , , <j —
-

Squaring ^2! we find

— 1

A(> = 0, ^j = /ce, ^1. = --

whence, on putting — f:g-K4x''2 = 1,

In " Harmonics " the definitions were
$.,i (|LI) = P.^ (p.) = 3 sin 0 cos 0,

S,1 (</>) = sin <> 1 - /3 cos 2<)i =

5 sin3 y8

(28).

sn

—
1 +P

Therefore, to make the two definitions agree, we must take

— cos

(29).

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 123

<>
Therefore — /~(/Wa = 3a (I + ft), and on multiplying (28) by this factor we have

/9i (sin) = ^M. 3(1 + 0), ........ (30)

agreeing with the result on p. 549 of ': Harmonics " with i = 2, s = 1, type EOS.

(5) The Sectorial Sine Harmonic.
This is defined thus :—

= sn (> cos <.

j

If in the last integral we had written \ir — 0 for (£, and |-TT — <£ for 9, and K' for *,
l^.,1 would have become ,£2S, and S.>' would have become P.,'-'. Therefore the result (28)
gives what is needed by merely interchanging K and K.

Therefore

/8>in) = 4^5^|?? . i ....... (32).

For the purpose of comparison I must put

32(<A)=0- v/' - 1 *' sin </. (^ - *'- sin2 ^)*, (31)

TO-/ • \ vr cos /3 cos y / i ,., .^ « ,,\ /..0\

and /as(sin)= 1 r (~ i/W^*) .... (33).

«J olll ^J

In " Harmonics" the definition was

= sin 2(> =2 sn cos <>.

In order to make the two definitions agree we must take

Thus — /Y/cV* = 22 . 3s ILJ^; introducing this in (33) we have

2(l+2^ + 2n . . . (35)

agreeing with the result on p. 548 of " Harmonics" with i = 2, s = 2, type EES.

B 2

124

PROFESSOK G. H. DARWIN ON THE INTEGRALS OF THE

THE HARMONICS OF THE THIRD ORDER.
In these the only coefficients are A0, AI} A«, As, and (2) becomes

' COS

sin

snr p

+ 2 (/C2 - /C'2) [i A0A ! - A /CVM A + & lA

+ -& (4 - 9/cV2) /V2 - yihr (12 - 25*V2) KV

A (^ ~

(* = 0, 1, 2, 3).

(1) and! (4)

These are defined thus :—

m/ (u.) = sin S (/c2 sin2 0 -

Second Tesseral Cosine Harmonics.

= « - K cos

2

- /c'2 cos2 <£)*, (s = 0, 2

• • • (36)

where g3 = f [1 + K2 ^ (1 — f K2 + «^4)i]5 with the upper sign for s = 0 and the lower

for s = 2. Writing

= i [3/c2 - 2 ± (4 - 7/c2 + 4/c4)'] = i [1 - 3/c'2 ± (1 - /c'2 + 4/c'4)4]

C3' ((/») = .
where f = - , g = 1.

K

Squaring ^./ we find

- K cos
^ + K'2 sin2

After some rather tedioiis reductions I find (for ,s = 0, 2)
' (cos) = 4^3?&3 «C°8 y - (t ^ - A (1 - SO «« + y§5 (4 - 25K'2

olil ^J

+ -^ (2 - 5^) K2K'2 <2 + 3-L Kv*} /

Now writing D = (1 — K/2 + 4«:/4)iJ

5<2 = 1 - 3/c'2 ± Z>,

52 «* = 2 - 7/c'2 + 13/c'+ ±2(1- 3/c'2) D,
53<° = 4 - 21/c'2 + 48/c'+ - 63/c'fi ± (4 - 19/c'2 + 31/c'*) Z>,

= 8 - 56/c"2 + 177/c'* - 314/c'c + 313/c'8 ± (8 - 52/c'2 + 136/c'4 - 156K78)!).

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 125

On substituting these in the above expression, and noting that Kzf-y~ will be unity
with the definition adopted, I find

(38).
/32 (cos) = the same with the sign of D changed.

If these expressions be developed in powers of K', and if the factor ic f-y" be re-
introduced, I find

= -?M (§)*(! -40 + ^0*). </V

ro/ \ 4irP COS /3 COS y /,/, » . 31

:! (C°8) = 7 sin" £ ' «5 ( K + ' "

= ^ M . ^ ^ ( I - 2/3 + ¥ /8':) . «

In "Harmonics" I defined,

P3 (/,) = P;s (/z) - i /8/V (/.) = sin 0 [f sin2 ^ (1
In order that our previous definition may agree with this we must have

Now

whence //c = f(l +I/3 + 5y82), and this value of /«• satisfies the second equation.

With regard to Cs(<j>) there is a mistake in the table (the only one I have detected
therein) on p. 556 of " Harmonics," for the coefficient of the second term should r.ot
be 3yS but f /8. The mistake obviously arose from my using the formula for j),2
instead of that for p'3 as given on p. 490.

With the corrected coefficient the definition is

C3 (<£) = (1 - /3 cos 2^)* (1 - f ft cos 2<£)

In order that the previous definition may agree with this we must have

<* = 1 + /8)* 1+1=1 + 3^8 4- f /32,

126
But

PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE
</- = 1 - <f = I (l + P + 4 /8~), and thence

This value of g will be found to give the correct value for g^.
Then fgK = (f )~ (1 + 5/3 + V P),

and /yV = (I)4 ( 1 + 1 0/3 + ^ 02).

Introducing this into the value of LA (cos), we find

L (cos) =

G/3+150*) ....... (41)

agreeing with the result on p. 549 of " Harmonics" for i = 3, ,s = 0, type OEC.
Airain in " Harmonics" I defined

¥s8 0*)= 15£7>3 (/i) + P32(/t) = 15Bintf[l-f/8-(l- |/3) sin^] . (42).
To make the former definition agree with this we must take

fa* =-15(1- P), >-3 = - 15 (1 - |/8).
In the present case

,/ = i [4 - ^ + ^(1 - ^ + 4^)] = 1 - ^ + f/c'* + -.V,

Omitting the term in /33 we find, with this value of <f,

fo= — 15(1 — ^/J — 3/82), and that the second equation is satisfied.
Again 1 defined

Hence to secure agreement we must take

Now * = 1 -

^= -2(1+0)*.

s = £ (1 - - 10 + |/32), and therefore

The second equation is satisfied.

SQUARES OF ELLIPSOIDAL SU11FACE HARMONIC FUNCTIONS. 127

We have then
/*0W* = 22 . 33 . 53 (1 -f /8) (1 - ft - V/33) = 22 . 32 . 5* (1 - 0 . /3 - V/82)-

Introducing this into the value of /32 (cos), we find

(44),

agreeing with the result in p. 548 of" Harmonics" for i = 3, .s = 2, type OEC.

(2) and (G) First Tesseral Cosine Harmonic and factorial Conine Harmonic.
These are defined thus : —

P8« (p.) = (K* sin* 0 - q*} (1 - *2 sin3 0)*, ]

> . . . . (45),
<£/ (<£) = cos $ (</'- — K'- cos3 (f,}, (s = l,B)J

where <f = ^(1 + 2/c2 ^ (1 — K3 + 4**)*), with upper .sign for .v = 1 and lower sign
for A- = 3, and </'~ := I — q'2.
Writing £'2 = *'3 - (7'2,

COS K + K COS

= r / - /c/3 sin3 < /c/3 - /c'3 sin3

where /= — 1,0= — — .

It is clear that [_P-'(p.)(^ss(<j>)J(n =1,3) has the same form as HM/'OO/ (<£)?(* = °- -)
when in the latter we interchange 9 with -.J-Tr — <^>, and K with «•'. The interchange of
the variables of integration clearly makes no difference in the result, and therefore we
need only interchange K and K, and replace t hy t'.

In the present instance

This shows that t''~ is the same function of K- that ^' was of /<'3, but that /y1 (cos) is
analogous with /32 (cos), and 733 (cos) with /., (cos). Thus we may at once write down
the results by interchanging K and K' throughout.

Let jy = (1 -- /c2 + 4*c*)*.

Then putting K''\f'2g~ = 1, we have by symmetry with (38)

73! (cos) = '^f^- • % DW - «> + *«*) ^ - (1 - K)(l - ^
/33 (cos) = the same with the sign of D' changed.

128

PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

If these expressions be developed in powers of */ I find, on reintroducing the
factor Kf~f2y2,

- — (1 - '' K'~ + 241 'i\ K'-2

3 * ' ^ 4 2 5 6

J 1

= y M.

- —

7 sin3 /3 3. 5* '
- S/8 +

os p cos •;
7 sin3 y8 9

, ,1 , A ,, , 8 , /4, „ « „

.5 K V 2 * T^ 256 K / • K J (I )

In " Harmonics" I defined

But we have defined it above by

i/

P,1 (p.) = / ( ] - ,c- sin- 5)5 (^ sin- 5 - ,/).

Therefore

. J +

+ a ^ + .^ ^

^

Whence / = '/ ( I + y /3 + ^ fr).

This value also satisfies the expression for //<-.
Again I defined

= COS <, - -

= cos 1

(!OS

cos

. . (49).

But we have defined it above by

(£•]' = (jKr cos (f> (K'~ cos2 (f) — ry'':).
Therefore

With the above value for (/ we have </- = f (I + J-^ -f -fa fP) ; whence

SQUARES OP ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 129

Therefore /VV3 = -3-~6"[l 4- *£P + -ft-/?2], and I find

A _

(50),

agreeing with the result on p. 549 of " Harmonics" with i = 3, /? = 1, type OOC.
In " Harmonics " I defined

+ 30) ^Y 0*) + JV

l+*0 + A0'-«n»tf(l+t/8-i|/8»)] . (51).

But P33 GU) =/(! - *2 sin2 0)* (
Therefore >* = - 15 (1 + £ ft - if/33), ./y3 = - 15 (I + |/3 +

*3 O 91

XT 0 i '-* / •"> i ' ' / 1 i ^ ' /i",

Now q~ = I - 22 K'- + afl K ' + ^ K'S...

Therefore q* = I - 1/8+ ?|/83, and

/=- 15(l+-V-/8+§i)8J).
This also gives the correct value toftc.

Again C:j3 (^) = f /3 ( 1 + f /3) cos ^> + cos 3<£,

= cos ^,[4 cos2./, -3 (I . ---i-^-333-/82)]

But Cj3 (<^)) = rX cos <£ (/c'J cos2 </> - r/'-).

Therefore (//c' = |, , and gr/c' . q''2 = 3 (1 - £0 - ^ /32).
/c

If we eliminate (JK', these equations give the correct value for <f2.

Therefore
Hence we find

7,3 (cos) = ^. 3GO (1+1/3+ f| /32) ...... (53),

agreeing with the result on p. 548 of " Harmonics" with i = 3, s = 3, type OOC.
VOL. com. — A. s

130

PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

(3) and (7) First Tesseral Sine Harmonic and Sectorial Sine Harmonic.
These are defined thus :—

¥8' (/,) = cos e (/<3 sin* e - 2s),

= sin $ (tf - *'3 cos3 <£), (s = l, 3),

..... (54)

where </ = i [2 + *3 T (4 - xV2)*], with the upper sign for s = I, the lower for
s = 3, and g/3 = 1 — </'.

Writing

i3 = /r — *,

||3' (M) = /. K cos 6 (t* - /c3 cos3 6),
S' <, = . K- 1 sin < *3 + *'2

(55)

where /=, 9 = -

Squaring ^;JJ we find,

^(J = o, ,1, = t\ A,= - 2t\ A, = l.
On substitution in the formula for harmonics of the third order I find

I3' (sin) =
If we write

f (K3 - K'2) KV2*3 + | KV4] ( - /

*° = * - ¥ ^'3 ~ ¥ «'6 ± (4 ~ ¥ *V3

^

*

£t

- 8 - 34/c'3

3i¥ «" ± (8 - 25/</3 + 5 IK'* -

On substitution 1 find, on putting — f'zg~K~Krz = 1 ,
r i / • \ 47T&3 cos 3 23

+ (Ks _ K't) (8
J33 (sin) = the same with the sign of the square root reversed.

. (56)

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 131

Developing these expressions in powers of «r'2, reintroducing the factor —
and reverting to the notation of " Harmonics," I find

rj (sin) =

(sin) =

In " Harmonics " I defined

To make our former definition agree with this we must take

A. «• = ¥(! +10 -A/88); >-r=;Hi+t/3-ii/32).

Hence /AC = ¥(l+¥/8 -Htm

It will l)e found that </e — ^-(1 — I /3 -\- -/',•. /3C), and tliat //c. ry- lias tlie above form.
Again I defined

S, ^ = sn <£ — £/3 1 - fyS sin 3<£,

= sin^[1 ---I8¥ + M/3C+ P(l --P)siu3^] . . . (58).

To make our former definition aree with this we must take

cV - 1 . t- = 1 - i85-y8 + 1I/8-,

It will be found that t~ = f (1 - ^-j3 + W/S2).

Whence .^K'^/ — 1=1(1+^/8 — H/S2), and (7/c'v/ — 1 . /c'2 has the correct form.

Therefore

y^V - i - 3 (i + W + W

-

Wlience

(59),

agreeing with the result on p. 548 of " Harmonics" with i = 3, s = 1, type OOS.

s 2

132 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

In " Harmonics " I defined

= 15 cos 0[- (1 - -

In order that this may agree with our former definition we must take
/«.«•= - 15(1 -- W + tf/32), /«.</* = -- 15(1 - f/3 +

Whence /* = -- 15 (1+ P - f^S9).
It will be found that

(60).

so that / K . '/3 has the correct form.

Again I defined

(61).

In order to make our former definition agree with this we must take

!/K ' \/ — 1 . £2 — 3 (1 -|- .I,/? — ssfiP)) f/K'\/ — 1 • K/2 — — 4.

Therefore 0«V - 1 . = - -*, .

It will be found that t~ = — %K'~ (I -f •££ — ^a2-/82), so that ^r/c'v/ — 1 . t* has the
correct form.

Then

345

and

Whence

^ VV3 = 33-42-53

K*

(62),

agreeing with the result on p. 548 of " Harmonics" with i = 3, s — 3, type OOS.

(5) The Second Tesseral Sine Harmonic.
This is defined thus :—
Pss(/i) = sin 6 cos 0 (1 -K2 sin20)* =/. K cos d (K"-KZ cos2 6»)5 (/c'2+/c2 cos2 0)* 1

*A*M*-**H» r(63)

where /= , ^ =

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS 133

Squaring P32 we have

A0=0, ^, = KV2, Az = **-,<>*, A,= -l.
Therefore
/3» (cos) =

A

Reducing this expression and putting — /2</W4 = I, we have

47r& cos i6 cos y

7

In " Harmonics" I defined
~

. (65).

1 — a2 ~

To make our former definition agree with this we must take

Again I defined

S32 (<£) = (1 - /3 cos 2<£)* sin 2<^ = 2 (1 + ^p (l - 2^ cos2 ^ )' sin </> cos f (6(5).

To make the former definition agree with this we must take

^V- 1 = 2(1 + £)*.
Therefore

/<//cV2 y - 1 • = 2 . 3 . 5 j [t |1 :Vnd

— /V*cV* = 22. 33. 52 ^ + ^' = 22. 32. 5s (1 + 3/8 + 4/32).
Hence in the notation of " Harmonics"

/32 (sin) = 47"M .3.4.5(1+3/8+ 4/82), ..... (67)
agreeing with the result on p. 548 of " Harmonics " with i = 3, s = 2, type OES.

[34 PEOFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

It may be convenient, as furnishing a kind of index to the foregoing investigation,
to state that the 1 + 3 + 5 + 7 integrals for the harmonics of orders 0, 1, 2, 3 are
given in equations 3, 5, 9, 13, 16, 24, 28, 32, 38, 47, 56, 64, corresponding to the
definitions contained in 4, 8, 12, 15, 23, 27, 31, 36, 45, 54, 63.

The definitions of the harmonic functions as given in my paper on Harmonic
Analysis are repeated in 6, 10, 19, 21, 25, 29, 34, 39, 40, 42, 43, 48, 49, 51, 52, 57,
58, 60, 61, 65, 66. Corresponding to these latter definitions the approximate
integrals are given in 7, 11, 14, 20, 22, 26, 30, 35, 41, 44, 50, 53, 59, 62, 67 ; and the
results confirm the correctness of the general approximate formulae for the integrals
given in § 22 of the paper on Plarmonic Analysis.

A mistake in that paper was detected in the value of the cosine-function for the
third zonal harmonic, and the corrected value is given in (40).

It must be obvious that the method exhibited here may be applied to higher
harmonics with whatever degree of accuracy is desired ; but it is also clear that the
labour of evaluating the integrals increases very much as they rise in order. It is
probable that the approximate results of the previous paper will suffice for most
practical applications.

APPENDIX.

On the Symmetry of the Cosine and Sine-functions ivith the P-functions.

In my previous papers I fiiiled to notice that the symmetry between the P-functions and the cosine and
sine-functions is not destroyed, but is only masked, in the approximate expressions for the harmonic

functions.

For example, (39) shows us that

therefore, in consequence of the symmetry which subsists, we ought to find

G,W=P. [( ••'f™2*)'

Whence

Cs (</>) = -7?-£4dP^ (1 - ft cos 2<f>)» (1 - £/? cos

This only differs by a constant factor from the expression (40).

It would be possible then to have only one type of function, viz. $ or p, and to express all the cosine
and sine-functions by means of the appropriate one of them. This would be found to be equivalent to
expressing the latter functions in terms of powers of sin <£. For the purposes of practical application I
do not think this would be so convenient as the use of cosines and sines of multiples of <f>, and the

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 135

advantage of using only one type of function would not compensate for the loss of convenience in the
result. Accordingly I do not think it worth while to undertake the very laborious task of revising all
the analysis of "Harmonics " from this point of view.

I may mention, however, that I have gone far enough in this direction to feel pretty confident that, if
this new form of developing the cosine and sine-functions were adopted, the remarkable coincidence,
mentioned in the footnote on p. 547 of " Harmonics," as to the form of the integrals of the squares of
surface harmonics would become explicable.

POSTSCRIPT.

[December 2<llh, 1903.]

Mr. HOBSON has shown me how these integrals may be evaluated by a simpler
method of analysis, without the intervention of elliptic integrals. As an example oi
the method he suggests I take the integral /., (cos) evaluated above.

The solid ellipsoidal harmonics are given, except as regards a factor, in § 3 of
" The Pear-Shaped Figure."

In (19) of that paper we find

S, = 193 (") $2 (M) ffj (ft = A ^fx- + (1 - -2q°) if - q*3* + £ </Y~J ,
where A is the factor to be evaluated so as to agree with the definitions

The ellipsoid over which we desire to integrate is defined by v — --.- , and the

K sin y

extremity of the c axis is defined by ju, = sin 0 = 1, ^> = ^TT.
Hence at this point

S.2 = (cosec3 y — (f] (/c3 — q"} q'2.

But at the extremity of the c axis

x — 0, y = 0, - =.f = -. .

K sin 7

Therefore

(cosec2 7 - cr) (K- - </) </- = 7^-1 f - -0- $~ - + <rl "•} = - A k~'l - (cosec- y - q*~).

\ K~ sur 7 KV / /c"

Therefore A = — T«— —, arid

A;"

f -, .T1 n « W3 /. o 3J

136 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE

Let us assume

.. k cos y

x -- at = - J i

j > k cos &
= b£= -.— =
K sin y

K sin y
Then when x, y, z is on the ellipsoid we have

f r

2 = c£ = - £.
K sin

Thus we may regard £, •>?, £ as the coordinates of a point on a sphere of unit
radius, or as direction cosines, if it is more convenient to do so. On substituting for
x, y, z their values in terms of £ 77, £ we find

S, = (cosec2r ~ </2) («2 - ?2) [- ?2£2 - (1 - 292h2 + <m

On performing the same operation to the points on the boundary of an element da
of surface of the ellipsoid, we find

, 7 7 /j3 cos B cos y ,

ndfr = cwcaa) = . ' _ --- ao>.

sin3 p

where dJw is an element of the surface of the sphere of unit radius, or an element of
solid angle.

Since on the surface of the ellipsoid ^., (v) — cosec- y — if, it follows that

2 0*)

Hence

Tl \ _ ^3 COS /3 COS y / 2 _ ?\2 fr 2« /i 9

It is easy to prove that

cw

_ f .3 3.3 , _ f >2 o 7

47T

5 '

47T

15'

Therefore

/.(cos) =

_ 3

On substituting for q* its value, viz., i (1 + «r2 - (1 - /cV2)*), and effecting
reductions we arrive at the result given in (16) above.

SQUARES OF ELLIPSOIDAL SURFACE HARMONIC FUNCTIONS. 137

It is obvious that this process is considerably simpler and more elegant from the
point of view of theory, but to carry these operations through for all the integrals
given above would entail a good deal of algebra. I think indeed that the work
might not be very much less than what I have already done.

Mr. HOBSON has further remarked that all the integrations may be avoided by the
following theorem :—

If F, (f, 77, £) be a solid spherical harmonic function of f, rj, £ of degree n,

Considering, however, how simple are the integrals involved in his first method, it
may be doubted whether this would save trouble.

VOL. com. — A.

[ 139 ]

VI Tin- Sncdfir. If i'n tx of McMh and tJic /.
INDEX SLIP.

Weight.— Part II!

Decific Heat to Atomic

By W. A. TILDEN, D.Sc.. F.R.S., Pn>f.

TILDBN, W. A.— The Specific Heats of Metals and the Relation of Specific
Heat to Atomic Weight.— Part III.

Phil. Trans., A, vol. 203, 1904, pp. 139-149.

Specific Heats of various Metals ; Changes with Temperature ; Belation to
Atomic Weights ; of Compounds and Alloys ; of Gaseous Carbon ;
Neumann's Law verified.

TILDES, W. A. Phil. Trans., A, vol. 203, 1904, pp. 139-149.

THE law of NEUMANN assumes that v.h.-i. .,:
retains the same capacity ?<>r heat a> \\\-

This genera lisa i it m is, ho'.M , -,•,.,, , i .

heats of elements ;ni(l theii .'oMijiounds !„

Attention was directed in I 'art If <.|'
found in the influence of temperatuie t"
aluminium on the one hand, and sikei ...
now about to he described were iindurt:j
extent these differences persist in tin- ••••

If the calculate :eats of rli-mei

stry in the Royal

temperatures through a oo tu-
rn the separate state at (\M- xtnn- U'mpn;
must he due to a fundanteiital dill^i.-nc.'
not to a difference in their >tat.-^ of ;t^"ir-.-ili,n!

The specific heats-of nirkel suljjhide a;,d hiivn
through a range of temj>-ra tin t- fVotu -18:1' t
melting-point of sulphur, ,-i.iid espefiaily (<•
modifications of this element in the solid .-{..T.
attempt determinations of its --pecitic heat at s^i

The only element which seemed to present ti

fur the purpose contemplated was tellurium. I

Wipply of tellurium which had already been ret

aud precipitation by sulphurous acid. It

-sium cyanide, solution in water, HIM!

-muds of silver, nickel and tin with tett

A«n. Chuji Fhy'n.' [3], I
t ' Pliil. : . ,r«». 801, p. 37 (1W

T 2

> chemical combination it
hied or elemental state.

'd for the mean specific

< *

to the great differences

' various metals, such as
itlier. The experiments
of ascertaining to what
tents.

n are equal at various
itomic heats of the same
rence between any two

elements concerned and

L_l

e already been compared

but owing to the low

e of several allotropic

thought worth while to

u res for this purpose.

: of characters required

to Dr. T. K. ROSE for a

•al dissolutions in aqua

purified by fusion with

by exposure to air.

nade by the following

Trans.,' 1865.

27.5.04

.1IJg

lo (loiteiofl itti baa «Uit.,M to
.III
,80S .Io» ,

oitio-Kj* <>rfT— >. W .«a.nT
JrfjreW oiuroJA 01

djiw •')5
to ;4^olIA biui

.'.'4 1-0«I .qq ,tOCI .80S .Io7 ,A ,.«njnT .Iit(1 .A .W .

[ 139 ]

VI. The Specific Heats of Metals and the Relation of Specific Heat to Atomic

Weight.— Part III.

By W. A. TILDEN, D.Sc., F.R.S., Professor of Chemistry in the Royal

College of Science, London.

Received March 9,— Read March 17, 1904.

THE law of NEUMANN assumes that when an atom enters into chemical combination it
retains the same capacity for heat as when in the nncombined or elemental state.
This generalisation is, however, based on the values observed for the mean specific
heats of elements and their compounds between 0° and 100° (!.*

Attention was directed in Part II. of this invest igationt to the great differences
found in the influence of temperature on the specific heats of various metals, such as
aluminium on the one hand, and silver or platinum on the other. The experiments
now about to be described were undertaken with the object of ascertaining to what
extent these differences persist in the compounds of such elements.

If the calculated atomic heats of elements in combination are equal at various
temperatures through a considerable range to the sum of the atomic heats of the same
in the separate state at the same temperatures, then the difference between any two
must be due to a fundamental difference in the atoms of the elements concerned and
not to a difference in their states of aggregation when separate.

The specific heats of nickel sulphide and silver sulphide have already been compared
through a range of temperature from — 182° to + 324° C., but owing to the low
melting-point of sulphur, arid especially to the occurrence of several allotropic
modifications of this element in the solid state, it was not thought worth while to
attempt determinations of its specific heat at various temperatures for this purpose.

The only element which seemed to present the assemblage of characters required
for the purpose contemplated was tellurium. I am indebted to Dr. T. K. ROSE for a
supply of tellurium which had already been refined by several dissolutions in aqua
regia and precipitation by sulphurous acid. It was further purified by fusion with
potassium cyanide, solution in water, and precipitation by exposure to air.
Compounds of silver, nickel and tin with tellurium were made by the following

* REGNAULT, 'Ann. Chim, Phys.' [3], 1, 129; KOPP, 'Phil. Trans.,' 1865.
t 'Phil. Trans.,' A vol. 201, p. 37 (1903).
(364.) T 2 27.5.04

140 PROFESSOR W. A. TILDEN ON THE SPECIFIC HEATS OF METALS AND

process. Tellurium and the pure metals were weighed out in the proportions
corresponding to the formulae Ag2Te, NiTe and SnTe2 respectively, with a slight
excess amounting to about 1 per cent, of tellurium. The materials were then fused
together in a stream of hydrogen at a temperature sufficiently high to volatilise the
excess of tellurium. The tellurides were obtained as black, crystalline, fusible
substances, and were cast into cylindrical form by melting in a glass tube.

The nickel used in the preparation of the telluride was in the form of soft wire
drawn from metal obtained by electrolysis. For this I am indebted to the kindness
of Dr. J. WILSON SWAN, F.R.S. Its specific heat was determined in the steam
calorimeter in order to compare it with the fused nickel made for the previous
experiments, but which was not found to be sufficiently ductile to admit of being
drawn into wire.

SOFP Nickel Wire.

Kunge of temperature.

Specific heat.

Mean.

0 C.

22 to 100

•1087]

20 „ 100

•1082 Y

•1086

23 „ 100

•1088J

The moan specific heat adopted as the result of the previous experiments on fused
nickel was *1084 for the same range of temperature.*

Alloys of silver and aluminium have also been examined. They were prepared by
melting together the exact proportions of the pure metals. In the first the silver
largely predominates, being in the ratio required by the formula Ag3Al. The second
contains aluminium in proportion corresponding to the formula AgA.\u, which
represents 75'1 per cent, of aluminium and 24 -9 per cent, of silver.

As in the results set forth in the previous paper, the specific heat adopted is the
mean of several closely concordant experiments made at each range of temperature.
The figures followed by E are estimated from the others which are the direct results
of experiment. It will be seen (Table I.) that the value for specific heat increases
with rise of temperature in every case except silver telluride, where the mean specific
heats found between 15° and 309° and 390° C. respectively are less than at lower
temperatures. This irregularity is attributed to the fact that during the later
experiments the mass cracked and it was found necessary to re-melt it several times.
This was done in hydrogen gas, and though no change in appearance was observed,
some slight change in composition or structure may have been produced. These
figures have, therefore, not been used in the subsequent calculations.

* 'Phil. Trans.,' A, vol. 201, p. 38 (1903).

THE RELATION OF SPECIFIC HEAT TO ATOMIC WEIGHT.

141

TABLE I. — Mean Specific Heats.

Range
of
temperature.

Tellurium.

Tin.

Silver-
telluride.
Ag2Te.

°C.
-182 to 15

•0469

•0499

•0516

15 100

•0483

•0557

•0672

15 180

•0486 E

•0577

•0686

15 200

•0487

—

—

15 227

•0488 E

—

—

15 300

—

—

—

15 309

—

—

•0670

15 315

. -0489

—

—

15 322

—

—

15 327

•0490E

—

—

15 380

•0500

15 385

15 390

—

—

•0663

15 410

—

—

—

15 427

•0508 E

—

—

15 437

—

—

—

15 495

—

—

—

Nickel-
telluride.
NiTe.

Tin-
telluride.
SnTe2.

Silver-
aluminium.
AgsAl.

Aluminium-
silver.

AgAl12.

•0588
•0670
•0689

•0471
•0494
•0489

•0620
•0696

•0703

•1477
•1802
•1861

•0690 E
•0691

•0486

•0704

• 1863 E
•1916

•OG95 E

•0496

•0705

•1939 E

•0703

—

—

•0708 E

•0725
•2015 E
•2026
• 2093

To these results may be added the mean spec! He heats of silver, nickel, and
aluminium taken from the previous series of experiments.

TABLE II. — Mean Specific Heats.

Range of temperature.

Silver.

Nickel.

Aluminium.

- 182 to + 15

•0519

•0838

•1677

15 100

•0558

•1084

•2100E

15 180

•0561

•1101

•2189

15 227

•0565E

•1120 E

• 2208 E

15 327

•0577 E

•1175 E

•2247

15 427

•0581

•1233 E

•2356

The mean specific heats thus determined have been used, as in the former paper,*
for the calculation of Q, the total heat measured in the calorimeter. The values of Q
for the two elements tellurium and tin, the tellurides of nickel and tin, and the two
alloys of aluminium and silver, have been plotted against absolute temperatures, and
the results are shown in fig. 1, in which the curves are for the most part hyperbolic,
those of tin, tellurium, and tin telluride approaching an elliptic form. In the case of
tin, which melts at 232°, this is most probably due partly to incipient fusion at 180°,
the highest experimental temperature, and is in accordance with experience.

* Part II., ' Phil. Trans.,' A, vol. 201.

142 PROFESSOR W. A. TILDEN ON THE SPECIFIC HEATS OF METALS AND

a

100

9(

60

70

ALUMIN

UM SI

VER k

%SILVE

7

60

50

40

NICKEL

30

20

10

7

NICKEL
SILVER

ELLUROE. O
ALUUMIlJUM. x

7

7

ELLUR

JM.

HIM

TIN TELLURIDE

0 2(

BSOLU

0 61
E TE

0 7
PERA

URE.

0 9<

o ipc

10

20

7

Fig. i.

THE RELATION OF SPECIFIC HEAT TO ATOMIC WEIGHT.

143

REGNAULT, for example, found the molecular heats of easily fusible alloys to be very
much higher than those of alloys of higher melting-point.*

The values of the true specific heats at successive temperatures on the absolute
scale are given in the following table, and are exhibited graphically in figs. 2 and 3 : —

TABLE III. — True Specific Heats.

Nickel-

Tin- Silver-

Aluminium-

t abs. Tellurium.

Tin.

telluride.

telluride. aluminium.

silver.

NiTe.

SnTe* Ag,Al.

AgAl12.

"0.

100 -0462 -0462

•0453

•0467 -0591 -1233

200 -0471

•0504

•0614

•0472 -0628 -1510

300 -0480

•0548

•0671

•0479 -0662 -1731

400 -0489

•0596

•0699

•0488 -0693 -1917

500 -0498

—

•0711

•0502 -0722 -2060

600 -0507

—

•0718

•0748 -2166

700 -0516

—

•0722

•0771 -2260

800

—

—

— —

•2340

It is obvious that the curves for the specific heats of the compounds are of the same
character as those for the metals aluminium, nickel, and silver given in the previous
paper, and that the inclination of each is determined by the principal ingredient.
Thus the curve for aluminium-silver containing 92 per cent, of silver is very near to
the curve for that metal, while the curve for the alloy containing 75 per cent, of
aluminium approaches the curve for pure aluminium.

The atomic heats of the elements are obtained by multiplying the specific heats by
the respective atomic weights, which have been taken from the International Table.

TABLE IV. — Atomic Heats.

/ abs.

Tellurium.
Te = 126-6.

Tin. Silver.
Sn=118'l. Ag=107'12.

Nickel. Aluminium.
Ni = 58-3. Al = 26-9.

°C

100 .

5-85 5-46 5-00

3-35 3-30

200 5-96 5'95 5-65

5-12

4-66

300

6-08 6-47

5-98

6-14

5-52

400

6-19 7-04

6-13

6-81

6-06

500

6-30

6-22

7-19

6-41

600

6-42

6-29

7-43

6-65

700

6-53

6-32

7-58

6-81

800

—

—

7-70

—

900

—

—

—

7-80

; Ann. Chim. Phys.,' [3], 1, 137 and 183.

144 PROFESSOE W. A. TILDEN ON THE SPECIFIC HEATS OF METALS AND

•Z7r

100 200

300 400 500

ABSOLUTE TEMPERATURE.

600 700

800

THE KELATION OF SPECIFIC HEAT TO ATOMIC WEIGHT.

145

•15

•13

NICKEl

•09

•07

NICKCI. TELI.URIDE

TE.LLI

R1UM. O

•05

•OS

•01

100 ZOO 300 400 500 600

ABSOLUTE TEMPERATURE.

700

800

900

Fig. .1

On the hypothesis that each atom in a compound behaves as it does in the solid
element, the sum of the atomic heats of the elements entering into the compound
should he equal to the molecular heat of the compound. The following table contains
a comparison of the sum of the atomic heats, A, with the molecular heats, B, of the
several compounds, that is, the product of the observed specific heat of the compound
multiplied by the molecular weight in each case.

VOL. com. — A.

146 PROFESSOR W. A. TILDEN ON THE SPECIFIC HEATS OF METALS AND

TABLE V. — Molecular Heats of Compounds.

t abs.

SnTeo.
A.

SnTeo.
B.

NiTe.
A.

NiTe.
B.

Ag,AL
A.

Ag3Al. AgAli2.
B. A.

AgAl12.
B.

100

17-10

17-33

9-20

8-38

18-30

20-58

44-60

53-01

200

17-87

17-51

11-08

11-35

21-01

22-38

61-57

64-92

300

18-03

17-77

12-22

12-41

23-40

23-00

72-12

74-42

400

19-42

18-10

13-00

12-92

24-45

24-14

78-85

82-41

500

—

13-49

13-15

25-07

25-15

83-14

88-56

GOO

—

13-85

13-28

25-52

26-05

86-09

93-12

700

14-11

13-35

25-77

26-85

88-04

97-16

800

—

—

—

—

—

— —

—

A = calculated from the atomic heats of the elements.

B = calculated from the observed specific heats of the compounds.

The figures contained in this table show that in the cases of tin-telluride, nickel-
telluride and the silver-aluminium alloy containing 92'28 per cent, of silver, there is a
remarkahlv close approximation of the values under B to those under A, the
differences between the two columns being throughout well within the limits of
variation due to experimental error.

With regard to the aluminium-silver alloy containing only 24 '9 per cent, of silver,
however, there are differences which are somewhat greater. The values for silver-
telluride must be regarded as open to suspicion, for reasons which have already been
indicated, and they are not included in the table. If the mean atomic heats of silver-
telluride are compared, it is found that the difference between the sum of the atomic
heats and the molecular heat of the compound increases considerably with the
temperature, as seen below : —

Temperature.

A.

B.

Difference.

°C.
-182 to + 15

17-06

17-59

0-53

15 „ 100

18-07

• 22-90

4-83

15 „ 180

18-17

23-38

5-21

This is perhaps due to some change taking place in the constitution of the solid.
This, however, does not seem to be the explanation in the case of the aluminium-
silver alloy, in which the differences between the two columns of figures, though not
constant, do not increase appreciably—

THE KELATION OF SPECIFIC HEAT TO ATOMIC WEIGHT.

147

B-

A at

100

__ Q

•41,

200

= 3

•35,

300

= 2

•30,

400

o

•56,

500

= 5

•42,

600

= 7

•03,

700

= 9

•12.

Remembering what a large factor, 429 '9, is used in calculating these figures, it will
be seen that the differences are really small, being about 1 5 per cent, of the molecular
heat at the lowest temperature and falling to about 3 per cent, at 300° abs. It will
be noticed that in this case B is throughout larger than A.

The results of these experiments show that NEUMANN'S law may be regarded as
approximately valid for the -specific heats at all temperatures. They also confirm the
view that the specific heat of a solid is not a measure of the work done in separating
the molecules of the substance, but that its amount is determined almost entirely by
the nature of the atoms composing the physical molecules.

All the facts at our disposal show that there is not a great difference between the
specific heats of elements in the solid and liquid states, but that in every case the
latter is the greater, as shown in the following examples :—

TABLE VI. — Specific Heats.

Name.

Solid.

Liquid.

Authority.

Lead

•034

•040

PERSON.

Bromine
Gallium

•084
•079

•107
•080

ANDREWS.
BERTHEI.OT.

Phosphorus
Mercury
Bismuth

•202
•032
•030

•204
•033
•036

PERSON.

KOI'F.

PERSON.

Tin

•058

•063

SPRING.

The atomic heat in the liquid state is thus in all cases greater than in the solid,
and in the above cases ranges from 5 '6 to 8 '5. In the gaseous state at constant
volume the atomic heat is, however, much smaller, being approximately for hydrogen
2 '42, for oxygen 2 '48, and for iodine, a solid at common temperatures and in some
characters approaching the metals, 3 '3. In respect to specific heat, therefore, the
liquid state is not intermediate between the solid and the gaseous states. This may
possibly be explained by the assumption that in the solid and in the gas at constant
volume every molecule in the mass remains in the same condition relatively to every
other molecule, for in the solid all are rigidly bound together by " cohesion," and in

u 2

148 PROFESSOR W. A TILDEN ON THE SPECIFIC HEATS OF METALS AND

the gas all are equally free. In the liquid state there is reason to believe that there
is a mixture of clusters or aggregates of molecules having different degrees of
complexity, and that the effect of rise of temperature upon these is to cause
dissociation of the more complex into simpler groups, a process which necessarily
implies work done.

Notwithstanding the validity of NEUMANN'S law, the attempts which have been
made to deduce the atomic heats of elements, such as oxygen, which do not admit of
experiment in the solid state, cannot, however, be regarded as satisfactory. It is
obvious that iii such calculations whatever change in the molecular heat of the
compound is induced by slight alteration of density, or of structure in the solid, is
concentrated upon one element in the compound of any two, when it is assumed that
the other enters into combination with the atomic heat it possesses in the elemental
state. Taking the figures for the compounds containing silver, for example, the value
deduced for the atomic heat of silver is found to vary considerably according to the
nature of the compound selected. To calculate the atomic heat of silver from the
mean molecular heat of the telluride— which is '0672 X 340'8 = 22'90 at the
usual temperature of experiment, 0° to 100° C. — the value for tellurium is deducted
and the remainder divided by 2. The result is 8 '39.

Similarly the atomic heat of silver in the silver-aluminium, Ag3Al, comes out as
G-19, and in the aluminium-silver, AgAlp, as 9'67. The variations are still greater if
a comparison is made over different ranges of temperature. Hence it appears probable
that the values which have been calculated for hydrogen, oxygen, nitrogen and
chlorine in the solid state are very far from the truth.

KOPP estimated the atomic heats of these elements in the solid state to be as
follows : —

Hydrogen 2 '3 Nitrogen 6 '4

Oxygen 4'0 Chlorine 6 '4

From its various compounds no approach to a uniform value for the atomic heat of
carbon is to be found, but KOPP preferred the value 1'8, which is deduced from the
specific heat of diamond.

Without here entering into a discussion of all these elements, it may be mentioned
that in the case of hydrogen gas at constant volume the atomic heat is practically
identical with that deduced from the specific heat of solid compounds, while that of
oxygen is less, and that of carbon in the form of carbon dioxide gas is greater than
the estimates thus made. JOLY found* the specific heat of air between 15° and
100° C. to be '172, that of carbon dioxide gas '173, and that of hydrogen 2*41, when
under approximately equal pressures. The atomic heat of hydrogen gas is therefore
2 '41. Assuming the specific heat of oxygen at constant volume very near to that of

* ' Phil. Trans.,' A, vol. 182, p. 73 (1892).

THE RELATION OF SPECIFIC HEAT TO ATOMIC WEIGHT. 149

air, as it was shown to be many years ago, when at constant pressure, by REGNAULT,
its atomic heat is about 27, which is a little greater than 2'48, the value deduced
theoretically from REGNAULT'S experiments at constant pressure. Lastly, taking
'173 for carbon dioxide and multiplying by 44, the value 7 '61 is obtained as the
molecular heat of carbon dioxide gas. If it be assumed that in the gaseous, as in the
solid, state the atomic heat of each element is preserved in the compound, the atomic
heat of gaseous carbon is left when the atomic heat of the oxygen in the dioxide is
deducted. We thus obtain the value 2 '65.

This is greater than 1'8, the value chosen by KOPP, but falls between 2'89 and
2'42, the atomic heats of carbon, in the form of wood charcoal and natural graphite
respectively, deduced from the experiments of REGNAULT between 0° and 100° C.

This deduction from data belonging wholly to the gaseous state is of interest
because it is in accordance with the theoretical view that specific heat in a gas is not
dependent on the temperature.

On the other hand, as the above experiments prove, atomic heats in the solid state,
and probably also in the liquid state, are largely dependent on the temperature, the
variation being abnormally great in solid carbon. It has also been shown that at the
same temperature the atomic heats are widely different for different elements in the
solid state ; but notwithstanding this fact it has been proved that the molecular heat
of a solid compound is approximately the sum of the atomic heats of its constituents
at each temperature.

In conclusion, I desire again to express the obligations I am under to Mr. SIDNEY
YOUNG and to Mr. LEONAED BAIKSTOW for their assistance.

in Enqu Sun-spot Frequency

and Terrestrial Magnetism.
INDEX SLIP.

By C. CHRKE:.. Or./)., LLD., Ftt.S.

Received F'.i-f.n1 ~-U«.-u! Ma>\-h 3 !904

CHRKK, C. — An Enquiry into the Nature of the Relationship between
Sun-spot Frequency and Terrestrial Magnetism.

Phil. Trans., A, vol. 203, 1904, pp. 151-187.

* • >\'i K\ rs.

Sun-spot Frequency — Relationship with Terrestrial Magnetism. ^"£e

CHEEE, C, Intrc Pail. Trans., A, vol. 203, 1904, pp. 151-1S7. 151

.... 152
Terrestrial Magnetism — Relationship with Sun-spot frequency.

CHBB'B, C. Phil. Trans., A, vol. 203, 1904, pp. 151-187. .... 154

. . . . 159

160

.... 163

id minima . 163

.... 164

.... 165

d smoothed

.... 166

ies . . . 168

.... 169

fee. ... 170

.... 171

.... 172

... 173

.... 175

.... 177

.... 178

. . . . 181

.... 184

.... 184

. . . . 185

iered the relationship
3t " days at Kew and
xnd German magnetic
r gone more fully into
i conspicuous at Kew
alt with.

* ' PhU. Train.,' A, n* JOS, p 386.

7.7.04

152 DE. C. CHEEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

Assuming the relation between any magnetic quantity R — such as the daily range
of declination — and sun-spot frequency S to be of the simplest type,

E = a + 6S .

(1),

I determined the values of the constants a and 6 in a number of cases. The chief
results were as follows : —

Supposing each month of the year treated separately, it was found that both
a and b were conspicuously lower in "winter" (the four months November to
February) than at the " equinoxes " or in " summer " (May to August) ; but b/a was
larger in " winter " than at the " equinoxes," and larger at the " equinoxes " than in
"summer." The b constants were generally fully larger at the "equinoxes" than in
" summer," but to this there were exceptions.

The values of b/a were distinctly larger in the case of inclination and horizontal
force than in the case of declination and vertical force ; generally they were larger for
inclination than for horizontal force, and larger for declination than for vertical force.

In all the elements b/a was larger when R represented the sum of the 24 hourly
differences from the mean value for the day than when it represented the range in
the diurnal inequality.

In (A) the letters D, I, II, V were used for declination, inclination, horizontal force
and vertical force respectively. This practice is continued here when it tends to
brevity.

§ 2. The first question now to be considered is as to the dependence of a and b on
the particular period to which the magnetic and sun-spot data refer.

This is not so simple as might appear at first sight. Very few observatories have
magnetic records extending over any large number of years, and in the few cases
where such long records exist their homogeneousness is seldom, if ever, beyond
dispute. There lias usually been change both in the' apparatus and its environment,
and it is difficult even for those in immediate charge of an old observatory to know
what allowance ought to be made to put old and new records on a common footing.
This is especially true of V and I. The element where least uncertainty should
prevail is D, but even here there is usually cause for doubt.

Milan Declination Ranges.

§ 3. RAJNA'" has recently considered a long series of data representing the mean
value for each year of the diurnal range of D at Milan. Since 1871 the range seems
to have been determined from regular daily observations at 8 A.M. and 2 P.M. ; but
previous to that date there seems to have been some lack of strict uniformity.
RAJNA has calculated values for a and b in the formula (1) from the data for the
59 years 1836 to 1894, and independently for the 24 years 1871 to 1894, employing

* 'Rendiconti del R. 1st. Lomb.,' Series II., vol. 35, 1902.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTKIAL MAGNETISM.

153

the method of least squares. The formulae he thus obtains, and one which he quotes
as obtained by WOLFER, are as follows : —

EAJNA . . .
WOLFER

1836 to 1894, E = 5-31 + 0'047 S
1871 „ 1894, E = 5-39 + 0-047 S
K = 5-67 + 0-040 S

. (II.),
. (III).

WOLFEB'S value 0'040 for b is based on data from Christiania, Prague, Greenwich
and Vienna, as well as from Milan. HAJNA compares the values calculated from each
of the three formulae with the ranges observed at Milan from 1836 to 1901.
Formula (I.) agrees rather better than (II.) with observation; formula (III.) seems
distinctly inferior. The difference between the observed values arid those calculated
from (I.) varies from — l''87 in 1838 to + 1''87 in 1866, the extreme values observed
in E, being 4'"21 and 12'"03. Since 1871 the agreement seems decidedly improved.
Over the 24 years for which it was originally calculated (II.) gives a " probable error "
of only 0'-21, the difference between observed and calculated values varying from
— 0''49 to + 0'-53.

EAJNA himself notices that there are several long runs of the same sign in the
differences between observed and calculated values. Thus from 1H37 to 1850 the
observed value is in excess of the calculated (from either (I.) or (II.)) 12 out of 14
times; on the other hand, in the 14 years 1854 to 1867, the calculated value is 13
times in excess. In the 11 years 1890 to 1900 (to which the Kew data treated in
(A) referred) EAJNA' s calculated value from either (I.) or (II.) has been in excess
8 times, including every year since 1893.

The two first specified predominances of one sign are certainly in excess of what
one would expect from pure chance. To throw some further light on the question, I
have calculated values for a and b for the above-mentioned series of years. Instead
of least squares, I grouped the years, following the method explained in (A), § 52.
The grouping of years and the corresponding mean values of E and S were as
follows : —

Period 1837 to 1850—

Years of sun-spot maximum, 1837, 1838, 1839, 1847, 1848, 1849 . .

„ minimum, 1841 to 1845

Period 1854 to 1867—

Years of sun-spot maximum, 1858 to 1862

„ „ minimum, 1854, 1855, 1856, 1865, 1866, 1867 . .

Period 1890 to 1900—

Years of sun-spot maximum, 1892 to 1895

„ „ minimum, 1890, 1899, 1900

Mean K.

9-270

10-950

7-552

6-586
8-176
5-245

7-163
8-753
5-587

Mean S.

68-9

107-7

L'5-4

41-4
76-1
14-3

41-7

75-0

9-5

VOL. CCII1. — A.

X

154 DR. C. CHBEE: AN ENQUIEY INTO THE NATURE OF THE RELATIONSHIP

The values found in the several cases for a, b and b/a, with the corresponding
values from RAJNA'S formula (I.), appear in Table I.

TABLE I. — Milan Declination Ranges.

Period of years ... 1837 to 1850.

1854 to 1867.

1890 to 1900.

f 1836 to 1894
\ (RAJNA).

a 6' -43

4' -62

5'-14

5' -31

1 -0413

•0474

•0484

•047

104 x (b/a) i 64

i

103

94

89

§ 4. It is certainly satisfactory that the values of a and b for the period 1890 to
1900 differ so little from RAJNA'S values for the long period 1836 to 1894. The
probable error, employing my values of « and b, is only some 4 or 5 per cent, less
than that found when employing KAJNA'S values for the long period.

In considering the results for the two earlier periods, we must remember the want
of homogeneousness referred to above. The mere existence of KAJNA'S formula (I.)
seems, however, evidence that, in his opinion, the want of homogeneousness is not
serious, and the similarity of formula? (I.) and (11.) to a certain extent supports this
view.

The period 1837 to 1850 gives a very high value for a and a distinctly low value
for b. The outstanding features of this period were the high mean sun-spot
frequency, and the largeness of the declination range in the years of sun-spot
minimum. Unless the results are very sensibly affected by heterogeneousness in the
data, we must conclude that values calculated for a and b from a period as long as
14 years may depart somewhat widely from those calculated from a different equal or
longer period. The range of variability would seem least in b and (naturally)
greatest in b/a.

The value calculated for b from the period 1854 to 1867 agrees well with RAJNA'S,
but the value found for a is distinctly lower than his. The sun-spot frequency
during this period presented similar features to those occurring in the 11 years 1890
to 1900.

Greenwich Declination and Horizontal Force Ranges.

§ 5. A second long series of data is that employed by Mr. ELLIS in two papers,* in
which he compares D and H ranges at Greenwich with sun-spot frequency.
Mr. ELLIS gives the observed D and H ranges from the diurnal inequalities for each
month of the period 1841 to 1896. These are based on all days, excluding those of

* ' Phil. Trans.,' vol. 171, for 1880, p. 541 ; ' Proc. Roy. Soc.,' vol. 63, 1898, p. 64.

BETWEEN SUN-SPOT FKEQUENCY AND TERRESTRIAL MAGNETISM. 155

large disturbance. Corresponding " quiet " day data appear in Mr. ELLIS' second
paper for the 8 years 1889 to 1896. Mr. ELLIS specifies several sotirces which may
have introduced some heterogeneousness into the earlier data as compared to the
later. Prior to 1848 there were only eye readings at 2-hour intervals, whereas
subsequently hourly data were available. Prior to 1864, when the magrietographs
were transferred to a new building, some uncertainty seems to have prevailed as to
the temperature correction for H, and the data for 1864 itself seem to be interpolated.
The data subsequent to 1864 would seem to be strictly homogeneous.

§ 6. Mr. ELLIS employed no formula, and, whilst his graphical method appeals
readily to the eye, it does not lend itself immediately to the present investigation.

I have accordingly calculated values for a and b for each month of the year in both
D and H for the following periods: 1841 to 1896, 1865 to 1896, and 1889 to 1896
for both "all" and "quiet" days. In treating the first period, use was made of a
group of sun-spot maximum years composed of 5 sub-groups each of 3 years, viz.,
1847 to 1849, 1859 to 1861, 1870 to 1872, 1882 to 1884, and 1892 to 1894. The
corresponding sun-spot minimum group consisted similarly of 15 years, made up of
5 sub-groups, viz., 1842 to 1844, 1854 to 1856, 1865 to 1867, 1877 to 1879 and 1888
to 1890.

For the period 1865 to 1896 the groups of sun-spot maximum and sun-spot
minimum years were composed in either case of the last 3 sub-groups specified above.

For the period 1889 to 1896 the groups were : 1892 to 1895 for sun-spot maximum,
and 1889 to 1890 for sun-spot minimum.

The values thus found for a, b and b/a appear in Tables II. and III. In addition
to values for the individual months, the tables give values for winter, equinox and
summer — each comprised of 4 months, as explained in § 1 — and for the year. These
seasonal and yearly values of a and b are simply arithmetic means of the individual
monthly values ; the seasonal and yearly values of b/a are derived from the seasonal
and yearly values of a and b. The tables also supply corresponding data for Kew as
given in (A), Table XL.

X 2

156 DK. C. CHEEK: AN ENQUIEY INTO THE NATURE OF THE RELATIONSHIP

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158 DR. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

§ 7. In the case of D, Table II. , we see on the whole a close resemblance between
the results from the two longest series of data. The later period shows, however, a
slightly increased value of b and a slightly decreased value of a in winter, equinox,
and the year as a whole. The last period, 1889-96, in the " all" day data shows a
further increase in b and decrease in a ; and the corresponding " quiet " day results
give a still smaller value of a, especially in the winter months. The "all" and
" quiet " day mean values of b for the year, from the 1889-96 period at Greenwich, are
practically identical and very close to the corresponding Kew value. The values of
b/a at Greenwich are in each season slightly larger for the " quiet" days than for the
corresponding "all" days; and comparing the "all" day data amongst themselves
we have an increase in b/a in passing from the longest to the mean period, and in
passing from the mean period to the shortest period. This would imply that b/a has
increased of late years.

At Greenwich, as at Kew, b is conspicuously lowest in " winter" ; but no one of
the four columns of Greenwich results gives so distinct an excess in the equinoctial
over the summer value as appears at Kew, and, on the whole, we should infer that
the equinoctial and summer values of b at Greenwich are practically equal.

At Greenwich, as at Kew, b/a is distinctly smaller in summer than in the other
seasons ; but the " winter" value of b/a at Greenwich, instead of markedly exceeding
the values for the other seasons, as at Kew, would seem to be if anything slightly
smaller than the equinoctial.

On the whole, the variation of b/a throughout the year at Greenwich is surprisingly
small.

§ 8. The H ranges in Mr. ELLIS' tables are expressed in terms of the value of H at
Greenwich. To make the results comparable with those for other stations, I have
expressed the ranges in terms of ly as unit. In doing so, I treated H as constant for
each period, and as possessing the following values :—

Period.

Vidue of H.

1841-96

C.G.S.
•179

1865-90

•180

1889-96

•1829

Not knowing the exact procedure followed at Greenwich, I may not unlikely differ
slightly from the exact values adopted there ; but, for the purpose of the present
enquiry, such small uncertainty as may exist is immaterial. The values of b/a are, of
course, independent of the unit adopted.

The 1841-96 and 1865-96 data in Table III. present some conspicuous differences;
a is larger in the former series than in the latter in every single month of the year,
but b shows exactly the opposite phenomenon in 9 months out of the 12. This

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM. 159

implies a very conspicuous difference between the phenomena in years prior and
subsequent to 1864. Not improbably the want of homogeneousness in the earlier
data, already referred to, may be partly accountable for the apparent change. At all
events, the results for the final period, 1889-96, show no distinct progressive
diminution of a or increase of I as compared to the period 1865-96. In fact, the
" all " day data for these two periods, and the " quiet " day data for the shorter
period, give almost identical values for the mean b for the year.

As regards the seasonal phenomena, each set of Greenwich data makes b con-
spicuously least, but b/a conspicuously largest, in " winter." The December value of
b is invariably the smallest. The "all" and "quiet" day data for 1889-96, as
already stated, give very nearly the same mean value of b for the year, but the
" quiet " days' value for b is much the larger of the two in winter, and the smaller in
summer. The ' ' quiet " day data at Greenwich present a remarkable similarity to
the corresponding Kew data, as is best seen by comparing the seasonal values for b/a.
The fact, however, that the absolute values of both a and b are some 10 per cent.
higher at Greenwich than at Kew is rather suggestive of some misapprehension as to
the scale values at one or both observatories.

§ 9. In cases such as the present, a comparison of calculated and observed values is
useful. The Greenwich data do not lend themselves very readily to this, as
Mr. ELLIS does not give mean values for individual years, and the interest attaching
to such a comparison for individual months of the year seems hardly sufficient to
justify the necessary labour. In (A), § 75, some results were given for individual
months of the year at Kew, but none for the year as a whole. This information is
accordingly now supplied in Table IV., so far as concerns the ranges and the sum of
the 24 hourly differences from the mean in the mean diurnal inequalities for the year
in the several elements. The calculated values are derived from the values given for
o and b in (A) Table XLIV. From the mathematical standpoint the nicety of
agreement is best judged by considering the last line in Table IV., which shows what
percentage the probable error in a calculated value is of the total variation exhibited
by the element in the 11-year period. For practical purposes the mean difference
between the calculated and observed values, and the percentage it forms of the
absolute mean value of the element, are, however, fully as important.

It will be seen that the agreement is about equally good for the ranges and for the
sum of the 24 differences. It is, on the whole, slightly better for D and H than for
V and I.

160 DR. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

• TABLE IV. — Kew (Units 1' for Angles, ly for Force Components).
Observed less Calculated Values.

Range from mean diurnal inequality
for the year.

Sum of the 24 hourly differences in
the mean diurnal inequality for
the year.

Year.

D.

H. V.

I.

D.

H.

V.

I.

1890 ....

+ 0-49

+ 0-5

+ 2-12

+ 0-5

1891 . . .

+ -40

+ 3-0 + 1-4

+ 0-21

+ 2-55

+ 16-5

+ 6-3

+ 0-86

1892 . . .

+ -24

- 1-4 !+ 1-5

•12

+ 2-22

6-8

- 0-2

- -54

1893 . . .

+ -32

+ 0-9 - 1-1

+ -05

•86

+ 2-9

- 0-2

+ -12

1894 . .

- -16

+ 0-8 - 0-9

+ -09

•49

+ 9-9

- 3-7

+ 1-12

1895

- -28

- 0-4 + 0-4

•01

. 1-05

6-4

+ 0-8

•61

1896 ....

•14

- 1-4 + 0-4

•08

•00

6-6

+ 7-7

- -76

1897 . . .

- • 53

1-1 1-8

•05

- 1-68

- 6-4

-11-4

- -24

1898

- -41

1-9 - 0-2

•13

1-11

- 14-0

+ 0-3

- -78

1899

+ -07

+0-3 + 1-2

- -01

+ -25 + 8-3

+ 4-8

+ -50

1900 . ....

+ -01

+ 0-6 - 0-9 !+ -07 - 1-93 + 2-3

- 4-2

+ -33

Mean difference calcu-

lated -»- observed . . 0-28

1-12 0-98

0-082

1-30

7-3

4-0

0-59

Probable error ... 0'23

0-95 0-78 0-071

1-09

6-1

3-8

0-46

Mean value of element . 7 "90

26-2 18-0

1-43

41-6 154-5

96-9

8-30

Range of element ... 3 • 54

15-5 7-5

0-95

21-0

108-1

37-1

6-35

Mean difference x 100 ,

4 5

G 3 5

4

7

mean value

Probable error x 100 „

6 10 7

5 6

10

7

range of element

Pawlowsl Data (59° 41' N. Lit., 30° 29' E. long.).

§ 10. Magnetic observations at tbe three Russian magnetic observatories at
Pawlowsk, Kathariuenburg, and Irkutsk are published very nearly on parallel lines
under the auspices of the Central Physical Observatory at St. Petersburg. The
results are very complete, and are kept well up to date. For the present enquiry I
have selected Pawlowsk and Katharinenburg. The former is, I believe, the furthest
north magnetic observatory which has been in continued existence for any length of
time. Its results include very full details of diurnal maxima and minima, and of the
amplitudes of movements in magnetic storms. They are not confined to " all" days,
but give in addition full particulars for selected " quiet" days, or " normal" days as
they were called by WILD, to whom the idea of their separate treatment is due.

I have treated the Pawlowsk data for the 11 years 1890 to 1900 by the method of
groups, taking the same combination of years as for Kew, viz., 1892 to 1895 for sun-
spot maximum, and 1890, 1899, and 1900 for sun-spot minimum.

Table V. refers to the ranges from the mean diurnal inequalities for the several
months of the year, both from " all" and " quiet" days. The seasonal and yearly
values of a and b are arithmetic means from the included months, and these
arithmetic means are employed in calculating the seasonal and yearly values of bja.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

101

5

fl
«

d
o

I

o

<D

o

i

^

M

=2

in

0)

'So
d

^

«s

d

ID

a

HH

P

^

H

CO

o

CO
IB

I

cS

I

[3

s

0

o

03
0)

s

o

I*

03

VOL. CCIII. — A.

s

>2

i

e'

1"

^_
o

-O

s.

g

o

a

o

2

' CO O »-H Oi C5

COT-HOOlOOi—COOOt^lOtO

I-HI— (<MI—I<M<MI— li— I^HP— i

CO t- O >0

•* O O3 O

O5 OO t- i-H
• IO
t-H I— 1 1— t

l-H 1C l-H

o o T!I

i— l<Ni— lr-l(Mi— Ir-lOO

a» to oo

CO «O Oi •

O '~) ^^ i— <

I-H CO O CO
50 00 1O «0

O CO 00 Ir-

CO O2 I-H OT

^ O< O) IM

hCS
M

O

'3

O"

CO C5 O •

o 10 o •

COCO-^OOCOl-Hl — 1 O

COOTCOOCSCOOOtOi— l

OO (M 7-1 t-

02 c-i -h -fj

CT CO C-l C-l

t- C-l T-H O

CO t- O t~>

o> o

CO T-H

1O C-1 CO l~ t- O O 00

0 OO

1 — 10

C-l CO

CO ^< l

CO O>

<?•! OO O O5 l-^ i

l^ X) O OO CT.

so t—

Oi o

0"

i— I CO (M CO <M <M i— I i— I (M O) i— I

eo O
o •*

i-H C-l

1-1 ao
co o

C-l C-l

(M CO (M CO CO <M <M C-l i— c

C-l CO

c-i -*

CO <M

ooii— ic-i-^ocoosi— 10-*

t-Hi— i CO CO CO CO CO <M C<1 i—i

c-i c-i

CO C-l

OOOOO-tcOCOOMlOiOr-iCS

00 00 <M CO IO C-l i-H C-l 00 C-l O I—

(MCOCOCOCOCOlMC-li-H

.2
'3

C?

OOlOi— i
COOC^lO

r-H CM i — 1

ICOCO(MCOCO(M

CO CO
(M

o oo

co co

-M C-l

CO Cl

c-i oo
m to

i-H to
OO t^

C-l tO
O CD

O
"3

G?

i— 1 -* C-l C-l

r-H OO CO •*

CO -* O •*

eoo»co(Mi

to Cft IO CO

01 o o •<*

CO IO IO -^

•* -* t- C-l

C-l i— I r- 1 IO

C-l t- O to

-* CO «O T-H

o t- to oo

fagfe

Ill

" 3

162 DR. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

S 11. Table V. presents several novel features. In the declination we see a
conspicuous difference between the variation of a throughout the year on " all " and
on "quiet" days. In December and January the "all" day value of a is more than
double the "quiet" day value, and the excess of the "all" day value is also
prominent in November and February. On the other hand, the " quiet " day value
of a is, in general, distinctly the larger throughoxit the equinoctial and summer
months. There is no such prominent difference between the "all" and "quiet" day
values of b for declination. There are, of course, conspicuous differences in one or two
individual months, but the seasonal and yearly values are closely alike. The rise of
b/a in winter and its fall in summer are conspicuous, especially in the "quiet" 'days,
where the phenomenon is even more prominent than at Kew. The mean values of
b/a for the year from "all" and from "quiet" days are in close agreement with one
another and with the corresponding Kew value.

In H the seasonal and yearly values of a in "all" and in "quiet" days are much
alike. The winter and equinoctial values of b in the two cases are also nearly equal,
but in summer the "quiet" day value is very decidedly the smaller. The mean
"all" day value of b for the year is distinctly larger than the " quiet" day value,
which is itself slightly in excess of the corresponding Kew value. The excess of b/a
in winter is conspicuous in both "all" and "quiet" days; in the latter case the
variation of b/a throughout the year is pretty similar to that at Kew.

In V the "all" and "quiet" day phenomena are vitally different. The fact that
the mean diurnal range during the 11 -year period for the "quiet" days was barely
40 per cent, of that for "all" days prepares one for a material difference between the
phenomena in the two cases, but hardly for the "all" day mean yearly value of b
being more than six times the corresponding " quiet" day value. The " quiet" day
value of a for summer is not much less than the "all" day value, but in the
equinoctial and winter months the latter greatly predominates. The value of b is
greatest at the equinox in both " all " and " quiet" days ; in fact, in " all " days the
summer value of b falls short of the winter value, notwithstanding a marked
depression in December and Jamiary. The variation of b/a throughout the year on
" quiet" days is somewhat irregular. In " all " days we have the fall in summer and
rise in winter seen at Kew, but there is a prominent depression in December and
January. As regards the absolute magnitudes of b and b/a, the " quiet " day data
are much closer than the " all " day to the Kew results.

Table V. gives only " all" day data for I, as no " quiet" day data for this element
seem to be published.

Here again the phenomena resemble those observed at Kew, b being conspicuously
small in winter and b/a small in summer. The values of a and b are, on the whole,
distinctly larger than at Kew (where the diurnal range of I is less than at Pawlowsk),
but the mean value of b/a for the year is 105 X 10 ~4 as compared with 111 X 10~*
at Kew.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

163

§ 12. The data ascribed to the year in Table V. are arithmetic means from the
12 monthly values. Table VI. supplies values of a, b and b/a for the ranges from the
mean diurnal inequality for the year as given in the Pawlowsk tables. They answer
to the range data for Kew in Tables XLIII. and XLIV. of (A), and I give the results
in the latter of these tables (calculated by least squares) for comparison.

TABLE VI. — (Units 1' for angles, ly for Force Components.)
Mean Diurnal Inequality for the Year at Pawlowsk.

Declination.

i
Horizontal force. Vertical force.

Inclination.

a.

104 x b.

104 x bla.

a.

103 x b.

104 x bla. a.

'

103 x /;.

104 x b/a.

a.

104x/;. 104xi/a.

All days . .
Quiet days .
Kew . . .

5-74
6-17
6-10

400
424
433

70
69
71

20-7
20-G
18-1

211
195
194

102 8-1
95 5-9
107 14-3

1

265

27

«

326

46
56

1-24
0-87

126 101

125 145

1

§ 1.3. The Pawlowsk tables give for each month the mean of the differences between
the daily maxima and minima, irrespective of their time of occurrence. The range
thus obtained is, of course, larger than that from the mean diurnal inequality for the
month, and is a quantity considerably more influenced by magnetic disturbances.
The mean of the 12 monthly means may be regarded as the mean for the year of the
absolute ranges in individual days. This is the quantity to which the results in the
first line of Table VII. apply. The figures in the second line refer to the mean of the
12 monthly ranges, a monthly range being defined as the difference between the
highest and lowest values recorded during the month. The third line in the table
refers to the annual range, i.e., the difference between the highest and lowest values
recorded during the year. Owing to occasional losses of trace, monthly and annual
ranges are sometimes under-estimated, especially at times of large disturbance. Both
quantities are mainly dependent on the amplitude of disturbances ; the mean monthly
range is the better measure of the generally disturbed character of the year.

TABLE VII. — Pawlowsk (Units 1' for angles, ly for H and V).

Declination. Horizontal force.

Vertical force.

a. 10s x b.

104xi/rt.

a.

10-' x b.

104 x b/a. a.

1

102 x b.

104 x b/a.

Mean daily range .
„ monthly range .
Annual range . . .

11-28 113
28-6 538
62-9 893

100

188
142

45-2
102
240

64
413
1226

141
406
510

17-6
74
338

52
412
698

295
556
207

Y 2

164 DE. O.-CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

§ 14. If we compare the results for the mean daily range in Table VII. with those
for the range of the diurnal inequality in Table VI., we see that, though the values
of a in Table VII. are about double those in Table VI. in the case of D and H, the
increase in b is relatively so much greater that the values of b/a in Table VII. are
some 40 per cent, in excess of those in the earlier table. This means that the
amplitude of magnetic disturbance is in general more enhanced relatively in years of
sun-spot maximum than is that of the regular diurnal inequality. The great rise in
b/a as we pass from the mean daily to the mean monthly range in Table VII. is
evidence of the same fact. That this rise in b/a is not continued, except in H, as we
pass from the mean monthly to the annual range, is at least consistent with the view
that the incidence of exceptionally large magnetic storms is determined by causes of
which sun-spot frequency is no exact measure.

§ 1 5. Table VIII. supplies information as to the degree of accordance between the
observed values of the ranges and those calculated from the values of a and b in
Tables VI. and VII. Table VIII. is constructed on parallel lines to Table IV.

TABLE VIII.— Pawlowsk (Units 1' for D and I, ly for H and V).
Observed less Calculated Values.

Ranges

from mean diurnal inequalities.

Absolute mean daily

ranges.

Year.

"Quiet "(normal) days. " All " days.

D.

H.

V.

D.

H. V. I. D. H. V.

1890 ....

4-0- -7

+ 3

+ i

+ 0'30

I

0 i 0-00 -^ n-rm 1-1 n-9

1891 .... + 0-3

4. 1

i

4- '15

_i_ O 4. 9 _}_ • 1 0

+ 0'71 + 2-5

+ 3-2

1892 + -6

- 4

+ i

+ -09

4- 1 ,+ 8 + -05

+ 1-51 +19-4

+ 17-5

1893 + -3

+ 2

+ i

4- -51

. i

11

•07

3-05

20 -5

21-0

1894 + -3

+ 5

•>

• ''8

i i

4. 1

•OR

4. n-33

4. 9- 3

4. 3-4

1895 .... -8

1

j

•08

i

0

4- -03

. 0-44 -- 6-2

4-7

1896 _ .3

_ 9

o

•02

i

_L A.

•00

+ 1'46 + 2-0

+ 3-3

1897 . . -a

2

. 1

• on

On

4. -no

4. n- 11 n -7

. i •«

1898

_ -3

2

]

. F;FC

n

4. 1

•09

4. 0 • 4.0 4- <")•(•) 14- 3 • fi

1899

- -2

_ 2

0 -20 + 1

4- 1

+ -04

+ 0-50 + 4-9 + 2-7

1900 ....

+ '3

4_ i

1 _L -AQ 1 1

Q

.no

i .01 7 .«

- 6-9

Mean difference calcu-

lated ~»- observed .

0-38 2-3

1-0

0-21 0-8

3-1

0-043

0-96

6-56

6-13

Probable error .
Mean value of element

0-30 1-8
7-94 28-7

0-8
7-0

0-19 0-7
7-41; 29-5

3-2
19-2

•037
1-77

0-91
16-0

6-60
71-8

6-27
39-3

Range of element . .
Mean difference x 100

3-6

20

4

3-62, 16

27

0-96

10-5

67-4

56-7

mean value

5

8

14

3

3

16

2

6

9

16

Probable error x 100

range of element

8

9

20

5

4

12

4

9

10

11

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM. 165

The calculated values of H and V employed in Table VIII. for the inequality
ranges were taken only to the nearest ly, because the Pawlowsk tables go no nearer
than this; but in the "all" days' D and I, both tables and calculation go to 0/-01.
The agreement between calculated and observed values is much closer in D, H,
and I than in V ; and in D and H it is considerably closer for the " all" day than the
"quiet" day results. Probably this only means that the fewness of the "quiet"
days (sometimes only two or three a month) introduces an element of uncertainty
which more than neutralises the effect of the greater regularity in these days.

§ 16. If the range of magnetic elements were largely dependent on influences
which did not proceed pari passu with sun-spot frequency, then what we should
expect to see in Table VIII. would be a notable occurrence of large -\- values in all
the elements in some years, and of large — values in other years. The same result
would follow if, while an intimate connection subsisted, it were not of the linear type
assumed in (l).

So far as the inequality ranges in D, H, and I are concerned, there is no
indication of such a phenomenon. There is indeed an excess of -4- signs from 1890
to 1894 and of — signs from 1895 to 1899, but the differences themselves are small,
and those for the " all " and the " quiet " days show no kind of regular relationship.
In the case, however, of the "all" day V inequality, and of the absolute daily ranges
for all the elements, especially H and V, the observed values are conspicuously in
excess of the calculated in 1892, and as conspicuously below them in 1893. This
phenomenon seems due beyond a doubt to the influence of the disturbance
element.

§ 17. With a view to further elucidation of the phenomenon described in the last
paragraph, I have placed side by side in Table IX. data as to the mean value for
each year of a variety of quantities which are affected in different degrees by
magnetic disturbance. The small figures in brackets attached to the annual figures
show the position which the year in question would occupy on a list which followed
the order of magnitude of the quantity in question. If two yearly items are equal, a
common number is attached. In the case of the years themselves, the attached
figures indicate the order when the arrangement follows sun-spot frequency. It
should, however, be noticed that the excess of sun-spot frequency in 1898 over 1897
was very trifling, and that the differences between 1899, 1900, and 1890 were not
large.

In the case of the diurnal inequalities in D and H, quantities but little affected by
disturbance, 1893 heads the list, just as it does in sun-spot frequency. In the case
of the mean daily range — a quantity more influenced by disturbance— 1892 and 1894
come to the front, and 1893 falls to the fourth place. Coming to the mean of the
monthly ranges, we see 1892 and 1894 still more in advance, while in the case of
H and V 1893 stands lower than 1898, a year of less than one-third its sun-spot
frequency.

166 DE. C, CHEEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

In the case of the annual range, 1893 has fallen to the eighth place in D and
ninth place in V, whilst 1898 mounts to the third or even the second place.

In the mean daily, mean monthly, and annual ranges, 1892 and 1894 are as
conspicuously in excess of what one would expect from sun-spot frequency as 1893
and 1895 are below it. Thus when we treat these four years as a unit, and compare
it with a similar unit made up of the three years 1890, 1899, and 1900, we may
arrive at a conspicuous connection between sun-spot frequency and amplitude of
disturbance ; but at the same time there is a marked absence of the close and regular
connection in individual years which characterises the inequality ranges in D, H,
and I.

TABLE IX.-Pawlowsk (Units 1' for D, ly for H and V).

Dinrna

inequality Jf (, ; y
mge.

Mean monthly range. Annual range.

Vnnr

D.

II. D.

II.

D.

H.

TT

D.

H. V.

i

1890 O1). • • 6-32 (

) 220") 12-11.0"), •«>('")

28-20')

118 (")

80 (") 42 -1 (»)

169 (")

179 (")

1S91 ("), . . 7-31 (") 30 (5) 10-01 ("')

70 (")

46-3 (!)

218 (')

233 (') 92-3 (6)

550 (»)

614 (4)

1892 (:1) . . . 8-75 (

) 37 (3) 21 -01 (')

in 0)

93-6 (')

G98 (')

575 (') 194-0 0)

2416 (')

1385 (')

1893 (') . . . 9-04 (

) 38 0) 17-82 (') 79 (4)

48-3 (J)

241 (4)

210 0) 87-1 (s)

514 («)

457 (9)

1.891 (-) . . . 8-58 (:1) 38 (') 20 '42 '(-) Q7(-)

84-1 (-)

493 (-)

493 O

145-6 (2)

1227 (2)

878 C)

1895 0) . . . 8-22 0) 33 (') 18 '07 (:1) 80 (:l)

47 -4 (5)

220 (6)

223 (") 73-9 C)

395 (»)

534 («)

18'.K> (•"') . . . 7 -39 (

) 29 (") 17 -46 (•') 74 (')

52-1 (*)

232 (')

236 (4) 88-7 (')

574 (4)

608 (5)

1897 (s)'. . . 6-79 (

) 2G (') 14-57 (s) Gl (s)

43-8 (s)

201 (s)

170 (s) 101-1 (4)

449 (") : 480 (8)

1898 (!) . . . G-25 (»') 2G (7) 14 "70 (7) 67 (!)

46-6 (")

276 (:i)

242 (:1) , 118-9 (3)

1136 (')

888 (2)

1899 ('•'). . . G-0201) 24 (9) 13 '11 (9)

58 (»)

38 -3 (!l)

178 ('•>)

150 ('•') 63-80")

382 ("')

527 (7)

1900("'). . . 6-200") 22 ('") 10-540') 44 (")

32 -8 ('")

134 ('") 89 ('")

94-2 (5)

457 (7) 365('")

Means . . 7 '41

30 : 15 -99

72

51 -1

i

274 246

100-2

752

629

§ 18. It was pointed out in (A), §§74 and 75, that whilst an intimate general
connection between sun-spot frequency and diurnal magnetic ranges is unmistakable,
it is open to doubt whether the mean values of these quantities for so short a period
as a single month can be regarded as directly interconnected.

If both phenomena proceed from a common cause whose intensity of action at a
given instant varies throughout the solar system, then it might possibly be better to
compare monthly magnetic ranges with sun-spot frequency for a longer overlapping
period.

As shown in (A), Table I., the mean sun-spot frequency for individual months of
the year from the period 1890 to 1900 varied from 35-0 in November and 35'5 in
March to 4.5 '4 in June and August. Clearly, if the connection between sun-spot
frequency and magnetic range is of the more general kind indicated above, the values
we have found for b and b/a at Kew and Pawlowsk will be too large in months such
as November and March and too small in months such as June and August.

To obtain an outside estimate of the uncertainty thus existing, I have calculated
values for a, b and b/a for the "quiet" day Pawlowsk data, employing WOLFER'S

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

1(57

smoothed sun-spot frequencies (Ausgeglichene Relativzahlen), each of which is a mean
from observed values for 13 months, of which the individual month forms the central
period. Table X. gives the mean seasonal and yearly values thus found ; these
answer precisely to the seasonal and yearly values based on observed sun-spot
frequencies which appear in Table V.

TABLE X.— (Units 1' for D, ly for H and V.)

Pawlowsk "Quiet" Days with WOLFER'S Smoothed Values (Ausgeglichene

Relativzahlen).

Declination.

Horizontal force.

Vertical force.

a.

104 x I.

10> x I la.

a.

103X//. '10'x /;/«.

a.

103 x //.

10 > x li/n.

Winter . . . 2 -17

312

144

8-0

152 191

3-6

23

63

Equinox . . 7 -29

428

59

26-2

216 82

7 • 4 55

74

Summer. . . 9-98

616

62

31-9

265 83

9-9 58

58

Year 6-48

452

70

22-0

211 96

7-0

45

65

So far as the mean yearly and winter values of a, b and b/a, are concerned,
Tables V. and X. are in practical agreement, but the equinoctial values of i> in
Table X. are decidedly lower, and the summer values decidedly higher, than the
corresponding quantities in Table V. The fact that the equinoctial values of b/'n for
D and H in Table X. fall slightly below the summer ones seems hardly likely <i priori
to be a natural phenomenon, and it is not in accordance with the results obtained for
Greenwich, in Tables II. and III., from the longer periods, where the variation of the
mean sun-spot frequency from month to month is naturally less than in 1890 to 1900.

§ 19. The effect of the substitution of the smoothed sun-spot frequencies on the
values of b and b/a from month to month is most easily followed by expressing the
monthly values as percentages of their mean for the 12 months. Table XL gives the
mean of the results thus obtained for D and H, employing smoothed and observed
sun-spot frequencies for the "quiet" days, and observed frequencies for the "all"
days. The employment of smoothed frequencies for the "all" days would alter the
results to about the same extent as it does in the " quiet " days.

The substitution of the smoothed for the observed sun-spot frequencies for "quiet"
days removes an isolated prominence shown by the b variation in March, and removes
slight depressions in June and August, but it produces a depression in September and
adds materially to an already conspicuous prominence in May. Also the June
depression and the March prominence are not apparent in the " all" day b variation
using the observed sun-spots, and if we used smoothed frequencies for the " all " days
we should have a marked depression in March and a largish prominence in June.

1(58 DR. C. CHEEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

The b/a variation throughout the year proceeds most regularly when we use the
observed frequencies.

TABLE XL — Pawlowsk.
Variation of b and b/a throughout the Year.

6.

b/a.

" Quiet " days.

" All " days.

" Quiet " days.

"All" days.

Smoothed

Observed

Observed

Smoothed

Observed

Observed

frequencies.

frequencies.

frequencies.

frequencies.

frequencies.

frequencies.

January .

50

50

43

116

117

85

February.

9G

95

97

188

187

180

March. . . .

110

159

128

89 135

124

April ....

123

116

119 66

63

76

May ....

171

151

123 90 78

75

June ....

126

114

142 59 55

90

July ....

124

120

137 63 63

89

August . . .

103

77

88

53 39

54

September .

71

85

89

40

52

69

October .

84

90

101

67

75

113

November . .

72

8C

95

121

147

170

December

04

5,

38 248

188

75

Katharinenburg (5G° 49' N. lat,, GO0 38' E. long.).

§ 20. Table XII. gives values of a, b and b/a for the range of the diurnal inequality
for each month of the year, and arithmetic means for the seasons and the year, at
Katharinenburg, corresponding exactly to the " all " day results for Pawlowsk given
in Table V.

In D, 7; appears decidedly less at Katharinenburg than at Pawlowsk, especially in
winter. The dip in the December and January values of b in Table XII. is
particularly striking. The summer and equinoctial values of b/a at Katharinenburg
are very similar to those at Kew and Pawlowsk, but the winter value is much less,
owing to the low values in December and January.

In H the mean b for the year is close to the Kew value, but the winter values
of b and b/a are distinctly lower than at Kew and Pawlowsk.

In V, Katharinenburg occupies an intermediate position between the " all " and the
" quiet " day results for Pawlowsk The mean value of b for the year in Table XII.
is little over half the corresponding " all " day value at Pawlowsk, but it is more than
thrice the " quiet " day value at Pawlowsk, and double that at Kew.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

169

The winter values of b and b/a at Katharineuburg are exceptionally large,
notwithstanding a marked depression in December and January.

TABLE XII. — Katharinenburg (Units 1' for D and I, ly for H and V).
Ranges from Mean Monthly Diurnal Inequalities.

Declination.

Horizontal force.

Vertical force.

I

ndination.

a.

b x 104.

104 x b/a.

a.

10* x/,

104x///«.: a.

103 x b. 104 x b/a.

104 x b.

104x/>/«

\

January .
February .

March . .

3-09

3-07
5-48

61
293
605

•20
95
110

7-0
9-8
20-6

91
115
231

130

117
112

6-1
4-8
9-5

95
249
224

156
514
235

0-55
0-69

1-21

61
60
138

112

88
114

April . .
Mav . .

9-18
9-62

483
509

53 24-5
53 32-0

308
185

126 15-8
58 16-5

121
121

76
73

1-32 181
1-77 111

143
63

June .

9-11

545

60 27-6

264

95 ;ll-8 159

135

1-54 144

94

July . .

9-12

438

48 26-1

296

113 9-9

235 238

1-39 177

128

August
September
October .

9-04
6-54
4-33

310
351
305

34 27-5
54 23-7
71 16-7

197 72
201 85
246 147

11-5 102 89
7-8 137 175
5-9 169 285

1-56 114
1-41 130
1-02 164

73
92
161

November

2-85

251 88

6-9

164 239 4-9 169 345

0-48 112

232

December.
Winter

2-68

114 | 42

7-7
7-86

38 , 50
102 130

5-4 96
5-32 152

177
286

0-54
0-565

38
68

71
120

2-92

180 61

Equinox .
Summer .

6-38
9-22

436
451

68 ,21-38
49 28-30

246
235

115

83

9-77 163 167
12-44 154 124

1-239
1-563

151
137

124

87

Year . .

6-18

355 58

19-18

195 101

9-17 156

170

1-122

120

106

In I, both a and h are distinctly smaller at Katharinenburg than at Pawlowsk ;
in summer and the equinox they are very similar to the corresponding values at Kew.
The winter value of b at Katharinenburg is decidedly less than at Kew, there being
specially low values in December, January, and February. The mean value of bja
for the year is very close to the corresponding values at Kew and Pawlowsk.

§ 21. Table XIII. gives results for the mean of the absolute diurnal ranges for
individual months, with corresponding seasonal and yearly values. The last mentioned
correspond nearly to the Pawlowsk data in the first line of Table VII. (see § 22 for
nature of difference).

In D the values of a in Table XIII., whilst invariably larger than the corresponding
values in Table XII. , are not very conspicuoxisly so, except in winter. The values
of b, however, in Table XIII., are conspicuously larger than those in Table XII., the
mean values for the year being roughly one double the other. The difference between
the values of b/a in the two tables, though less prominent, is unmistakable.

VOL. coin. — A.

170 DE. C. CHKEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

TABLE XIII. — Katharinenburg (Units 1' in D, ly in H and V).
Monthly Means of Absolute Daily Ranges.

Declination.

Horizontal force.

Vertical force.

a.

104 x b.

104 x I>/a.

a.

103 x b.

104 x b/a.

«.

10»x6.

10* x b/a.

1

January

6-34

247

39

25-2

200

79

11-9

139

116

February .

5-62

920

164

21-7

502

231

10-2

384

377

March .

8-44

974

115

31-7

468

148

15-6

383

245

April . , .

10-38

604

58

35-4

392

111

20-7

225

109

May ....

10-97

615

56

45-7

242

53

22-8

159

70

June

10-00

666

67

36-1

' 397

110

17-9

215

120

July ....

9-61

706

73

32-6

538

165

15-0

324

215

August .

9-92

503

51

37-9

304

80

16-9

203

120

September . .

7-61

803

105

32-8

365

111

15-3

219

143

October . . .

6-68

6G1

99

28-9

365

126

11-4

255

223

November .

4-67

910

195

18-7

440

236

8-6

313

365

December . .

5-15

427

83

21-5

194

90

10-1

155

152

Winter . . .

5-44

626

i
115 21-76

333

193

10-20

248

242

Equinox

8-28

761

92 32 -21

398

123

15-76

270

172

Summer. . .

10-13

623

61 38-08 '•> 370

97

18-15 225 124

Year. . . .

7-95

670

84 30-68 367

120

14-70

248 168

i

1

In II the values of b in Table XIII. are on the average about double those in
Table XII., but owing to the large values of a in Table XIII. the excess in its values
for b/a is not striking, except in winter.

In V the values of a and b are again much larger in Table XIII. than in
Table XII., but the seasonal and yearly values of b/a in the two are closely similar.
In Table XIII. the December and January values of b/a are conspicuously low in all
the elements as compared to the values for November and February.

§ 22. Table XIV. gives results for the range from the mean diurnal inequality for
the year (corresponding to the Pawlowsk data in Table VI.), for the mean of the
absolute daily ranges for the year (corresponding _to the first line of Table VII.), for
the mean of the 12 monthly ranges, and for the yearly range. The results for
the second of these quantities, though practically accordant with those in the last
line of Table XIII. , are not absolutely identical. The figures in Table XIII. represent
arithmetic means of a and b resulting from applications of formula (l) to individual
months of the year. Table XIV. assumes the 12 monthly mean ranges for each
year to be meaued, and these means dealt with by a single application of formula (1).
The last two quantities dealt with in Table XIV. do not in reality accord very closely
with the linear formula (1), but the figures at all events supply, as in the corre-

BETWEEN SUN-SPOT FREQUENCY AND TEREESTEIAL MAGNETISM. 1/1

spending case at Pawlowsk, a rough measure of the amplitude of the fluctuation
throughout a sun-spot period.

In all the elements included in Table XIV., as we pass from the range of the
diurnal inequality to the mean absolute daily range, and thence to the mean monthly
range — quantities increasingly influenced by magnetic storms— we see that whilst
a increases, b increases in a greater ratio, so that b/a notably rises.

The fall of l>/a as we pass from the mean monthly to the annual ranges in D and V
may not improbably possess no real significance, but a similar phenomenon, it should
lie remembered, presented itself in the corresponding Pawlowsk results in Table VII.

TABLE XIV.— Katharinenburg (Units 1' in D and I, ly in H and V).

Declination.

Horizontal Force.

Vertical Force.

Inclination.

1

a.

10<i.

104 6/0.

a.

Itf' X I.

104 x i/o.

a.

10:i x I,. 10* x 1,'a.

a.

\
JO'x J. 104x !>la.

Mean diurnal in-

equality for the

'
i

year

5 -20

342

65

IB -8

1S2

109

8-6

117 , 137 ' 0-93

105 113

Mean of absolute

daily ranges for

the year ... 8 -(X)

052 82

30 "7

366

no 14-6 I 248 171

Mean of absolute

monthly ranges

for the year . .

18-5G 2552 ; 137 70 '3

1080

220 40-:} 1770 382

1

Absolute yearly

range ....

•11 -85

4750 j 118

1 it; -:i .1590

314

178 •!) 3600 201

— —

1

§ '23. Table XV. shows the excess of observed over calculated values at Katharinen-
burg ; it answers to Table VIII. for Pawlowsk.

The results for the diurnal inequalities in 1), H, and I in Table XV. are similar to
the corresponding "all" day results in Table VIII., but on the whole show a slightlv
less close agreement between theory and observation. In V, however, the agreement
is distinctly better at Katharinenburg than at Pawlowsk.

In the case of the absolute daily range the agreement between observed and
calculated values is particularly good in D, and it is closer in all the elements than at
Pawlowsk. This may be ascribed to the fact that magnetic disturbances are larger
at Pawlowsk than at Katharinenburg.

The difference between the observed and calculated values in the monthly range
is somewhat large, and there is now clear indication that sun-spot frequency is not
by itself a sufficient guide. The observed values in 1893 are conspicuously below,
and those in 1892 and 1894 conspicuously above, the calculated. The deficiency in
the observed values in 1895 and the excess in 1898 are also marked. Even in the
absolute daily range in Table XV. there is a distinct depression in the observed
values in 1893 and corresponding enhancement in 1892, though not to the same
extent as in the corresponding case at Pawlowsk (see Table VIII.).

z 2

172 DR. C. CHEEK: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

TABLE XV. — Kathavinenburg (Units 1' in D and I, ly in H and V).
Observed less Calculated Values.

Year.

1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900

Mean difference calcu-
lated -»- observed
1 Probable error .
Mean value of element
Range of element .
Mean difference x 100

mean value

I'robable error x 100

range of clement

Ranges from mean diurnal
inequalities for the year.

Mean of absolute
daily ranges.

Mean of 12 monthly
ranges.

D.

II. V.

I.

D.

H.

V.

D.

H.

V.

+ 0-30

0 0

+ 0-04

+ 0-03

0 0

- 2-4

- 7

- 11

+ -35

+ 2 !+ 1

+ -07

+ • 45

+ 3 :+ 2

- 1-3

9

6

- -04

+ 2 + 2

+ -02

+ -fi2 + 6 +4 +6-5

+ 54

+ 56

+ -64

_ _ ••>

•03 - -94

-8 - 8 -10-2

- 69

- 90

•151- 1

- -03

- -07

0 .+ 1 + 7-0

+ 54

+ 94

•18 ) 1

+ -03 -19-1 1 - 6-7

- 44

- 60

•21 + f 2

+ -01 + -39!+ 2

+ 2

+ 0-2 - 3

18

•17! 0

•05

- - 03

0

+ 1

+ 1-9

+ 3

6

•44 - 1

- -02 + -23 + i

+ 2 +5-0+18

+ 28

•37+1 +1

+ -06 + -04+2

0 - 0-1 + 11

+ 15

+ -27

.1-2

•10

- -51

- 4

-3 - 0-1 - 8

- 3

0-28

1-0 1-2

0-042

0-32 2-4

2 • 2

3-8

26

35

0-23 0-8 1-0 0-035

0 • 30 2-5 2-2 3 • 5

25

34

6-71 24-4 13-r

1 • 373

10-72 4f> 25 29-2 146

120

3 • 50 : 15 11

0-87 5-27: 33

23

27-5 180

230

4 4 9

3

359

13 18

29

7 5 9

4

6

8 10

13 14

15

§ 24. Table XVI. supplies disturbance data at Katharinenburg, corresponding to
tbose given in Table IX. for Pawlowsk. If we compare tbe 11-year means in the
two tables we see convincing proof of the remark already made that Pawlowsk is
more disturbed than Katharinenburg, the mean ranges in Table IX. being roughly
double those in Table XVI.

The small bracketed figures in Table XVI. have the same significance as those in
Table IX.

According to the mean monthly range— probably a better criterion than the
annual range — 1893 would seem to have been relatively less quiet at Katharinenburg
than at Pawlowsk, but it stands much below 1892 and 1894. Whilst, however, all
the Pawlowsk data made 1892 more disturbed than 1894, the monthly ranges at
Katharinenburg give the first position to 1894. In all the columns 1890 appears as
the least disturbed year. The monthly ranges — though not the annual ranges —
assign to 1900 and 1899 the two next lowest places, the same positions as they
occupy according to sun-spot frequency. But, as at Pawlowsk, 1895 is less disturbed,
and 1898 much more disturbed than they should be if sun-spot frequency were the
sole criterion.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.
TABLE XVI.— Katharinenburg (Units 1' in D, ly in H and V).

173

Mean monthly range.

Annual range.

I).

H.

V.

D.

H.

V.

1890 . . .

18-0 (H)

82 (ii)

48 (H)

29-3 (H)

163 (»)

117 (ii)

1891 . . .

26-4 (»)

127 (")

101 («)

47-1 («)

205 (i°)

257 (6)

1892 . . .

43-8 (2)

253 (-)

232 0

116-9 (i>

837 (»)

591 (2)

1893 . . .

30-0 (4)

150 (3)

106 (4)

45-5 (••')

296 (6)

172 (1°)

1894 . . .

45-5 (')

261 (')

278 (!)

90-0 (-')

674 (2)

849 (i)

1895 . . .

28 -2 ('')

140 f)

100 (")

47-4 (<)

205 ('•')

184 (»)

1896 . . .

29-4 (5)

143 (*)

102 (*)

48-6 (6)

320 (4)

244 («)

1897 . . .

27-2 (~)

124 (»)

87 (»)

67-1 (••)

233 (7) 210 (~)

1898 . . .

30-4 (3)

139 («)

122 (»)

S3-8 (3)

338 (3)

471 (3)

1899 . . .

21-6 (••')

10S (•••)

S3 ('•')

40-8 ('")

208 (8)

333 (4)

1900 . . .

20-9 (10)

84 (!")

60 H

01- (5 (•')

237 («)

191 (»)

Means . .

29-2

146

120

61-6

338

329

Batavia (6° 11' S. lat, 106° 49' E. long.).

§ 25. Prior to the introduction of electric tramways in 189!) the magnetic results
fit Batavia Observatory were treated with great completeness in the annual Batavia
1 Magnetical and Meteorological Observations.'

Up to the end of 1900 the I) and H results seem to have suffered comparatively
little, but the V results even then were too disturbed for publication.

The Batavia magnetic records go back to 1882, but are incomplete until 1884.
Inspection of the vertical -force data for 1884 and 1885 created some misgivings,
which gathered force from an editorial statement that the scale value in that element
was at first very variable and remained so until a new magnet was introduced.

After considering all the circumstances, I decided to confine myself to the results
for the 12 years 1887 to 1898, coming down to the latest time at which all the
elements were free from electric-tram effects. This period has the advantage of
supplying a sun-spot minimum group of years 1887 to 1890 equal in length to the
sun-spot maximum group 1892 to 1895.

The Batavia tables give not merely the hourly values, but also the sum of the
24 differences from the mean, in the monthly diurnal inequalities.

In (A) I found the sum of the 24 differences in D and H to show the sun-spot
connection even more prominently than the ranges. Accordingly I decided to use
the sum of the 24 differences at Batavia, in preference to the ranges, when dealing
with the diurnal inequalities for the individual months, and to employ the sum of
the 24 differences as well as the ranges when dealing with the mean diurnal
inequalities for the year.

§ 26. Before giving the results, I would draw attention to a feature wherein Batavia

174 DR. C. CHEEE-. AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

differs widely from European stations. At Kew, for instance, D, H, V, and I all
show a large variation in the amplitude of the diurnal inequality throughout the
year. The range is three or four times as large at midsummer as at midwinter, and
the way in which the range, or the sum of the 24 differences, varies throughout
the year is pretty similar for all the elements. Thus, assuming that the mean diurnal
inequality for the year were derivable from a potential, one could obtain a fair first
approximation to the mean diurnal inequalities for individual months by multiplying
this potential by appropriate numerical factors.

How exceedingly far this is from being the case at Batavia will be seen on
inspection of Table XVII. This gives the sum of the 24 differences in the diurnal
inequalities for each month of the year, with their mean, and the sum of the
•24 differences in the mean diurnal inequality for the year. Batavia being in the
Southern hemisphere, May to August are the " winter" months.

The D data in Table XVII. proceed, on the whole, like European data. In V, too,
the lowest value occurs in the winter months, but there is likewise a low value in
December. While the average value for I from the four winter months is below the
mean for the year, the lowest values of all occur in November and December. In H,
three out of the four winter months show values above the average, while the four
summer months are all below the average. Thus no two elements behave alike, and
the phenomena exhibited by H are more nearly opposite than parallel to those
observed in high latitudes.

TAHLK XVII. -Batavia, 1887-1898 (Unit 1' in D and I, ly in H and V).
Sum of the ~24 Hourly Differences in the Mean Diurnal Inequality for the Month.

DC

olination. Horizontal force.

Vertical force.

Inclination.

1

January j
February
March i
April i
May
June i
July j
August i
September j

22-04 313 -8
22-55 310-4
16 -87 357-0
11-96 373-0
12-34 346-3
10-26 322-5
12-14 347-1
15-25 373-7
18-56 396-2
20 • 57 348 • 5
22-06 281-9
22-01 263-4

276-7
289-7
282-1
231-3
189-5
189-8
202-6
187-4
214-7
249-8
215-8
196-7

31-77
32 • 36
33-72
30-87
26-85
26-02
27-84
27-87
30-08
30-51
25-82
23-82

October

November j
December . ....

Arithmetic mean from 12
months

17-22 336-2

227-2

28-96

From mean diurnal inequality
for the year ...

13-73 334-5

216-6

28-36

.

BETWEEN SUN-SPOT FKEQUENCY AND TERRESTRIAL MAGNETISM. 175

§ 27. The results pointed out in the last paragraph help to explain some novel
features in Table XVIII. , which gives the values obtained for <i, b and b/a by
applying the method of groups to the sums of the 24 differences in the diurnal
inequalities for the several months. As in similar tables, the seasonal and yearly
values of a and b represent arithmetic means for the included months.

In D and V the lowest values of a are found in winter, but in H and 1 the lowest
values occur in November and December, that is, at midsummer.

In D and V, again, b is distinctly below the average in winter, though not nearly
to the same extent as in Europe, and the winter value for F is less than the summer
value ; but in H the winter value exceeds the summer.

In D, b/a is distinctly smaller in summer than in the other seasons ; but in 1, H
and V the summer value is a trifle the highest. The winter and equinoctial values
of b/a are almost identical in all the elements. As compared to northern stations,
the variation in b/a throughout the year is exceedingly small.

TABLE XVIII. — Batavia. Sum of 24 Hourly Differences, L 887 1)8 (Units I' in

D and I, ly in H and V).

i

Declination.

Inclination.

Horizontal force.

Verticil 1 force.

a.

103 x b.

10'xi/K,

,i. 10:txk

104 x l>j«.

c.

10L' x/).

104x//,«.

u. 10-x/f.

10'xA/V/.

January .

18-09

109

GO 1 25-37

178

70

244-7

192

78

225-4

142

63

February

19-11

87

45 25-70

168

66 : 232-4

197

85

23.S-8 128

54

March

13-52

105

78

26-65

222

83

266-8

283

106

232-6

155

67

April . .

9-94

55

55 ! 25-74

139

54

309-6

171

55

196-0

96 i 49

May . .

9-60

08

71

20-00

171

85

264-2

204

77

143-0

116

81

June . .

8-62

40

46

19-87

149

75

241-1

197

82

151-5

92 61

July . .

9-24

69

74

22-29

131

59

259-2

208

SO

171-8

60

38

August .

12-16

69

57

22-27

125

56

281-3

207

74

156--I

69

44

September

14-04

108

77

22-51

181

80 303-9

221

73

161-4

127 70

October .

17-87

73

41 21-73

156

63 265-7

224 8 1

215-6

93

43

November

19-52

78

40

19-20

203

106 i 212-3

214 101

165-7

154 93

December

18-32

95

52

19-12

121

63

207-5

144

70

163-2

86

53

Winter .

9-90

61

62

21-11

144

68

261-5

204

78

156-4 86

55

Equinox .

13-84

85

61

24-91

174

70

286-5

225

78

20L-4! 118

58

Summer .

18-76

92

49

22-35

167

75

224-2

187

83

198-3 128

64

Year . .

14-17

80

56

22-79

162

71

257-4

205

80

185-4

110

60

§ 28. In applying the method of groups, it is evidently desirable that one group of
years should fall near the middle of the period dealt with, and that part of the second
group should precede, and part follow it. This arrangement helps to eliminate any
long-period variation, or any gradual change in the conditions. The period 1887 to
1898 being by no means ideal in the above respect, in dealing with the diurnal

176 DR. C. CHKEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

inequality for the year, I have employed both the method of least squares and the
method of groups. If large differences had presented themselves between the results
from the two methods, it would have become necessary to reconsider Table XVIII.
As the question of the reliance to be placed on the method of groups is important,
I give the results from both methods in Table XIX. The agreement, it will be seen,
is least good in D, but it will, I think, be allowed that even there it leaves little to
be desired.

TABLE XIX. — Batavia (Units 1' in D and I, ly in Forces).
Mean Diurnal Inequalities for the Year.

Declination.

Inclination.

Horizontal force.

Vertical force.

Total force.

Groups.

Least
squares.

Groups.

Least
squares.

Groups.

Least
squares.

Groups.

Least
squares.

Groups.

Least
squares.

(a . . . .

2 -455

2-470

3 -61

3 -60

38 -74

38 -74

30-13

30 -11

20-94

20-90

Ranges. .4 104 x b. . .

183

179

215

218

2738

2739

1550

1559

1541

1552

[ 10* x A/a . .

746

725

597

605

707

707

514

518

736

743

Sum of fa....

10-30

10-34

22-21

22-19

258-5

258-0

173-3

173-1

145-9

145-4

24 1 103 x 1 . . .

88 -H

87-6

159-1

159-8

1967

1980

1121

1127

1190

1204

differences [ 105 x Itja . .

862

847

710

720

761

767

647

651

816

828

I

§ 29. In the case of I, H and V the values of a, b and b/a given in Table XIX. for
the sum of the 24 differences in the mean diurnal inequality for the year do not differ
much from the yearly mean in Table XV1I1. In the case of D, however, the results
in the two tables are widely different, the a in Table XVIII. being nearly 40 per cent,
in excess of that in Table XIX. The closeness of the values of « in the two tables in
I, H and V, and their divergence in the case of D, is what we might anticipate from
the figures for the sums of the 24 differences in the last two lines of Table XVII.
The real inference to be drawn is that in I) the hours of maximum and minimum vary
somewhat widely from month to month, though apparently to a smaller extent in
years of sun-spot maximum than in years of sun-spot minimum.

The data in Table XIX. for Batavia correspond exactly to those given for Kew
in (A), Table XLIV.

In D the Batavia value of b for the 24 differences is almost exactly the third of the
Kew value ; in. the case of the ranges the Batavia value is relatively larger, but still
less than half that at Kew, Greenwich or Pawlowsk.

The D values of b/a, however, at Batavia and Kew are nearly equal.

In I the Batavia value of 6 is 70 per cent, larger than the Kew in the case of the
ranges, and almost exactly double in the case of the 24 differences. The Batavia

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM. 177

values of a, however, so much exceed the Kew that b/a is more than twice as big at
Kew as it is at Batavia.

In H the Batavia values of b are both roughly 50 per cent, in excess of the Kew,
but the Kew values of b/a are more than 50 per cent, in excess of those at Batavia.

In V the values of b are again very much larger at Batavia than at Kew ; in the
24 diiferences the Batavia value of b/a is somewhat the higher, but in the ranges it is
slightly the lower.

§ 30. The relation between the values found for b/a is probably the best measure of
the relative importance of the sun-spot connection in any two cases. Applying this
criterion to the 24 differences and range results obtained by least squares
Tables XIX., we obtain the following values for the ratio of

(b/a) from sum of 24 differences : (b/a) from ranges :—

in

Declination.

Inclination.

Horizontal force.

Vertical force.

Total force.

1-168

1-191

1-085

1-257

1-115

The mean of the first four of these ratios is 1'18 and the corresponding figure for
Kew (as deducible from (A), Table XLIV.) is 1'19. Thus the greater variability of
the sum of the 24 differences with sun-spot frequency observed at Kew is also seen
at Batavia, and to approximately the same extent.

§ 31. The Batavia publications record the values of the constants in the 24-hour
and 12-hour terms of the Fourier series

j sin (t

c., sin (2t + «2)

for the mean diurnal inequality for the year. Here clt c2 replace the Batavia notation
AI; A2.

Table XX. gives the values which I have found for a, b and b/a in this case from
the method of groups. The results should be compared with those given for Kew
and Wilhelmshaven in (A), Table XLII., though the slight difference in the method
of obtaining the Kew results should be noted.

In I, H and V the values of b/a in Table XX. are nearly alike in c± and c.,, and
they approach fairly closely to the corresponding values in Table XIX. applicable to
the ranges of the mean diurnal inequalities. In declination and total force, however,
the values of b/a in Table XX. are decidedly higher for cx than for c3. This
phenomenon was observed at both Kew and Wilhelmshaven in the case of the
declination and the westerly component.

VOL. coin. — A

2 A

178 DR. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

TABLE XX.— Batavia, 1887-98 (Units 1' for Angles, ly for Forces).

Amplitudes of 24-hour and 12-hour Terms in Fourier Series for Mean Diurnal

Inequality for the Year.

Declination.

Inclination.

Horizontal force.

Vertical force.

Total force.

f'\-

C2.

Cj.

fin-

c\.

C-2.

Ci.

Cjj.

Cl-

Cg.

Moan value of

amplitude (for
12 years) .

0-748

0-778

1-793

0-837

20-89

9-07

13-62

7-27

11-71

5-45

a 0-548

0-614

1-427

•663

16-13: 7-13

11-14

5-90

8-84

4-43

10'x/; . . .

52

42

95

45

1233 502

641 353

744 263

10lx/'/rt. . .

94

69

G6

68

7G

70

58

60

84

60

§ 32. Disturbances have special attention paid them at Batavia. Following a
practice, of which SABINE was an advocate, a reading at Batavia is regarded as
disturbed when its difference from the mean reading at that hour during the mouth
reaches or exceeds a certain limit. The limiting values adopted at Batavia are
l'-3 in D and lly in H and V.

The arbitrary nature of such criteria, and the difficulty of justifying one limiting
value in preference to another, have been more than once pointed out. It is arguable
that the limit should vary with the season of the year, and even with the sun-spot
frequency. In a European station, for instance, the range of the regular diurnal
inequality near sun-spot maximum at midsummer is very large compared to that near
sun-spot miniimim at midwinter, and a good deal might be said for a limiting value
which bore a fixed ratio to the range from the mean diurnal inequality for the
month.

The disturbed values which exceed the hourly mean, and those which fall below it,
are termed respectively "positive" and "negative" disturbances; they are in the
first instance treated separately at Batavia, tables being given of the sum of the
values of the disturbances and of their number. A final summary gives the aggregates
of the positive and negative totals treated* numerically. Table XXI. gives these
aggregate values and numbers as published in the annual Batavia ' Observations.'

The number of disturbances in D is less than half that in V, and little over
a quarter that in H. We cannot, however, draw any safe inference as to one element
being absolutely more or less disturbed than another. If we calculate the ratios
borne by the disturbance limits accepted at Batavia to the mean ranges of the
diurnal inequalities for the year in the respective elements, for the period 1887 to
1898, we find the following results :—

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

179

D.

Disturbance limit _
range

0-41

H.

V.

0-30

If instead of the ranges from the mean diurnal inequality for the year we had
taken the arithmetic mean of the 12 monthly inequality ranges, we should have
obtained a somewhat smaller fraction in the case of D. But the figures are at least
suggestive that the explanation of the great difference in the number of disturbances
in D, H, and V may be largely due to the disturbance limit being less exacting in
one element than another.

TABLE XXL— (Units for " Values" 1' in D, ly in H and V.)
Aggregate Values and Numbers of Disturbances at Batavia.

Year.

Declination.

Horizontal force.

Vertical force.

Values.

Numbers.

Values.

Numbers.

Values.

Numbers.

1887

339 • 3
237-3
237 • 5
354-0
425-4
1020-8
730-8
840-2
616-0
458-2
464-5
434-2

210
149
153
252
262
571
427
462
360
286
286
262

17,1 GO
16,339
11,686
6,227
22,605
40,582
23,731
37,239
23,595
19,983
14,187
18,605

1023
933
700
346
1208
1786
1286
1666
1380
1139
815
1030

9,612
12,709
11,581
4,781
16,394
20,295
11,021
13,418
11,441
5,790
5,995
8,485

671
807
783
301
1016
1095
715
751
757
409
408
548

1888

1889
1890

1891
1892
1893
1894
1895
1896

1897

1898
Means

513-2

307

20,995

1109

10,960

688

§ 33. On examining Table XXI. it will be seen that the number and aggregate
value, though generally increasing or decreasing together, are far from being in a
constant ratio in any of the elements. As to which is the better measure of
disturbance, opinions may well differ ; but the aggregate values constitute probably
the nearest parallel to the Pawlowsk and Katharinenburg data in Tables IX. and XVI.

According to both numbers and aggregate values, 1893 was less disturbed than
1892 or 1894, but its relative quietness is not so conspicuous, especially in D and V,
as it was at Pawlowsk or even Katharinenburg.

Table XXI. must, of course, receive contributions — at least, in the case of H

2 A 2

180 DR. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

and V — from a number of days which are not days of large disturbance ; but if this were
the true explanation, we should expect the position of 1893 at Batavia and Pawlowsk
to diifer more in the case of H than in that of D, which is the reverse of what
happens.

§ 34. Table XXII. gives values of a, b and b/a calculated for the data in
Table XXI. The value of b/a answering to the " aggregate value" is in each case
greater than that answering to the " number" ; and, except in the case of V, both
values of b/a are considerably higher than the corresponding yearly values in
Tables XVIII. and XIX.

If we compare Table XXII. with Table XIV. for Katharinenburg, we see that in V
the Batavia disturbance values of b/a are much less than the lowest value of b/a at
Katharinenburg, viz., that for the diurnal inequality. In H the Batavia disturbance
values of b/a are similar to the value of b/a in the absolutely monthly range at
Katharinenburg. In D, however, the Batavia disturbance values of b/a are much in
excess of any corresponding value at Katharinenburg.

The way in which disturbance influences the records at the two places are thus
widely different.

TABLE XXII.— Batavia " Disturbances," 1887-98 (Units for " Values" 1' for D,

ly for H and V).

Declination.

Horizontal force.

Vertical force.

1

Aggregate
values.

Numbers.

Aggregate
values.

Numbers.

Aggregate
values.

Numbers.

(I ....

217-7

153-7

10,312

657-9

8425

578-9

!> . . . . 7-65

3-96

277

11-7

65-6

2-84

10'x//> . .

351

258
.

268

178

78

49

§ 35. Table XXIII. compares observed and calculated values in the mean diurnal
inequality for the year at Batavia, and in the aggregate value of the disturbances.
The values employed for a and b in the case of the ranges are those calculated by
least squares.

The nicety of agreement in the case of the ranges is very similar to what has been
already observed at Kew, Pawlowsk and Katharinenburg; and, as at Kew, the
agreement is practically the same for the 24 differences as for the ranges. As has
been generally observed elsewhere, the agreement is least good in the case of V.

In the case of the aggregate value of the disturbances, the agreement is pretty
similar to what was found for the mean of the absolute monthly ranges at
Katharinenburg; and, as elsewhere, the failure of the formula to account satis-

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM.

181

factorily for the phenomena observed in 1892 and 1893 is conspicuous. Unlike
Pawlowsk and Katharinenburg, Batavia shows, however, no abnormal excess of
disturbances in 1898.

TABLE XXIII.— Batavia (Units 1' for D, ly for H and V).
Observed less Calculated Values.

Mean diurnal inequality for the year.

Aggregate values of

disturbances.

' Year.

Ranges. 24 differences.

D.

H.

V. D.

H.

V.

D.

H.

V.

1887

-0-25

+ 0-5

- 2-7 j- 0-77 + 2-2

18-6

-I- 0 1 • i

4- 3 9^5

4- S^T

1888

- -07

+ 0-9

+ 0-9 • - -28 i + 5-0

+ 0"2 - 3'7-J.

+ 4 146

+ 3 838

1889 ...

4- -16

- 0-3

+ 3-0 + -36 - 1-1

+ 13-7

- 28-4

- 368

4- 2,743

1890

4- -09

+ 0-4

+ 1-3 + -21

+ 8-4

4- 18-0

+ 82-0

6 049

- 4110

1891 ....

- -22

4- 2-4

+ 5-3 - -88

+ 15 -2

+ 36-9

-64-6

4- 2,448

4- 5,633

1892

+ -16

+ 0-6

+ 2-8 4- -40

+ 12-3

4- 17-9

4-244-6

4- 10 089

4- 7 OSO

1893

•02

+ 1-1

+ 0-6 ! -00

+ 8-9

4- 3-9

-136-4

-10,061

- 2,975

1894 . . .

•04

1-6

1-8 - -03

9-8

9-3

4- 25-7

4- 5,355

125

1895

•08

4- 1 '4

+ 0-6 -53

0-3

0-7

- 91-4

- 4 418

1 184

1896

+ -03 - 3-8

- 5-1 + -41

- 29-1 - 34-2

79-3

1 890

5 378

1897 ....

+ '19

- 0-6

- 3-4 + -66

6-3 - 23-0

4- 46-3

- 3,311

- 4,149

1898

+ -04

1-0

1-6 + -44

5-5

5-1

4- 12-2

+ 909

1 699

Mean difference calcu-

lated -»~ observed

0-112

1 '22

2-42 0-414

8-7

15-1

72-1

4,361

3,270

Probable error .

0-096

1-09' 2-02 0-344

8-1

13-4

61-2

3,734

2,730

Mean value of element

3-16

49-3

36-1 13-73

334-5 216-6

513

20,995

10,960

Range of element .

1-52

22-9

14-9 7-12

165-6: 104-0

783

34,355

15,514

Mean difference x 100

4

2

73 3

7

14

21

30

mean value

Probable error x 100

6 5 14 5 5 13

8

11 ! 18

range of element

Mauritius (Lat. 20° 6' S., Long. 57° 33' E.).

§ 36. Owing to the novelty in some of the features of the Batavia results,
examination of the data from a second tropical station seemed desirable. I have
accordingly made use of a number of tables of magnetic results* at Mauritius,
published in a convenient form in 1899. In D, data are given for the period
1875 to 1890, in H for 1883 to 1890, and in V for 1884 to 1890. The shortness of
the two latter periods, and the fact that the data are not contemporaneous with
those for most of the other stations, are drawbacks, but there is small choice of
magnetic data in low latitudes.

* ' Mauritius Magnetical Reductions,' edited by T. F. CLAXTON, F.R.A.S., Director Royal Alfred
Observatory, Mauritius, 1899.

182 DE. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

On examining the tables, I found that the mean D ranges in 1881 and 1882 — years
of fairly large sun-spot frequency — showed a remarkable depression, being only about
half those in 1880 and 1883, and in the preface I found the following editorial
reference to some readjustment of the declination magnetograph in December, 1882 :
"In the latter part of the year 1882 the effect of torsion on the magnetograph is
very pronounced." As the phenomenal smallness of the range seems to have ceased
with the readjustment, and as the Milan and Greenwich records show no parallel to
the reduction of the ranges in 1881 and 1882, I have omitted these years entirely
from the calculations.

The Mauritius publication gives in special detail the mean value for each month
and year of the absolute daily ranges. These ranges seem based entirely on hourly
readings, and so are not absolutely equivalent to the Katharinenburg ranges dealt
with in Table XIII.

As the variation in these daily ranges throughout the year has exceptional features,
I give particulars in Table XXIV. There is a resemblance to phenomena at Batavia.
In D the variation in the range, though much less conspicuous than in Northern
Europe, is well marked ; the vahies for the three midwinter months — May, June,
and July — are well below the average. In H the variation is small, and somewhat
irregular ; on the whole, the range is smallest in winter, from May to August, but
the next lowest values appear in February and December.

There is a distinct reduction in the V range near midwinter, but a very similar
reduction occurs at midsummer.

TABLE XXIV.— Mauritius (Units 1' for I), ly for H and V).
Monthly Means of Absolute Daily Ranges.

Declination,
1875-80 and 1883-90.

Horizontal force,
1883 to 1890.

Vertical force,
1884 to 1890.

January ....

C-93

37-9

17-1

February

7-79

35-0

19-5

March

7-11

30 -2

20 -1

April

5-75

37-6

17-3

May

4-87

35 • 0

16'5

June.

4-03

34 -i

15-5

July .

4-36

3V8

17-1

August .

6-00

34-5

22 '0

September ....

6-28

36 -6

go -7

October ....

G'71

37-4

1Q-4

November

6-99

37-8

16-7

December

6-78

35-3

15-2

Mean ....

G'13

35-0

18-2

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM. 183

§ 37. Table XXV. gives the values of a, b, and b/a applicable to the monthly
means of the absolute daily ranges at Mauritius. The method of groups was
employed, the groups being as follows : —

Years of sun-spot.

ForD.

ForH.

For V.

Maximum ....
Minimum

1880, 1883, 1884, 1885, 1886
1878, 1879, 1888, 1889, 1890

1883 to 1886
1887 1890

1884 to 1KHG
1 SK7 1 X<lO

The seasonal and yearly values of a and b are arithmetic means from the included
months, and these means are employed in calculating b/a.

The values of b in Table XXV. fluctuate somewhat erratically from month to
month even for D, where 14 years' data are utilised. The exceptional features
presented by the September and October data in V had better be regarded pro-
visionally as accidental.

In D and V we have b largest in winter and least at the equinox, a very remarkable
feature.

Comparing Tables XXV. and XIII., we see that whilst the mean values of a for
the year are fairly similar, the values of b and of b/a at Mauritius are roughly only
from a half to a third of the corresponding values at Katharinenburg.

TABLE XXV.— Mauritius (Units 1' for D, ly for H and V).
Monthly Means of Absolute Daily llanges.

Declination.

Horizontal force.

Vertical force.

a.

I04b.

IQ*b/a.

a.

10%.

104i/«.

«.

10%.

Wb/a.

January .

6-02

381

63

35-1

|

89 26

14-0

110

78

February . . .

7-34

186

25

29-7

164 55

17-8

55

31

March ....

6-70

158

23

29-7

200

67

16-8

104

62

April ....

5-06

278

55

33-5

119

35

14-1

115

82

May ....

4-10

354

86

30-1

165

55

13-7

97

71

June ....

3-70

130

35

26-9

209

78

13-2

82

62

July ....

3-65

290

80

28-3

159

56

13-9

113

81

August . . .

5-20

362

69

28-6

214

75

18-1

157

87

September

5 '93

152

26

30-8

214

69

20-6

9

4

October . . .

6-41

145

23

31-8

221

70

20-2

[- 47]

[-23]

November . .

6 '35

331

52

29-2

376

129

16-2 33

21

December. . , 6 '56

120

18 29-7

226

76

13-5

96

71

Winter . . .

4-16

284

68

28-5

187

66

14-7

112

76

Equinox . . .

6-03

183

30

31-5

188

60

17-9

45

25

Summer . . .

6-57

254

39 30-9

214

71

15-4

73

48

Year ....

5-58

241

43 30-3 196

65 16-0

77

48

1 1

184 DK. C. CHEEE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

§ 38. Table XXVI. deals with the mean of the absolute daily ranges for the year,
and with the ranges and the sums of the 24 differences in the mean diurnal
inequalities for the year. The grouping of the years is the same as for the previous
table. The values of a, b, and l>/a for the yearly mean of the absolute daily ranges
do not agree quite so closely with the corresponding means of the 12 monthly
values in Table XXV. as was the case at Katharinenburg (cf. Tables XIII. and XIV.).

In H the values of a and b for the absolute daily range are about double those for
the range of the diurnal inequality ; in D and V the differences between the two sets
of values are smaller, but still considerable. In all three elements the values of b/a
for the two species of ranges are fairly similar.

In the case of the mean diurnal inequality in D, the values of b are lower even
than those given in Table XIX. for Batavia, and the values of b/a are the lowest we
have yet met with. The values of b for the ranges of the mean diurnal inequalities
in H and V are much lower than at Batavia, but the values of b/a at the two places
are fairly similar.

The values of b/a for the 24 differences do not show that decided excess over the
values for the ranges that was seen at Kew and Batavia.

TABLE XXVI.— Mauritius (Units 1' for D, ly for H and V).

Ranges.

24 differences.

Mean of absolute daily
values for the year.

From mean diurnal
inequality for the year.

From mean diurnal
inequality for the year.

a.

b x 104.

(I I a) x 104.

1

a. b x 104.

!

(bja) x 104.

a.

b x 103.

(ft/a) x 104.

Declination . .
Horizontal force .
Vertical force . .

5-53

30-4
16-2

255

1859
840

46
61
52

4-06 164
15-0 956
11-9 685

j

40
64

58

15-96
116-0
66-6

79
695
292

49
60

44

§ 39. Table XXVII. gives the differences between observed and calculated mean
yearly data at Mauritius. Comparing the figures in the last line of the table with
the corresponding figures in Tables IV., VIII. , XV., and XXIII., we conclude that
the agreement is not quite so good at Mauritius as at the other stations. The
agreement is closest for the mean of the daily declination ranges, where it is very
fair ; it is on the whole better for V than for H, which is exceptional.

BETWEEN SUN-SPOT FREQUENCY AND TERRESTEIAL MAGNETISM. 185

TABLE XXVII.— Mauritius (Units 1' for I), ly for H and V).

Observed Less Calculated Values.

Mean of absolute daily

Mean diurnal inequality.

ranges.

Ranges.

Year.

24 differences.

D.

H. V.

D.

!

H.

V.

D.

H.

V.

1875 +0-1

+ 0-4

+ 0-7

1876 + -1

•o

4- 0-2

1877 - -2

- -2

__ ' _

1-8

' _

1878 + -3

—

i

+ -6

+ 1-5

1879 -1

- -2

- 0- S

1880 - -3

1883 -f- -3

H- 0-3

+ "5

- 0-9

+ 0-4

- 2-8 —

1884 + -3

4-1

+ 0-9

+ '3

- 3-6 + 1-1

+ 1-2 - 22-9 f 6-9

1885 : -2

- 0-4

- 2 '2

+ -1

+ 1-4

- 1-5

+ 0-7

+ 4-9

- 5-4

1886 -2

+ 42

+ 1-3

•6

+ 3-1

+ 0-5

1-4

+ 20-7

1-2

1887 - -2

+ 1-8

1-7

•G

+ 0-8

- 2-3

1-2

+ 9-0

- 6-7

1888 -1

+ ^-l

- 0-1

. 0

+ 1 '4

0'4

- 0-6

4- 8 ' r>

4- O'fi

1889 - -1

- 1-4

+ 0-7

+ -2

- 0-9

+ 1-4

+ 0-8

8-0

+ 4-3

1890 4- -1

- 2-4

+ 1-0

— • 2

+ 1-2

- 0-2

9-6

+ 1-9

Mean difference calcu-

lated ~r observed . 0-19

2-09

1-13

0-31

1-67

1-20

0-79

10-8 3-9

Probable error . . . 0'14

1-80

•94

0-25

1-41 0-98

0-67

9-2

3-3

Mean value of element 6 • 1 1

35-9

18-2

4 • 44

17-8

13-6

17-8

137

74

Range of element. . 1 1-90

13-2

6-8

1-90

7-0

6-8

7-0

46

28

Mean difference x 100

6

6

7

9

9

4

8 5

mean value

Probable error x 100 -
range of element

14

14

13

20 14 10 20 12

I j

Summary.

§ 40. A slight progressive decrease in a and increase in b in the sun-spot formula
(1) is suggested by the Greenwich D and H data from 1841 to 1896, but this does
not meet with support from Signor RAJNA'S analysis of D ranges at Milan from 1836
to 1894. In both cases there is an element of uncertainty, arising from want ot
homogeneousness in the data.

According to RAJNA'S earlier data, values of a, and to a lesser extent values of b,
calculated from periods as long as 14 years may differ very sensibly from those
calculated from longer periods, but differences of this kind seem to have diminished
since observations became more homogeneous and are probably ascribable in part to
observational uncertainties.

VOL. CCIII. — A. 2 B

186 DE. C. CHREE: AN ENQUIRY INTO THE NATURE OF THE RELATIONSHIP

Results calculated for Milan from the period 1890 to 1900, which is the period
chiefly utilised in the present paper, differ but little from those found by RAJNA for
the periods 1836 to 1894 and 1871 to 1894.

The tendency for b to be small in winter and large in summer, described at Kew, is
also, in general, conspicuous elsewhere ; but there are exceptions, especially at tropical
stations.

The tendency in b/a to be large in winter as compared to summer, so prominent at
Kew, is also, in general, prominent at other northern stations, but the phenomenon is
comparatively inconspicuous in the case of the declination range at Greenwich. At
the tropical stations the seasonal change in b/a appears much reduced and is some-
what uncertain.

There is no conspicuous difference between the "all" and the " quiet" days' mean
yearly values of b and b/rt for the ranges of the D and H diurnal inequalities at
either Greenwich or Pawlowsk ; but at Pawlowsk there is a somewhat notable
difference between "all" day and "quiet" day D results in winter, and the difference
between " all " and " quiet " day V results is very large throughout the whole year.

If we exclude Mauritius, the values of W*b/a for the ranges in the mean diurnal
inequality of declination for the year at the several stations vary only from 65 to 73.
The corresponding values of b show also a pretty close agreement at the northern
stations, but the values for the tropical stations are much smaller.

In H there is no very conspicuous difference in the values of b or b/a for the ranges
from the mean diurnal inequality for the year at the northern stations ; but the values
found for b/K, at Batavia and Mauritius are considerably smaller, while the value
found for b is smaller at Mauritius, but very materially larger at Batavia.

When the formula (l) is applied to any ordinary measure of magnetic disturbance,
it gives much too high values for 1893 — the year of sun-spot maximum — and much
too low values for 1892. Thus the application of (l) to disturbances has not the same
justification as its application to ordinary diurnal inequalities. It may, however,
serve a useful purpose in giving a greater degree of definiteness to the comparison of
contemporaneous disturbance phenomena at the same or at different stations.

In the case of results obtained by the application of (l) to individual months of the
year a considerable latitude must be allowed to chance, especially in winter months
when the diurnal range is small, unless an exceptionally long series of observations is
available. liesults obtained from arranging months in seasons are much less exposed
to numerical uncertainties, but they are insufficient for the reason that there are
conspicuous differences between months which have to be grouped under the same
season. Tins remark applies more particularly to winter and equinoctial months in
higher latitudes.

[June 8, 1904. — The following additional data — all obtained by the method of
least squares — apply to the ranges of the mean diurnal inequalities for the year at
Irkutsk (" all" days) and Colaba (" quiet" days), and to the mean difference between the

BETWEEN SUN-SPOT FREQUENCY AND TERRESTRIAL MAGNETISM. 187

absolute daily maximum and minimum at Zi-ka-wei. The units are 1' for angles,
ly for force components.

Place . . .

Irkutsk (Siberia).

Zi-ka-wei (China).

Colaba (Bombay).

Latitude . .

52° 16' N.

31° 12' N.

18° 54' N.

Longitude . .

104° 16' E.

121° 26' E.

72° 49' E.

Period of years

1890 to 1900.

1890 to 1900.

1894 to 1901.

Element . .

a.

104/>.

104A/«.

«.

I04b.

104Z>/«.

«. 1046.

104i/a.

D ....

4-815
0-971
18-18
6-49

358

87
1896
710

74
90
104
109

4-369

303

69

2-373 ' 66

31-65 2814
19-35 723

28

89
37

I

H

V

At Irkutsk the values of b for D and H are similar to those at Katharineuburg ;
the values of b/a for these elements are similar to those at Kew ; in V the values of
b and b/a, are decidedly less than at Katharinenburg.

At Zi-ka-wei bja is decidedly less, and b much less, than the corresponding values
for Katharinenburg (second line of Table XIV.).

At Colaba the ("quiet" day) values of b and b/a for I) are notably less than the
corresponding (" all " day) values at Mauritius, the smallest occurring in the paper ; but
the value of b for H exceeds that at Batavia, the largest previously noted. The
value of b/a for V is conspicuously small.]

2 B 2

VIII. On some. Physical f'oiustantx «/'

Solutions.

liy tkc fi.\iti. OK BEI;KEI,KN

INDEX SLIP.

Communicated /«/ F II.

BKIIKKLKV, Earl of. — On some Physical Constants of Saturated Solutions.

Phil. Trans., A, vol. 203, 1904, pp. 180-215.

Saturated Solutions, Physical Constants of.

BKEKBLKV, Earl of. Phil. Trans., A, vol. 203, 1904, pp. lfS!)-215.

THK follow! i •_; -*!"-
application of V.\
probable that it' t ;
of VAX OKI; AV \
solutions tliiit is.
Saturated s»!'i' .
greatest osmotic
concentrated si-'
the relative dis^
and tempera tun

data for the tentative
itions. It is evidently
te solutions, then that
apply to concentrated

y presumably have the
'ti to believe that, in
concentration the less
its of volume, pressure

This term of iL-
saturated solm !•.•<, an
Part I., I ijive the deii
the methods nnd ii

s of the density of a

ing temperatures. In

with a description of

HARTLEY, in testing a

• dd the method fail, I

I solutions at different

motic pressures. The

[K will be trivet:

20.7.04

.'IIJ& X3UZI

h-ilHiuiug In ttaatenoO Jfni«YJl1 -MHO* nO — .1o h«3
.qq ,*0fit ,£OS .loy .A ,.«n«iT .lull

.40UI ,K(tt .k>7 ,A ,.

ti

[ 189 ]

VIII. On some Physical Constants of Saturated Solutions.

By the EARL OF BERKELEY.
Communicated by F. H. NEVILLE, F.R.S.

Received March 28,— Read May 19, 1904.

INTRODUCTION.

THE following work was undertaken with a view to obtaining data for tlie tentative
application of VAN DER WAALS' equation to concentrated solutions. It is evidently
probable that if the ordinary gas equation be applicable to dilute solutions, then that
of VAN DER WAALS', or one of an analogous form, should apply to concentrated
solutions — that is, to solutions having large osmotic pressures.

Saturated solutions were taken for investigation because they presumably have the
greatest osmotic pressures, and also because there is reason to believe that, in
concentrated solutions at a given temperature, the greater the concentration the less
the relative dissociation. For the purpose in view, measurements of volume, pressure
and temperature are required.

Volume.

This term of the equation is deducible from observations of the density of a
saturated solution and of the solubility of the salt at varying temperatures. In
Part I., I give the densities and solubilities obtained, together with a description of
the methods and apparatus used.

Pressure.

I am at present engaged, with the collaboration of Mr. E. G. HARTLEY, in testing a
method of directly observing large osmotic pressures. Should the method fail, I
propose to determine the vapour pressures of the saturated solutions at different
temperatures and from these calculate the corresponding osmotic pressures. The
observations and details will be given in Part IT.

(366.) 20.7.04

190

THE EARL OF BERKELEY ON SOME

Temperatures,

-The temperatures at which the densities, solubilities and osmotic pressures were
determined are given with those quantities respectively.

Part III. will be devoted to the application of the results to theory.

The selection of the particular salts whose solutions were examined was governed
by the following considerations :—

(1.) Fairly soluble salts should be used, so that differences between the ordinary
phenomena of dilute and those appertaining to concentrated solutions
may be the more marked ;

(2.) They should have as wide a range of molecular weights as possible, so as to
bring into prominence any effect the interacting masses may have on the
space occupied by the molecules ;

(3.) For the purpose of comparing members of the same family of elements the
salts should be isomorphous, the presumption being that isomorphous
salts give similarly constructed molecules in solution.

i. (A).

Determination of the Constants.

The densities were obtained by the following method : An approximately saturated
solution was kept in contact with crystals of the salt at a definite temperature by
means of a thermostat, and continuously stirred. When a sufficient length of time

had elapsed, a pyknometer, whose capacity was known,
was immersed in the solution and filled to a mark, then
washed and dried by means of pure alcohol, and weighed
against a counterpoise which had been similarly washed
and dried. The solubilities were determined by washing
the contents of the pyknometers into platinum crucibles
and weighing them after evaporating to dryness. It
was found in the course of the work that, in the case of
very soluble salts, this method was not satisfactory,
because a crust of salt formed on top of the solution in
the platinum crucible, and the accumulation of steam
under it, on finding its way out, carried particles of
solution with it. Glass bulbs, represented in fig. 1 and
made of Jena glass, were therefore substituted for the
platinum crucibles and the solution evaporated to dry-
ness in them. This was effected by passing a current of dry air through the tubes
while they were being heated to 110°-170° in an air oven, the air current and the

nat. size.

PHYSICAL CONSTANTS OF SATUKATED SOLUTIONS.

191

m.m

120

no

100
90
80

f-

heating being continued until the bulbs had attained a constant weight. The air
current was obtained by means of a Fleuss pump. The same filling of a pyknometer
gave, therefore, both the density and the solubility. As a check on the latter, the
contents of a pyknometer were occasionally analysed.

Pyknometers.

At first Sprengel pyknometers of various shapes and sizes were tried, but were
found .to be unsatisfactory. This was because it was almost impossible to prevent the
solution from crystallizing in the capillary
during the time the level of the liquid was
being adjusted to the mark.

The following was the form finally
adopted and found quite satisfactory. A
pear-shaped bulb, of about 5 cub. centims.
capacity, terminating above in a stem
composed of a graduated capillary 120
millims. long, and below in a finer capillary,
bent as in fig. 2, was used for salts of
medium solubility. For somewhat insoluble
salts a similar pyknometer, but of about
11 cub. centims. capacity, was found to be
more suitable, while for very soluble salts,
such as sodium sulphate, which have great
differences in solubility at different tem-
peratures, it. was necessary to have similar
pyknometers made of thicker glass, so that
when the crystals formed and practically
filled the whole of the bulb, the latter
would withstand the pressure. It was
also found necessary to make the capillaries
of a larger internal diameter, so as to be
able to fill quickly. And the shape of the lower capillary (see fig. 3) was altered and
its end fitted with a glass cap to prevent the solution from " creeping" out when 011
the balance. The stem was also fitted with a cap to prevent evaporation.

Determination of the Capacities of the Pyknometers.

Before determining the capacities, the pyknometers were heated rapidly and
repeatedly to 200° C., being allowed to cool to the temperature of the room between
each heating ; by this means it was hoped that the gradual shrinking in volume
would be accelerated. The graduated capillaries were then calibrated by the usual

.'.0

10

Fig. 2.

Fie. 3.

192

THE EARL OF BERKELEY ON SOME

method of running a thread of mercury along the bore, measuring its length, and
then weighing it.

The capacities were found by weighing the pyknometers filled with water at 0° C.
and at 90° C. respectively ; the volume occupied by the water was taken to be that
o-iven in LANDOLT and BORNSTEIN'S tables for water which is air-free. The difference

to

between the capacities thus determined gave the expansion from 0° C. to 90° C., and
for intermediate temperatures it was assumed to be proportional to the temperature
interval ; this assumption was tested with one of the pyknometers, and it was found
that the resulting difference was within the experimental errors. With the 11 cub.
centim. pyknometers, however, it was deemed advisable to examine the error more
closely, and for this purpose the capacities were determined at five approximately
equal intervals of temperature between 0° C. and 90° C. The numbers obtained were
plotted against the corresponding temperatures, and a bent-ruler curve passed
through the points ; the capacities for intermediate temperatures were taken from it.
The maximum difference between this curve and a line joining the penultimate
observations represented a difference of '0015 cub. centim. This is a quantity which
is barely greater than the experimental errors, as will be seen from the following
numbers obtained with one of the pyknometers :—

Tempe-
rature.

Capacity.

Tempe-
rature.

Capacity.

Tempe-
rature.

Capacity.

Tempe-
rature.

Capacity.

Tempe-
rature.

Capacity.

°C.

cub. centims.

°C.

cub. centims.

"C.

cub. centims.

'C.

cub. centims.

°C.

cub centims.

91-85

11-4406

68-60

11-4321

45-40

11-4239

25-85

11-4181

0-70

11-4111

91-35

•4397

68-40

•4317

45 • 30

•4239

25-80

•4186

0-60

•4113

91-10

• 4394

68-00

•4313

45-15

•4239

25-40

•4176

0-50

•4111

—

—

—

—

45 • 30*

• 4238*

—

—

—

On re-determining the capacities after an interval of several months no change was
apparent.

As the table used for the expansion of water gives numbers derived from air-free
water, and as the pyknometers had been filled with water which had not been freed
from dissolved air, it was thought possible that an error had been introduced in this
way ; a pyknometer was therefore filled, in a vacuum, with water which had been
boiled in that vacuum for three-quarters of an hour ; it was then withdrawn and
brought to a constant temperature in the thermostat and weighed in the usual
manner. The results of three observations carried out thus did not differ from those
obtained with ordinary water by more than the latter differed among themselves.
Taking into consideration that the solutions themselves are not air-free, it was
considered unnecessary to pursue the matter any further.

* This observation was one made with air-free water.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.

193

In weighing the pyknometers care was always taken that they should be slung on
the balance in such a way that the end of the lower capillary was at a higher level
than the level of the liquid in the stem ; this, as a rule, was sufficient to prevent any
loss of weight by evaporation, but such observations as did show a loss were rejected.

Stirring.

The stirring was obtained by means of a small platinum rod, fitted with a
two-bladed screw, suspended vertically in the solution, and rotated by a cord and a
small electric motor. The speed of rotation could be varied from 2 to 20 revolutions
per second. During the last two years of the course of this work the separate motors
were replaced by a shafting driven by an electric motor ; pulleys of various sizes were
fixed on it and driving cords taken to the stirrers as required. This shafting also
worked the Fleuss pump mentioned above.

Constant Temperatures.

At 0° C. the beaker containing the solution was surrounded by ice and water. At
15° C. it was placed in a copper vessel, in which was suspended a thermostat, and
through which a current of cold water passed. The former actuated a gas burner

Thermometers

Tap for closing
CSf"~~ pyknometer.

• -Pyknometer.

—-^- -Solution .

r thermostat,
ih.

Platinum beaker.
-SaJb.

Fig. 4.

and kept the temperature constant. From 30° C. to 90° C. the solution was placed
in a platinum beaker of 300 cub. centims. capacity which formed part of a D'Arsonval
thermostat. The platinum beaker (see fig. 4), having a flange welded on to it three-

VOL. CCIII. — A.

2 c

194 THE EARL OF BERKELEY ON SOME

quarters of an inch from the top, was passed through the top plate of the thermostat and
soldered in position. The body of the thermostat was of copper and held about three
litres of water, and the expansion and contraction of this was enough to actuate the
flexible diaphragm (the iron plate of the receiver, of a telephone) sufficiently to keep
the temperature of the thermostat constant to about 0°'2 C. ; that is to say, that for
2 or 3 hours before taking an observation, the temperature in the beaker would only
show, at the utmost, a change of 0°'05 C., but in the course of 24 hours after setting
the temperature of the thermostat, and consequently that of the solution, might slowly
rise or slowly fall to the extent mentioned, and then remain constant to 0°'05 C.

Means employed for Determining the Point of Saturation.

After numerous experiments the following general method was found to be the
most satisfactory. The thermostat was set at 90° C. , and water, together with a
quantity of salt more than necessary to saturate it, placed in the platinum beaker,
and the mixture stirred very rapidly until it was thought that saturation had been
attained ; an observation of the density was then made, and the stirring continued for
a further period of 2 or 3 hours, and then another density determination made. If
the two observations agreed within the experimental errors, it was considered that
saturation had been practically reached ; if the two observations did not agree, the
stirring was continued and the density taken at intervals until it became constant.
The temperature of the thermostat was then lowered 2 degrees, and after stirring
2 or 3 hours the density again taken. The mean of this and of the constant
density previously mentioned was considered to be the density of a saturated solution
at the mean of the respective temperatures. The temperature of the thermostat was
then lowered to the next point of observation, and after 2 or 3 hours' stirring the
density was taken ; water was then added to the solution and the stirring continued
until the density, taken at intervals of from 4 to 12 hours, was constant — the mean of
the first and the last observations, which usually differed by an amount slightly
greater than the experimental errors, was taken a£ giving the true density. The
process was then repeated for the other temperatures. It should be noted that
whether working with supersaturated or an unsaturated solution, the liquid is always
stirred in contact with a large excess of solid salt.

In the case of salts whose solubilities decrease with an increase of temperature,
the process is reversed ; with Na2S04, for example, which has a maximum solubility
at 32°-5 C., the thermostat was set at 33° C., and stirring continued until constant
density was obtained ; the temperature was then raised 1° C. and the density again
determined, and the mean of this and of the constant density above mentioned was
taken as the true density of a solution saturated at the mean of the respective
temperatures. The temperature of the thermostat was then raised to the next point
of observation, the density taken, boiling water added and a constant density

PHYSICAL CONSTANTS OF SATUEATED SOLUTIONS. 195

obtained, the numbers being " meaned " as before. The process was then continued
for the next higher temperature, and so on. The object of adding boiling water
is to make sure that the solution is unsaturated, for if cold water were added the
temperature of the solution would fall, and if the rate of attaining saturation be
greater than the rate at which the solution comes back to the constant temperature,
you get a solution supersaturated with respect to that temperature.

It will therefore be seen that the method adopted resolves itself into this : at any
given temperature, two observations of density and solubility are taken; one is
obtained by stirring a supersaturated solution in contact with the solid salt, the other
by stirring an unsaturated solution in contact with an excess of salt — and the true
density or solubility is considered to be the mean of the two observations.

In the earlier part of this work it was found that, in many cases, a very long time
elapsed before the densities obtained, when starting with an unsaturated solution,
approached sufficiently closely to that derived by starting with a supersaturated one —
this was partially remedied by increasing the speed of stirring from 2 to 20 revolutions
per second — but even then there was generally a difference in the two densities of
some few units in the 4th decimal place. The cause of this discrepancy was
eventually traced to the fact that a considerable length of time was also required for
the point of saturation to be attained by a supersaturated solution, even when stirred
in contact with its salt. It was owing to this that some 300 density and solubility
determinations had to be discarded — for preliminary observations had shown that
concordant results could be obtained by merely covering the top of the beaker with a
glass perforated for the stirrer to pass through, and removing the plate while the
pyknometer was being filled. At the higher temperatures the removal of the plate
caused a fall in the temperature of the solution and a consequent supersaturation.
This, however, was not suspected (because when the results were plotted the curve
was regular) until I was dealing with very soluble salts, which, on the removal of the
plate, tended to form crusts of salt on the surface of the solution. The difficulty
was overcome by closing the beaker by an india-rubber stopper, which was perforated
for the stirrer, the thermometer, and the pyknometer. The latter was closed at the
upper end by a tap attached by rubber tubing. The tap served two purposes : it was
kept closed on immersing the pyknometer, so that no liquid could enter during the
time that the pyknometer was attaining the temperature of the solution, and it was
closed after filling the pyknometer, so that no liquid could flow back during the
withdrawal of the rubber stopper.

As an extreme example of the necessity of giving an unsaturated solution plenty
of time to attain saturation, and also as showing the importance of having a sufficiency
of salt in contact with the solution, I extract the following numbers from my note-
book. An unsaturated solution of thallium alum, together with a quantity of the
salt, was placed in the beaker, which was at the constant temperature of 61°'0 C.
This was stirred for 12 hours at the rate of 10 to 20 revolutions per second; at the

2 c 2

196 THE EAKL OF BERKELEY ON SOME

end of this period the density was found to be 1*2539 (temperature 61°'00 C.).
Having reason to believe, from previous work, that saturation had been reached, the
temperature of the thermostat was lowered 1° C. and the solution stirred for another
3 hours; its density was then l-2546 (temperature 59°*85 C.).

The next day it was 1*2555 (temperature 59°*90 C.).
„ 1-2572 ( „ 60°-OOC.).

„ 1-2591 ( „ G0°*00 C.).

Between each observation about 10 hours' continuous stirring was given to the
solution, and all the time there had been about 5 cub. centims. of solid salt in contact
with the solution ; another 20 cub. centims. of salt was then added, and the stirring
continued for 12 hours, with a resulting density 1*2810 (temperature 60°'00 C.).
And a further 12 hours gave T2813 (temperature GO'OO C,).

On the other hand, the following shows the reverse phenomenon, i.e., that a
considerable time must elapse before a supersaturated solution attains its true point
of saturation. A solution of NaeSO4, saturated at GO0 C.. was heated to the constant
temperature of 75° C. (it must be remembered that Na^SO^ is more soluble at GO0 C.
than at 75° C.) and stirred at the rate of 13 revolutions per second in contact with
the anhydrous salt for 3^ hours; the density of the solution was found to be 1*2738
(temperature 7o°"00 C.). The next day, after 12 hours' stirring, the density was
1-2729 (temperature 75°'00 C.); 20 cub. centims. of boiling water was then added
(if cold water had been added, as before explained, the solution would have become
supersaturated), and after 12 hours' stirring its density was 1*2727 (temperature
75°-00 C.).

Where it was suspected that the solutions, when at the higher temperatures, might
decompose non-reversibly, the observations for the lower temperatures were first
recorded, in the manner already outlined, and those for the higher temperatures were
obtained by heating to the constant temperature required and stirring the solution
until the density was constant ; the temperature was then lowered by 1 or 2 degrees,
and after a sufficient length of time the density again determined ; the mean of this
last observation, and of the constant density first obtained, was considered to be the
density of a solution saturated at the mean of the corresponding temperatures.

The following are the important points to be observed in obtaining a saturated
solution : —

(1) A sufficiency of solid salt should always be in contact with the solution ;

(2) A thorough stirring should be continuously kept up ;

(3) A sufficient length of time should be allowed to elapse before taking the

required observation. This last condition seems to depend on the nature
of the salt, the speed of stirring, and on the temperature.

I had hoped to have been able to determine both the rate of attainment ot
saturation, and the time at which it is attained, by observing the change in the

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.

197

electric conductivity of the solution while it is becoming saturated, and I have made
a few somewhat unsatisfactory experiments on the method, but hope to be able to
return to it shortly.

Measurement of Temperature.

Thermometers whose graduations were sufficiently open to allow of an estimation
to 0°'01 C. were used. They were standardized at Kew, and the rise of the zero
point was determined after an interval of 18 months. The rise of the zero point
was assumed to be proportional to the elapsed time, and all observations are corrected
on that assumption. Corrections were also applied for the emergent column by
suspending an auxiliary thermometer half way up the exposed stem and calculating
in the usual manner ; in no case did this correction amount to more than 00-37 C.

The temperatures given in the tables are corrected to the hydrogen scale of the
Bureau International at Paris.

Experimental Errors.

On page 192 I have already given an example of the results obtained in determining
the capacities, and it will be seen that the largest difference between any two
observations at the same temperature is '0012 cub. centim., which is roughly '01 per
cent. To give an idea of the order of accuracy of the observations of density and
solubility, the actual figures obtained with NaCl are appended below. NaCl was
selected because the differences between the two sets of densities are fairly typical,
while, on account of the small change in the solubility of the salt, those between the
solubilities, besides being typical, show the experimental errors without the necessity
of correcting for small changes of temperature.

Starting supersaturated.

Starting unsaturated.

Temperature. Density.

Solubility.

Temperature.

Density.

Solubility.

0-35 1 ' 20900

35-75

o'-35

1-2Q896

35-74

15-05 1-20209

35-83

15-35

1-20193 35-85

30-05 1-19556

36-22

30-05

1-19555

36-19

45-30 1-18908

36-62

45-50

1-18902

36-59

61-80 1-18221

37-30

61-60

1-18227

37-26

75-85 1-17644

37-86

75-45

1-17637

37-80

90-50 1-17009

38-53

91-25

1-16971

lost

The largest difference between two densities at the same temperature is '00038 at
91° C., which, if the observation at 90°'5 C. be corrected to 91°'25 C., is reduced to
a difference of '00020, and this corresponds to an error of about 0'02 per cent. In
solubilities, however, the largest difference is '06 at 75° C., and this corresponds to

198

THE EAKL OF BEEKELEY ON SOME

an error of 0'16 per cent. This large difference in the two percentage errors is
remarkable, and I have not yet been able to account for it — it is manifested in most
of the salts hitherto worked with. A fact which may possibly throw light on the
subject is noticeable in the above table, and is one which most of the salts also show,

namely, that the solubilities obtained when
starting with an unsaturated solution, tend to
be slightly less than those obtained when
starting with a supersaturated one, and this
although the corresponding densities are prac-
tically identical. I hope to investigate the
matter while determining the electric conduc-
tivities of these solutions.

Tap for closing
pyknometer.

Thermometer.

Platinum wire for_
pulling off the filter.

I.R-Stoppex FU

Solution.-
Filter.

Sa.lt.-

Pyknometer.

Side tube 0
graduated
incjn.

Fig. 5.

w

The Densities and Solubilities at the
Boiling-point.

Attempts were made to determine these in
a Beckmann apparatus, but without success —
the difficulty of keeping a constant temperature
being too great — so recourse was had to a method
first suggested by, I believe, BUCHANAN.

In the apparatus shown in fig. 5, the outer
glass tube A contains water, and the inner
tube B the salt and solution ; by boiling the
water vigorously and closing the side tube C,
steam, passing through the tube D, is forced
to bubble rapidly through the solution (D is
graduated in centimetres so that the level of
the solution may be estimated while the
pyknometer is in the solution). The steam, if
passed rapidly enough through the solution,
stirs it thoroughly, and the temperature rises
up to the boiling-point of the saturated
solution and remains constant at this point as
long as there is enough undissolved salt left.
The constancy of the temperature therefore
indicates that saturation is attained.

Determination of the Density.

When it is seen that the steam is passing freely through the solution, an india-
rubber plug, through which the thermometer and pyknometer pass, and which is also

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS. 199

perforated to allow the steam to escape, is inserted into the top of the inner tube.
When the temperature becomes constant, the pyknometer, with the tap closed and
with the end of the lower capillary covered by a filter, is forced through the stopper,
so that the bulb and capillary are completely immersed ; when the temperature is
again constant, the tap is opened and the pyknometer quickly filled by gentle
suction, and the tap closed. The filter is then removed from the end of the capillary,
the level of the solution in the stem is read, and both thermometer and pyknometer
are taken out of the solution by the withdrawal of the rubber stopper ; the pykno-
meter is then washed, dried, and weighed in the usual manner.

It was found advisable to use the pyknometers described on p. 191 and shown in
fig. 3, not only on account of the pressure set up when the salt crystallized out, but
because they could be more quickly filled, and therefore less condensed steam formed
in the stem ; a further reason for using these pyknometers was that the larger bore
of the stem and lower capillary enabled them to be emptied with less difficulty.

Determination of the Solubility.

When the pyknometer had been weighed, its contents were emptied into a beaker,
and the solution washed into a Jena glass bulb (described on p. 190 and shown in
fig. 1), and evaporated to clryness as before.

Great difficulty was experienced in emptying the pyknometers when filled with
the solutions of rubidium nitrate, thallium nitrate, or caesium alum, and the only way
of doing so was by alternate heating and cooling when completely immersed in nearly
boiling water — the operation taking in some cases as long as G hours. It is
interesting to note that on testing the boiled saturated solutions of the nitrates of
sodium and rubidium for nitrites by means of fuclisine, the conversion of a small
quantity of the nitrate into the nitrite was distinctly indicated.

Modification of Apparatus Necessary to Meet the Case of Extremely Soluble Salts.

In the case of the nitrates of rubidium and thallium, which are extremely soluble at
the boiling-point, the apparatus described on the foregoing page was found to be
unsuitable, because a constant temperature could not be maintained for a sufficient
length of time to allow the pyknometer to be filled. Two things are essential for
maintaining the solutions at their boiling-points : that thorough stirring should take
place, and that there should be a sufficiency of undissolved salt left in contact with
the solution ; with extremely soluble salts the larger quantity of steam necessary for
thorough stirring dissolves so much salt that by the time this stirring is attained the
solution is nearly clear, and shortly after, all the salt is dissolved and the temperature
begins to fall.

200

THE EAEL OF BEEKELEY ON SOME

. A modification of the method was adopted in which steam, generated in a boiler A
(see fig. 6), is forced through a tube B and delivered at the bottom of the large test-
tube C, which contains the solution. The test-tube is immersed in an oil bath D

Thermometers.

Platinum wire
for pulling off-
filter

Stirrer. -

Fig. 6.

maintained at a temperature close to that of the boiling-point of the saturated
solution, the oil in the bath being vigorously stirred by a stirrer driven from the
main laboratory shafting. When the temperature of the oil bath was below the
boiling-point, salt dissolved ; when above, salt was thrown out of solution. By care-

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS. 201

fully adjusting the temperature of the oil bath, two densities could be obtained, one
while the temperature of the solution was close to the boiling-point, but slowly rising,
and the other when it was above the boiling-point, but close to it and slowly falling.
The former gave the density of a slightly unsaturated solution, and the latter that of
a slightly supersaturated solution when referred to a solution saturated at its boiling-
point. The mean of these two observations was considered to be the density of the
solution saturated at the boiling-point — and similarly with the resulting solubilities.

The results obtained at the boiling-point were found not to be as concordant as
those at the other temperatures ; doubtless the greater part of this is due to the
exceptional difficulties of the experiments. The following are some of the sources of
error. It was impossible to prevent the condensation of steam in the stem of the
pyknometer, and it was therefore necessary to estimate the length of each drop, and
add this length to the reading of the level in the stem. There was also an error
introduced by the fact that, for the purpose of reading its level, the solution had to
be sucked into the cold part of the stem which projected through the indiarubber
stopper ; on reaching this colder part, salt immediately crystallises out ; the total
volume thus changes, and the observed level is not that which the solution would
otherwise have attained. The maximum error possible from this cause was calculated
for the case of rubidium nitrate, and was found to be such as to give an error in the
density of O'l per cent.

Another and a much more important source of error was that the reduction of
pressure on the surface of the solution in the pyknometer, unavoidable when filling
by suction, very often caused steam bubbles to form. As it was essential to fill whilst
a large excess of undissolved salt was still being stirred by the steam, the solution
surrounding the pyknometer was often semi-opaque, and consequently it might
happen that part of the space inside the pyknometer was occupied by an unseen steam
bubble, and might thus be an unobserved source of error. Numerous fillings had to
be rejected on this account, the steam bubbles showing themselves on the withdrawal
of the pyknometer from the solution. The numbers tabulated below are those
derived from fillings in which there were no observed steam bubbles.

Determination of the Temperature.

As mentioned above, the boiling-point of the saturated solutkm was considered to
be the constant temperature which the solution and salt reached when steam
was rapidly bubbled through them ; this temperature was indicated by mercury
thermometers, and is given in the table of results. They are, however, unconnected
for emergent column, because it was found to be practically impossible to apply a
satisfactory correction. It is hoped that, later on, when determining the osmotic
pressures,* these boiling-points will be accurately ascertained by means of platinum

* These experiments are in progress, but not complete.
VOL. CCIII. — A. 2 D

202

THE EARL OF BERKELEY ON SOME

thermometers. In the expectation of this the total pressures under which each
solution was boiling when its density was taken was noted, and is given in the table.
This total pressure is made up of the barometric pressure, together with the pressure
due to the height of the boiling liquid. To ascertain the effect of the latter,
observations were made on the boiling-point of water of varying depths, and through
which steam was being rapidly blown. The results showed that the boiling-point was
increased by an amount equal to that which a pressure equal to half the depth of the
liquid would create ; this, of course, was what was to be anticipated, provided the
stirring was thorough. It was assumed that a similar result would hold for the
solutions, and the total pressures given are those calculated on this basis.

Results.

The first table gives the results obtained by means of the apparatus shown in fig. 5.

Column I. gives the approximate boiling-point, which is also the temperature at
which the py kilometer was filled.

Column 1 1. gives the total pressure in millimetres of mercury, at the time of filling.

Columns III. and IV. give the corresponding densities and solubilities; the latter
are in parts of anhydrous salt dissolved by 100 parts of water.

TABLE 1.

I.

II.

III.

IV.

Ijoiling-point.

Pressure.

Density.

Solubility.

NaCl

107;5

740-1

1-1634

39-57

,,

107-8

749-1

1-1629

39-72

KOI

107-4

738-7

1-2118

58-09

jj

107-4

739-4

1-2118

58-12

RbCI

112-9

756 • 6

1 -6146

146-65

,,

112-9

756-6

•6149

146-65

CsC!

119 -3

754-2

2-0855

290-04

M

119-5

757-6

2-0863

289-93

T1C1

99-2

730-5

•9787

2-42

' „

99-6

746-9

•9786

2-40

Na-jSOj

101-9

750-6

•2451

42-15

n

101-9

751-0

•2449

42-2-1

K..SO,

101-0

752 • (i

• 1 206

24-23

„

101-0

752 • 6

•1207

24-18

RboSO,

102-4

742-4

•4752

82-57

„

102-4

742-4

•4753

82-56

CajSOj

108-5

737-2

2-0927

224-24

•*•

108-6

737-6

2-0942

224-75

T12S04

99-7

748-7

1-1164

18-45

,,

99-6

747-5

1-1165

18-45

KNOa

114-0

745-5

1-6266

311-79

,,

114-0

745-1

1-6272

311-48

CsN08

106-0

747-6

1-8642

219-29

"

106-3

749-2

1-8664

1

221-12

These observations are derived from pyknometer fillings which were considered to be particularly good ;
they are therefore given double weight when taking the " means" for the tables at the end of the paper.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS. 203

The second table gives the results obtained with the apparatus shown in fig. 6.

Columns I., II,, III., and IV. give the temperature of filling, the total pressure,
the density, and the solubility, respectively, when the temperature of the oil bath
was below the boiling-point, but close to and rising; while columns V., VI., VII.,
and VIII. give the same, when the oil-bath temperature was higher than the boiling-
point, but close to and falling.

»

TABLE II.

i

1

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

Temperature.

Pressure.

Density.

Solubility.

Temperature

Pressure.

Density.

Solubility.

i •

i

!

NaNOs

119-0

737-7

1-5369 ' 208-27

118-9

734-8

1-5379

209-42

RbN03

118-1

739-6

2-1867 614-27

118-4

729-2

2-1867

619-94

T1N08

104-1

756-8

3-1725 583-39

104-8

768-1

3-2086

604-46

CsAlum

100-3

757-8

1-1278 22-47

100-4

758-0

i

1-1292

23-21

Purity of the Salts — the Chlorides.

The alkali chlorides were obtained from Messrs. MKRCK, and were sold as the
purest they made ; the thallous chloride came from Messrs. KAIILBAUM, and was also
sold as pure.

The solutions of the potassium and sodium salts did not require filtering, and were
tested for purity by an analysis of their chlorine contents. The sodium salt giving
60'58 per cent, (calculated 60'5'J per cent.) and the potassium 47'60 per cent.
(calculated 47 '54 per cent.). The rubidium chloride was tested spectroscopically for
the presence of potassium and caesium by first locating the chief lines of these metals
by observation of their spectra on the graduated circle of the spectroscope, and then
exploring the rubidium spectrum for them. No definite evidence of impurities was
obtained. An analysis of the chlorine content gave 29 '34 per cent, (calculated
29 '32 per cent.). From the appearance of the caesium chloride it was thought
necessary to filter the solution and recrystallise several times ; the mother liquor of
the first recrystallisation was distinctly yellow, that of the second faintly so, while
the third was colourless. A spectroscopic examination, similar to that mentioned
above for the rubidium salt, revealed, it was thought, a trace of rubidium. An
analysis of the chlorine content gave 21 '13 per cent, (calculated 21 '06 per cent.).

The thallous chloride was found to be free from lead, and an analysis of the
thallium content gave 8 5 '40 per cent, (calculated 8 5 "21 per cent.). Owing to the
insoluble nature of this salt, the solubility determination cannot be relied on to as
great a degree of accuracy as in the other determinations.

2 D 2

204 THE EAEL OF BERKELEY ON SOME

The Sulphates.

The alkali sulphates came from Messrs. MERCK, and were sold as their purest ; the
thallium salt came from Messrs. KAHLBAUM. Neither the sodium nor the potassium
salts required recrystallising, nor did their solutions require filtering ; analyses of
their sulphuric acid contents gave for the sodium salt 67 '37 per cent, (calculated
67'57 per cent.) and for the potassium 55'20 per cent, (calculated 55'12 per cent.).
Not having purchased enough of the rubidium salt, the balance was made good by
treating pure rubidium carbonate (also purchased from Messrs. MERCK) with pure
sulphuric acid in just sufficient quantity to neutralise the solution, and then crystal-
lising out. The two quantities of salt were then added together and recrystallised,
and the crystals examined spectroscopically, in the manner before stated, for potassium
and caesium, but with no definite indication of either. An analysis of the sulphuric
acid content gave 36'05 per cent, (calculated 35-99 per cent.).

The caesium sulphate was recrystallised three times, and the spectroscopic examina-
tion gave no definite indication of either potassium or rubidium. An analysis of the
sulphate content gave 26 '62 per cent, (calculated 26'55 per cent.). The thallium
sulphate was recrystallised three times and found to be free from lead. An analysis
of the thallium content gave 80'96 per cent, (calculated 80'95 per cent.).

The Nitrates.

All the salts were Messrs. MERCK'S purest, except the thallium salt, which came
from KAHLBAUM. The alkali nitrates were all recrystallised two or three times, and
were examined spectroscopically and found to be free from impurities. The thallium
nitrate, however, was found to contain some lead ; it was freed from this by repeated
recrystallisation. An analysis of the thallium content gave 76'89 per cent, (calculated
76-69 per cent,).

On account of the difficulty of obtaining accurate analyses of the alkali nitrates
they were not analysed, but after the first recrystallisation a series of densities and
a corresponding series of solubilities at different temperatures were obtained, and
these series were compared with similar series obtained from the solution of the
crystals of the next recrystallisation. The two differed by no more than the experi-
mental errors.

During the evaporation to dryness in the Jena glass bulbs for the purpose of
determining the solubilities, it was found that a trace of nitrate almost invariably
came over with the condensed water, and those observations in which more than a
trace came over were rejected. It was also noticed that, except in the case of
caesium nitrate, the dried salt remaining in the bulbs contained a trace of nitrites.
The quantities in both cases were so small that it was not considered necessary to
apply any corrections to the resulting solubility.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS. 205

The Alums.

The potassium alum purchased as pure from Messrs. MERCK was found to contain
a small quantity of both iron and ammonium, and repeated recrystallisation did not
purify it. Pure aluminium and pure potassium sulphates were therefore purchased,
and the pure alum made from these. An analysis of the sulphuric acid content of
this salt, when recrystallised, gave 40 '33 per cent, (calculated -40 '49 per cent.). The
remainder of the alums, purchased as pure from Messrs. MKRCK, were recrystallised
several times, and the spectroscopic examination showed no impurities. Analyses
gave, for the rubidium salt, 36-G9 per cent. SO (calculation being 36'89 per cent.),
and, for the caesium salt, 33'67 per cent. SO (calculation being 33'80 per cent.).

The thallium alum was analysed by determining the thallous sulphate content,
giving 39 -29 per cent., the calculated value being 39 '43 per cent. It will be noticed
that the tables of results give no values for potassium alum above 60° C., for the
rubidium alum above 70° C., and for thallium alum above 60° C. — this is because it
was found that the prolonged heating at (>8° ( <., at 80° C., and at 7f>° ( '. respectively
decomposed the solutions, and a white insoluble precipitate was formed. The c;t'siuni
alum, however, could be heated to the boiling-point without decomposition.

The solubilities of the potassium and rubidium alums could not be determined
to so close a degree of accuracy as that of the other salts, for it was almost
impossible to dry the contents of the pyknometers to a constant weight, without
decomposing the salt. The method finally adopted was to evaporate to partial
dryness in the Jena glass bxilbs at 115° 0., then raise the temperature of the oven
gradually to 175° C. (dry air passing the while), and, when all perceptible moisture
had been driven oft', to heat the bulb gently with a naked flame, care being taken not
to heat to a temperature high enough for the glass to give a sodium flame. With
caesium alum a constant weight was obtained by keeping the oven at 130° C. to
140° C.

The Observed Densities and Solubilities.
In the following tables the numbers in each column are obtained as follows :—

Column I. gives the temperatures to which the observation recorded in the
remaining columns refer. These temperatures are corrected for emergent
column, and are, as before stated, the mean temperature corresponding to
the meaned density and the meaned solubility.

Column II. gives the density of the saturated solution, obtained as already
stated. It summarizes over 600 observations, excluding the 300 men-
tioned on p. 195.

206 THE EARL OF BERKELEY ON SOME

Column III. gives the corresponding solubilities in parts of anhydrous salt
dissolved by 1 00 parts of water, and is also a summary of some 450
observations exclusive of the above mentioned 300.

Column IV. gives the number of gram-molecules of salt in i litre of solution
saturated at the temperature recorded in column I. The numbers are
obtained by dividing the weight of salt found in the litre by the molecular
weight of that salt.

Column V. gives the number of gram-molecules of water in the litre. The
numbers are derived by dividing the weight of the water in 1 litre of
saturated solution by the molecular weight of water. Throughout this
work the atomic weights used are those based on hydrogen as unity and
oxygen as equal to I5-88.

Column VI. gives a measure of the concentration — it is the ratio of the
number of salt molecules to the sum of salt and water molecules in the
same volume of solution.

Column VII. gives the solubilities, taken from COMEY'S ' Dictionary of
Solubilities,' of such salts as have already been investigated.

SODIUM Chloride.

I.

II.

III.

IV. V.

I
VI. VII.

Number of

Number of Solubility,

Temperature.

Density.

Solubility.

gram- gram- Con- from
molecules molecules centration. COMEY'S

of salt. of aqua. dictionary.

0-35

1-2090

35-75 5-484

49-81

10-083 35-7

15-20

1-2020

35-84 5-462 49-49

10-061 35-9

1-2025]

30-05

1-1956

36-20

5-473

49-09

9 • 969 36 • 3

1-1960 [

45-40

1-1891

36 -GO 5-488

48-69

9-872 36-8

1-1895 [-*

61-70

1-1823

37-28 5-529

48-17

9-712 37-4

1-1827 |

75-65

1-1764

37-82

5-560 47-74

9-586 38-2

1-17691

90-50

1-1701

38 • 53

5-606 47-24

9-427 39-1

Boiling- 1
point J1

1-1631

39-65

5-688

46 • 58

9-189 40-2

1

The solubilities were determined by evaporating to dryness in Jena glass bulbs.

1 The numbers in this column are the densities of the saturated solution of NALL obtained by ANDRIA
(' J. Prakt. Chem.,' [2], 30, 305), and reduced to the temperatures given in column I.

PHYSICAL CONSTANTS OF SATUKATED SOLUTIONS.

207

POTASSIUM Chloride.

1

I.

II.

III. IV.

y

VI. VII.

Number of Number of

Solubility,

Temperature.

Density.

Solubility. g,ram;
J molecules

gram-
molecules

Con- from
centration. COMEY'S

of salt.

of aqua.

dictionary.

0-70

1 • 1540

28-29 3-438

50-31

15-633 28-7

28-231

19-55

1-1738

34-37 4-057

48-84

13-038 34-6 34-06 U

32 • 80

1-18.39

38-32 4-432

47-87

11-801 38-2 38-05 f

59-85

1-1980

45-84 5-088

45 • 95

10-031 45-5 45-47J

74-80

1-2032

49 • 58 5 • 389

44-99 9-348 49-6

89 • 45

1-2069

53-38 5-676

44-01 8-753 53-6

Boil»f \ 108-0
point J

1-2118

58-11 6-018 42-87 8-124 58-5

; I

The solubilities were determined by evaporating to dryness in platinum crucibles,
those at boiling-point in Jena glass bulbs.

Uumnii'M ( 'blonde.

I.

II.

III.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

i

" c.

0-55

1-4409

77-34

5 • 238

45-44

9 • 675

76-4 at r C.

18-70

1 • 4865

90 • 32

5-881

43 • 69

8-429

82-9 „ 7 C.

31-50

1-5118

98-61

6 • 257

42-58

7-805

44-70

1-5348

106-24

6-590

1 1 • 63

7-317

60-25

1-5558

115-63

6 • 955

40 • 35

6-802

75-15

1-5746

1 24 ' 52

7-280

39 • 22

6 • 3S7

89 • 35

1 -5905

132-73

7-562

38 • 22

6-054

Boiling- ~l
point J

114-0

1-6148

146-65

8-003

36-62

a • 575

The solubilities were determined by evaporating to dryness in platinum crucibles,
those at boiling- point in Jena glass bulbs.

* The numbers in this column are the solubilities given by ANDRIA (' J. Prakt. Chem.,' 137, 468)
reduced to the temperatures given in column I.

208

THE EARL OF BERKELEY ON SOME

CJESTUM Chloride.

I.

II.

III. IV.

V.

VI.

Number of

Number of

Temperature.

Density.

****»*' modes

gram-
molecules

Concentration.

of salt.

of aqua.

°C.
0-70

1-8458

162-29 6-836

39-36

6-758

16-20

1-8984

182-24 7-337

37 • 62

6-127

29-85

1-9359 197-17 7-688

36-43

5-739

45-55

1-9702 213-45 8-030

35-16

5-379

60 ' 20

2-0012 229-41 8-342

33 • 98

5-073

76-10

2-0286 245-76 8-630

32-81

4-802

89 • 50

2-0500 259-56 8-858

31-88

4-599

Boiling- 1119.4

point J

2-0859 289-98 9- 283

29-92

4-223

The solubilities were determined in platinum crucibles, except those at the boiling-
point, which were done in Jena glass bulbs.

Tir.uj.ors Chloride.

1.

II.

III.

IV.

V.

VI.

Number of

Number of

Temperature.

Density. Solubility. g.ram;
molecules

gram-
molecules

Con-
centration.

of salt.

of aqua.

0 I1.

0--1

1-001:1

0-17

•00707

55-91 7916

15-6

1-0017

0-29

•01199

55-86 4660

30-05

•9996

0-47

•01930

55-65 2930

45 ' 20

•9964 i 0-72

•02976

55-33 1860

59-80
75-65

•9922 1-03
•9870 1-48

•04266
•06060

54-92
54-40

1288
896-6

89 • 65

•9821

1-96

•07964

53 • 87

677-4

Boiling ~l
point J

99 • 35

•9787

2-41

• 09684

53-45

552 • 9

VI.

VII.

Solubility,

Con-

from

itration.

COMEY'S

dictionary.

91(5

0-19

660

0-27

930

0-40

860

0-52

288

0-74

896-6

1-03

677-4

1-32

552 • 9

1-55

The solubilities were determined in platinum crucibles, except those at the boiling-
point, which were done in Jena glass bulbs.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.
SODIUM Sulphate.

209

I.

II.

III.

IV.

V.

VI.

Number of

Number of

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

of salt.

of aqua.

1

°C.
0-70

1-0432

4-71

•3327

56-05

1
168-46

10-25

1-0802

9-21

•6456

55-32

86-63

15-65
20-35

1-1150
1-1546

14-07
Lost

•9747

54-67

57 • 07

24-90

1-2067

27-67 -

1-8534

52-86

29 • 57

27-65

1 • 2459

34-05

2 • 2425

51-99

24-18

30-20

1-2894

41-78

2-6926

50-86

19 -88

31-95

1-3230

47-98

3 • 0400

50-00

17-45

33-50

1-3307

49-39

3-1174

49-82

1C -98

38-15

1-3229

48-47

3-0608

48-47

17-28

44-85

1-3136

47-49

2-9980

49-81

17-62

60-10

1-2918

45-22

2-8507

49 • 75

18-45

75-05

1-2728

43-59

2-7383

49-57

19-11

89-85

1-2571

42-67

2 • 604.",

49-28

1 9 • 50

Boiling- 1101. 9

point J

1-2450

42-18

2-6175

48-97

19-71

VII.
Solubility,

from

COMEY'S

dictionary.

5-0

9-18

14-12

27-6
34-1
41-8
47-6
50-5
49-3
47-7
45-3
44-0
43 • 1

42-3

Most of the solubilities were determined by evaporating to dryness in platinum
crucibles, the remainder in Jena glass bulbs. On plotting out the results of the
density and solubility determinations against the temperatures, it will be seen that
both curves give the transition point at 32°'5. The direct estimation of the melting-
point of hydrated sodium sulphate made by Messrs. RICHARDS and CHURCHILL* gave
32°'379 on the hydrogen scale.

POTASSIUM Sulphate.

I.

II.

III.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

i

of aqua.

dictionary.

1

°C.
0-40

1-0589

7-47

• 4253

55-11

130-66

8-5 7-42]

15-70

1-0770

10-37

•5849

54-58

94-31

10-4 10-31

31-45

1-0921

13-34

•7429

53-89

73-52

12-5 13-22 U

42-75

1-1010

15-51

•8551

53-31

63 • 35

14-5 15-25 I

58-95

1-1086

18-01

•9792

52-53

54-65

17-6 17-99J

74-85

1-1157

20-64

1-1036

51-72

47-86

20-8

89-70

1-1194

22-80

1-2019

50-98

43-41

23-8

Boiling- \
point J

101-1

1-1207

24-21

1-2621

50-47

40-99

26-4

i

The solubilities were determined by evaporating in platinum crucibles, those at the
boiling-point in Jena glass bulbs.

* ' Zeit. Phys. Chem.,' 26, 690 (1898).

t Reduced to the temperatures in column I., from ANDRIA'S solubilities, see ' J. Prakt. Chem.,' 137, 471.

VOL. CCIII. — A. 2 E

210

THE EARL OF BERKELEY ON SOME

RUBIDIUM Sulphate.

I.

II.

III.

IV. V.

VI.

VII.

Number of Number of

Solubility,

Temperature.

Density.

Solubility.

gram- gram-
molecules molecules

Con-
centration.

from
COMEY'S

of salt. of aqua.

dictionary.

°C.
0-50

1-2740

36-66

1-2903 52-14

41-41

36-3

15-80

1-3287

46-04

1-5810 50-89

33-19

45-3

31-60

1-3704

54-25

1-8193 49-69

28-32

55-9

44-20

1-3998

60-75

1-9970 48-70

25-35

65-5

57-90

1-4232

66-59

2-1475 47-78

23-24

71-2

74-75

1-4480

73-25

2-3111 46-74

21-22

74-5

89 • 45

1-4649

78-61

2-4337 45-87

19-84

77-5

Boiling- "1 102.4

1-4753

82-57

2-5185 45-19

18-94

80-2

point J

The solubilities were determined by evaporating to dryness in platinum crucibles
those at the boiling-point in Jena glass bulbs.

CAESIUM Sulphate.

I.

ii.

III.

IV. V.

VI. VII.

Number of Number of

Solubility,

Temperature.

Density.

Solubility.

gram- gram-
molecules molecules

Con-
centration.

from
COMEY'S

of salt. of aqua.

dictionary.

0-70

1-9766

167-55

3-4467 41-32

12-99

158-7 at -2°C.

15-00

1-9992

176-02

3-5499 40-51

12-41

—

30-40

2-0202

184-35

3-6469 39-74

11-90

—

44-90

2-0365

192-49

3-7318 38-94

11-44

—

59-50

2-0512

199-35

3-8035 38-32

11-08

—

75-70

2-0664

207-89

3-8850 37-54

10-66

—

89-75

2-0774

214-82

3-9471 36-91

10-35

—

BplT}108-6

2-0932

224-50

4-0323 36-08

9-95

—

The solubilities were determined by evaporating to dryness in the Jena glass
bulbs.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.

211

THALLOUS Sulphate.

|

I.

II.

III. IV.

V.

VI. VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility. g,ram;
molecules

gram-
molecules

Con-
centration

from
COMEY'S

of salt.

of aqua.

dictionary.

0-15

1-0248

2-72 -0541

55-80

1032-5

2-8

15-60

1-0384 4-32 -0858

55-67

649-9

4-6

29-80

1-0512

6-13 -1214

55-40

457-3

6-7

44-95

1-0652

8-39 -1647

54-97

334-7

8-8

60-40

1-0795

10-96 -2130

54-41

256-4

11-2

75-90

1-0941

13-84 -2654

53-76

203-5

13-8

90-05

1-1071 16-54 -3138

53-14

170-3

16-8

Boiling- J99.?

1-1165 18-45 -3474

52-72

152-7

18-7

point J

The solubilities were determined by evaporating to dryness in the Jena glass
bulbs.

SODIUM Nitrate.

I. ! II.

in.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature. Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

0-30 1-3530

73 • 30

6-776

43-660

7-443

73-4

15-45 1-3769

84-48

7-466

41-74

6-591

85-7

30-00 1-3992

96-15

8-121

39-89

5-912

95-0

44-50 1-4210

109-10

8-779

38-01

5-329

106-5

60-00 1-4446

124-56

9-489

35-98

4-792

124-3

76-15 1-4701

143-15

10-248

33-81

4-300

142-1

90-25 1-4920

161-61

10-917

31-88

3-920

163-5

Boiling- j 7

208-84

12-310

27-84

3-262

5220

point J

The solubilities were determined by evaporating to dryness in Jena glass bulbs ;
and it is to be noted that a small trace of salt was always found in the distillate from
the bulbs.

2 E 2

21-2

THE EAKL OF BERKELEY ON SOME

POTASSIUM Nitrate

I.

II. III. IV. V.

VI

VII.

Number of Number of

Solubility,

Temperature

T~V -^ i i -1-4. gram- gram- Con-
Density. Solubility. m»lecules m*lecules centration.

from
COMEY'S

of salt. of aqua.

dictionary.

j

0 C.

|

0-40

1-0817 13-43 1-276 53-34

42-80

13-5 13-53"

14-90

1-1389 25-78 2-326 50-64

22-77

25-9 25-75

30-80

1-2218 47-52 3'921 46-32

12-81

45-7 47-28 *

44-75

1-3043 74-50 5-547 41-80 8-536

73-5 73-67

60 • 05

1-3903 111-18 7-291 36-83 6-051

111-1 110-02

7G-00

1-4700 156-61 8-936 32-04 4-585

159-0

91-65

1-5394 210-20 10-391 27'75 3'767

212-6

Boiling- j

1-6269 311-64 12-269 22-10 2-801

327-1

point J

The solubilities were determined in platinum crucibles, those at the boiling-point in

Jena glass bulbs.

RUBIDIUM Nitrate.

1

I.

II.

III.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

" C.

0-60

1-1389

20 • 39

1-318

52-91

42-66

20-1 at 0°C.

15-85

1 • 2665

44-28

2 • 656

49-09

19-49

43-5 „ 10° C.

31-55

1-4483

86-67

4-592

43 • 42

10-45

45-85

1-6216

139-38

6 • 450

37 • 90

6-875

63 • 40

1 ' 8006

217-06

8-423

31-76

4-770

75-60

1 • 9055

284-06

(J • 630

27-75

3-881

90-95

2-0178

382-89

10-932

23-37

3-138

Boiling- "1

118-3

2-1867

617-11

12-858

17-05

2-326

The solubilities were determined by evaporating to dryness in platinum crucibles,
except in the case of the observations at the boiling-point ; these were done in Jena
glass bulbs, and the distillate always showed a trace of nitrate as having come over.

*• These solubilities are calculated from those of ANDRIA at slightly different temperatures, ' J. Prakt.
Chem.,' 137, 474.

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.

213

CAESIUM Nitrate

I

II.

III.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

0

•35

1-0701

9-54

•4815

54-64

114-35

10-58 at 3°-2C.

15

•95

1-1345

19-46

•9555

53-11

56-56

30

•45

1-2219

34-50

1-6205

50-81

32-36

45

•15

1 • 3306

55-58

2 • 4572

47-84

20-47

—

59

•90

1-4565

83-50

3 • 4260

44-39

13-96

—

76

•40

1-6068

124-64

4-6082

40-00

9-68

—

90

•55

1-7307

165-18

5-5724

36-50

7-55

Boiling- 1
point J

•2

1-8657

220-32

6-6351

32-56

5-91

—

The solubilities were determined by evaporating to dryness in platinum crucibles ;
those at the boiling-point in Jena glass bulbs.

THALLOUS Nitrate.

i

I. II. III. IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility. ^, a
J molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

0-65

1-0346

4-07 -1532

55-60

363-92

15-40

1-0653

7-93 -2963

55-20

187-50

9-7 at IS C.

30-60

1-1150

14-63 -5385

54-40

102-06

—

44-65 1-1891

24-98 -8995

53-22

60-13

—

57-30 1-2986

41-31 1-4369

51-40

36-77

43-5 at 58° C.

64-95

1-3957

56-33 1-9036

49-93

27-33

. —

76-00 1-6096

91-93 2-9183

46-90

17-07

—

87-80

2-0258

174-02 4-8780

41-23

9-45

—

3-1906

593-93 10-3366

25-72

3-488

588-2 at 107° C.

point J

The solubilities were determined by evaporating to dryuess in platinum crucibles ;
those at the boiling-point in Jena glass bulbs.

214

THE EARL OF BERKELEY ON SOME

POTASSIUM Alum

I.

II. III.

IV.

V.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density. Solubility.

gram-
molecules

gram-
molecules

Con- from
centration. COMEY'S

of salt.

of aqua.

dictionary.

0-40

1-0292 3-01

•0586

55-88

954-6

3-05

15-30

1-0461 5-09 -0989

55-67

564-0

5-06

28-10

1-0661 7-83 -1510

55 • 28

367-1

7-50

43-20 1-1044 13-31 -2530

54-51 216-5 13-40

60-45 1-1835 25-06 -4624

52-93

115-5

25-7

1

The solubilities were determined in the Jena glass bulbs. Column IV. gives the
number of grain molecules in the litre calculated on the assumption that the anhydrous
salt is K2AL (S04),.

RUBIDIUM Alum.

I

I. II.

III. IV.

V.

VI. VII.

Number of

Number of

Solubility,

Temperature. Density.

SuluLilitv. srm;
molecules

gram-
molecules

Con- from
centration. COMEY'S

of salt.

of aqua.

dictionary.

0°0.
0-40 1-0072

0-73 -0121

55 • 92

4607 0-73

15-20 1-0112

1-28 -0211

55 • 84

2645 1-33

32-20 1-0165

2-38 -0391

55-53

1420 2-44

45-80 1'0267

4-13 -0673

55-14

820-6 4-33

59-65 1-0466

7-27 -1241

54-57

440-8 7-97

69-75 1-0804

12-23 -1947

53-84

277-5 13-42

Solubilities in Jena glass bulbs.

Column IV. gives the number of gram-molecules
calculated on assumption that the anhydrous salt is Rb.,Al.j

PHYSICAL CONSTANTS OF SATURATED SOLUTIONS.

215

CAESIUM Alum.

I.

II.

III.

IV.

v.

VI.

VII.

Number of

Number of

Solubility,

Temperature.

Density.

Solubility.

gram-
molecules

gram-
molecules

Con-
centration.

from
COMEY'S

of salt.

of aqua.

dictionary.

0°C.
0-40

1-0017

0-21

•0030

55-91

18890

0-19

16-60

1-0022

0-35 -0050

55-85

11230

0-36

29-15

1-0010

0-58 • 0082

55-66

6773

0-59

45-25

•9994

1-07 -0151

55 • 30

3658

1-07

60-60

1-0004

2-05 -0287

54-83

1911

2-04

75-35

1-0107

4-32 -0599

54-19

905-6

4-39

83-05

1 0250

6 • 86

•0934

53 67

575 • 6

5-87

90-85

1-0328

11-20 ; -1524

52-92

353 • 8

Boiling- "I
point J

1-1285

22-84

•3002

51-38

172-2

—

Solubilities in Jena glass bull).

Numbers in IVth column are derived from
assumption that anhydrous salt is ('So, A12 (SO,),.

THALLIUM Alum.

I.

II.

III.

IV.

V.

Number of Number

Temperature.

Density.

Solubility.

gram- gram-
molecules molecule

of salt.

of aqua

o°c.
0-45

1-0299

3-22

•0382

55-81

16-10

1-0503

5-61

•0664

55-62

29-85

1-0808

9 • 33

•1097

55-29

37-50

1-1090

13-09

•1527

54-85

45-20

1 • 1500

18-50

•2136

54 • 28

52-40

1-2051

25-39

•2904

53-75

60-05

1-2812

35-43

•3988

52-91

VI.

Con-
centration.

1460
839-3
505-0
360-2
255 • 1
186-1
133-7

Solubilities in platinum crucibles. Numbers in IVth column are calculated for
N2A18 (SO,),.

In conclusion, I am glad to have this opportunity of thanking Mr. E. G. HARTLEY
for his help in the observations on the densities at the boiling-points, and
Messrs. NEVILLE and WHETHAM for«the kind interest they have taken in the work,
and for several suggestions.

[ 217 1

IX

Third K

mic PrMem.

A. G. (iKKENHlLL. F.

Received IVremlicr :."-'. 1903,-- Read January

INDEX SLIP.
THK ABEL ( entenma eremony, held ni < l<nsti

the attention of tn ithem:itieimi« to the greit inH
and to the history of ellipl ic f'unct inn* and of
"Journal fiii die K-me ..ml nMgewan In-- Math, r.

GBEBNHIU,, A, G.— The Tliird Elliptic Integr*! and the Ellipsotomio
Problem. Phil. Tr»n«., A, TO!. 203, 1904, pp. 217-304.

" UflxT dlt' I lit ei^i'iitloii "iff I )'ll''i fii t la i- 1 '' '! iin
Division-values of the Elliptic Functions.

aKKBXHir.r,, A. &, Phil. Trans., A, vol. 203, 1904, pp. 217-304.

Mechanical Applications of the Elliptic Integral of the Third Kind.

GBEENHILL, A. G. Phil. Trans., A, vol. 203, 1904, pp. 217-304.

Pseudo-elliptic Integrals.

G»Ki!fltn,i,, A. G. Phil. Trans., A, vol. 203, 1904, pp. 217-304.

intended to show ti|(. ],:'

Provided v>:~" ~~ -•(•-> :.,.-,

tucteuding .is ('.: :.s :">.-..-. i!,!t-. tli--
ti>e complete solut i'in ct' iM.ni\'
unfinished stale; at ihe ^mu- ti:ne
;<mgress is ejected of the M,-:ii-i-l ;in;il
glided in a path likelv to :i!ri\(j .if '..--> 'ni

The prohleill «f the l»i\'-ni:i ;tr.-i I ii-ni.'f lisn

fictions .is solved incidentally; as also the ditti
ber to ABEL, of Januarv I'l, 1S"JH:-

)er, 1!)02, lias directed
j on modern analysis,
>n hv CKKLLE of the

(A).

nee as indicating the
the present memoir is

0 mechanical theory.
the simplest case arid
js will he ahle to effect
now abandoned in an
the simplest lines of

thematical research is
the theory of elliptic

isformation, of elliptic

1 hy LEUKNDKE in his

nctions elliptiques.

•.tions de 3"1C espece a

ulaire, savoir que les

•li-s, facilement reductible

done par le fait quatre

rieme sera it hien pins

xamine et mis an clair.

• votre investigation ef

3.8.04

.•lug xanrci

•)ini<>1o«qilia 9ili him (tiy>lol sitqillS h-iiilT erIT — .f) -A

.<|q ,4-0«I ,£f»S .Io» .A ,.«n*iT Mill .nreido-il

WI6-TIS

oilqillH oril lo
,80$: .lor ,A ,.«n«T .Jill? .O .A

.l)inil [niilT ijfllo Ui^tJnl oiiqilia ei(J 'lo tnoiteoilqqA Uoi
.ROE .!OT .A ,.«D«iT .liri? .O .A

.107 .A ,.tn*iT .liflH .O .A ,.

[ 217 1

IX. The Third Elliptic futff/ml. and the KUipsotomic Problem.
By A. G. GREENHILL, F.R.S.

Received December 22, 1903, — Read January 21, 1904.

THE ABEL Centennial Ceremony, held in Christiania, September, 1902, has directed
the attention of mathematicians to the great influence of AHKL on modern analysis,
and to the history of elliptic functions, and of the foundation by CRELLE of the
"Journal fur die reine und angewan Ite Mathematik."

ABEL'S article in the first volume of ' OKELLE'S .Journal,' 1826.
" Ueber die Integration der Differential- Formel

P </f

wenn R und p gauze Functionen sind," is of great importance as indicating the
existence of what is now called the pseudo-elliptic integral; the present memoir is
intended to show the utility of this integral in its application to mechanical theorv.

Provided with a list of these integrals, proceeding from the simplest case and
extending as far as possible, the student of applied mathematics will be able to effect
the complete solution of many interesting mechanical problems now abandoned in an
unfinished state; at the same time, the exploration along the simplest lines of
progress is effected of the general analytical field, and mathematical research is
guided in a path likely to arrive at useful development in the theory of elliptic
functions.

The problem of the Division, and thence also of the Transformation, of elliptic
functions is solved incidentally ; as also the difficulties raised by LEUKNDHE in his
letter to ABEL, of January 16, 1829 :—

" Mais de la nait une difficulte sur la division generale des fonctions elliptiques.

" Admettrons-nous cette difference enorme entre les fonctions de 3"1C espece a
parametre logarithmique et les fonctions a parametre circulaire, savoir que les
premieres peuvent s'exprimer par une fonction de deux variables, facilement reductible
en table, et que les autres ne le peuvent pas ? II y aurait done par le fait quatre
especes de fonctions elliptiques au lieu de trois, et la quatrieme sera it bien plus
composee que la troisieme. C'est un point qui merite d'etre examine et mis au clair.
Je le recommande a votre investigation et a celle de M. JACOBI."

VOL. ccm.— A 367. 2 F 3-8M

218 PROFESSOR A. G. GREENHILL ON THE

But a mere change of sign in ABEL'S results is sufficient to pass from the
logarithmic to the circular form of the third elliptic integral, and as it is the circular
form which is of almost invariable occurrence in dynamical problems, we shall adopt
it as our standard form.

Applications of this third elliptic integral are introduced in the course of the
memoir to these problems, such as POINSOT'S herpolhode, and the spinning top or
gyroscope, the spherical catenary, the velarium, and the elastica under uniform
normal pressure.

References to former articles on the same subject in the ' Proceedings of the London
Mathematical Society,' vols. 25 and 27, are given in the sequel in the abbreviated
form, L.M.S., 25 or 27 ; frequent reference is made also to KIEPERT'S articles on
the theory of elliptic functions in the ' Mathematische Annalen ' ('Math. Ann.,'
vols. 26, 32, 37).

1. Working then with the third elliptic integral in the circular form, when the
elliptic parameter v is a fraction of the imaginary period w3, we change the variable
in the standard form of WEIERSTRASS,

by putti

= t , - ..- (1),

(w-f r)

n

Pit — V'r = .s — cr, f"~U = b (2),

*>V=V/-SJ v=fw, (3),

where / is a real fraction, so that the integral changes to

i /_X ,/., ..(fr-'r)

(where the elliptic arguments u and v may be supposed for a moment to be normalised
to the Jacobian form), and * is an elliptic function of u which we may denote by s (?/),
differing from WEIERSTRASS'S Vii by a constant, so that

while

cr =

Considering that (HALPHKN, ' Fonctions Elliptiques,' 1, p. 222)

is an algebraical function of x when v is an aliquot part of a period, we take

__rV(v)(s-<r)-±^/-Z ds

- -

THIED ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 219

where IP (?>) = ^ /» — £y> /7)%

and denote this integral hy 1 (r), and work with it as our standard form of the
elliptic integral of the ITT. kind.

The function employed hy HALPHEN (' F.E.,' I, p. 230) is now

</> (", '") = x '(* ~ o-) exp { inP (•/.) - /! (r)} (8),

</> (", - r) = x/(x - o-) exp { - iuP (r) 4- i\ (,>)} (<)),

and is a Lame function of the first order, satisfying his differential equation

\(^£-29u-&v=0 (10);

<f> du-

thence the Lame functions of higher order may he derived hy differentiation, as shown
hy HERMIT.K, ' Comptes "Rendus,' 1877, and these can he employed in the prohlems
considered hy Professor G. H. DARWIN, in the ' Phil. Trans.,' 197, 1901, " Ellipsoidal
Harmonic Analysis."

In the Hermite-Jacohi notation we may take

, , 000 (u — v) 00R (u — i')

$(u'v>- 0U0.0 /exP (<«*'')' or 0;/H/, 'expfcznv) (11).

'2. Next introduce the ./• and ?/ employed hy HALPHEX (' F.E.,' 1, p. lO-l), which
may be connected with the a, h, and /> of ABEL'S notation (' (Euvres/ II., p. 155) hv

and put

.s'- („) = S = 4* (x + .r)- -{(!+?/) x + xyY (-2),

*(u) = Vu+-x (3);

then if we |>ut

ft- o- = x (n) — x (r) = x + x (4),

„-•.(.)„-* ,w»_(' + *+-4« (5),

»'(,•) = ;/(,•)= v/-S=:«

The multiplicative values

can now all he expressed rationally in terms of x and y, with the y functions of
HALPHEN ('F.E.,' 1, p. 102; also ABEL, '(Euvres,' II., p. 159; MONTARD-PONOELOT,
'Applications d'analyse et de geometric,' t. I., Paris, 1862), by means of the recurring
formulas

s (mr) _ ., („ ,,,) = x* y- + :;yT", y (nr) = x y*i (9).

ym~y,i~ y«

2 F 2

2-20 PROFESSOR A. U. GREENHILL ON THE

Thus
.,(») = -*, tV( r) = .r (10),

,s(2r) = 0, »V(2r) = .ry (ll),

«V (3r) = ,/ - ;K - ?/- (12),

x (y — x) • / / , \ •"' (y — *' — v2) — (y — x)2 /-, r,\

.s'(4?') = --•' 2 ' — y, w(<iv) = x y s' (13),

.xv (?/ — x — if] • / / c \ '/2 (a?V — a;~ — ?/3) — x (y — x — y~)~

s(ov)= •' ^ ^ J ' — x, iti'(5v) = x-'~ — 7^— v< (14),

(y - *)'- (y - x)

,y (6t.) = y (^ ~ x}:{x(!> ~ x~ }~.} ~ ('ry ~x'~ f] (15),

y,r
s (nr) = x* ZiiL^- 1 _ lT) ?V (>jr) = a;^ (16).

y«" 7"

.3. For the determination of P (ra;) we h.ave the formula (HALPHEN, ' F.E ,' 1 , p. 1 02)

^3 = *-(») = (F'fply. (i);

whence, by logarithmic differentiation,

,,p (,,)_. p (,,,')= i(n£t>- C'//r)
_ W2 _ i $»".,. i r/y//
''

- ~ (i 4. w\ 4. " 4.

Sn ^ J} n7,\SxcM Sydw

Now for every homogeneity factor, and as elliptic functions of degree zero,

2J- — xy (1 + t/) _ x ( 1 + y)
xy x

5-8(1+*) (4),

»/

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 221

and _(p2i'-pr)8 ,,,,

>*

iv'r

_ 0
x- c/n " V2r — Vi' " iy'r

= I — 8// (6),

so that

NIO = 2**

-!- - j.t'O — 8y) /" + (4,r — 3i/ — 3?/-) .'
«y« I 005 d.; j

Thus, putting ), = 2, 3, 4, . . . ,

2P(r)-P(2r) =J(l+y) (8),

3P (i>) - P (3r) =1 (9),

4P(r)-P(4r) =1(1 +//) + ' (10),

5P(«)-P(5r) =l+v^c <'l),

6P («) - P (Gr) = 1 (! + ;'/)+ I + ? ^ (~_;< 'Jo ( ' ")'

7P(/,)_P(7/.) =2 + ?/ + ?/('/" (13),

/ 7

8P(») - P(8t.) =i(i + y) + 2 + ^ x)"(^ ''' ~ r) (14)'

9F(r)-P(9r) = 3 + '*' ('y " C5)>

7;i

fO + Nlu (lf))'

Tiu

+ .r(-2?/' + 4//'+r) -/'-?/,

(17),

- 2

222 PROFESSOR A. G. GREENHILI, ON THE

4. When r is an aliquot /Ath part of the imaginary period 2w:!, so that

or

2ra).,

(1),

and fjir is congruent to a period, then

P (/xr) = oo , P (w<>) + P (p. - m) v = 0

so that p.P (o) is given by equation (7), § 3, by putting n = p. — 1, with the
additional condition that

= oo , s (mv) — f> (ju, — w) v — 0

or

Tims

(3),
(4).

xt/

y -

X

y

7]1 = o, P(f»r) + P(Gr) = 0, so that adding (11) and (12), § 3,

and, similarly,

,, = o.

y — x y — x — if

y (v ~ x)z i • (v ~

' - -

y,7 = 0, 17P(v) = 5

lfl = 0, 19P (v) =3

7iu

7io 7n

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 223

The ellipsotomic problem of the determination of the division-values (Thettu'erihe)
of the elliptic functions resolves itself thus into a consideration of the curve in x and
y given by this rational equation (4), which may thus be called the alUpsotomic
equation, by analogy with the ci/clotomic equation for the circular functions; and the
problem may be considered solved when ;c and // can be expressed, rationally or
irrationally, in terms of a parameter; according to POINCAUK (• Bulletin de la Societe
Mathematique de France,' 1883, 11, p. 112) this can always be effected by uniform
functions of an independent variable.

The details have been carried out in the ' Proceedings of the London Mathematical
Society' (L.M.S.), 25, for values of /A up to 22 inclusive, omitting 19, which still
remains awaiting solution.

When fj. is an even number, 4// -4- 2 or 4n-, x (i/x/') — ••< (I'M.,) is a root of S = 0,
so that the cubic S can be resolved into factors, and we can employ the functions of the
Second Stage of LEGKNDHK and JACOBI, as required in most dynamical applications.

But when /u, is odd, this resolution cannot be effected algebraically, and WKIKU-
STRASS'S functions of the First Stage must bf employed.

The ellipsotomic relation (L.M.S., 27, p. 405)

are equivalent, so that

/.even, *?='*"* "I V = r*f ; , ... (7).

•y ^ (p. — 2) y ', ((*. -

Writing r for " 3 , and a> for w:i,
/*

^,;/ (r) = x*{"' '"y,, (8)
(HALPHEN, ' F.E.,' I, pp. 102, 198) ; and changing n into ^ — u, and dividing,

"--' -- r^-m.'.X^ K---M /<A

— LO-CX^A j (J).

But

a- (^ _ „.) v = a- ('2<a - nv) = 61""" '" <rnv ( I U),

so that

_i'' -'

c. M rrv = .r~'' X ''

and generally

• '

and this is KIEPERT'S function T( — ), defined in ' Math. Ann.,' 32, p. G.

V /

224 PROFESSOR A. G. GREENHILL ON THE

According to HALPHEN (' Math. Ann.,' 15, p. 359 ; ' Comptes Rendus,' March, 1879)

H(nr)_H^^

H(2(')J(1"-1)

5. The integral employed l>y ABEL, changed to the circular form, can he written

Z = z - (Az* + Kz + C)- (2),

and, as pointed out in the ' Archiv der Mathematik und Physik,' 3. Reihe, I., p. 72,
this is exactly in the form required in the problem of LEVY'S ' Elastica,' with z = r-/a~.
We reduce it to our standard form in (4), § 1, hy putting

. _•"•(") — *(3*>) _. , // /ox

' ,s (,,)-,x(r) ' s + x

Z = ,-(A^ + B2 + Cr- = -- (4),

so that

(A*-' + B*

_ - ---
x -f ./• (s + ,r)4

- x)+ x^-

4 (., + i j*

X

A-.- + B: + C = 1 - + (6),

2 (x + x) ^ -2 (x + xf

and therefore

A - -r R - — 2r + .'/ + //" ( ' - x — y + v~ t7\

i!r ' -2>r t!f

A + B + C = I (8),

and : — 1 is a factor of Z corresponding to * = ao .
More general ly, with

so that 1/M is the value of z corresponding to x = oo ; then in ABEL'S notation,
changing the sign of z and Z to obtain the circular form,

(10),

_ --8 JL

M x - * (wv) 4 [s - x (nt') }* "

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 225

on putting

8-s(nv) = t (12),

and

N = 4p«3 \t — a (3nv) + * (nv)}

- 4 MQ {« + *(»«>)} U + .s(m>)-f.Cp
+ MQ {(1 + >,) t + (1 + y) * («y) + a?//}5

/MQ_ w'(w)

V p s>nv)-s(w)

which also satisfies (16).

Then

Mz = ! _ *j3m>)_-

t

and

2 v/K - X/MQ S4H < + '

'

o

-

= P-v „„,.„

to \fjivir~ i v \vw * ^' y " " /

VOL. CCIII. — A. 2 G

'(3m1) — ,s(m>) -j- ft

+ MQ j (1 + ;//)- - 8x - 12,s- (nr) {• «3 - 2MQx" (/4r) - MQ./- 1
which is a perfect square in the form

= 1 2 i/pt* - \/MQ |rS + x/MQ/V (nr) 1 (14),

implying that

,0 , , , MQ /MQ *"(>,r)

•(M -•(««)+ ^; = V p £7M

MQ <n 4. \3_ «r l".s(n-r^ = M(^ K'('i£ll' 4. 4 /A1(^ ,-/(„,•)

requiring the relations

1 = l
2

_ __

= 3- v -s, (w) ~(3w-v) _ s (w) - | ., (3w) _ >s (nv)

T (21),

226

PKOFESSOR A. G. GEEENHILL ON THE

»- \/^-t^i

s" (nv) 1

is' (nv) s (3nv) — s (nv)

+ iyQ

is' (nv)

{s(Snv) — s(nv)}*

s (2nv) — s (nv) _ s (4m>) + s (2nv) — 2s (nv)
is' (nv) 2is' (nv)

/r> s (4nw) — s (2nv)
2is' (nv)

No

w

so that

and taking
so that

&
_x/Q

(3m') — s(nv)

M

?'«' (m;)

(2nv) — s (nv)}2

we obtain from (17), (21), (22),

p =: 4?V (2m')

_ ,s" (2nv)
~ if' (2nv)

.s (2nv)

and ABEL'S integral

•~— A-

d* _ H 'v (nw)

v/Z J 1 ,s (3nu) - « (nv)

. f
-I

2P

c?,L _ T , v

""

on taking

k = - Y- '°-- ' / x - 2P (nv)
s (3m>) — 6- (nv)

= P (2nv) — P (4m;) = £a — P (2nv)

Putting 2nv = w,

(22).

(23),
(24),

(25),

(26),
(27),

(28),

(29),
(30),

(31);

(32),

(33).
(34),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 227

so that ABEL'S recurring formula for </„, (' CEuvres,' 2, p. 157),

s. + s^,»le?, + -9B (35),

becomes

' "

(36),

p"v<>

¥'/'"-! 5" '/»'-!

and on comparison with the Weierstrassian formula,

P (m + 1 ) n> + P (m — 1) 10 — 2P?0 = , pW|P , + -w (37),

— 3

we infer that

p?«7 (38).

From another formula,

(39),
— — w)a

ABEL'S relation

c*_j _ </,„</„,_! _ {SJ ('//i + 1) zv — P/y} (tfmw — Pw)
p,,,-\ p i^'w

(40)j

2 tp'to Pm»; — P//'

and

, • P'HIZO — pV ..,, -Y -r

-A-? =: iL(m -\- 1 ) ir — limn- — ILW

2

= P (wt + 1 ) »r - Pm/r - P/r (41) ;

so that

(;„_! = a - C'"-J = 1 a - P (m -f 1) w + P (WJ.M-) + P («•) (42).

j''™ - 1

Writing /A for ABEL'S n -\- 2, his k is given hy (2, p. 161)

M* = G» — 1) « — (jfo + ft + • • • + ^-a)

= (/A — 1) « — (/A — 2) Ja + P (/u, — 1) w — (p. — 1 ) Vw

= lMa-/xP(w) (43),

as before in (33), since

(44).
With

x y y

k = - 2P (w) + * (46).

»
2 G 2

228 PROFESSOR A. G. GREENHILL ON THE

But with

n = 1? (fj. — 1), Wiv = Wv,

™ X t) = 4x a = lA-v b—--x

i \ i 10' r^ ~~^ ^i*', w/ ^^ j. | w, i/ ^^ tA/

k = P(v)-P (2v) = P (v) — P (/* — 1) t> = 2P (i>) (48).

G. In the simplest case of

/t = 3, « = foi3, 03 = 0 (1),

tlie form (G), § 1, is illusory; but we take (L.M.S., 25, p. 210)

o -

2.s!
and putting s = i2,

Writing w for co3,

In ABKI/S form (' (Euvres/ 2, p. 1G3),
J=f ^+i

and A\rith

2 = y\ p= -

J = - ^

Or, in the circular form,

= _ 2

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 229

In the next case (L.M.S., 25, p. 213),

H = 5, r = | rw, s (5v) = 00, y = x (1),

I (v) = f AlL+»)

a

2 ,v

cos-1

J. • _ "1 O ' " JLr

Sln

2s*

•> O \ f \ \ / '

to = - I + 6x - x", 19' § w = X~, P-i-ta=l~ 3X, (~*^ a- * a = x~> (a),

10

and x1 is the icosahedron irrationality, and KIEPKRT'S _/"' = ./•.
In ABKF/S form, with y = x, Z has the factor z — x,

2

and putting z — x — -, ABEL'S integral

^dz = 2l(2v), with k-

V/Z ;j

Putting ,s + .r = />',

! /, A - 2 ! (* + I ) v/ { 2^:! - ( L - *) ^2
5 2«5

- 2 si ri (* ~ 1) x/i2^3 +_(!- «) <3 ~ 2^ - "I
5 2«J

and the degree of the expressions is halved, with great gain of symmetry.
The degree is halved with greater ease by putting s = f1 in I (2'c), and now

I (2.0 = 2 cos-1 <* ~ x) v/{2t* + (1 + *) t2
5 2^s

= 2sin-i.(* t^A/l2*!- (! +^1
derivable from the preceding I (w) by writing - for t, and — for x.

230 PfcOFESSOR A. G. GftEENHlLL ON THE

8. This suggests that in the general case of /u, = 2n + 1 it is simpler to work with

and to put ,s = tz ; and then with

T! = 2*;i + (1 + .//) t- + 2:rf + xy (2),

T., = 2f:i - (1 + y) /,2 + 2«tf — ary (3),

we si i all have

"] "'" + ' ''"" ' " +

(4),

according as w is even or odd ; and the results are of one-quarter the degree that
would be given hy ABKL'S method of the periodic continued fraction ; and since

(t"-1 + V/'~3 + ... )2T1 + (t"-1 - hj"-* + . . .)2T2 = 4£2'<+1 (5),

the determination of h{, />.,, . . . can be carried out by a consideration of the reduites
(HALPHEN, ' F.E.,' 2, p. 576) in preference to continued fractions, once the coefficients
of / in 'I', and T., have been assigned.
9. Thus (L.M.S., 25, p. 222) for

H = 7, v = ™ , 7l — 0, or xy — x~ — f = 0 (1),

is a unicursal C3, in which

x = z(l-zy, y = z(l-z) (2),

and

TL T, = 2£3 ± (1 -f- 2 - «2) t2 + 2z(l - zf t ± z* (1 - 2)» (3)

P (2v) = 3 - +^2 (4),

(6).

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 231

Introducing a normalising homogeneity factor M, so that the substitution

1 z — I \
, - — , - ) should correspond to (v, 2v, 4?>),

1 ~~ Z %> i

M = **(!-*)' (7),

derived from (L.M.S., 27, p. 453),

M3 = MM, M = ?'s' < *™L- -- = .x» -VZL- 8

1 2!Pv _ — 1 — 6? + 923 — 223 — 2* r>V _ 1 P (r) _ 5 — 2 — z-

M3 25(L --s); M3 : M = Uz'(l --2);;

2v _ — I + 62 — 1523 + 1023 — 24 ip'Zv _ P (27') __ 3 — 9,z + 5z° , ,

M3 z> (l-z)r M3": M : 142»(1 :.«)«

12^4?: _ - 1 + 62 — 323 — 22s — 24 7>'4-c _ P (4r) _ - I + 3: + :j;3

"M3 2» (1-2)5 M3" - 1' M 142*(1-2)S

JL»V"T-» ' t~r , — - "/ i ( 1 __ K) T fiVll - rr ~ *' ' i 1 — — " 1 7

1Q 7 " ^ ' ' 4Q "/ '

4r i/ / /

— I 8wi) GOJ _., /n
6XP 49 °"Y: '(

In the notation of Krj-HN-FiuCKE, ' Modulfunctioneu,' 2, p. 39!),

Pt? + *»2»'-h V'4r _ (!, /rn

M3 M~

T = L~^i+_5|_+^==5+2+1^2 + 271 (14).

10. With (L.M.H., 25, p. 232)

/.-»; -=7 o.

= 0 is a unicursal (/-, in which

and from the relation

P(4r)+ P(5-«') = 0 (4),

(5),

L8P(2W) = 4 + 2(1 + y) + 4^ + - - J) (1 +

= 1 + 0 - 3jr + 7/>3

232 PROFESSOR A. G. GREENHILL ON THE

dt*

I(2t

2

2*» x 2'

l-l>+^)J /<,=P(i(l-X>0 -P+r) (8).

The substitution ( p. ^ } will correspond to (v, 2v, 4-c) with a nonnalis-

p 1 — pi

ing factor

M = (MjMoM^)5 =p(l-p) (y)f

and

P .(3-i>) _ . - 1 + 0 + 3p- -p* __ r, _ £ + 3
M 6p(l— p) "6 G

'Math. Aim.,' 32, p. GG).

> - - y __ - , ,

p-p}* W l~P

12P4«_ -1 + l-2^-80/r + :54y>:i-21/.1+&/^-y>(; <9'4r (1 -;>)(! -
Ma p3(l -/')': M:i ' p2

while

//-(I -|>)- M ' M3

unchanged by the substitution.

Also

p _ Vr + 92 1- + fr»3 r + ^4 r _ ( - 1 + p - p:f , , ,,

G'~ M^ - :^('-^)2 ( }'

a quantity required in the transformation of the ninth order; and

P(v)_5 + 0 + 3pt-p< , }

M ""

P(2t/) L
"M

P (4v) __ — 7 + 18p —
~

THIRD KI.UITtC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 238

(L.M.S, 25, p. 241 ; ' Math. Ann.,' 52, p. 484), leads by the substitutions

to the bicursal C.

.,

o

*0 — 2) + <•('• + 1)~ = 0 (2),

1 2z = Y/C, C = 4, (c + 1)' + | /q\ .

v / ^o; ,

^-~ic(l+C)(l + 2C + v/0) (4),

y = " ' ~ IV-M" " v/° ' L + y = 2 + c ~ ^t iV'-) ~ " v/( ' (5)-

We now find

_- c
-c

" M S 2i" ' v//fJ ' ^7^'

10t-2+ 4c:i + (2 + 3c) ./C

2(1+,) (8),

/ V J \ /*

2

(t + r + iy + 22) + 5)./Cl (II)

using detached coefficients.

We find also

where

P1= i4 + 19c+ 0 - 2c3, Q,= 4+ c,

P2 = 6 + ^7c + 44c- + 18c3, Q, = 8 + I3c,

P3=- 2 + 13C+ 0 Gc\ Q3= 12+ 3c,

P, = 12 + 43c + 44c- + 14c:i, Q4 = - G - 9c,

P.= 4+ 7c+ 0 -10c3, Q.= - 2+ 5c (13)

and the values of ar and br are given in L.M.S., 27, p. 455.

VOL. CCIII. — A. 2 H

234 PROFESSOR A. G. GREENHILL ON THE

The normalising factor is

M = (M1MSM8M4M5)» (14),

but so far this has not made evident any symmetry of results ; it will be noticed that
our parameter c here is an elliptic function of (L.M.S., 27, p. 429)

,, rX
(15),

which is 2"5-th of a period out of phase with that required to lead to KLEIN'S results.

12. ,1= 13, v = '2™, ylt, = 0 (L.M.S., 25, p. 251; 'Math. Ann.,' 52, p. 484)
1 o

by the substitutions

,2

x = y(l-z), z — y=~, z-c(p-}) (I),

leads to a C, with class p = 2, in which

2p = 1 - o» - c» + x/C (2),

0=1 + 4c + 6c2 + 2c:! + c4 + 2^ + c11

= (1 + 2c - c2 - o:i)- + 4c- ( I + c)- (3),

and we find, using detached coefficients of ascending powers of c,

, , C, + I 2 - 'J - 33 + 4 + 8 - 18 - 1 1 + (4 + () - 1 5 + 7 + 1 1) V/C , ,v

'-'

/T"T

1 + -2 - 3 - 9 + 1 + 2 - 5 - 3c7 + (1 + 0 - 4 + 2 + 3c4) /C

9 /i _i_ ,.y^ v /'

2(1 + cf '

Aa= — 1 — 4 — 1 + 16 + 15 — 26 — 28 + 28 + 14— 38 — 20 + 6 — 5 — 10 — 3c'*,
B3= — 1 -- 2 + 4 + 9 — 7 — 15 + 12 + 14 — 7 - 2 + 7 + c" (7),

Ao = 1 + G + 7 - 2G — 64 + 24 + 154 — 6 — 222 + 32 + 266

+ 10 - 10!) + 104 + 143 + 22 + 4 + 32+ 21 + 4o1!),
B3 = 1 + 4 - 2 - 25 - 10 + 61 + 27 - 97 - 28

4. 90 - 7 — 82 — 12 + 15 - 15 — 17 - 4c16 (8),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 235

^-1 + 5 + 3-21-21 + 56 + 4(5 - 114 - 39 + 158-3-135

+ 52 + 91 - 37 - 22 + 35 + 14-5 + 2 + 4 + c~\

^ = 1 + 3 — 4 — 15 + 14 + 35 — 41 — 37 + 65

+ 12 - 61 + 8 + 31 -- 16 - 14 + 5 + 1 - 3 - ^ (9),

,,n

A- = I + 7 + 1 2 — 22 — 74 + 38 + 223 - 7G - 448 + 205 + 014 - 403 - 551
+ 555 -1- 305 - 442 — 68 + 353 + 47 - 99 + 57 + 70+7 — 1 + 10

?n

B5 = 1 + 5 + 1 — 28 - 16 + 91 + 35 - 205 - 2 + 301 - 97 - 290

+ 169 + 120 - 176 - 79 + 90 + 0 - 54 — 7 + 7 — 5 - 5 - <:5S (to).

By means of an appropriate homogeneity factor M, we can express

52 P (re) p n ,p 12^/v , /, ,

M %''x/ ' M-

in such a manner that the substitutions

'c, — -, — | correspond to ('', 3r, !)/•)

and

(v/C, - v/0) to (r, 5r)

(L.M.S., 25, p. 255 ; 27, p. 416), and

M = (M,M.,M.,M,M,M(;)' or (M.M,)' (12).

13. /i=15: r = 2-**, (L.M.8., 25, p. 258; ' Msitli. Ann.,' 52, p. 485),

y,5 = 0 is reducible to a bicursal Cv

Changing c in L.M.S., 25, p. 259, into 1> — I, and normalising by a homogeneity
factor

- |

T,, T2 = 2<:i ± Q«2 + 2R« ± S (3)

2 H 2

230 PROFESSOR A. G. GREENHILL ON THE

I

(fc_l)(^-2¥-6i-/>3 + ^+0 + l)-(^+lH^-3^+363-6+l)v/B

~2b\ (&2 + />Tl)

E = (/>- + fc + 1 ) (/>2 - 36 + I ) (5),

(/,_ i) r(/,« -4+3+3-2-3+2+2 -- 1 ) v/(/r + ?,+ 1)

a;

M2

,,,_
~M:!

<(/,!•-' _ f, + J 0 + 1 - 10 - G + 1 2 + 0 - 9- 5 + 4 + 2 - 1) v/(/r + k + 1)

(7),

M
Diiferentiating (2) and equating coefficients of/ we find

H,= -i(i5P + Q)

_ (6- 1)(- :!//•+ 7 + 2 - 1 - 5 + 0 + ^) + (~ '^r' + 7 -1-3 + 0 + 2) X/B ,(), .

26;5 v/ (/>2 + I, + 1)

HO- -iI! + i(i5P-:;Q)H

_(b-l) (M, + N. X/B)
2//(/^ + /, + i)

M., — — (5/>1:! + 3:3 — 49 + 0 + 8 + f)8 — 22 — 3!) + 0 -f 24 + 4 — 8 — 1 + 1,

N, = - M" +27 — 28-13 + 9 + 28-7 — 18 + 1 + 7 + 0 — 1 (10):

M, = 4//!) - 3 I + 80 - 6 L - 21 - 54 + 1 a!) + 16- 113 - 85

+ 97 + 90 — 41 — 64 + 7 + 29 + 2 — 8 — 1 +1,

N, = + 4//7 — 27 + 57 — 23 — 33 — 34 + 84 + 30 — 65 — 46

+ 42 + 40+15-21+3 + 7 + 0-1 (11);

H| = (^--l):MM,. + N4v/B)

M+ - 5//23 - 49 + 175 - 247 + 48 + 8 + 408 - 380 - 435 + 93 + 684

+ 48 - 576 - 234 + 350 + 252 - 124 — 150 + 18 + 54 + 3 - 1 1 - 1 + 1,

Nt = 5?;21 — 44 + 136 - 145 - 39 + 28 + 303 — 145 — 325 — 19

-f- 362 + 69 — 267 — 120 + 131 + 93 — 38 — 41 + 5 + 10 + 0 — 1 (12) ;

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMTC PROBLEM. 237

__ -_, v + 1+JL+ NB v/(fr - 3fc + 1)1

2&"(6^+V,+ l)-

M5 -— -- &85+13 — 68 + 176 — 209 + 43 + 20 + 305 — 359 — 235 + 298+413

— 308 — 437+191+386 — 70— -260— 3+130+21— 44— 11+9+26— 1,

N- = — &-•"'+! 1—4(5 + 84 — 13 — 23 — 100 + 215 + 59 — !97 — 210 + 231

+274— 143— 297 + 34+230+40 — 127 — 54+45 + 32— 7 — 9?>3+0+l (13):

S , _ (/>-!)<•> {M, v/(/>2 + & + IL+ Nn v/(/r - 3& + 1)J

ft 26i!!(64 + 6 + l)s

Mfi = — //'" + 15 — 93 + 299 — 494 + 298 + 70 + 387 — I I 51 + 1 28 + I 2 1 (I
+ 370 - 1762 - 619 + 1720 + 982 - 1:349 - 1086 + 77:! + 89 I

— 288 — 540 + 36 + 232 + 25 — 65 — 13 + 11 + 'll — \ .

Nfi = - &2n+13 — 67 +165 — 166 — 10 — 56 + 513 — 285 — 588 + 26 + 1062

+ 152 — 11(55 — 5(50 + 1 014 — 827 — 594 -815+1 69 + 562 + 64

— 266 — 100 + 75 + 49-9- 11/r-f 0+1 (14).

These calculations, as well as for ^ = 11 and 13, and their verification, were carried
out for me by Mr. J. W. HICKS, of Greenwich Observatory.
Putting

M- " 2//' (//' + /> + I )
then, since

we find

«.3 = — /;u+ 14 — 43 + 28 + I 9 + 22 — 54 - 1 2 + 30 + 22 — 17 — 8 + 5 + 2—1 (I 7).

b3= — (b — 1) (hn - 1 2 + I !) + 1 1 - 9 — 1 9 + 5 + 1 5 — 1 -- 5 + 0 + I ) (1 8).

/ 1 \

The substitution (b, - ) changes v into 4w, 2r into 80, . . ., so that «* and />s are
obtained from a2 and />., by writing the coefficients in reverse order.
tta - _ /,i i + 2 + 5 - 8 - 17 + 22 + 30 - I 2 - 54 + 22 + 19 + 28 - 43 + 1 4 - I (1 9),
6S = _ (b - l) (6n + 0 - 5 - 1 + 15 + 5 - 19 -9 + 11 + 19-12 + 11 (20).

Again, since

12fW ,-v2 (T) /.7i\

— , = — (Jr — 4K \^l)>

we find

a1= — bu + 2 + 5-8-5-2 + 18 + 0-18-2 + 7 + 16-7 + 2-1 (22)
6, = - (b - 1) (bu + 0-5-1 + 3+5-7-9-1 + 7 + 0+1) (23),

and in a~, b^ the coefficients run in reverse order.

238 PROFESSOR A. f4. GREENHILL ON THE

So also

2-8-0-11)

,. .

M" 6Q~b\/(b* + b + I)

and I' (4r), P (Hr) are obtained from P ('•), P (lir) by writing the coefficients in
reverse order.
We find also

an = - 61* +2 + 5 — 8 — 5 — 2 + 6 + 12+6 — 2 — 5 — 8 + 5+2 — 1,

1,., = - (7,2 - 1 ) (/,< - b:' - Ir -6+1) (//• + 0 - .%' + W - 3b° + 0 + 1) (25) ;

ftfl = — &4 + 2 + 5 + 4 — 29 — 26 + 18 + 60 + 18 — 26 — 29 + 4 + 5 + 2 — 1,

hn = - (b- - 1 ) (IS - !>"• - b- -6 + 1) (//' + 0 - W - 1 1 b* - 3/r + 0+1) (26) ;

M 20//! v/(62 + />+!)

M :i06:ix/(6'- + 6+ L)

and these are unchanged by the substitution (/>, ) .
Also m

P (f,,,) _ - (&s + | ) (//. _ b" _ w _/;+,)_(/,_!) (/>i. + //, _ 6« + /, + 1) v/B
M 12^^(^ + 6 + 1)

and

M- M (30^-

With

(32),

^ j

the expressions for M/?, are lowered in degree ; for

M'

_ (// + o _ 2 - 2 + 0 + 1) /(//- - 36 + 1)] (33),

= (6 - l)p, v/(/>- + />+!) + />r/, y(// -86+1)
M'- 26»6-l* "

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 239

Pl = - 26° + 3 + 3 + 1-9-9-2, Vl = 3 (- fc5 + 0 - 2 + 2 + 0 + 3) (35),

^4= -2 - 9-9 + 1 + 3 + 3-2, ^=3(3 + 0 + 2-2 + 0-1) (36),

p., = - 2 +15-21 + 1 3 - 9 + 3 - 2, <h = 3 (3 - 4 + 2 - 2 + 4 -- I) (37),

pg = - 2 + 3 - 9 + 1 3 - 21 + 1 5 - 2, </s = 3 (- 1 + 4 - 2 + 2 - 4 + 3) (38),

Pi + Pi + Pi + ft = - • 4 (26- - b + 2) (// - U< + 3/r -6+1) (39).

?i + Sf4+fc4 &== l^(/>5 + I) (40);

^8 = -2 + 8 + 3-11+ 3 + 3 - 2, </, = 3 (/, + I) (- /,' + I + I + 1 - I ) (41),

pfi = -2+3 + 3 + 13 + 3 + 3- 2, 7i; = 3 (6 + 1) (- 6* + 1 + 1 + 1 -- 1) (42),

so that

P(k- — V'3r = M'° (43);

pa =-2 + 8 + 3 + 1 + 3 + 3 - 2, 7, = 3 (/, + 1) (- // 4- 1 - 3 + 1 - - 1) (44),

14. The case of /x = 15 can also be derived hy a trisectiou of \L = 5, and so
generally when /u. = 3/6 is any multiple of 3.

For when

S = 4.s(.s + ./•)' - !(1 +//)••< + r^/;- (1)

is expressed in the form

S = 4(* + 0:!-(A.s-+B)^ (2),

this implies that

s (| »)=-*, w'(|o,) = A«-B (3);

and now

A^ = 12« + (1 + //)- - 8,r = - 12V1 w (4),

so

that

2AB = l-2t- -- 4x-
B; = 4«:l + .r/y

reducing to

3«* + {(1 + yy - 8x] t* + 3x(2x-y- //-) f: + 3.

a Jacobian quartic for ^, which can be resolved.

For

n = 5, ;?/ = a.1,

(9),

and putting t = ex,

(c _ i)3 tf + 3c- (c - 1)- x + c:i = 0 (10).

240 PKOFESSOR A. G. GKEENHILL ON THE

To identify with the preceding results in § 13, put

, = "+£+ i.c_i = &_-;)•-, 3c + l = ('' + ')\3c_4 = ''!-f+' (u),

and then the associated octahedron irrationality o — \.

15. For /A = 17, the quartic for q in terms of c is given in L.M.S., 25, p. 204,
obtained by the substitution in y,7 = 0 of

, .

This irreducible quartic is made reducible by putting q = — <• (e +1), uud with
t; = /> — 1 becomes

e (,, + L) 6* + c (:>• + *• + 1 ) If' + (<''• - *;! - 3e* - 3c - l)l>-

b 2(e+l)

The alternating function

,(Kr) - .s(2r) _ (/> _])(& + c + 1) - x/r + x '(^ + 9'' + 4)
,,(4r)_.s.( ,,)- ,/, -J(e+ 1)C3

and the division values are associated with elliptic functions of an argument

1

<<<'

= ' , e (io>) = 1 , e (a)' + io>) = - - 1, o = -^^ (G),

)8 + e/=0 (7).
By the transformation

4e + 9 + 4 = ^, *-(«±^ = «•-!, 4^'-1)e = ^-17 (8),

and a comparison can he made with the equations of KIEPERT (' Math. Ann.,' 37, p. 386)
1G. The next case of /j. = 19 presents difficulties not yet surmounted, although it

was hoped that the analogy with /n = 11 would give the clue required.
In the case of ^ = 11, the substitution of

(1)

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 241

in (2), § 11, makes the ellipsotomic relation yu = 0 equivalent to

a-cz — 2ac - a — c — I = O (2),

an addition equation of the elliptic functions a, and c of the argument

C=l + 4c(l+c)* (3),

a and c differing in phase by one-fifth of a period ; and the five division values of the
arguments 2rr, v = -nr&)3, are derivable from each other when considered as elliptic
functions of •»/ + irw> r — 0, 1, 2, 3, 4.

The connexion with KLEIN'S parameter T is made through

K=-llr, K'' = 4K~(K -- 11) + (10K + 11)'-== 121r'- (4),

and the quintic transformation

TT 1 + 4c

* . c=,+

--;r=(^+ 1)* (^ +!)' c ^x

TT/' _ (2 + 8c + 12<~ + 9c3 - c* - 3cs - c'1)- C

c«(l+e)«
and then

H=K3 - 110HK - 121 (H + K) + 1331 = O (6),

an elliptic-function addition equation (L.M.S., 25, p. 244; 27, p. 428).

If analogy is to help us in passing from /x = 1 L to 19, the ellipsotomic equation
y,9 =r 0 should he reducible to the relation

H3K3 - 152HK -- 3G1 (H + K) = O (7),

where, in terms of FBTCK'S T and T' (' Math. Ann.,' 40),

K = - 19r, K'- = 4K3 + (8K + 19)- = 361^ (8),

with the addition of the cubic transformations

__ a3 _ 5a2 _|_ 2a + 1 w, _ (a3 - 2a - 2)2 {4a3 + (2a + I)2} /yx

-il — " } -Li V / >

a2 a°

and K, K' the same functions of 6.

This combination of (7) and (9) leads to an equation of the 12th degree between

a = a(u) and 6 = a I u -f : - ) , functions of the elliptic argument

\ y /

A°L. (10).

i/r»ii\')i » *

VOL. CCIII. — A. 2 I

242

PROFESSOR A. G. GREENHILL ON THE

The nine division values of the argument 2rv, v = W3 should now be functions of

I . '

an argument u + , and thence derivable from each other, being grouped in sets of

three

r, 8v, 7v

v, 23?>, 2°v
2v, 2*v, 27r
, 25?>, 28v

or

2v, 3-)', 5f
4v, Gv, 9r

(11).

In passing horizontally in these sets, the substitution connecting a = a (w) and

(12).

(13)
(14).

7 / , £iTtO \ •

b = a ftt + — j is

a?bz — 2ab — a — 6 = 0

The additional cubic transformation

_ c3 + 15c2 +_57c
(3c + 19)2

leads to a multiplication relation of the 9'h order, connecting

H = H (u) and c = H (4 u)

The relation of the 12th order between a and b is of the Gth order in p = a + 6 and
^ = a&, representing a Cfi in the coordinates p and q.

But so far the various transformations of y]9 = 0, as given in L.M.S., 25, lead to a
C7 of the fifth degree in each variable, and the reason of this is a mystery still.

17. For /j, = 21, applying the trisection equation (8) § 14, with the relations of § 9

x = z (1 — z)~, y = 2(1 — z) (1),

Put

+ 3:~ (1 - z)*(l - 82 + z~) ta~ + 3^(1 - 2)6/, + £«(1 - z)' = 0 (2).

_

t = Z~(- (3),

- I)3 = 0 (4),

(5),
(6),

2 — M;
+ 0 - 1) z
3-ios + 0 -

- ?»* - 2it;3 + 310s8 + 0 - 1 +

— ,' =

W = ™8 -8 + 22 — 24 + 11+4 — 6 + 0 + 1

= (^2 — w+ 1) [(ter« - Stt-2 + 0 + I)2 - (w2 - w) (w8 - 3w2+0+ 1) + (w2- w)2] (7).

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 243

The substitution (iv, } gives (z, -}, and

1 — w w I * \ 1 — z z )

\ \ - \ ) \"/>

12P61' = - (1 + z - z2)2 + 8z (1 - - z)2 (9),

s (3v) - —2(1 — z)'\ is' (3r) = z (1 - z)2 (10),

,9 (6v) = 0, is' (6r) = z3 (1 — z)3 (11),

x (9r) = s3 (1 —z}, M (9r) = s13 (1 — =) (12),

(13).

z — w v (z — t)s

18. For ^ = 23 the class of the modular equation is 2, so that simple relations
cannot he anticipated; but 'as the class is zero for fj. — 25, it is possible that the
ellipsotomic equation y.,- =; 0 may he susceptible of reduction (L.M.S., 25, p. 275).

19. As mechanical applications of the preceding integrals of the First Stage we
may cite the case of LEVY'S " Elastica," mentioned in § 5 and discussed in the ' Math.
Ann.,' 52, the Spherical Catenary (L.M.S., 27), and the Velarium surfaces considered
in F. KOTTER'S ' Inaugural- Dissertation ' (Halle, 1883).

Take an umbrella with straight ribs, and hold its axis vertical, as an illustration of
a velarium.

If the gore laid out flat forms a sector of a circle, then it is obvious that for any
other angle between the radii formed by the ribs, the edge and its concentric lines
form spherical catenary curves, as shown by KOTTKR'S equations.

But with triangular gores (Kih'TKK, ' Diss.,' p. 38) the projection of the edge on a
horizontal plane is given by

with t = ( ) ; and this is reducible immediately to our standard form (l), § 5, by
putting

= 16M6T = 4M-* (Wt - 2M-B)- + 16MW (M*t - M-)
= 4,- (s - 2M-B)3 + 16M1A- (« - M2) (3),

and equating coefficients,

2 I 2

244 PROFESSOR A. G. GREENHILL ON THE

(A3 + B3) = 2x* -xy(l+y) (5),

4AM3 = xy (6),

so that

256M*A2 = 64.x2 — 32xy (1 + »/) — [(I + yf — SxJ

(l-y)-(l+yf] (7),

Now denoting the elliptic argument by it,, where

ds

0 + 7«P(2r)=I(2r) (10),

and the preceding integrals can be utilised for the construction of solvable cases.

The chief interest is in the purely algebraical case, obtained by putting P (2r) = 0.
Thus we find for /A = 5, putting // = x = •§-, in (3), § 7,

'2r> cos *0 = (r + a) v/(2r's - 4ar- + Gcf-r + 3as) (II),

•JH sin I 6» = (»• - a) v/(2?-3 + 4«r2 + Ga-V + 3a3) (12).

20. The expression of the pseudo-elliptic spherical catenary, discussed in L.M.S.,
27, p. 127, can be halved in degree by changing to the stereographic projection, with

tan£0 = *, 2 = cos0=| (1);

i -\- t

Z = i - *-,•- A- - 2

:4**(1XJ-+"-A-I^"-(l+«T = T1Ti .(3),

[^ _ 2t (^~'-\ -f ^ + 1

* \ A A

\ -ii. Xi.

a-
In a pseudo-elliptic case, with

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 245

we put

4P (v) r dt*

h y + 1 ' v/TVfjj =

and now the expression for the catenary can be reduced to

N (tan £ 0c*')" + * = (B + E,t + B/- + . . . + B^2''-') ,/T,

+ { (B - B!< + B2«2 - ... - B2(,_/-i) v/T8 (8).

Thus for

/*=?, o: = z(l-z)3, y = z(l-z) (9),

P (r) = 5 — *4— = 0 for z = - -I + 1 v/21 (10) ;

and we can calculate M, A, h, and the six B's in

N (tan|0e*')= = (B + BL« + . . . + B/) V/T,

+ t-(B-Bi< + .. • --B/)X/T.: (11);

this catenary has been drawn stereoscopically by the late Mr. T. I. DEWAR.

21. ABEL'S integrals are applicable immediately to the construction in polar
co-ordinates r and 9 of algebraical orbits or catenaries under a central attraction of
the form (Huco GYLDEN, ' Kongl. Svenska Vetenskaps-Akademiens Handlingar,"
1879)

(1);

with p. = 0 the orbit can be realised by two balls, connected by an elastic thread,
whirling round each other in the air ; and the addition of a term to P, varying

inversely as r3, merely has the effect of qualifying the angle 9 by a factor.

i
Putting r = -, and denoting the velocity in the orbit by v and twice the rate of

\Ju

description of area by h,

(2),
4 + 2 Aw.3 + Bit2 - 2Cw - D = U (3),

and now put ;t = z — ^ to identify with ABEL'S results.

246 PROFESSOR A. G. GREENHILL ON THE

Thus- for /u, = 3 in § 6 (5), putting u — z + \a, and a = 36,

= cos-

A=-6, B = 362, 2C=-4&3-_p, D = 6(4&3+p) (7).

For instance, fr = 0 makes fj. = 0, H = 0, /"j = 0, and the orbit is

r» cos | 0 = c* (8),

described under a constant central force.
For fj. = 5 in (7), § 7, putting

w = 5z — 1 — 2x (9),

0 = cos- -

~5~ v/'!~ '"3 + (/ "" 'r) ?y2 + 8 (~ 1 + H* + x2) w

+ 4(4 + 3x)(-l + lla; + a52)} (10),

and to make /JL = 0, put x = — 3, so that with w = 5w,

2 _,»-- — M

, cos

o b

2 _,»-- — M + 4 / / ,x
0 — , cos - v/ (M + 2)

= - sin l - Y/ ( — ?/,3 -|- 2«'~ — 8u -{- 4) (H)>

an orbit described by a body attached to an elastic thread, which is led through a
fixed origin, which can be written

Gr* cos J 6 — (4r~ — or + a-) N/ (2?" + a) (12),

GH sin f 0 = (/• + a) ^/ (4r3 — 8«re + 2a~v — ft3) (13).

So also for /M, = 7.

THE SECOND STACJE.

22. But in the dynamical applications, such as POINSOT'S herpolhode and JACOBI'S
associated motion of the top, the integral of the Second Stage is required, corre-
sponding to an even value of p., and S can now be resolved into its factors

a — A /« . \ /.. * \ (v 9 \ /ll

O — 4 ^,S — Sl) ^6 — f>2) (f> — S<i) (L),

while in most cases of the applications

s1><r>si>s> ss (2),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 247

so that

v = CD, + /Ju,,, u = f'ca, -\- o)» (3)

1 J .J' ./ i i i V"/>

where f, f denotes real fractions.

To* obtain this resolution of S we find it convenient to put (L.M.S., 27, p. 449),

x= m3ot « - - (1 ~_2m) "

(a - m)2 ' a - m

and with s = £2,

TL T, = (* ± — m— \ \ 2t~ T - "" h ~~, (5).

\ a — m/ I a — m (a — m)*

Denoting the roots of S = 0, irrespective of order of magnitude, by a~, lr, c~, we
can put

(6),

__
2(a-m)'

'Ma. (a. —
L_

a+b =

a-m a-m

' — '

(8).

With % fj.v congruent to a half-period a>, we can take

o 2

and this leads to a relation between a and m, by means of which they are expressible
theoretically in terms of a single variable.
Also, as shown in L.M.S., 27, p. 450,

S(W)-S(F)= m*a (10),

v ' a — m

s (uy- s (3 r) = (i^71)-* (12),

m2 Ul - 2m) a — mil — m)}2

v, , -^~~
(1 — 2w)2 (a — m)2

m-ot. \(l -2m) a — m2 (I — m)\~
= __ - 2m -- - _-

a — m \D

248 PROFESSOR A. G. GREENHILL ON THE

where

D5 = (1 — 2m) « — m (1 — m)2,

and N5 is obtained from D5 by a change of m into 1 — m, and a into — a.

•s (w) ~ s (6v) = 7-

D6 = 2 (1 - m) (1 - 2m) a — m (1 — m)2,

NG = (1 — 2m + 2m2) (1 — 2m) a — m (1 — m) (1 — 3m + 3m2) ;

/ \ /« \ ™2« /N7\2 /, „,

,(«)-, (7*)=-- -=J (16),

a — m\D7

D7 = (1 — 2m)3 a3 - m(l - m)(l - 2m) (1 — 3m) a — m4(l — m)2,
and N7 is obtained by a change of m into 1 — m and a into — a ;

\2

D8 -= {(1 _ 2m) a — m(l -- m)} {(1 -- 2m) a - w(l - m)(l — 2m + 2m2)},

N8 = (1 - 2m)3 a3 - m(l — m)(l - 2m)3 (1 + 2m — 2m2) a2

+ 4m;!(l - ///)3(1 - 2m) a - m4(l - m)4;

D9 = (1 -- 2m)s(3 - Gm + 4m-) a3 — 2m (I — m)(l -- 2m)2 (3 — 7m + 5m2) a2

+ m8(l -- m)-(l - 2m) (4 - 10m + 7m2) a — m3(l — m)6,

and a change of a into — a and m into 1 — m gives N9 ;

Dio = {(1 - 2m) a - m(l - m)2} {(1 - 2m) a - m2(l - m)}

{(1 - 2m)* a2 - m (1 - m) (1 - 2m) (1 - Gm + 6m2) a - m3 (1 — m3)},

N10 = (1 - 2m)c a4 - 2m (1 - m) (1 - 2m)3 (2 - 7m + 7m8) a3

+ 2m2 (1 - m)3 (1 - 2m)2 (3 - llm + 13m2 — 4m3 + 2m4) a2
— m3 (1 — m)3 (1 — 2m) (4 — 17m + 27m2 — 20m3 + 10m4) a
+ m4 (1 — m)4 (1 — 5m + 10m2 - 10m3 + 5m4).
A simplification is effected by putting

(1 — 2m) « = m (1 — m) ft, and 1 — 2m = p (20),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 249

and now the change of m into 1 — m and a into — a is equivalent to a change of
sign of />, leaving ft unchanged.
Thus we find, with 2ft — I = e,

Nn=

NM - Dn = 2>{/>* - e(e* + !);>« + ^ - e + I)},

Nn + DM = 2e|(e- 2)(e2- e- I)jp++e(c2_ a)p2 + e**it
= 2e](e — 2)y72 -f- ej • -}(e3 — e — l)pa + ef.

Thus, if fji •=. 22, .s(l I*1) = s(a>), and Nn = 0, which replaces in a much simpler
form the relation y32 = 0.

So also DH = 0 makes ,s-(llr) = 00,7- = ""— , and so enables us to connect up the

results for /JL = 11 ; and other values of [L can he treated in the same way.

23. There are three cases to consider, relative to the half-period 01, to which ^v is
congruent : —

T / 1 \ / \ i 3 72 /i\

and introducing WEBER'S function fat, or KTEPERT'S equivalent function L(2),

K4 1 6 (,s-, s. )'2 1 8(1 2m) a

(^/ == (yiw)~ = ^ /e ~= 7~ ~w \ == ^7 w i 1 \ \^/'

and to the complementary modulus «:',

_1_ 1 1 + 4a ± y/{ 1—8(1— -2m) a ,gv

cn2/K" dir/'K' = 2

cn2/K' = ]) , dn2/K' = -, sn (1 - 2/) K' = 6 (4) ;

c c ot

II. O) = at,,

16 16i£,-*,)8 _i + 8(l-2m> =_.L(2)M

W ~ (s, - SZ) (*, - *,) " a2 (« - m) (a - »n + 1)

, , (8);

VOL. CCIII. — A, 2 K

250 PROFESSOR A. G. GREEN HILL ON THE

III. w = ws, s (far) = s (&)3) = s.A = c2, s, = a3, *2 = .&2 (9),

--— , ,-
j — S3) (s, — «3) a2 (a —

a —

1 »f3 _ 1 _-- 4« db_v/ll — 8 (l — 2m) «} , *

/c'2sn2/K" " /c'2cn2/K' " 2

*

cn2/K' = C , dn2/K' = 6 , sn (1 - 2/) K' = ^ (12).

« a o

A. fourth case, where 1 — 8(1 — 2m) a. is negative and «, /> imaginary, need not be
discussed here.

The Weierstrassian Sigma Functions of the Division-values (Theilwerthe) can now
he expressed in terms of the Jacobian Theta and Eta Functions by the relations on
p. 52 of SCHWARZ, ' Formeln der elliptischen Functionen,' so that

^ = 4n+2, t,= K + ^£T (13),

e K/

-- ^{(*, - ^ ("* -- *,)} ar*X-Wi (14),

H 2K/

- = ^{(*, - ,3) («a - «3)} «r*x-5ss (15);

2K'

2n+ 1 7/ x -ix-r4r, 2K'

00 ^v/^-^^X— us2n+i

(16).
a — m

(17),
(18),

(19),

(20).

®0 a — m

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 251

24. When p. is of the form 4n + 2, so that, as required in dynamical problems,

then ^ p.v = (2n -f- 1) v is congruent to co, or <a.,, according as r is even or odd ; and
the integral I (r) is expressible in the form

I (cr — s) + § x j&

<T — .S ^/S

e» -1- P «»-' -I- -1- P

1 " //(- — .v)

and when the quantities a, wi and therefore a, b, c, cr, x, and P (r) have been
determined as algebraical functions of a parameter, the calculation of P,, . . . PM,
Q,, . . . , Q,,, is effected readily by a differentiation and verification.
It will be found that

Q, = (2«+ 1)P(«) (3),

also

P (w, + /O).,) / f.^7, /\ P ( /Wo) ,-Tr/

// \ = zn (./ K ' * ' ~// \ = zfi/ K (4)>

v/(A'i - *s) \/(si ~ *s)

znw denoting JACOBI'S Zeta function Zu or log <s)?t, while

du

ZMI = znit + log sim = log llx. (5)

in GLAISHEH'S notation ; also

cr — x (w) "a — -in

Here already the degree of the results obtained by ABEL'S method have been
halved in (12) ; and the degree can be halved again by the substitution

o cr — x /-\

and now

1 .1 s" + P,*"-1 + ...

- cos ' — ' - .-•

2n + 1 (cr — s)» y

- cos

+ 1 r

2_- sin-i rlr^/"^±^±^ (8).

2n "

2 K 2

252 PROFESSOR A. U. GREENHILL ON THE

Denoting by a'2, b'2 the values of yz corresponding to a2, 63 of s,

reducing to

/•7 7 /•! •"

+ 4a ± ^/{i _ 8 (1 - 2m) a}
so that

a' = en/K', V = dn/K', in Region I. (11);

a' = sn/K', 6' = dn ^ R/ , in Region II. (12).

Changing the variable to y in the integral

i^)==r P («') r - Q ("')
)>,«*• y8 '

where

Y = 4(r- !)(?/- a'2) (r-//e) (14),

P (,') = P (^~ V"> , W> = m*a (» ~ ^ + 0 (15),

Q(,/)=i^^ = a'6' = G2 (16);

/"( 0 ^X'1?,/-' / 1 >7 V

:"

20., = «'•' + 6'- = a

4a(a — m + 1)

1 - a'- ,, V- — I

-

A/1 — 8 (] — 2y/i)a

25. To connect up ABEL'S results for even values of ^ we take n = 1 in (9), § 5, so
that

2a; — y — y~ 1 — x (u — x — ii~\
a- , & = - _A2_ 2_ - ^ ' , p = 4a;^/ (1),

Z = - (22 - «2 + Vf + _p2

= - (z- - az + b + 2s)3 + 4 (Az + B)2 (2),

where

A- = s, 2AB = — a,s> + ajy, B2 = s (s + 6) (3),

aud therefore the resolvin cubic becomes

(4),

or

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 253

S = 4.9 (8 + *f _ [(1 + y) S + XyJ = 0 (5).

ihen, with

m?a (1 — 2m) a m2a?

l — v ' ?/ == ~ , s = , (6)

(a - m)- a - m (a - m)~

a — m I ( L - 2m) (a — m)

and Z breaks up into two quadratic factors, Z, and Z.,,

Z = 22 _ m (X ~ 2m + 2m2) » _i_ _m? (l — mY_

' (I — 2m) (a — m) '* h ( L — ~2w)4 "(a ' — ~m)-

L (1 - 2m) (a - m)J L (1 - 2m) (a

- m)J

\°)>

_ ^ , »i [4 (1 — 2m) a — 1 + 2m — 2m2} <
(L -2m)(a-m)

W2 [2(1 — 2w) a — y« (l — in
(\ — 2m)~ (a — m)-

)]': (»);

1 " I " (I — 2m) (a — m)J

(l — 2/n)
z
a. — ?yi

(10),

2 _ [- _i_ 2?" (1 — 2m) « — wl~ I1 -

L ( 1 — 2m) (a — m)

- ni)~l- m(l — 2iu) _
a — in

(ii);

and by means of a homogeneity factor

7/1

replacing z by ^ . , we may replace Z; and Z., by

Zt = [z - -HI (I - m)j-' - (1 - 2m)- 2

26. The resolution of T in the spherical catenary for p, = 4w + 2 is not the same
as for p. = ~2 a + 1 ; we must put

rp rp —r- / ^ | »>\y i y2\2 |^ o/ / 1 /2\ / 1 \

and then in § 20
so that

" '" ' "' ""' :^^n = » (3>,

254 PROFESSOR A. G. GREENHILL ON THE

a quartic for X, having a root X = 1, which was used for p. = 2n + 1 ; the remaining
cubic, putting X = 2B — 1, becomes

B3 _ W + V---1 - l B - *-± 1 + 1=6 (4)

. A.

or as in 20 with

2c
so that B = reduces this to

?/ + *

2r3 - (y + 1) c2 + 2a;c — a = 0 (7),

as in (3), § 8 ; and with

m3a (1 — 2m) a (2a — l)m /R.

x = — , y = — v 7 , ?/ + 1 = v (8).

a — m a. — m a — m

c= -7//a- B= 2a X==2a + 1 (9).

' - ' -

w,

2a - 1 ' 2a -

We find that CLKBSCH'S k (CEELLK, 57, p. 105; L.M.S., 27, pp. 146, 185) is

given by

P = 4a - 1 (10),

so that

and
so that

and in quadratic factors

(15).

Thus, as shown in the ' Bulletin de la Socie'te' Mathematique de France,' 29, the
algebraical spherical catenary for p. = 10, drawn stereoscopically by the late
Mr. T. I. DEWAR, can be represented, by taking k = y/f, h= — ^x/V, ni
symmetrical form

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 255

N (tan £&*•> = (B + B,« -|- B/ + B/) v/T,

+t (B - B,« + B2«* - B,«») yr, (16),

B, B;! = ± /SI + 7, B,, B2 = ± 3v/51 + 19, N = 192 ^3 (17).

27. These expressions for I (w) are applicable immediately to the construction of a
series of quasi-algebraical Poinsot herpolhodes, in continuation of the one invented
by HALPHEN (' F.E.,' 2, p. 279).

Making a digression on the motion of a rigid body about a fixed point under no
forces, as illustrated by the motion of a body about its centre of gravity when tossed
in the air, POINSOT'S polhode and herpolhode are obtained by rolling the momental
ellipsoid on the invariable plane.

The equation of the momental ellipsoid can be written

Ax2 + B*/2 + Cz> = D8* (1),

where 8 denotes the, distance of the invariable plane from the centre of the momental
ellipsoid ; and then the polhode will be the intersection with the coaxial quadric

AV + B-//- + G-z- = D'-S3 (2),

and the direction cosines of the invariable line will be

A£ B?/ G£ ,„.

DS' T)8' DS

Denoting by p, nr the polar co-ordinates for the herpolhode in the invariable plane,

*2 + r + z3 = p2 + s2 (4),

and by solution of these equations (l), (2), (4),

= p- — p,~ suppose, &c. (5),

and then

where

R = 4 (p* - p~) (p? - p3) (p" - p2) (7).

Denoting the component angular velocities about the principal axes by p, q, r, and
about the invariable line by h,

£__? -v -h-ri (8),

— —— "" ^ " 7 \ /'

x y z o fc

256 PROFESSOR A. O. OREENHTLL ON THE

and DARBOUX'S equations (DESPEYROUS, ' Cours de mecanique,' II., note 17).

O n o £> O O

r+?+r^ ?>+?+- = ! (9)>

a b c a? W c-

are identified with our notation by putting

A« = B& = Cc = DA (10),

and

A.p* + B?2 + Or2 = D/?,2 (11),

+ BY + cv3 = D2/>2 (12),

are two integrals of EULKR'S equations

A^ = (B-C)^... (13),

or, as they may be written,

Then

2 = / 4-

_ ^B - 0 0 - A

A T> I

- C'> (° - A) (A - B) ^ __ n

ABC yfc jfc^

supposing

P/ > p~> p,~ > 0 > p,2 (17),

and inverting the integral,

ps = /o22sna (K - m«) + PsW (K - »i«), -= \/ P* 7*^ (18).

11 \r A*

Again, projecting on the invariable plane the areas swept out on the co-ordinate
planes

o dnr Ax I dz o

and

dz _ z dy _ n /'A_— JB ^fl _,_ A^- C
dt dt

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 257

so that

„ rfw _ n8 /A — D A o B - D ,. , C - D , ,

p dt ' ' k \ BC CA AB

n 8 A3*2 + By + C"za - D:i S2 , .

~k ' ABC

and this reduces to

-<fo_n8 j , (A-D)(B-D)(C-I))S4
p dt k \p ABC J

so that

S?, + (A-D)(B-D)(C-D),;

ABC

Putting

2 •' ° "O Q

p cr — x p,~ — p x — ,s',t 1\ D

A- " M- I2 "M2" jyi ~= M"

and p = 0, with ,s = cr,

-_
k \0udr

4- i (Q^"-1 + • • • + Q«) v

VOL. CCIII. — A. 2 L

RO= w*v 4V VAB(! j - --M»

so that

(A - B) (B - D) (C - D) ,, _ iP /•

ABC; M:i

and from (G) § 1

TTT = Tit — \

k Js cr — x V/O

- (! - M ) "( - ' (r)

or

pt — v=*I (r) (2S),

where

1> S Pr (•>'))•

also

nt = f ]" ^ (30).

Then

f'"l('>=«^l1 <31)'

0 (w + v)

and

M2 ^-, •=• <r — f< = F»' — 9u

_ o- (•« + r) o- (n. - r) _ ff-00 (n + '') 0 (« - '')

where A. is a homogeneity factor, X = ni/n ; so that as a quasi-Lame function

258 PROFESSOR A. G. GREENHILL ON THE

in the pseudo-elliptic case ; and cancelling the secular term pt by making p = 0
gives a purely algebraical herpolhode.

With the degree halved by the use of the variable y, as in (8) § 24,

M P_

*

? = /- (35).

Ir y - 1

In the associated motion of a top, the projection on a horizontal plane of a point
on the axis describes the hodograph of a herpolhode traced out by the axis of
resultant angular momentum of the top ; and thus EULER'S angles 6 and i/i for the
top are connected by a relation of the form

1 M2 sin ft> = - i d (pe™') (36)

(' Science,,' Dec. 20, 1901 ; 'Annals of Mathematics/ vol. 5, Series II., 1903).

We may dispense with POINSOT'S rolling surface and consider the polhode as a
material line, or as the edge of a material cone, as in L)r. FR. SCHILLING'S model,
constructed by MARTIN SCHILLING, Halle a. S., and this is rolled on a fixed plane.

According to M. DK LA GouRNElttE, the polhode is also a line of curvature, the
intersection of two confocal surfaces, an ellipsoid and hyperboloid of two sheets ; and
9 will now be the angle between the generating lines of the hyperboloid of one sheet,
one generator being perpendicular to the fixed plane, so that the other generator
moves parallel to the axis of the top in the associated motion (DARBOUX in
DESPEYROUS, ' Mecanique,' note 18).

28. To utilise for ^ = -in -\- 2 the preceding results for odd order 2n + 1 of /^,
where

ma. ,,x

t = 11 (1),

a — -n J v ''

put

« (a>) - s (4n + 2) v = co , Da,+1 - 0 (3),

and thence express «, in, . . . in terms of a new parameter, and thereby express the
integral I (4r) of p = 4;i + 2.

Returning then to the v of p. = 2 a + 1, which, when normalised to LEGENDRE'S
form, can be written

and replacing y^ = 0 by its equivalent D2),+1 = 0, the series of functions

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEItf. 259

—
— (6),

m — 1

— 1

— l\/T)o,,
N "

v-L™ 2«

a__~-

<* »/»/ \-l->2;|

is such that

(10),

so that JACOBI'S division-values of the Second Stage can he calculated.

The Weierstrassian Sigma Functions of the First Stage in § 4 can now he converted
into Jacobian Theta Functions of the Second Stage by the relations on p. 52 of
' Formeln und Lehrsatze ' ; and forming the series of functions

(12).

«• — Til

(13),

a — m

-ii») A/- 7, (14),

7 V a — 7/t

_.,-;/', ma N,,

C.,r = 05 *X 2ll+1- -r>""

a — m Dr

c _ jp-Jx-^T A/ a N2'-+i y (16),

2f+1 " " V a — m1 Dir+1 72

we find that

®rfK' Hr/K' /17x

c— 80 °r HE' (17)'

or one of its linear transformations ; and <\J - is always rational in a and m, m

consequence of the relation

s (w) _ ., (WV) =; S (fl)) - S ('/I + 1) V (18).

2 L 2

260 PEOFESSOR A. G. GREENHILL ON THE

The function c is the one that should be tabulated numerically, as

HM /ew H (K - u) l®u , _ B (K - w) /®M , .

= HK/0K' HK /ed' OK 700

Also

or one of its transformations, depending on the region.
29. ,1=6, /=£ or f.
Derive from p. — 3 in (3) § 6, by putting c = a3 (2a — 1), so that

_ 2 , y(« + a) ,/^ - (2a - _l) .« _+ 2_a* - «}

3 2«*

= 2 gin_] y (< - a) v/] 2e~ + (2a -^1+ 2^- a }

3 ' 2i3

nt* + Za'_-ct dt

~]a t ' x/{4/;'i-(r- + 2«3-a2)2}

Next put

6 + 1 , /r - 1 1> - I

/1 = ^ + 3' '» == * 6T4TJJ ' ; -/>- + 3

(6- l)3(^ + 3) „«„ (6 + !)»(- > + 8) 4

1G?>

T /•")\C?'i

Then from (8), § 23, in the region 3 > b > 1,

so that there results the well-known relation
and in LKUKXDKE, 1, chap, vi., p. 27,

/ 1 \7~ an J. "~~ ZitXj /n\

x = sin rf> = sn 4- K = --- - , K~ = (8).

6+1 — 2.7T + x*

The following table shows the six linear transformations of the division-values for
fji = G, obtained in accordance with the preceding theory by the substitutions

6 + 3

the accents are omitted from K by changing to the complementary modulus.

The numerical value b = ^/3 gives a modular angle 75° ; while b = ^/5 corresponds
to the functions required for /A = 10.

A

A

r0

A

CO

A

8

O

'So

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM 2(J1

M

-

S,

M

§

M

dm

M M M Ml. M

«« m H«

_|^»_j

8

co

.A

M M ^

rt ^ ^ Cl

V "r* ^ IT T {

1 M W M, M'W «^
a g" -H© fflP ©1®

i 1

A

i c i '°°

^ .*l^ -i*s< •' | |

<M

co
1

1 1

co'

1

.A

M M w *

«|M PH ,_,!„

1 M M w Mw «

\ \

A

i— (

^ v T3 «!« cc r"'"J r-c|T> *^H » *•'

« § f .a .8 .! *<* K| £,

1 1

1

I— i

CO

1

1|a,wMMMWMM'M M. + +

• A

o; '« "^ <N|

IM|« FH|« i-1!" S HH ^ a|M /p ^ ,^

rO

| o a S »

3 53 53 W 1 _1 0 <£

> \,

A

| 1

^ • v • v 1 1

^ \

o

1

«

o

0

+

70

+

co]

i

J.

, — ^
col

+

rO
'

4- Cd

CO

OH

' — 1

+

o

262 PROFESSOR A. G. OREENHILL ON THE

30. =10,= '•'--

To identify with the results in L.M.S., 27, put

l _ ! TO_C+JL T m _ c — J
then

and from (9), § 7,

2t:>

= 5 sm-

,C8

Put

/ nrl 9/2 — <S ^ ' ~

' -

(2).

dt*

v/I\T2

with

s(v) = 8c(c+iy(c-l), «(») = (o + l)»(o-l)* (7),

and we obtain !(•»•) in the form employed in L.M.S., 27, p. 601, derived by putting

N5 = 0, with

P (V) = i (C + 3) (- c* + 4c + 1), Q 00 = 4f (c + I)3 (c - 1) (- c' + 4c + 1) (8) ;
and in Region II. , /= i or f,

«i = 4 (c2 + v/C)2, Sa = (c+l)»(c-l)4, ^ = 4(^-^0)^ (9),

where

C = c3 + c3 - c (10),

and

L/2v>4_ , ,

-2_52_

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 263

This c is connected with FRIOKE'S T and t (' Diss.,' Leipzig, 1886) by the relation

_ - Cs + 4c
"~

The following table, analogous to the preceding one for /* = 6, gives the division-
values for /j. •= 10, the six transformations being derived from the substitution

and it may be compared with the corresponding results given by F. MULLER.
' Archiv der Mathematik und Physik,' p. 161, 1884, ROHUE, ' Archiv,' 1886, p. 138,
and by W. GORING, 'Math. Ann.,' 7, p. 311, 1874, working on GAUSS'S unpublished
memoirs.

For the application of this integral to a case of motion of the top, consult the
' Annals of Mathematics,' vol. 5, 1903.

The numerical value c = 2 will serve for verification ; this corresponds to the case
of ten cusps at intervals of JTT in the associated motion of the top (L.M.S., 27,
p. 602) ; and c — 2 (sin 36° ± sin 18°) will make K'/K = x/5 or 1, cases of Complex
Multiplication.

264

PROFESSOR A. G. GREENHILL ON THE

Region.

A.
> c > v/5 + 2.

B.
v/5 + 2 > c > 1.

(2c - 1 + ye) (2C + i - yc) (ye +

2c

c3 +

2c
v/C + 1

x/C - 1
v/C + 1

V/C -Jl

c2 +

c3-
2c

(c + 3)(-c3 + 4c4-
20c ^ C

(3c- l)(-c2+ 4c+

3c3 + 7c2 + c
20c

(c + 1)» (c -

cnfK

cnfK

sn|K

nd |K

@0

snfK

cn|K

dnfK
cnfK

dn|K

znfK

zs|K

zsfK

©K

_
@K

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 265

0.

I).

E.

F.

2 *

o>c>-y5+2.

-/> + ,>,>-..

v/5 + 1
^

o

1

^,,

K*

'•2
K '

«-

1

2

2

I

K*

j

«••

RSiK

dn | K

nd ? K

— nc ;, K

nsfK

-dnfK

- nd i K

-ncfK

- dn ? K

- ns .V K

- nc 4 K

- nd i K

- en ? K

ne i K

ns i K

sn 1 K

dn | K

.) T7-

— ns 1 K

- no f K

nd ^ K

cnjK

2T?

— nc : K

- ns f K

— sn :\ K

-

K

?' o Tr

, XII r K

iznfK

* xc ^ K

- T xs-fK

»C

? Xll ^ K
/c

•/XM 1 K

? xc •* K

1 zs * K

K

? O TT

xs : K

K

7' zs ?, K

/, o TT"

xs r K

K

-zsfK

K

K

- / xs ^ K

* xs i K

K

HK

80

« -» K
BK

H ? K
HK

H|K
HK

8|K

BO

BK

HK

VOL. CCITT. — A.

2 M

PROFESSOR A. G. GREENHILL ON THK
31. ^=7, 14;= .'.*•»•/•*•«;

D7 = (1 - 2m):! a'- - m ( 1 -- 2-H?) ( I -- 3w) « - «t' ( 1 - w)3 = 0 (l) ;

and on putting

I - 2<m = ^, (l -- 2wi) « = »»• (1 -- "<) & 2/8 = 1 + (2),

the relation becomes

]>~<f 4- pfj - P - <1 = ° (3),

an elliptic-function addition equation, connecting

t> = p (u) and 7 = p (n -f fw) (4),

-3;;+ 1) (5),

and

p(&) = 0, p(&)=l (fi).

Thence

a-m

The connexion with KiEi'Kin'ss results in 'Math. Ann.,' 32, p. 87, has been given in
L.M.S., 27, p. 4:58 ; here

f = 86=:<» + 1H'J"1>s (10),

"

a transformation of the Third Order; and

suppose, when- w is the real half-period of the. elliptic function P=JO(M) of the
elliptic argument //, and now

f, = f(«-iw), J =f(«-i«) (13),

bt

f2=l, p»-2p»-|) + l = 0 (14),

THIRD ELLIPTIC INTKUKAL AND THK KLMl'SOTOMIC PROBLEM. 267

the roots of which are

-2ra>

O, J

7
+ 1=0

(15),
(16),

the roots of which are derived from the preceding, by the substitute

/ •>

P+ I.

Thence, the contour of the period parallelogram is divided into segments of },w, and
a similar table of division-values for /JL = 14 can In- constructed for the corresponding
regions of p and compared with the results of (}ORIN<; and MULLKR; tins must be
reserved for the present.

The, associated r j)arameters of GfKRMTKK (' Math. Ann.,' I 4) have simple numerical
values :

T — T — T — - T

ris — J ri: -- '•> T:> — :i> Tc, - -i- i - > ':; — -27' -: — i -2*

The parametei' ;, employed in ij !) ('!). is ^iven in terms of /> hy the relation

thence //,, //., in § (J ((i) can l>e determined, and Th T., in factors, as required tor the
Second Stage of p. = 14, in terms of ft and x 'R

Changing from the variable t to y, in § -J8 ( 1 ). \\>- find

I -, . .7 ' v' /

, _ \ M '///

1 1 1 T I

c . ,,c - 2£K3ErLLd^/E

7 '" + I-*

•2 M 2

268

PROFESSOR A. G. GKEENHILL ON THE

A, =

p-

p (v) _ 5;/1' + 1 2ps - 4p2 + 2p - I + 2 ( p* - p + l)
M 140>

- 2m

(23),
(24),
(25).

Test by

For the elliptic-section values the region may be selected in which, in accordance
with § 28,

1)} (.,

V^

1)| , .
, .

With

/(H) f K' \7 _ 7 _ 7/(a- /a —
\ BO C] " 1 — 2«t \ a

H

BO

( 1 — 2m)7 \xa —

(32),
(33),
(34),
(35).

THIRD ELLIPTIC INTEGRAL AND T1IK KLLII'SOTOMIC 1'KOBLKM. 269

32. ,.= 18, /srlOLl^J.

The relation Dy = 0 becomes by tbe substitutions

a; + 1

y + 1 2V (g^ + 7

Now the p employed in (2), § 10, distinguished here as /;,„ is given by

•^ = ^/(1-2). Z-V = ~P

returning for a moment to the original x and y of § 10, so that

the same as those employed with DH in § 22, a new relation

representing a C5 in (p, e) ; and putting

1 1

this (J5 reduces to a (J.j. in (,r, ?/) with deficiency p = 2,

,,f __ (tf + -2x - 2) //' • - (x + 4) .nj + x~ (.,: + 2) = 0 (4),

and this with y = (q + 1 ) ,»•, as in L.1N1.S., 27, p. 4U3, I)ecoiues

= 7 v

2(V+ 1)^
= ,/•• + 2,/' + 5,y' + I0<f' + 107- -H 4V + I (8),

-

v - y3 + v3 - 29 - - 1 + v^ do)

2(3 + 1)

270

;ii id

PROFESSOR A. (4. GitEENHILL ON THE

•

i) ' -

z — y y (y - x - r)

lira
— in

'lira iii.-'fx.'

'< — 111 (a — ?/(,)-
-Hi- i(l — 2w)_a — in(] —
( I — 2m)'1 (a — m}~

_

( I -- -Ini a — iii: ( I -- iit)~ ~ ft — 1 + m

:;7:; + :i</-: + I'-/ + I 4-

e + I '/'' + '/" + -''/+ 1 + v

c - /' - ('/ + 1 )''

(18) J

(I -I),

(18).

This vuluc of !>,,, em])loyH(l in the previous equations (7), § 10, for ^ = 9 will lead
to the functions of the Second Stage for p, — I H ; the result, in a modified form, has
been given alreudv in the ' Arcbiv der Matlienmtik und 1'hysik,' :!, lleihe, 1, }». 74.

Put

(19),

to agree with the r< suits in the ' Arcliiv,' and in the modified form of § 28

T - f P.'/ -
J '

(20),

THIRD ELLIPTIC INTEGRAL AND THE KLLIPSOTOMIC PUOHLEM. 271

(21),

(22),

'

-2^ + (5 4 I 1 + 13 + U+1+ (2,r 4 4n + 3) /A
- 23

_ „.<•• _ 4 _ <) _ i (; _ -20 — 1 4 — 5 — (a:! 4 :>, + 4 4 3) V/A

2(« 41)'

_ ,,o _ 5 — 15 — :!4 — 58 — ?:? — 0:1 — 31 — (i 4

, +(_,,'-._4-y- 14- .14-8- 1)V/A

2 (a +!)>* + « + 1)

f — 1 -5-11 —64 I44:!l 4 2!) 4 I74C4 1
i +(- i --4-5 -<;-<; --4-- I), 'A

4a+(a+ l)a(«i + a+ 1)

+ (1 + 2+1- -2-- l)x A

4V (a 4 1 )•(«.- 4 '* + i)

Expressed in terms of ct, tlie parameters employed !>y KIKI-KKT (' Math. Ann..' :'>2.
p. 1 28) are ^iven liy

l=,r4^-2,-14v/A (2g)i

fc-i -" ('< 4- I )

TIS l>eiiit>' (liKRSTKit's T ('Math. Ann.,' 14. p. 540),

T^ =4"ll= 2a(a+ I)

.. — f.v' — <r 4 2«( 4 1 4
?:! " 4a (u 4 1 )

a+_8o + q-- 1 - v (32)

•2 (a-' - 3,, - 1 )

272 PROFESSOR A. (4. GREENHTLL ON THE

_ ,,s _ 9,,* _ 6a +1 + 3 V/A
s

Then for the section-values

._ o8 - a2 - 4« j- 1 -f- /A ,

~ "

,
(37).

* K, _ „ _ + ('/'' -2-11-18-14-6-1) v (

9 '"

(39),
4 + 9 + 16 + H + 6 + 1 •*• 3 + 3= +4+1

«7° + 1 + 5 + 28 + 66 + 83 +- 73 + 29 + 8 + 1

..
2 (7 + I) (Vs + 7+1)

. 5 K/ _ 1 1 + 4 + 9 + 16 + U + 6 + 1 .- (1 + 3 + 4 + 1) v

,, rv — „ , ., \^ l /•

3

_ _v»_;;_;i_r,_4-7-ll-9-4- I +(_v«-2-l +2 + 2 + 2 + 1) X/Q , ,

°

^=y-X = z = p(l-p) (44),

/ Li t/

(47)

«0 *

. ^ + ./-2(y- 1 4 v/Q
HO — 4g (q + 1 )

- l - (//2 - l)' (48)

-rls + 2- 2

from the table for p. = 6, § 29.

TRIED ELLIPTIC INTEGKAL AND THE ELLIPSOTOMIC PROBLEM. 273

33. n = 22, f= -
11

The relation Dn = 0 in (21), § 22, gives a Cs in (p, e) ; but putting

e = ^Ti' '> = ?/+i 0).

it becomes

(x + 1 )2 y& + (4x2 + 9a. + 4) ^ _ (_r + 2) ^ _ 3a. _ 2) ^

+ a: (a- + 2) (a* - 7.r - 2) f + 4.r'> (x + 2) y - ^ (.r + 2) = 0 (2),

a C7 in (x, ij) witli triple point at the origin ; so putting y = — sx,

.S.5X4 + (2.,'1 - 4.s^ - 3* - 1)*V + (.s3 - 9.s'4 - S.s" + 5,s2 + 4.v + l).x-2

- 2 (2,s' - 4.s;i - 8*8 - 4.s - 1) x + 4,s" (.s + 1) = 0 (3),

a quartic for x, which can be resolved

[2^2 + (2.s;! - 4** - 3x - 1) x - 2 (2.s- + 2x + 1 )]' = {(-' - 1 ) a' - 2 j 3 8 (4),

S = 4*(s+l)3+l (5).

Writing it

[2**(Aj-2»- l) + (2s2 + 2s+ L)K«-- l)x--2|.]2

= {(s_l)a!_2}2S (G),

and putting

(8),

zj.>

so that, with

rx ,1 .

(9),

« = «(tt.-t«) (10).

This is the elliptic integral of which the ikosahedron irrationality 77 of KLEIN is
unity ; it is curious also that this integral occurs us one of ABEL'S numerical exercises
('CEuvres,' 1, p. 142).

Then to connect up with ^ = 11, as in L.M.S., 27, p. 469,

, , s ('Zv) — s (V) (x — iir ,, N

l+cn=_pu = Y\ / \ = / w~~ "o~TH \ ' ''

after reduction ; and

_ s (3v) — s (2v)

VOL. COIII. — A. 2 N

274

PROFESSOR A. G. GREENHILL ON THE

reducing to

2) (x -

(y

y - x

(13),

where x and y are given in terms of s by equation (64).

Substituting these values of c and ^/C in terms of a; and y in the expression for
I (v) in (42), § 11, for p. = 11, we obtain a result applicable to fj. = 22, and associated
functions of the Second Stage.

34. The next two cases of //, = 13, 26 and //, = 15, 30 still present analytical
difficulties not yet surmounted, although /JL = 30 could be treated by the trisection
method of § G, applied to /j. = 10.

35. When /j. is of the form 4n, so that

2r + 1 TT i 2r + \v,. f 2r + 1

T = o), -4- -Wo, or K + Ki, / =

2/i 2*i 2n

then J/j,v = 2?ty is congruent to «j, as in Case TIL, and T (/.') is given by

T (v) - x- 1 r1 pis""1 + P2S>"2 + ' ' ' + P» /fs ,s 1

2n (a- — isY V(' '3>

?i

(»• .

J]vS.»-2 _|_ _
((T - S)"

cr — .sj

Here the degree is halved by putting

so that

and

f * -f

»/s

dx

X =

j/u, odd

C0 + C^ - x2

(Xj — X) (X2 +

C0 — CjO; — x2

also

and putting

= (ajj + as) (x2 — x)

even

= C0 + (> + a;2
= (a?! + a) (asg + x)

-^ WQ ^™* \^^OC ~p X
= (Xj — x) (X2 — X)

S = a- - «3 = (,Sl - as) dn2/K' = a^2 dn2/K'

(1),

(2),
(3).

(4),

(6),

(8),
(9).

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 275

and

P(w) = a;1zn/K/ (10),

s - xa~

n (8 - X3)*"

= ls}n-1Iio-IV- + . - - +(- tJ-'R.-,*-' /r-
n (8-,T-)i« vt^i

Thus for the vector of the corresponding herpolhode

M p = ^/(S - a;3), x = x.sn (K - nit) (12),

M £ **-->' = {(R0 + lllX + ...) v/^X, + i (R,, - H,x- + ...) v/iXj- (13),
and in the associated hodograph described by the axis of a toj)

sn

= {(A0+A1x+...A2/(_1^-'),/J-X1+i(A0-ArT+...-A;!,,_1^'-|)v/iX,}' (14).

36. With /u, = 4n, the resolution of rf into factors for the spherical catenary must
be such that Tj and T2 are quadratics in t~ ; and thus

T=

(15),

Tl = _ (k _ 1)* + 2 2 ~ - F + i ^3 - (k + l)2^1 (17),

T, = (k + IY-2 (2 ^l - A* + l) $ + ^ - I)2 ? (18).

—

— h

2 N 2

276 PROFESSOE A. G. GREENHILL ON THE

37. The simplest cases may be cited before proceeding to the general theory.

,1=4, v = K + ±K'i (1),

(I -

_ , / - * - Bin-i

V 2(K-<)

and in the associated dynamical applications

M P = X/(K - x"\ x = KSD. (K - mt) (3),

/c

M k e(vt-"» = x/i (1 - .x) (« + .x) + i V\ (1 + ^l/T^T) (4),

reducing to HALPIIEN'S algebraical herpolhode when the secular term pt is cancelled

by putting p = 0.

In the associated motion of the top (' Annals of Mathematics,' vol. 5)

m

(5).

But the quadric transformation

(1 — K)X r/, 2X/

'„ - en |(1 + K)i>it,y\, y = --V

makes

thus effecting the identification with the result, p. 147, ' (Euvres/ 2. when the sign of
ABEL'S K is changed so as to obtain the circular form.
For the division-values

j-^ cn|K'= Vr-^f, dniK's=--/K (8);

iK' = Z(liK' = 1(1 -- /c), zsiK'= -zc£K' = i(l + K) (9);

iK'_(l+w)t HiK'_(l-K)» HiK'_ _K*
00 2V ©0 2^ HK' 2*(H-«)*

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 277

38. p. = 8, v = K + -1- 3 K'i,

8 - s3

where the octahedron irrationality o = ^/K is given in terms of a by

The result is obtained by putting

/ \ / 1 \ '" ( I — '")

.< (o>) - ,s (4v) = 0, a = ; , /3 = I

I — ZWl

and the values of .s-,, ,s-.,, .s'.{ are rationalised by putting

Ii
+ a

(I),

o = A- ( a

\a

unchanged by the substitution (a, — j, which interchanges /= } and |.
Also, in the region v/2 > a > v/2 — 1 , f— },

4a

P (v) = * = xn | K' (4)

Q(r) = K/-S = (1 +a)°(l

16as

(5).

a

\ / — -
,.1—81— 2w/) a =
<t - n- 1 + 2(t - <i~

To normalise the results, a homogeneity factor M is introduced, such that

. - h = M^,, M = v/(,, - ,,) = (I + o) (f; 2a _ a)) . (8).

The division-values are shown in tabular form, and the numerical test of
= \ (v/5 — 1) is required for yu, = 15.

278

PROFESSOK A. G. GREENHILL ON THE

Region.

A.
=°>a> x/2 + 1.

B.

C.

1 ^^^ n "N*%. /Q 1

J- tt^ *M ^P A/ ^ ""™ L .

i /I \2

i — a

\a 1

1

-sc2|K

0

K"

-cs2|K

/C2

dn2|K

(1 + -<)3 (1 - tt)

4rt3

*(s— )

-nd|K

- dn i K

dnJK

(1 + 2a + 3a3) ( — 1 + 2a + a2)

-1, zs i K
/c

zs^K

2niK

16a8

(3 — 2ft + a2) (— L + 2f« + a2}
16«2

i-IK

.

zsf K

zn|K

*(£ + ")'

^zn|K

znJK

zn^K

(l+a8)*(-l+2a+~aa)»

H|K

HK

H4K

©IK

HK

a"- (1 + f/)'"«(l — ft)''"

HK

H|K
HK

BfK
HK

/I V
— ct

\a

B^K

HK

»1K

00

"S

Test by a

v/5 + 2

-""

— "

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 279

D.

E.

F.

C1

_ 1 ^^^ /•/ "*^ _ /f? 1

L ,X*^ w ^^ */ ^ ^^ I .

H.

1

i

^

^

1

_K~

~/c'2

^

K2

g
K~

~/c/2

nd2 i K

- sc2 -J K

-cs2f K

dn2|K

nd2|K

nd2|K

- sc2 i K

-cs21, K

dn'JK

ndHK

ndJK

- nd l K

- dn l K

dn IK ndlK

1 „ ^

. zn f K

K

K'

-zsfK

- zn J K T. zn 4 K

K

- l zn 1 K
/c'

->*K

-Z8iK

1
1 T7" '^ T7"

— zn ] K zn f K

K

/?Zn*

1 zn 1 K

zn|K

zuiK WK

rv

©|K
HK

HK

HfK
HK

0fK ©|K
HK HK

HK~

HfK
HK

H*K

llK

@|K

"HK

Q^K

©K

@k

00

©k

^5-2

- x/5 + 2

-„.-.,

-"•+*

--"-

280 PROFESSOR A. G. GREENHILL ON THE

39. ABEL'S form in 'CEuvres,' 1, p. 142, 2, p. 1GO, for n = 2 becomes the same
as in 2, p. 148, by replacing his x + ^a by x, his a by 2 (q + q'), and b by

1 / '\9

-H?-?)3-

But now in our notation for /A = 8, equation (14) § 25 becomes

Z, = [z + »i (1 - m)]2 - (1 -- 2m)2 z (1) ;

and putting

1 + ft ft (I — ft) 2 / — 1 + 2a + a2\2 /„..

m = „,!— »i = ' „, 1 - 8m + 8m2 = ( r - i (2)

] _)_ 2a — cf- 1 + 2« — rt- 1 -f- 2a — a2 /

and replacing z by * where M = 1 + 2<t — «2, the roots of Z = Z{Z.2 = 0 are

(3).

A quadric transformation is required in ABEL'S result to obtain our I (r), namely,
with the 8 above, and the sign of Zl changed to obtain the circular form,

_ (1 +«)"(! -«)(1-O

8 - re2 ( >'

_(l + rr)(-l + 2a + «2)3 .

Z' ~- 3 " ( j>

"3 ~ 3 - x-

and

Z _ /, _ Zw2 .„ 0 _ - - - ( }

" ^ ":'"^ (8 — tc212

so that Y/Z., is rational.

Afterwards the degree is halved, as ABEL'S integral is 21 (v).

A similar quadric transformation and halving of degree will be required for all
even values of ABEL'S n, to reduce his results to our form, involving heavy algebraical
work.

ABEL'S integral (' CEuvres,' 2, p. 148), changed into the circular form, can be utilised
for the construction of an algebraical orbit described under the central attraction

THIRD ELLIPTIC INTEGRAL AND THE ELLlPSOTOMIC PROBLEM. 281

in the form

N COS 26 = (u + q 4- 2(j') ^/(u* + 2qu - <f - 2qq') (11),

N sin 20 = (u 4- 1q 4- q1) x/(- «2 - '2q'u + 2^' + r/2) (12),

N2 = (,,'- r,)3 (</ + ,y) (13),

so that the condition

a [udu

O = \ u (14),

U == (ir 4- 2qu — r/2 — 2qq')(— «2 — 2q'u + 2qq' + q'2) (15),
is satisfied by taking

t=a_ _, H_ 2, /2 ft, = ._8 '/ , A

A degenerate circular orbit if — f/ is obtained by taking

q + q' = Q, it = 0, ii0 =0, H — 2<//r,

P = //'-</, ^ ~ ^'/ (l7)-

But the integral in ' (Euvres,' 2, p. 147, putting .'-' + '/ = n, requires /x = 0, /it,, = 0.
so that we merely obtain a conic for P = /A,/1.

40.

IM_. f - P («) (8 - «*) + Q (») cte

-

(a + «2 + a3 - x2)'

. . i 1 + a + a2 — (1 — a) x — x~ , l Y / 1 \

= 5 Sln 2 l 3 2V V t "** ^/'

(a + a2 + ad — a;*)'

L X2 = a3 (1 4- a + «2) ±1(1+ «);i (1 ~ «) * ~ ^ (2)'

P (v) = JLa- (1 - a) (5 + 3a + 3a2 + a") = ar,zn/K' (3),

Q („) = i (i _ «4) (a 4. fl> 4. as) = jCjS^n/R'cn/K' dn/ K' (4),

VOL. CCIIT. — A. 2 O

282 PROFESSOR A. G. GREENHILL ON THE

results obtained by putting

m(l - m)(l — 3m + 3m2) « 1 — 3m + 3m2 ,,

s (to) — s (6v) = 0, a = -j— s\ > P == T~ — a \n/-

(1 — 2m) (1 — 2m + 2m2) 1 — 2m + 2m2

M = m =' l (6).

1 -a

Treated by the trisection of/x = 4, putting y = 0 in equation (8), § 14,

3/4 + (l -- 8x) tA + Gx2f- - tf* = 0 (7),

and putting

'=:•

T> ~S.r- ' \h (ll)'

and

(

so that

and 1) ix the equivalent of y in KLEIN -FmCKE, ' Modulf'unctionen,' 1, p. G88, while

x = - Tlg - 2 (12),

2r(i + 8 == (T|S + 2);i = y- = (T,S - 3)-

and GTEKSTER'S

3G 3G

Also

so that

6 = ^4-1 + « (17).

In KIEPERT'S notation, 'Math. Ann.,' 32, p. 104,

f i = 1 - , &c. (18).

Putting

A = (14-«2)(l44a +

the section values are shown in the table :—

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 283

a a~

_
2« y (a 4- a2 4- a3)

(l+r()3 (l -«) + (!+ a2), /A

a

(1 _|_ a) /l 4 4a + a- — (1 - a) v/1 4
(1 4 a) /I 4 4a 4 a2 + (1 ~ «) \A +
(I 4 «) v/1 T4^"+^a - (1 - ^) N/l +

as

2(1

4. rt)
4- a)

4

-f-

X/F+ a2 — (1 - «) \/l
2(1 + a) \/l +a*

2(1

(1 - a) (5 + 3a + 3as + i3

(1 + a + «*)
'

a22 (a 4 a3 + a8)3
(a4«2 + Q3

0 <a < 1.

dn|K'
o

dn i K'

en * K'

sn § K'

dn § K'

en * K'

sn :\ K'

dn 1 K'

Zn i K'

zn|K'

HK'

^fK
HK'

1 <a < oo.

l

K

dn i K'
o

nd K'

sn f K'

en * K'

nd | K'

en * K'

zn i K'

HK'

HK
HK'

2 o 2

'284 PROFESSOR A. G. GREENHILL ON THE

41. In the general case a distinction must be made between £ /a odd or even.
I. Jfi = 2n + 1, ft = 8n + 4.

r i 2r -4- 1 vf- _ r<

X, = Cn + 20^ — x2 = (05, — x) (x, + a;)

X2 = C0 — 2Crx — xz = (.r, + x) (xz — x)

jCn ' — KJC i

GO = V/O?, — S3) (Sj — -^) = .S' (2«, + 1) V — S

20, = /(*, - .3) - /fo - «3) = 2P (2n + 1
2C0C, = I ?>' (2n + 1) r = Q (2» + 1) u

1

-

O

(i);
(2),

(3),

(4),
(5),
(6),

(7),

(8),

where o denotes the octahedron irrationality X/K; so that if the expressions are
normalised by the homogeneity factor M = v/0,,,

X,, X; = 1 ± Or - x-, 0 = ' - ,>

,T, = , ,r., = o, .c = osn(K — mf.)

/"K' = ° 'S3 = " = K D = «- D, zn/K' =

M

(9),

(10);
(11).

No

w

I(«) = J

M M3

D - x"

2n + I

cos"

(D - x*)

. , Rn 4- R,

" P (2n

/i x

= (- l)",

(12).

Q (w) _

M3

(13).

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 285

Next putting

and with

.!-/>_, <J* 1+c*

2 1 -c+" 21 -c4

(1 -c4) 2

6a3^2 _|_ fe4 a _ ft-2

(a2 + fr~) " 2?>(l — a)

_ _ (1 — 2ni) a _ ±pot

m,(\ — m) j^ — 1

s) (3as + as + (14. 3a) 62}
2a(l -?,2)(«2 + ft2)

8 positive, v = oj, + /w3, 8 negative, v =/wv
p . m2a 1 _ S

MD — — "a ~ 2

a — m <r c2

so that when normalised hy M = ^/Co, D becomes c2 and R0 becomes <r"+l.
In the sequel it is convenient to put

_ a - b dn/K' _ o , .

a+h o dn(l-/)K'

p = l-2W = 6«-_^ (19),

--^W^^^^^T" (20);

so that

^ (a - by

- =

a «- « (24)

a(l _6z)(as + 6»)

(1 + a) («2 - &') (a + 62) (25)

8 »

286

PROFESSOK A. G. (JREENHILL ON THE

_

s .-

6)*

2C =

a (I- 62) (a? +

{3«2 + a8 4- (1
(l-a)3&2

- a)3 fe* (a -

3a)

C,2

1 —

o _ , 2

— _. — — it — G

C0

.

4 (a — m)8

4 (1 — 2m) a — 8a2

— j— '—r —

4a (a — m)

4a

(a- -

h

Haft (a3 + //-) 2ab (l - a) (n2 + /r)

(1 + 3a) fe2}

0

i/ o)o = i^=

'^!) + 0 (a4 - a5) &*_(! + 3a) 6s - 8as6s (a3 -

1 /I , Ov: __ a « a - «-

3a" + a" + 6 (a* - a5)

Thus m the quadric transformation

2 (a3
(l + 3a) 6s - 8a362 (a3

-- - -

D = (a* - 7'4)3 {(3a2 + a3)2 - (1 + 3a)- 6*}
involving powers of b* only ; this will be useful in the sequel.

( }

,

, }

,
*) (

//) . .
+

(35),

(36),
(37),

THIED ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 287

To connect up with the order /A' = 4n -f 2 = £ /j., denoting p (/) by p',

P' + I _ A K) - s (6v) _ (1 - 2m + 2m2) ft - (I - 3m + 3m2)
p' - 1 v »(«,,) - *(.V) »«(! - m)(2/3 - 1)

_
a4 - 1 + 2,-< 64

/= -e .«-

(e - 1) />- L — 6* . 2a3 + n* - ( I +

We have found

dn/K' = t._«-6_«(l-/)K'

o a + b H (fK.')

and proceeding with the series, writing f for fK',

dn3/_l-m _l-l>

m

b

Working these out with the assistance of the analysis given subsequently,

dn5/_ «3 - (1 -f a — <r) IS + nk (a - n- — «3 + l>]) , .

o ~ a? — (1 + a— a-} />' — nl> (H — a- — <? -f //)

and generally

dn(-2»-+ 1)/_.E,,.+ , +6F,r+1 a ,

" E,+1 - fiF^,
where F is a rational function of a and //', and E derivable from it by the substitution

0,1
Thus for

^ = 8^ + 4, ^(2^+l)/=1; FB+I = O (45).

The results in the sequel give

F7 = a4 (1 + a - 2a2 - «3) - (a + «2 - a3 - :5"4) // - a2l>* (46),

and therefore

E7 = a5 - (3«3 + «4 - a5 - a8) b* + (1 + 2a - a2 - «3) 6s (47),

F9 = a° (3 + 0 - 3a2 - a3) - 3«3 (1 + 2a - 2a2 - 2a3) 6*

0 + 0 - a6) // + a3^12 (48),

288 PKOFESSOR A. G. GREENHILL ON THE

E(J = a6 + a3 (- 1 + 0 + 0 - 6a3 + 0 + 3a5 + a6) //

+ 3n3 (2 + 2a - 2a2 - a3) 68 - (1 + Sa + 0 - 3a3) 612 (49),

Fn = A0 + A^ + A268 +A3612 + A,616 + A6620 (50),

En == B5 + B4i* + B3fc8 + B,/>12 + B^16 + B0620 (51),

where the A's are found in § 45 (3), and the B's are derivable by the substitution (a, — ).

\ <*/

These expressions have been tested by putting

F5 = 0, b4 = - a + a2 + a3 (52),

which makes

E7 = a2 (1 - a8)3, F7 = a (1 - a2)3 (53),

E9 = a-' (1 + a)5 (1 - a)4, F9 = a2 (1 + a)5 (1 - a)7 (54),

Eu = a5 (1 + «)8 (1 - a)7, F,, = «4 (1 + *)8 (1 - a)7 (55),

and thus verifies

dn7/_ o 1+6 dn9/_ dnll/_ a + b

o dn3/"" I -b' o o " ~ a-b

42. M = 20, /=1'3'07'9.

^4 = - a + a2 (- a3, a3 — 6* = a (1 — a) (1),

Co _ + 64a*6* ,«x

- (1 + a2) (1 + 2a - 6a2 - 2a3 + a*) - 8a262

16a^(l + 6)" (3)

(1 _«)-2(a_^)«

64a*(-a + a2 + «3) ,,

" (1 - a2)5 (1 + 4a - a2)

1' 2 i~\ I 9\9 / -i t f~\ y» 9 *-*" *l i 4A9

_i_ \ \ (1 + a ) (1 4- 2a — 6a2 — 2ad + a*r /K\

X / /I — a2^5 n -I- 4a — a2) ' ''

Now, as in the < Archiv,' III., 1, p. 75, with the 6 there replaced by - l + a and

1 — a?

I - «'

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 289

I («) = \

*/ X

„,

D - x2 X/X,X2 '

= icos-1 ft±-J* + ^ + Bad±*? y^x, ,

_

Xt> X. = 1 ± C.c - x2 (7),

_,_1(, . v

'-2(o

/ ., \

(!!l= (12),

M3 v

— X

(13),

while Kj and Rc can be calculated and the whole result verified by differentiating (6)
and equating coefficients of powers of x.

A change of b into — b will change v into !)v.

The c employed for p — 10 is --,, so that, putting

l+o />' + 1
we find

so that, with

u^f"-^-, A = 46* (16),

a = a (?*), a' = a (u — ^w)

(KlEPERT, ' Math. Ann.,' 32, p. 107), and then, with a = a20,
VOL, CCIII. A. 2 P

290 PROFESSOR A. G. GREENHILL ON THE

1

*7i = - - .

Then, in § 28,

= a? _ 4« _ i '

- (18)-

43. ,z = 28, r = K + - , r = 1, 3, 5, 9, 11, 13.

D7 - cN7 = 0 (1).

a N7 + D7 ,2v

6"N7-D7

A change of b into — h changes v into 13'»\

Now, with 1 — 2/u = p, pa. — in (I — m) ft, 8pm = (l — /r) (e + l),

2/1 vl /e+1V 3/) - 1 e + 1

D7 = m' ( I - »)• { /> ( - 2-) • ^- -3-

= I me (i _ ,n)J { _ ^ + e (e _ 1} {t + e} (3) ;

and changing p into — p,

N7 = i- «r (1 - /u)2 { - p* - e (e - 1 ) p + *} (4).

op£(e-l) + 6(6-pa) = 0 (6).

Snhstituting for /) and e in terms of a and b, and cancelling a factor (a3 — 6'2)3,
there results

ab* + (i + a - a- - 3a;i) // - as (l + « - -Ja3 - a3) = 0 (7),

a quadratic for 6+.

To obtain a rational expression for C, we put

a3 - 6* = as(l -a)c2 (8),

and now the relation (7) becomes

(aV + a)2-(l +a)*c2 = 0 (9),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 291

«V + ric 4- a + c = 0 (10),

- 1 a 4.

A == (1 + «)2 -- 4a;i - (1 - a) (1 + 3« + 4as) (12) ;

so that, with

a = a(«). « = J"^ (13),

then

c = «(w — ^ w) (14)

(KIEPKRT, ' Math. Ann.,' 32, p. 108).

Introducing these values into (35), § 41,

,/l x\2 - 128^(1 + a- 2a2-«3) ,1M

1 U " (1 + ^ (1 - «>! [7« (1 ~ «2) + (2 - a + a2) /A]

to be compared with the expression for £, in fj. = 14, L.M.S., 27, p. 440, putting

r = [ - n, 1 + 2c = l (16).

2'f 't

Writing q or j/ for pu,

_ as 4. (_ «2 4. rr + rt+) ^ + ( L + a - «-) 6+ - ab[> . .

(1 + ft) (a - b~)(<r - //)
and after reduction

. > / . \ / . ,, (18)

or

,yJ — a(t+ 1)7 + fit. = 0, t = n (11 — § <») (19),

to be interpreted as an elliptic function relation.
Rationalising again

(1 _|_ fA- ,/' — 2« (l + 3«) (/y + 2aa (l + a) (/2

+ 2a- (1 - a) r/ - «2 (1 - a2) = 0 (20),

or arranged in powers of «,

a4 4- 2« ((/ — 1) a3 + ('/'' i ^f/3 "I" 2</2 -f 2^ — 1) a2

a94_r/(9_l)a4-?2 = «v/Q (22)'

Q = 4tf3 + (<? - I)2 (23)>

2 P 2

292 PEOFESSOK A. G. GREENHILL ON THE

44. ^ = 36, v = K + 2r +-^ K't.

18

The relation

{•(»)-8(9v)}> = (*-»»)(•»- 9>) (1)

leads to

N9_. 1+6 ,,

DB 1 - b

ft-.N9 + D9_ *[(cT_2)j»» + e]

N,-:D,"jp|y.-f(^:-f + fj

ami putting

r -1 = 9, £ " 1 = »', 2r - 1 = «, e ~ l = t, t + 1 = u (4),

-

and thereby the superfluous factor a2 — /r is cancelled, and finally
aW + ( 1 + 3'< + 0 - G«3 + 0 + 0 - «(1) 1s

- :3 (1 + i« - -2rr - 2«:{) a:!//' + (3 + 0 - 3«- - «:i) «(i = 0 (6),

a cubic equation for b{.
Putting as before

' «3-"=t(v~;) (?)-

this relation (G3) becomes

[V<. + 2(1- «-) r]- - ( ! - tt? 1(2 + a) (.- + (1 + aff = 0 (8),

<:•'• + (i ._ „) (2 + „) c» + 2(1- o-) c + (l - a) (1 + «)2 = 0 (9),

and jmtting r = (1 + ^) ry,

(1 + a) <f> - (1 - n) (2 + '<) 7- + 2 (1 - «) q + 1 - « = 0 (10),

and arranged as a quadratic in a,

,,•-„* - ((/' - ,f - 2? - 1) « - (,/ + 1) (f + ,, + 1) = 0 (11),

Q = /' + 2r/ + 5</' + 10</:i + 10r/ + 4</ + 1 (13),

as in /A = 18, § 32 ; and

c = q* + f-2g-i

2q

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 293

45. ,1 = 44, r = K+fK'i, f=2r+l.

£tM

Here the relation to be reduced becomes

Dn __ e [(e - 2)p2 + e] [(e2 - e - 1)^ + e]
-

« " N,, - D,, - ,> [>* - e8] [>» - 6;! + c2 - e]

The algebraical work has been carried out by Mr. 11. H. MACMAHON, and he
reduces the relation to a quintic in b[,

A0 + A,M + A.&* + A,//- + A^10 + A5fc20 = 0 (2),

A0 = a1" (1 - 2a — 5a~ + 2":! + 4a' + a5),

A1 = a7(— 1 + 2 + 15 +8 - 10 - 10a5),

Aj = a(i (-15-27+14 +45 +0 -11+0 + 3 + as),

A, = «:! (G + 22 + 5 — 40 - 23 + 20 + 5 - 8 - 3as),

At = — 1 — 5a - 5a- + <) + J3 5 -- 10 + 6 + 3as,

A, = - a6 (3).

As before, putting

the relation can be halved in degree, and becomes

C'Vf1' + c- (c + 1) (2f° + 2r + L) ft3 + (»-' + 3r4 + Gr! + 8f- + 4c + 1) a-

- c* (2t-- + 5c + 3) /t - (t- + 1 )' = 0 (5),

a quartic in a of similar structure to the one in /x = 22, § 33 (3), and capable of
resolution in a similar manner into

[2cV + (<• + 1 ) (2c- + 2r + 1 ) a - (<• + I )]- = ( ! [(<• - - 1 ) a + (c + 1 )]' (fi),

C = 4c(c+l)8 + l (7).

Put

2cV + (c+l)_(2Cl+2c+lla-(C+l) = 2^_2c_1 ( }

(c_ ! a + (c+ 1)
and then

if

c = c(u), « =

and

c2«2 + (c3 + 3c+l)« + c + l

(o - l)~a~+ (o"+l)

a relation which requires interpretation, connecting c, x, and a, elliptic functions of it.

294 PROFESSOE A. G. GREENHILL ON THE

It is now possible theoretically to determine the coefficients in

r

\

= J.

M

,

"

11 (8 - a

. . . - o /r-x

- 8*

sin-i

U (S-Z2)*

and to construct an algebraical herpolhode and associated top motion complete in
44 branches.

This is as far as we can go at present, as the next cases of p. = 52 and 60 must
await the solution of p = 26 and 30, not yet accomplished.

46.11. ip. = 2n, /x = 8n, v = K + fK'i, f = .

47i

Xt = C0 + C^ + x* = (x, + x) (x, + x) (I),

X2 = C0 - C^ + xz = (x, - x) (x, - x) (2),

G0 = s (ws) - s (2nv) = ^/(sl - ss . .s-2 - *3) (3),

G! = y(*! - %) + y(s2 - *3) = 2P (2m.) (4),

G0Gt = ^y (2m') = Q (2nv) (5),

(o+^-?J2 = C2 (6),

°o
and normalised by M = ^/Gy,

X,, X2 = 1 ± Ca: + x\ x, = I, a;a = o (7),

x = osn (K - »»«)• " = U (8),

o M

and the associated integral

So
a

M M3 c/a;

~2

I*/

~R w _L _1_ T? /V.SM— 2 I "R /y.2«— 1

= - sin- ~ ,, • ^ - *-' y| X2 (10)

2»i (D — x2)"

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 295

R,2 = D2», R,,,_, -= - [2nP (v) + P (2ww)] R^, R2,_, = (- 1)« (11),

9w = (£ " °2) sn/K'CI1/K' dn/K' <12>-

In this case II. put

(a - m) (a — m + 1) = «2a4 (13),
then

=

e+

e =

fe' 1+6

and the relation (13) becomes

a pa. b

F5 a -a4)

~

6s

4a2et2 a-® ct

-a4] _. i^*4 „ 2_

(a — m)4'
S = s (•?;) — s (ws) = (positive) (15).

Ot ^^ 771

Now, with

1 — 2m = p, (1 — 2m) a = m(l — m)^3 = m(l — m) € (16),

Z

,_,,t++'- = 0 (20),

1 - n* , 1 -P'2 ^(1 + ?') /->n

04. m - m* = -r = (i+2&f^ (21)>

- (22)>

296 PROFESSOR A. G. GREENHILL ON THE

' '»--;< (26),
. (27,

We now find, as in § 41, writing f for fKf,

dn2/_ a- — ss __ 1 (m

o2 C0 ' " '

a

0

and then, from the relation

* v/26 + 1 - a2 + v/1 - a* v/2fc 1 + a2

-"a2 - v/1 -^

dn_(2n + 1 ) /• N,,,
dn/ * D,,

(30),

writing c for a2, and putting

1 +c) = B2 (31),

- c2) + & (&3 - <?) (2b~ - 1 - c2)

E7 ,= // + -2W, - - Vc* - <• (3 + r-1) // - 6V + c (1 + c2) 6 + c4

= (/^ _ p»)a (ft-: + (.o) + /w. (fte _ 1} (2ft3 _ j _ c3))

o
and F7 is obtained from E7 by changing c into — c.

47. ^ = 16, v = K +/KV, /= !• 38^,

and now the condition

s (a,) - ,s (4r) = y (S] - *3,. s2 - Ss) = Cn (l)

s
reduces to

a = b (2),

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 297

262

= dny [V (6* + 1) /(63 - 26 - 1) - (6 + 1)

£) — — J __ LV v i .*•/ v \" I V" ~r • -; v/ v" /K\

,-2 A 7_ '-t \ /»

o 4 o

I (w) = l cos - ! o + if V + •«' y ^ Xls

^_u -- a; ;

= isin-lK°" i* Jf" '^VjX, (6),

^ — ar;

•/• 4- r- /'7^

X -|- X ^^,

RO = D2 (8),

_ 4 _ 4 _ 2

Ki - ' ' o/l4 (9),

R, = -

26*

'2b + ] + v/6

P (7-) _ zii/K' 4 (6 + 1 ) x/ (^' - 1 ) (b- - 21 - 1 ) - (//- + -2b + 3) (Ir - -2b - 1 ) (ll}
M o 166-

Q(")_(&+l)v/(6*- l)(/r-26- 1)
M3 262

Bisection formulas for /A = 8 in the region /.' = £ K'i will lead to the same results,
with ffls = «,

v/ ^2 + 26 _j + x/ /,? - 26
<

62 + 26 - 1 - v/ 63~- 26 - 1

6^1+a0 (15),

« — ct"

having the substitution

Hence the following table of section-values for /A = 16, holding in the region

00 > b > v/2 + 1 ; tested numerically by 6 = ( -v^- ) , o = %, K = %, as on p. 278,

\ Z /

region C :—

VOL. com. — A. 2 Q

298

PROFESSOR A. G. GREENHILL ON THE

72

6

&2 + L± v/(&l- 662 + 1)
2 v/26

v/(62 + 2 x/26 + 1) ± v/(&3 - 2 v/26 4- 1)
2 vV2 v/6

26'

(6 - IjVft8^ l+_v/|F+ 1) (62 -"26"- l)

26*

(6 4- l) v/P^'26"--! - (6 - l) v/62 -^26 ^T
(6 4-1) v/62 4- 26 - 1 + (6 - 1) v/62 - 26 - 1

(6 -J) v/62 4 26^1 4- (6 4 1) v/62 -26-1
" 2 v/64^^!

(641) v/6* 4- 26 - 1 - (6 - 1) v/62 -26-1
2 -/&*- 1

(6-1) y/62 4- 26~-^"i -(641) v/62 -26-1
(6-1) v/62 4- 26 - 1 4 (6 + 1) v/62 -T~2&"^"i

(6 - l) v/Z^TTfe ^1 - (6 4- 1) v/P^'26^- l

2 v/64 -I

(6_±_1) v/6M^26 - 1 +(6_- 1) v/62 - 26 - 1
2v/F^t

62 4- 1- V64^^66ir4- l
62 + l 4- v/F- 662T 1

62+ l 4- ^6*^668"+!
2 (62 - i)

6s + 1 - v/64 - 662 + I
2 (&a - l)

v/o

— \/o

dn|K'

sn^K'
onJK'
dnfK'
cnfK'
snfK'

cn^K'

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMlC PROBLEM. 29y

v/6^-26— ]

1G62

1662

•1[4(6-l)x/64-l-(362-2fc+l)\/62-26-l]

"-- [4 (6+1) v^4-! — (62+26 + 3) x/62-26-l]

R6 + 1)

7,25

26*

2 ± l)(P-26-

)T"

(6 - 1)7 (6 + l)la (6Z - 26 - I)4 (6* + 1)B
C*S =(tf- i)3(68"+Tp

48.

and this leads to

o o o

-(«-mj*

(1 -- 2w + 2»i'-) /3 - (1 - 3m + 3wr)
+ (2/3 — 1 ) (tn — m2) a = 0

m (1 — in] [3 — « — 2 (1 — a) y8] + ^3 — 1 = 0

( L + 6) [a (1 — 6) + 1 + 36] - (1 + 26)- + «4 = 0

«(1 _ 6-) - 6- + «4 = 0

,,, a + a4 ., ,

62 = - =« — «- + aj

a

/, , 0x2_

"

derivable from the results for p. = 12 by putting a12

1 (a - I)2

- = v — L

= c,

c a

2 Q 2

znjK'

BO

HK'

(0.

(5),
(6),

(8).

(U),

300
and now

PROFESSOR A. G. GREENHILL ON THE

aV(l -«8+2 %«

a'+2y/a"-a«4:«8)

(10).

In KIEPRRT'S notation ('Math. Ann.,' 32, p. llf>),

Treating /u. = 24 by the trisection method of § 14 applied to p. = 8, we put

z (1 — 2z) 1 — 2z2

35 = 2(1-22), y= Y-2 ' 2/+lr i_z

so that, putting- f = 22y) in the efjuation (8), § 14, it may be written

/. Ox2= (1-2^)' ;>3(¥ + 4)

4221 -22 4 + l3

(12);

(13).

To agree with the notation in KLE1N-FRICKE, ' Modul-Functionen,' 1, p. 688, put

(14),

=*y

_

256(1 -

^ + 1/2 = 1

and denoting the tetrahedron-irrationality by f,

x = — TIS - 2, y = T12 — 3

8 = (

18

and GIEUSTER'S r12 (' Math. Ann.,' 14) is given by

36

36

_ ™3 Q

t/y ™ O

So also ^ — 48 can be discussed by a trisection of p. = 16, by putting

(16),

(17);

(18),

(19):

(20);
(21).

(22).

THIRD ELLIPTIC INTEGRAL AND THE ELLTPSOTOMIC PROBLEM.

49. p = 32, V - K +/K't, /= !• 3. 5. 7,9.^11.13. 15

and this leads to

£3 - (1 + 2m

" (a - m)*

2m2) /82 + 4 (m - m2) £ - (m - m")

- /3 (ft - 1) (/3 - 1 + 2m - 2m2) a = 0

so that

2£2 (1 + n) — 2/3 (2 + «) + 1

6(1 + «&)
(1 + 6)(1 + 2b - U2 + 2«6)

= 0

a C6 in (a, b).
Put b = ac,

a quadratic in a2,

{(1 + 26)2 - a4
|3 — a (a* + 1)'

«*c — (c4 + 2c:3 + 0 + 0 -- 1) «-' + c = 0

c4 + 2(-'' + 0 + 0—1

a

(« - 'T =

a = X/C'3 + l • C* + 2c - 1 + (r + 1

6 =

. C2 + 2c - 1 + (c + L) y/c2 -

(c+l)»(c-l)r(c8+l)(c* + 2c-l)

4c

(c2 - I)4
4c4

-]

301
(1).

(2),

(3),

(4),
(5),

(6),

(7),
(8),
(9).

(11),

(12),
(13),

302

PROFESSOE A. G. GEEENHILL ON THE

(14),

as in fj. = 16, so that the bisection formulas can be carried one stage further.
We find now

.AK' V
V

dn:i-6-iV _

1 +

+

x/(

-V

(15).

50. fji = 40, v = K
The relation

leads to

or putting m — m? = n,

N

10

With

and with

= 0

+ «(/32 - j3 + «)[(! - 4?i)/32 - (1 - &»)£ - »] = 0
/3 = ; , and arranged in powers of w,

- 1 /72 /<7 i \~\ 2

- [563 + 362 + 6 — 1 + 2a (26 + 1)] bn
+ 62 (62 + a) = 0

6 _ (1 + 26)2 - a4
* ~ (1 + 6)

this becomes

(b + I)4 [562 - 1 - a (A* - 46 - 1)]

- (6 + 1) [563 + 3/>2 + 6 - 1 + 2a (3b + 1)] [(26 + I)2 - a*]
+ (&2 + a) [(26 + I)2 - a*]2 = 0

A factor 1 — a may be cancelled and

0 . 65 + A264 + AS63

06

(1)
(2),

;

(3).

(4),
(5)

(6).

- a4 (1 + a + a2 -|- a3 + a4) = 0
A2 = 3a (1 + « + a2), A3 = 0, A, = - a (1 + a + a2 - a3 + a* + afi + a6)

(8).

THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 303

Put b = ac,

C* + 3 (a + l + ") C*
1 1 1

a," a

1

a? c2

(9).

Put - + a = t,

ct

(I + z3) (1 + 22 - 622 - 2e3 + z4) +
as in (2), § 42, for ^ = 20.

_ (js + ^ _ 2t _ 3) C2 _ t* _ t + j = Q
Put c2 = x— \,

x* + 3to2 - t (t* + < + 4) x + «3 = 0 (11).

Put x = yt,

-( + « + 4)y + «s=0 (12).

Put

(14),

~ 52 - z3 + t3 + 4,/Z

3

.. 2 == VZ (v/Z + 1)

---

(20)>

304 ON THE THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM.
To connect up the results, //, = 20 and 40,

s a> — s <>• /r,lX

\ ' = a3 in u. = 40 (21),

s (w) — s (2v)

s (w) — s (v) \e — pi
are equal, so that

%) + &20 /90\

— T- (Wh

£> — e

and thus, with a30 = a, 620 = b,

-~»

,

-20

(26)

'Math. Ann.,' 32, p. 119), and the preceding results are thus merely
bisection formulas for /A = 20.

We arrive at the conclusion that it is the quotient of two theta functions,
6u and 6 (u — 7;), with constant phase difference v, which is required in dynamical
application, the functions a, /8, y, S for instance employed by KLEIN in top-motion ;
but the separate theta function 6u has no mechanical interest.

This quotient, qualified by the constant factors 00 and 6v, is an elliptic function
of u when v is a half-period, dn u for instance when v is the half-period K, and the
quotient is the /A*'' root of an algebraical function of the elliptic functions of u
when v is an aliquot /ith part of a period ; in this way we express the result of ABEL'S
pseudo -elliptic integral.

The formation of this algebraical function for the simplest values of /A has been our
chief object, and in the course of the work the ellipsotomic problem has been carried
out of the determination of the Division- Values of the Elliptic Function.

The Transformation problem may be considered solved at the same time by means
of symmetric functions of the division-values ; but as Transformation has no
dynamical utility, it has not been developed in this memoir.

[ 305 ]

X. On Ihe Rexistrtnce mid Elwtram<it'n

Electric Arc.

INDEX SLIP.

'>y 1'roj-

M.S.

i, W. — On the Resistance and Electromotive Forces of the
Electric Arc. Phil. Trans., A, vol. 203, 1904, pp. 305-342.

Arc, Electric, Resistance and Electromotive Forces of.

DCDDSLL, VP. Phil. Trans., A, vol. 203, 1904, pp. 305-342.

Electrolytes — Measurement of Resistance of.

DTTDDBLL, W. Phil. Trans., A, vol. 203, 1904, pp. 305-342.

SINCE DAVY'S discov,-;

Power Factor — Measurement of, with small high frequency currents.

DirnnsLL, W. Phil. Trans., A, vol. 203, 1904, pp. 305-342.

conducts electricity, \>:\\<- U-IMI <i-
experiment. In <.nl -i to t x
and length apye •> -
assumed that the *<•• |.. ^ . ,. , ...
more important in ••!>*>? met ini:
considered, has be^t,. a.:nl i-, •-
is hoped, will he tin •.;.• -..-d !A
cation.

A priori i\ is highly p.-ob.-i ';.)••
if they exist, will bo functi'ins
consider the definitions oj'tln.sf .in.-, :i:tir
adopted whethi-r tfit- m->- cat, U- s;u.i •

• ordinary t«'\t-lMmk definiti'ins »i' •.
assumption that they arc constant uiiant it'i< -
if possible variation is generally de\> im. •; ..
_r the state »>r- nature of the b", !v
i.te being the primary cause in !' > ••h.m^- I:. r«-s!H
•re is much experimental rvi<k/av to siipp";: t
it, A, (lows i! roiii;!) -•/"/ conducting appiratu^;
•rmifials can l»e written V = E -f- KA w!
only depend on £he uatm-e, state ami mn1
is of the current A or the pott-
tor any

tolier 2, 1903.

n to the present time,
lechanism by which it
.•upted discussion and
nnecting P.D. current
n, experimenters have
liich of the two is the
letlier botli must be
jttlement of which, it
)ed in this communi-

's of the electric arc,
ire necessary to first
end on the definitions
in E.M.F., or neither,
erally start with the
: current flowing, and
set due to the current
1, these alterations in
d E.M.F. observed.

that when a steady
;ential difference, V,
id E, the E.M.F. and
>f the apparatus, and

so that the equation
ly constant conditions
9.8.04

304 ON THE THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM.
To connect up the

= i

are equal, so that

and thus, with «20 = a,

'Math. Ann
bisection formulas for p

We arrive at the (
6u and 6 (u — v), with
application, the functio
but the separate theta f

This quotient, qualifi
of u when v is a half-pe
quotient is the fith roo
when v is an aliquot //,th
pseudo -elliptic integral.

The formation of this
chief object, and in the
out of the determinatioi

The Transformation f
of symmetric function!
dynamical utility, it hat

>r(J 1o riyio'
£t*-r.()6 .qq

qq ,

qq

.4ua xacwi

V**/i

(23),

nO — .

.oiA •liilo'ii

,8()S JOT ,A ..".ami' .HA

.lo traa
,808 .IoT ,A ,.an«iT .ti

.77 .

(25),

i(>(irf HBUIP. illiw ,lo
,80S .Io7 ,A ,.

i-twol

results are thus merely

• i"»ti<-nl of tw<> theta functions,

.bioli is required in dynamical

: up]- >yed by KLKIN in top-motion;

Mid Mr, is an elliptic function

t.-n >' i.s the half-period K, and the

'•it of the elliptic functions of u

: we express the result of ABEL'S

i simplest values of p. has been «>ur
••lii|*H>toniic problem has l>een carried
>f the Elliptic Function.

solved at the same time by m-
but as Transformation has no
>ir.

[ 305 ]

X. On the Resistance and Electromotive Force* of the Electric Arc.

BIJ W. DUDDELL, Wh.Sc.
Communicated by Professor W. E. AYRTON, F.R.S.

Received and Eead June 20, 1901,— Received in revised foi-m October 2, 1903.

[PLATE 2.]

SINCE DAVY'S discovery of the electric arc, a century ago, down to the present time,
the nature of the physical processes going on in it, and the mechanism hy which it
conducts electricity, have heen the subject of almost uninterrupted discussion and
experiment. In order to explain the fact that the equation connecting P.U. current
and length appears to contain a large practically constant term, experimenters have
assumed that the arc possesses resistance and E.M.F., though which of the two is the
more important in obstructing the flowr of the current, or whether both must lie
considered, has been, and is still, a matter of controversy, the settlement of which, it
is hoped, will be furthered by the experimental results described in this communi-
cation.

A priori it is highly probable that the resistance and E.M.F.'s of the electric arc,
if they exist, will be functions of the current ; it is therefore necessary to first
consider the definitions of these quantities, as it will largely depend on the definitions
adopted whether the arc can be said to possess a resistance, an E.M.F., or neither.
The ordinary text-book definitions of resistance and E.M.F. generally start with the
assumption that they are constant quantities independent of the current flowing, and
their possible variation is generally developed as a secondary effect due to the current
altering the state or nature of the body or apparatus considered, these alterations in
the state being the primary cause in the change in resistance and E.M.F. observed.

There is much experimental evidence to support the view that when a steady
current, A, flows through any conducting apparatus, the potential difference, V,
between its terminals can be written V = E + RA when E and R, the E.M.F. and
the resistance, only depend on the nature, state and movement of the apparatus, and
are not directly functions of the current A or the potential V ; so that the equation
connecting the P.D. and current for any apparatus under perfectly constant conditions

VOL. coin. — A 368. 2 R 9.8.04

306 ME. W. DUDDELL ON THE EESISTANCE AND

only contains the current raised to the powers 0 and 1, and the coefficients of these
terms are considered as constant specific properties of the conducting apparatus under
the given conditions. It is, however, quite conceivahle that conductors might exist
for which the equation might contain other powers of the current, and in this case
their coefficients would be equally justly considered as specific properties of the
conductor for which at present no names exist. Is the arc such a conductor ?

On the assumption that when the conditions are maintained constant the P.D. can
be represented by an equation of the form V = E + PA, for all values of the current,
then the power spent or furnished by the apparatus VA = EA + KA2, that is, it
consists of two parts — the one depending on A2, and therefore irreversible, so that the
sign of the power does not change with change of sign of the current, and the other
depending on the first power of the current, and therefore a reversible phenomenon,
so that if the apparatus absorbs energy when the current flows in one direction, it
will give back energy when the current flows in the opposite direction.

This idea of distinguishing E.M.F.'s from resistances, according to whether the
dissipation or absorption of energy is a reversible or irreversible phenomenon, is by
no means new, as it underlies the views expressed by Professor FITZGERALD* and
GRAY! in the letters they contributed to the 'Electrician' in the discussion of
Messrs. FRITH and RODGERS' paper,! and has also been suggested by Professor
S. P. THOMPSON. It seems to afford a satisfactory basis for a definition of resistance
and E.M.F., which will be adopted in this communication.

Definition..

Suppose any apparatus under any given set of conditions, through which a certain
current is flowing, and that it is required to determine its resistance and E.M.F.
under these particular conditions and for this particular current. The energy
transferred electrically between the source and the apparatus can be divided into two
parts : the one an irreversible part, so that if the direction of the current be conceived
reversed the direction of the transfer of energy remains unchanged, and the other a
reversible part. If it be found that the irreversible transfer of energy is proportional
to the square of the current, and the reversible to the first power of the current,
when in some way or other perfect constancy is maintained in all the conditions of
the apparatus, such as the size, shape, nature, temperature, temperature gradients,
relative movements, &c., of the different parts of the apparatus which are existing
with the particular current and under the given set of conditions, then the irreversible
rate of transfer of energy divided by the square of the current will be defined as the
resistance, and the reversible rate of transfer of energy divided by the first power of

* 'The Electrician,' 1896, vol. 37, pp. 386, 489.
t 'The Electrician,' 1896, vol. 37, p. 452.
\ 'Proc. Phys. Soc.,' 1896, vol. 14, p. 307.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 307

the current will be defined as the E.M.F. of the apparatus, under the given set of
conditions and for that particular current which was initially supposed to be flowing.

In this definition the qualification, " if the irreversible transfer of energy is
proportional to the square of the current and the reversible to the first power of the
current," predetermines that the apparatus must obey OHM'S law over whatever
range it may be possible to vary the current without in any way changing the state of
the apparatus, in order that it can be said to have a resistance at all. If, however,
the qualification is in any case not fulfilled, it will become necessary to consider the
terms in the conceivable equation between V and A other than those in which A
occurs to the powers 0 or 1.

So far nothing has been said of the signs which the two quantities resistance and
E.M.F. can have, as their signs are more or less a matter of convention. If we call a
current flowing round the circuit in the same direction as the E.M.F. of the source
would tend to make it flow a + current, and a transfer of energy from the source to
the apparatus a + transfer of energy, then their signs are determined and agree witli
ordinary practice, so that the resistance and the E.M.F. of the apparatus which
oppose the flow of the current will have -J- signs. It is to be noticed, however, that
this definition does not preclude in any way the possible existence of a negative
resistance ; for if, instead of an irreversible transfer of energy from the source to the
apparatus, proportional to the square of the current, there were found (the conditions
of the apparatus being, of course, maintained constant as before) to be a transfer in
the opposite direction, i.e., from the apparatus to the source, then the coefficient of A2
would have to be negative, so that in this case the apparatus would possess a true
negative resistance. Although in what follows it will be shown that this is not the
case with the arc, it is as well to draw attention to the matter, as a considerable part
of the controversy on the negative resistance of the arc under certain conditions arose
from some of those who took part defining resistance as an essentially positive
quantity, and then trying to prove that it could not be negative in the case of
the arc.

A single value of V corresponding to a single value of A is evidently not sufficient
to determine whether any conductor fulfils the above definition of resistance and
E.M.F. To determine this the current must be varied over some range, SA, and in
such a manner that the conditions of the conductor remain unchanged, and a series
of observations must be made within this range.

The essential stipulation, that the test must not alter the body tested, is the main
difficulty in the experiments on the resistance and E.M.F. 's of the arc. For it is well
known that, corresponding with each steady value of the current, the size and
configuration of the vapour column and craters are different in spite of the fact that
the length, the nature of the electrodes, and the other conditions may be kept
constant, so that the arcs corresponding with any two different steady values of the
current, however nearly equal they may be, are really two distinct and different

2 B 2

308 MR. W. DUDDELL ON THE RESISTANCE AND

phenomena. Therefore all methods which depend on the steady change 8V in the
potential difference V produced by a given steady change SA in the current A, that
is to say, which depend on an excursion on the steady curve between V and A,
however small it may be, simply measure the difference between the P.D.'s required
to maintain an arc with a current A and a distinct and different arc with a current
A Jt: SA, which is evidently no measure of either the resistance of the arc with
current A or with A ^ SA.

If the measuring current SA is only applied for a short time 8t, it is necessary that
the energy supplied to or removed by it shall be so small as not to appreciably alter
the thermal conditions of the very small mass of gaseous and other material which is
taking part in the conduction of the current. How extremely short the time that
may elapse is will appear later ; for the present it is sufficient to point out that it
has been found that even in - 7-5-^0 second a change of 3 per cent, in the arc current
has appreciably altered the thermal conditions and the light emitted by the arc.*

It thus appears that the only available methods of experimentally determining the
resistance and E.M.F. of the arc must be based on making the necessary change in
the main current, i.e., the measuring current, as small as possible, and on completing
the test so soon after making this change that none of the conditions of the arc will
have had time to appreciably alter before it is completed.

The first method tried consisted in sending the oscillatory discharge from a
condenser through the arc, and recording by means of an oscillograph the variations
in the P.I), between the terminals of the arc and in the current through it. If the
frequency of the oscillatory discharge can be made so high that the conditions of the
arc are not in any way altered by it, then the wave-forms of the oscillatory part of
the P.D. and current will be similar curves and in phase if the arc possesses a true
resistance. This was not found to be the case with the oscillations used, which had
frequencies up to 5000 -^ per second, the current oscillation always lagging behind
the P.D. oscillation.

At low frequencies and with solid carbon electrodes the oscillations were 180° out
of phase, and this difference was gradually reduced with increase in frequency to
below 90° at 5000 -r per second, and there were indications that this lag would
finally disappear if a much higher frequency were used, so that the conditions of the
arc were not altered by the oscillations.

A largo number of experiments, some of which have been published in a paperf
before the Institution of Electrical Engineers, were made to determine the effect of
small rapid variations in current on the conditions of the arc itself. The conclusion
drawn from the above experiments was that a very much higher frequency than
5000 — per second was necessary in order that the arc might not be affected by the
measuring current.

* 'Journal of the Institution of Electrical Engineers,' 1901, vol. 30, p. 236.
t Ibid.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 309

Owing to various reasons, the above method was not suitable for these higher
frequencies; consequently a new method was devised similar to that used by
Messrs. FRITH and RODGERS,* based on the R.M.S. values of the superimposed P.D.
and current, and not on the instantaneous values of these quantities, with this
difference from Messrs. FRITH and RODGERS' method, that there was a criterion, when
a result was obtained, as to whether the arc was behaving like an ordinary resistance
or not.

Basin of Method Adopted.

Consider any apparatus A, fig. 1, which may have resistance and E.M.K. but no
self-induction or capacity, through which a steady direct current may be flowing, and

Fig. 1.

let there be mixed with the direct current an alternating testing current of R.M.S.
value C.

Let VA be the R.M.S. value of the alternating part of the P.D. between the
terminals of A, and let rA and c be the instantaneous values of these latter quantities.

The impedance of the apparatus A = \/ (rr ->\~dt} \/ (-,„( c*dt)=~VJG = IfL.

\1 -II / / V \ 1 J ()

Suppose that the frequency of the alternating current can be made such that the
conditions of the apparatus are not in any way changed by the alternating current,
then if the apparatus has a true resistance it will be a constant, so that the
instantaneous values rA and c will have a constant ratio, i.e., will obey OHM'S la\v.
Then the wave-forms of VA and ( I will be similar curves and in phase, and the true
resistance of the apparatus = vjc = VA/C = IA.

A criterion is now required that rA and c do obey OHM'S law, ami this is supplied
by the power-factor of the apparatus A, the power-factor being denned as

For it can be proved that the necessary and sufficient condition that the power-
factor may be unity is that the wave-forms of VA and C are similar curves and in
phase, so that VA and c obey OHM'S law.

Therefore if, when the current C and its frequency are such that the conditions of
the apparatus are not changed, it can be proved that the power-factor of A is unity,
then A has a true resistance numerically equal to IA. Further, in any apparatus in

* 'Proc. Phys. Soc.,' 1897, vol. 14, p. 307.

310 ME. W. DUDDELL ON THE RESISTANCE AND

which the resistance is a function of the conditions, the possibility of obtaining the
power-factor unity is a proof of the constancy of the resistance and consequently of
the conditions, so that if the apparatus A is an arc, and if it can be shown that a
sufficiently high value of the frequency can be reached for which the power-factor is
unity, then the conditions of the arc are not being altered by the alternating testing
current, and the arc has a true resistance numerically equal to IA.

It is assumed above that the arc or apparatus A has no self-induction or capacity ;
to prove this it must be shown that not only can the frequency be increased till the
power-factor becomes unity, but also that it remains so for a considerable further
increase of frequency.*

Finally, therefore, in order to prove that the arc has a true resistance, and to find
its value, it is necessary to show : — First, that it is possible to find a value of the
frequency of the alternating testing current for which the power-factor of the arc
with respect to this current is unity ; second, that the power-factor remains unity
and the impedance constant even when the freqiiency is greatly increased above this
value ; third, to determine the value of the impedance of the arc under these
conditions, which will also be its true resistance.

Method of Measuring the Impedance and Power-Factor.

At 'first sight it would seem as if there were a considerable number of available
methods for accurately measuring these quantities. But the number of methods
becomes exceedingly limited when it is considered that it is necessary for the
alternating testing current C to have as small a KM.S. value as possible (O'l ampere
was that generally used in the experiments), and that the effects due to this small
current have to be sorted out when it is mixed with a direct current of 10 amperes
or more. Added to this, to make the difficulties greater, it was finally found
necessary to use frequencies up to and even over 100,000 — per second. Watt-
meters and dynamometers were tried and abandoned, and finally the well-known
3-voltmeter methodf was adopted.

A non-inductive resistance R was placed in series with the apparatus A (fig. 2),
through both of which the main direct current flowed ; to this direct current there
was added, as before, an alternating measuring current of R M.S. value C.

Let VA, VB, and V be the E.M.S. values of the alternating part of the P.D.'s as
shown in fig. 2. The impedance of A is IA = VA/C = RVA/VE. Power factor of
A is PA = (V2 - VA2 - VB2)/(2VAVE).

It seems possible that the power-factor of a conductor which did not possess self-induction or resistance
in the ordinary sense of these terms might still depart from unity at very high frequencies, owing to the
time taken by the carriers of the electric charge to hand it on becoming comparable with the periodic
time of the testing current.

t See AYRTON and SUMFNER, ' Roy. Soc. Proc.,' vol. 49, p. 424.

ELECTEOMOTIVE FORCES OF THE ELECTRIC ARC.

311

The two quantities, the impedance and power- factor, are therefore determined in
terms of a resistance R and three R.M.S. voltages quite independent of any

--v-

Fig. 2.

assumptions as to the wave-form of the alternating testing current. If the same
voltmeter is used to measure each of these voltages, then it will be noticed that the
results only depend on the relative calibration of one instrument, a consideration of
great importance owing to the difficulties in the way of accurate absolute calibration
with the very high frequencies used.

Circuit and Apparatus Used.

In order to measure the impedance and power-factor by the method just considered,
several different arrangements of the circuit were tried ; that finally adopted for the

I/0'.

Fig. 3

experiments is shown diagrammatically in fig. 3. The main direct-current circuit
consisted of: —

312 MR. W. DUDDELL ON THE RESISTANCE AND

B, a battery of from 50 to 90 accumulators which supplied the direct current to

the arc,

L, a self-induction,
p}, p.,, adjustable resistances,

A, a Weston ammeter which indicated the direct current through the arc,
R, the standard non-inductive resistance, consisting of 12 coils of about 0'5 ohm

each, described later,
tl and £2, the terminals of the arc lamp,
£3 and t4, terminals which could be moved along R,
I, 2, and 3, mercury cups in a wax block,
fi,fz, and y^, fine fuses in the connections between the above points.

The direct P.O. between the terminals of the arc lamp (called " P.D. arc lamp")
was measured with a Weston voltmeter (not shown) which was connected between
the points 1 and 2. The same voltmeter, which could be connected between points
2 and 3, gave direct P.I), between the terminals t.2 and 1?> (called " P.D., R"), from
which the resistance of 11 was obtained in terms of the readings of the voltmeter and
ammeter, both of which were carefully standardised.

An image of the arc was projected by means of a lens on to a screen, divided and
marked so as to read the actual distance in millimetres between the ends of the
carbons ; this distance is called the " arc length."

The circuit for adding the alternating testing current to the direct current
consisted of:—

D, a special high-frequency alternator which supplied the testing current, capable

of producing small alternating currents having frequencies up to 120,000 -»-

per second,

p3, a variable resistance,
S2 and S3, plug switches,
f±, a fine fuse,
T, an Ayrton and Perry reflecting twisted-strip ammeter, having a sensibility of

400 scale divisions for O'l ampere at a scale distance of 1700 divisions

(1 division =: -fg inch),
F, a condenser to prevent any direct current from flowing through the alternator,

the capacity of which was 1 mf. for the frequencies from 120,000 to 50,000 -*-

per second ; 2-5 mf. for frequencies from 50,000 to 2000 -»- per second; and

increased up to 15 mf. at 250 -»- per second.

The alternating current supplied to the arc circuit was kept at a constant R.M.S.
value as read on T by means of the adjustable resistance pz ; this current flowed
through the arc and R in series, and was practically prevented from flowing through
the battery by the self-induction L. At the higher frequencies of from 10,000 to
120,000 — per second this self-induction behaved almost like an insulator; at the

ELECTROMOTIVE FORCES OP THE ELECTRIC ARC. 313

lower frequencies a small percentage of the alternating current flowed round the
battery side of the circuit ; but this did not affect the accuracy of the results, as the
expressions for the impedance and power- factor are independent of the current, so
that it is only necessary to maintain the testing current constant during each test.

The circuit for measuring the alternating part of the P.D. arc lamp VA, the P.D.
between the terminals of K, VB, and the P.D. total. V, consisted of :—

M, a thermo-galvanometer, whose deflections were practically proportional to the
mean squared value of the current through it, and which, though practically
non-inductive, gave a deflection of about 500 scale divisions for 1 milliampere
of alternating current. The deflections of this instrument were read on the
same scale as those of T, so that both could be observed at one time ;

p4, an ordinary resistance box,

k, a key,

S1( a switch, consisting of mercury cups in a wax block,

G, a condenser which allowed a current due to the alternating part of the P.D. to
flow through M, but prevented any current due to the direct P. I.), from
flowing through it,

4 and 5, mercury cups by means of which the measuring circuit could be connected
to either the points 1 and 2 ; 2 and 3 ; or 1 and 3 ; to measure VA. VR, and V
respectively.

The impedance of the measuring circuit, which consists of the thermo-galvanometer
M, the resistance pt, and the .condenser G, need not be accurately known, as it is only
the relative values of VA, Va, and V which are required to a high degree of accuracy
and not their absolute values. As the frequency in each experiment was kept
constant to well within 1 per cent., the impedance of the measuring circuit need not
be quite independent of the frequency.

In all arcs neither the direct P.D. nor the current keep quite steady, owing to the
necessity of feeding the carbons together, and to the impurities, cracks, &c., existing
in them, so that the comparatively slow variations produced must be prevented from
sending any appreciable currents through M ; for this reason the capacity of G was
made as small as compatible with the impedance of the measuring circuit, not
depending too much on the frequency. A standard j mf. condenser was used for G
for all frequencies from 120,000 to 10,000 -»- per second inclusivs ; down to 3,000
1 mf. was employed, and for all lower frequencies 5 mf. Even under these conditions
it was absolutely essential that the battery B should not be in use for any other
experiments, or spurious currents were obtained through M, and such a thing as the
arc giving a small hiss sent the spot off the scale.

The key k was so arranged that in its up position the condenser G was always kept
charged to the correct. P. D., so that on depressing it G was neither suddenly charged
nor discharged through M, for owing to the delicate nature of the latter any

VOL. com. — A. 2 s

314 . MR. W. DUDDELL ON THE RESISTANCE AND

considerable sudden change in the P.D. between the armatures of G sent sufficient
current through M to burn it up. Even this precaution did not prevent M from
being burnt up several times, owing to the battery connections being broken or the
arc going out while the key k was depressed.

The switch St was to enable M to be completely disconnected when taking its zero,
as it was thought that at these high frequencies, so long as one pole was connected,
there might be a small current flowing into the instrument, due to the capacity of the
instrument with surrounding objects, but this effect was not observed.

Owhig to the very high frequencies used, very great care had to be taken in
arranging the circuit so as to avoid self-induction and capacity errors in the leads and
connections. All the leads through which the alternating current flows were carefully
twisted and bound together.

To reduce any possible error caused by the lead tj) between the arc and R, the
drop along which had to be included either with the arc or with R, this lead was
made by twisting together 12 No. 23 double cotton-covered wires, to avoid possible
skin effects, and its length was reduced to GO centims. As first constructed, each
wire of the lead t.J> was twisted with the corresponding wire of lead t^, which was of
the same length and made in the same way, and then the 12 pairs of wires were
twisted together. It Avas found on testing these leads that at a frequency of
18,000 -»- per second, and with a P.D. of 33 volts between the two leads, an
alternating current of about 1'9 X 10~:! ampere floAved between them due to their
capacity. To reduce this capacity current, tAvo IICAV leads were constructed, exactly
the same as before, only that instead of twisting the individual wires belonging to
each lead together in pairs, all the Avires belonging to each lead were stranded
together so as to form tAvo separate leads. The lead tj) Avas then bound over with a
layer of silk tape and the t\vo leads were twisted together. On re-testing in
the same way as before, the capacity current was found to be reduced to about
0'53 X 10~3 ampere with 33 volts between the leads. As the alternating P.D.
between the leads Avas under 0'5 A^olt in most of the experiments, this capacity
current Avas negligible compared Avith the working current of O'l ampere.

As the self-induction of II had to be determined and allowed for, the' lead tjb was.
included with II, so that its small self-induction could be corrected for at the same
time.

The lead c, d Avas brought back along the connections b, c between the 12 coils
of R, so as to neutralise as well as possible the magnetic field of these connections.

In arranging the circuit the capacity of those parts of the main circuit between the
measuring points t{, tz, t3, as Avell as of the whole of the measuring and alternator
circuits, to surrounding objects and to earth, was kept as small as possible, so as to
avoid what might be called capacity leaks. The alternator itself was practically
insulated from earth by being fixed down to a wooden frame, and the field circuit of
the alternator was well insulated and removed from earth.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 315

To form some idea of the magnitude of these capacity currents or leaks, the four
points a, b, c and the arc, were opened in pairs. One of the opened points was
re-connected through the thermo-galvanometer used as an ammeter to measure the
capacity current supplied to that part of the circuit between the instrument and the
other point that was opened. The direct current circuit was disconnected at the
points a and d. The alternating currents observed are given in Table I. for a "R.M.S.
P.D. of 3'65 volts and a frequency of 100,000 ^ per second.

TABLE I. — Capacity Currents in Leads.

Thermo-
galvanometer

Points.

Alternating
current

at—

a.

b.

c,.

Arc.

in 10~l ampere.

a

—

closed

closed

open

:: • s

a

—

open

closed

1-0

b

closed

closed

open

2-9

b

)*

—

open

closed

1 '2

c.

»

closed

—

open

:i • 9

C:

D

open

closed

1-2

The maximum value of the capacity leak observed is 3 '9 X 1.0-' ampere at the
highest frequency used in any series of experiments, and at a P.I), about seven times
as high as that used, so that if the capacity current is proportional to the P.U., it
should not exceed O'G X lO"4 ampere, or about O'OO per cent, of the working current
of O'l ampere, and may therefore be neglected. Even if this capacity leak had been
many times larger, it would not have appreciably affected the P.U.'s measured, since
at these high frequencies the arc behaves like a non-inductive resistance, and
therefore the measuring current and the capacity current would add approximately
as vector quantities at right angles.

As it is only the relative values of VA, VK, and V that are required very accurately,
any small self-induction in the leads connected with the points 1, 2, and 3, or in the
measuring circuit itself, is of no importance. Nevertheless, all the leads were
carefully stranded together to prevent any E.M.F.'s being induced in this circuit,
caused by magnetic induction. The only wire which could not be stranded with a
corresponding wire was about 30 centims. of the lead between the fuse f3 and the
movable contact t3. Experiments were made by varying its length and position to
see if it introduced any error, but none could be detected.

The condensers F and G were placed some distance apart, so as to prevent any
direct electrostatic action between them. This, as well as any mutual induction, both
electrostatic or magnetic, between any part of the main or alternator circuits and the
measuring circuit was examined for, but none could be detected. Experiments were

2 s 2

31(5

MR. W. DUDDELL ON THE EESISTANCE AND

also made to see if the condenser G really prevented M from being deflected by steady
direct P.D.'s ; for this purpose steady P.D.'s of from 100 to 180 volts were applied
to the terminals of the arc, the carbons being separated, and M was connected in the
ordinary way as if to measure VA. A deflection of M might have occurred due to
leakage or electrostatic forces, but no such deflection was observed.

The Arc Lamp.

A hand-fed arc lamp, enclosed in a,n iron case, was used in all the experiments.
The sliding contacts were shunted by flexible leads, so as to avoid any uncertainty in
the resistance of these contacts. The resistance of the contacts and of the carbons
was determined by adjusting the carbon holders to 10 centims. apart as in use, short-
circuiting them with a brass rod and the different carbons in turn, and then measuring
the drop in volts between the terminals of the lamp when 10 amperes direct current
flowed round the frame and holders, etc. The results are given below : —

TABLE II.

Carbon holders short-circuited with

Brass rod

11 millims. solid "Conradty Noris" carbon.

11 ,, cored ,, ,, ,

11 ,, solid " Lc Carbon" electrographitic carbon

Resistance.

ohms.

0-0017
0-13
0-22
0-048

The self-induction of the loop formed by the frame of the lamp with the carbon
holders 10 centims. apart and short-circuited by the brass rod was approximately
determined by passing an alternating current rpund the lamp and measuring the
P.D. between its terminals, at a frequency of 30,000 — per second. The value
obtained was 2'4 x 10~7 henry, which was used as a correction. All the above tests
were taken with the lamp in place in its case, and with the carbon holders, etc., in
exactly the positions they occupied when commencing a test on the impedance of
the arc.

The. Standard Non-Inductive Resistance, R.

The essence of the test of the arc consisted in comparing its behaviour to alternating
currents of various frequencies witli that of the standard resistance R in series with
it. It was, therefore, necessary that R should be as free as possible from self-
induction and capacity. The type of resistance adopted was that described by
Professor AYIITON and Mr. MATHER before the Physical Society* in 1891.

As both the self-induction and the capacity depend on the size of the resistance, it

Philosophical Magazine,' 1892, vol. 33, p. 187.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC.

317

Top
Carbon H( 'Icier

was decided to make the resistance as small as possible and to allow the temperature
of the strips to rise considerably, which necessitated slightly modifying their design.
The resistance consists of 12 platinoid strips, each about 170 centims. long, 2 '5 centime,
wide, and 0'076 millim. thick. Each strip is folded back on itself and has its ends
soldered to two brass blocks let into the top of the frame, and is stretched tight with
a tension of about 4 Ibs. by means of a brass spring attached to a small glass tube,
about 5 millims. diameter, at the bottom of the loop formed by the strip. Between
the up and down sides of the strips is placed a sheet of asbestos millboard, about
-3-2- inch thick, and the strips are pressed together against this by glass rods from
side to side of the frame.

The resistance of each of the 12 strips was roughly adjusted to 0'5 ohm, and the
strips could be used in series or parallel by connecting up the brass blocks forming
the ends of the strips with copper links and set-screws as required. Owing to the
considerable heating of the strips by the current, their resistance depended on the
current; thus the resistance of all the strips in series, which was (3 '00 ohms with
1 ampere flowing, rose to 6'25 ohms with the current
of 10 amperes which was used in many of the experi-
ments. For tin's reason, and because it formed a
check on the instruments, the resistance of II was
determined during each experiment from the known
values of the direct current and P.I), between its
terminals, and the value so obtained was used in
calculating the results.

The apparent self-induction of 11, including the
connection between it and the arc lamp already
mentioned, was measured by comparing it with a
non-inductive resistance put in place of the arc.
This latter resistance (see fig. 4) was made to imitate
an arc possessing non-inductive resistance localised
between the ends of the carbons. Tt consisted of
158 millims. of No. 38 platinoid wire bent back on
itself, the two extremities being soldered to the ends,
previously copper plated, of two solid " Conradty
Noris" carbons; these carbons were held in the
carbon holders of the lamp so that the resistance wire
occupied the position the arc would when burning. A piece of mica was interposed
between the ends of the carbons which served to keep the loop of wire taut. This
method of determining the correction to be applied to 11 really converts the test of
the arc into a substitution test, for having determined how an ordinary metal
resistance behaves when localised between the carbon tips, the behaviour of an arc
substituted for it under exactly similar conditions was compared with it.

Mica _._V-- '

FT"

\Vir<-

Ikittom
Carbon lio

Fig. 4.

318

ME. W. DUDDELL ON THE RESISTANCE AND

To compare R with the wire, they were adjusted to have practically the same
resistance. A current, either direct or alternating, having a R.M.S. value of
O'l ampere as indicated by T, was sent through R and the wire in the place of the
arc in series, fig. 3. The P.D. between the terminals of R and between the terminals
of the arc lamp (i.e., wire) was measured by means of M, in the same way as in the
experiments on the arc, the condenser G being short-circuited. The amount by
which the impedance of R exceeded the impedance of the arc lamp and wire is
tabulated below for the strips 1 to 7 having a resistance of 3 '50 ohms. Each of the
results is the mean of at least 1 2 comparisons. The self-induction of the loop formed
by the frame of the arc lamp has been found to be about 2'4 X 10~7 henry, and the
self-induction of the wire itself was calculated to be about 3 X 10~8 henry ; so that
the total self-induction of the lamp and wire is 27 X 10~7 henry. Assuming that
the alternator gives a sine-wave form, which was approximately the case, and
allowing for this self-induction and for the difference in resistance 0'03 per cent.
between the lamp and R, the true power-factor, cos 77 of R, has been calculated,
from which TJ the lag of the current in R behind the P.D. and the time constant
have been deduced.

TABLE III. --Test of R, Resistance 3'50 ohms.

Frequency -f- per second.

0 (direct current)
32,400
41,000
50,700
60,000
80,000

100,000
120,000

Impedance of

Time

Impedance

11 > impedance of

COS 1].

rl-

constant in

of R > its

lamp and wire.

10^7 second.

resistance.

per cent.

per cent.

0-03

1-0000

0

0

0-17

0-998:,

3 8 2-

0-15

0-25

0-997,j

5 58 2-7

0-24

0-39

0-996!

5 4

2-8

0-39

0-51

0 • 994S

5 5! 2-7

0-52

0-90

0 • 990(i

7 5,

2-8

0-94

, . 9 of mean of
\ 24 tests

| 0-9863

9 30 2-7

l-3r

1-85

0-980!

11 27 2-7

l-99

Method o/" Experiment.

The arc length, the direct current, and the frequency of the alternating testing
current having been decided upon for any experiment, the test was carried out in
the following manner. The carbons having burnt into shape corresponding with the
required current and length, a rough experiment on the impedance of the arc was
made, and the value of R adjusted by moving £3 (fig. 3) to that contact on R which
made VA and VK most nearly equal, as this gives the greatest accuracy in the power-
factor by the 3 -voltmeter method.

The main direct current was now interrupted and the arc short-circuited by pressing

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 319

the carbons together. The testing current was adjusted to its working value, in
most cases O'l ampere, as read on T, and the resistance pt in series with M was
adjusted until VB gave a deflection of 100 scale divisions on M. The reason that this
adjustment was made without the direct current flowing was that it formed a check
on the satisfactory working of the measuring circuit, since any apparent change in
the sensibility of M when the direct current was re-established would have indicated
an error somewhere.

With solid carbons the positive or upper electrode was adjusted to project G centims.
from its holder and the lower or negative 4 centims. With cored carbons these
lengths were 7 and 4'5 centims. respectively. The object of adjusting these lengths
was to make the mean resistance and self-induction of the loop formed by the frame
of the arc lamp and carbons as nearly as possible the same in every experiment.

The arc was now re-started, and the length and direct current having been adjusted,
the carbons were fed together as they burnt away, so as to keep the direct current
constant during the whole of the time (about half an hour) that VA, VR, and V were
being determined. This kept the P.D. arc constant, as long as the P.I), of the
battery remained constant ; any slight drop in this P.I), was compensated for by
adjusting plf any considerable drop necessitated recommencing the experiment. The
arc length with solid carbons also remained constant ; but with cored carbons it
constantly varied about a mean value according to the amount of material from the
core present in the arc. In all cases, readings were oidy taken when the length was
observed to be correct as well as the direct P.D. and current.

As soon as about 5- millims. had burnt off the end of the positive carbon, the
deflections of M corresponding to VA, VR, and V were observed in turn, the zero of M
being taken after each reading, until in most cases five consecutive sets were obtained
which were reasonably consistent with one another, the li.M.S. value of the testing
current as read by T and its frequency being kept constant. It was easy to obtain
individual deflections corresponding with VK which differed from the mean by less
than 0'3 per cent. The deflections corresponding with VA and V were not so definite,
VA being within 1 per cent, and V within O'G per cent, of the mean, except in a few
exceptionally unsteady arcs, such as long-cored arcs and small-current arcs.

The values of the direct P.D. arc lamp and direct P.D. R were noted, and the
drop of volts in the frame of the lamp and carbons was found by pressing the carbons
together, the direct current being so adjusted that when the carbons were in good
contact its value was that used for the experiment. This observation was repeated
until consistent results were obtained with the carbons hot as in use. By deducting
this value from P.D. arc lamp, P.D. arc was obtained.

The relative calibration of the thermo-galvanometer M was then determined by
means of direct currents. This completed the observations required for a single
experiment.

Mean deflections corresponding with VA, VB, and V, were calculated and corrected

320 MR. W. DUDDELL ON THE RESISTANCE AND

*

for the relative calibration of the thermo-galvanometer. From these values the
impedance of the arc lamp IA = IB, VA/VE, and the power-factor of the arc lamp,
PA = (V* - VA2 - VE3)/2VAVU, were calculated.

In order to obtain from these values the impedance and power-factor of the arc
itselj, a small correction had to be applied to IA for the resistance and self-induction
of the loop formed by the frame of the lamp and the carbons, and also to PA for the
self-induction of R which had previously been determined. To make these small
corrections, it was necessary to assume that the alternating current had a sine-wave
form, which was approximately the case.

As a check on the method of experiment and on the calculation and correction of
results, the impedance and power-factor of the platinoid resistance, described on p. 317,
which had a resistance of 3'499 ohms and a self-induction of about 3 X 10~8 henry,
were determined, the experiment and calculations being performed in the same manner
as for the arc. The values obtained were : impedance 3'50 ohms, power-factor 0'999,
which show that the method was satisfactory in this case.

Results Obtained by Varying the Frequency.

The fundamental experiment of this investigation into the resistance of the electric
arc consists, as has already been explained, in varying the frequency of the super-
imposed alternating testing current, in order to determine whether with a sufficiently
high frequency the condition of the arc will remain unchanged, the value of the
resistance being then measured at this frequency. The criterion that the conditions
of the arc remain unchanged has been shown to be that the power-factor of the arc
as measured with the superimposed alternating current must be unity. The true
resistance will then be equal to the impedance.

The results of the experiments on the effect of varying the frequency on the power-
factor and the impedance for solid and cored* arcs are represented graphically in
Curves I. and II. (Plate 2).

With solid carbons the power-factor at 250 — per second is — 0'91. On increasing
the frequency it decreases numerically until it vanishes and changes sign at 1,950 ~»-
per second, the waves of superposed alternating P.D. and current being then 90° out
of phase. With further increase of frequency the power-factor increases rapidly at
first, then more and more slowly, becoming asymptotic to -f- 1, and finally practically
attains this value at a frequency of 90,000 -»- per second ; above this frequency the
power-factor is, within the limits of experimental error, equal to + 1 up to the
highest frequency attained, namely, 120,000 -~ per second. The impedance of the
solid arc increases with increase of frequency from 0'97 ohm at 250 — to 3'8 ohm at
a frequency of 90,000 -*- per second, above which it remains practically constant.

"Solid" and "cored" arc mean respectively arc between two solid carbons and between two cored
carbons.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 321

At frequencies above 90,000 the power-factor is + 1, therefore the excursions of the
P.D. and current obey OHM'S law, and the impedance of the arc is equal to its true
resistance. So that the true resistance of an arc, 3 millims. long, between 11 millims.
solid " Conradty Noris" carbons, and through which a current of 9 '91 amperes is
flowing, is 3*81 ohms.

The P.D. between the terminals of the arc, accounted for by ohmic drop in the arc,
is therefore 37 '8 volts out of an observed P.D. arc of 49 '8 volts, so that there appears
to be a red back electromotive force opposing the flow of the currents in this arc of
12 volts.

Considering next Curve II. for both cored carbons, the power-factor at the lowest
frequency of 250 — per second has a positive value of + O'f>7 and increases
asymptotically, as in the case of solid carbons, until it is practically +1 at a
frequency of 15,000, and remains unity within the limits of experimental errors up to
the highest frequency tried of 50,000 -^ per second, the impedance becoming
practically constant, as with solid carbons.

Therefore tlie, true resistance of fin arc 3 millims. !</>/,</, bet tree it I 1 mil limn, cored
" Conradty Noris" carbons, and t/ir.nif/li tchic/i, a current of 10 amperes is flowing, is
2'54 ohms, and the back E.M.F. is IG'i) volts, calculated in the same way as for solid
carbons.

Finally, therefore, arcs between either solid or cored carbons have both back
E.M.F. and resistance, and the true values of these quantities differ greatly from
those usually assigned to them.

In order to test whether the II. M.S. value of the added alternating current affected
the values obtained for the impedance and power-factor, the testing current was
varied over the range 0*030 ampere to 0*130 ampere, and the impedance and power-
factor were found to be constant within the limits of experimental error.

It is of interest to enquire how the results obtained by Messrs. FRITH and RoDGERS
can be explained by the aid of these curves. The quantity that they measured and
called the resistance of the arc was really the impedance of the arc and a certain
resistance together, less the impedance of the resistance part alone. From fig. 2,
p. 311, using the same notation as before and assuming II non-inductive,

PVC = VBC + PAVAC, or PV/C = U + P.JA.

The quantity measured by Messrs. FRITH and RODGKRS was V/C — II.

In the case of solid carbons at their frequencies, about 100 ^ per second, PA is
roughly — 1, and a little consideration shows that under these conditions P was
practically + 1, so that the quantity they called the resistance of the solid arc was
PAIA, or the product of the power-factor into the impedance of the arc, which is
evidently a negative quantity for frequencies under 1950 — per second with the
solid arc investigated in Curves I. As they did not use the same make or size of
carbons as those used in this paper, it is impossible to make an accurate comparison

VOL. cm. — A. 2 T

322 MR. W. DUDDELL ON THE RESISTANCE AND

between the results obtained. Taking arcs of tbe same lengtb and current between
"Apostle," "Brush" and " Carre " carbons, the mean of the value of what Messrs.
FRITH and RODGERS called the resistance of the arc is about 0'79 ohm, and this
agrees very well with the value of PAIA obtained by extra-polating Curves I. back to
about 100 -r- per second. So that this curve explains both the sign and the value of
the so-called negative resistance of the solid arc.

With cored carbons, PA will only be about + 0-5 at 100 frequency. The value of
P is unknown, but is certainly less than unity, so that the quantity measured by
Messrs. FRITH and RODGERS for cored carbons must be less than PAIA. By extra-
polation from Curves X., the value of PAIA at 100 frequency for cored " Conradty
Noris " carbons is about + 0'5 ohm. Owing to the indefinite nature of cored carbons,
this value cannot be expected to agree very well witli those obtained by Messrs.
FRITH and RODGERS for carbons of other makers. Taking arcs of the same length,
and current as that used in Curves II., the values given by Messrs. FRITH and
RODGERS for different makes are: — "Apostle," 0'03 ohm; "Brush," 0'59 ohm;
" Carre," 0'55 ohm. The two latter values agree with the value 0'5 of PAIA deduced
from the curves for " Conradty Noris " carbons to a higher degree of accuracy than
might reasonably be expected, whereas the disagreement in the case of " Apostle "
carbons, as measured by Messrs. FRITH and RODGERS, seems to indicate that their
result is in some way abnormal, possibly due to some accidental impurity in the core.

The critical frequency of 1 '8 ^ per second observed by MESSRS. FRITH and
RoDGERS for the cored arcs, at which the quantity they measured changes sign,
evidently corresponds with the power-factor changing sign, so that the curve between
power factor and frequency would cut the zero line at a frequency about 1'8 per
second, and it is evident that at lower frequencies it would become negative, an
increase of current being accompanied by a decrease in P.D. The curves for the
cored arc, if they could be obtained down to sufficiently low frequencies, would
therefore lie similar to those obtained for the solid arc. The great difference
between the solid and cored arcs lies in the much greater quickness with which the
conditions of the former can vary, corresponding to any change, however small, in
the current through the arc. The relative sluggishness of the cored arc is probably
due to the presence in its vapour column of saline matter derived from the core.

The fact that the solid arc lias a negative power-factor at frequencies below the
critical frequency of 1950 -»•• indicates that the arc is under these conditions supplying
power to the alternating current circuit, and that this is the fact can easily be shown
experimentally by connecting a wattmeter so as to measure the power supplied to
the solid arc by the alternating current, when it will be found that at low frequencies
the solid arc is actually supplying poiver to the alternate-current circuit, ivhile at
frequencies above the critical value the alternate-current circuit supplies power to the
arc. This observation is of course not in any way at variance with the principle
of conservation of energy, since the alternating energy given out by the arc is

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 323

derived from the direct-current energy supplied to it, the arc acting as a converter.
This fact, that the solid arc is capable, under suitable conditions, of automatically
transforming energy derived from a source of direct current into alternate-current
energy of any frequency over a wide range, is the explanation of the transformation
of energy observed in the Musical Arc recently shown for the first time at the
Institution of Electrical Engineers.*

Effect of Varying the Direct Current.

Having found that with a sufficiently high frequency it was possible to measure
the true resistance of the arc, this resistance was measured for arcs of a fixed length

120

S B 7

Direct Current through Arc.

* ' Journal of the Institution of Electrical Engineers,' 1901, vol. 30, p. 248.

2 T 2

324

MR. W. DUDDELL ON THE RESISTANCE AND

of 3 millims. with different values of the direct current. The results of these tests
for solid and cored arcs are plotted in Curves III. and IV. The frequency of the

•j'wiH^a pu*

•sqiQA ft _J| SL

I

_3

•3

1

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ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 325

testing current was 100,000 — per second with solid arcs, and 26,000 ^ with cored
arcs. In every experiment the power-factor of the arc was also determined, and
found equal to + 1 to within the limits of experimental error, except for a very few
small current arcs whose resistance was so high that the power-factor could not he
measured with any certainty.

In the Curves III. and IV. there is plotted, besides the resistance of the arc and
the P.D. between its terminals, the back E.M.F., calculated as the difference between
the P.D. and the product of the direct current into the resistance, or ohmic drop in
the arc.

The resistance of both the cored and the solid arcs increases with decrease of the
direct current, apparently tending to become infinite for current zero. The back
E.M.F. of the solid arc first decreases with increase of current and then slightly
increases again, having a minimum of 11 '3 volts at about (! amperes. With cored
carbons the back E.M.F. increases with increase of current, from 12 '2 volts at
1 ampere to 18 '5 volts at 20'8 amperes. It is curious to note that the back E.M.F.
of solid arcs is larger than that of cored arcs for .small currents, the reverse being
the case with the larger currents.

As the back E.M.F. does not vary much for any of the ares, the whole of the
values observed being between 11 "2 volts and 18'f> volts, the high P.D.'s required to
maintain very small-current solid arcs is mainly due to the resistance of the arc, and
not to the change in its back E.M.F.

The connection between the resistance r and the current A for the cored arc, length
3 millims., between 11-millim. " (Jonradty Noris " carbons, can be approximately
expressed over the range from 1'5 amperes to '20 amperes by the very simple relation
(r + 0'25)A = 29.

For the solid arc, length 3 millims., between 11-millim. " Conradty Noris" carbons,
no such simple relation seems to exist; but the curve may be approximately repre-
sented over the range I'D amperes to 11 amperes by the relation r = 33'5A~' -|-42A"~~.

Effect of Vdryinc] the Arc Lenyfh.

The direct current through the arc being kept constant, the connection between
the back E.M.F., the resistance, and the length, is given in Curves V. and VI.

With both solid and cored arcs the effect of increasing the length is to increase the
resistance, though not proportionately to the length, the curve between resistance
and length being very similar to that between P.D arc and length. This latter
curve is generally assumed to be a straight line, but such is not the case over
the wide range of lengths 1 millim. to 30 millims. used in these experiments
(see Appendix I.).

The back E.M.F. of the solid arc is nearly independent of the length, dropping
slightly to a minimum at 4 millims. and then rising again. With the cored arc the
back E.M.F. decreases with increase of length.

326

ME. W. DUDDELL ON THE RESISTANCE AND

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ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 327

Effect of Varyimj the Nature of the Electrodes.

The results of experiments to determine whether the resistance and back E.M.F.
depend on the size and nature of the electrodes are given in Table IV. The effect
of the size of the electrodes is not well marked in the case of " Conradty " and
" Apostle " carbons, the changes in size being probably too small to make the effect
very evident; the resistances of the arcs are, however, slightly larger with the
smaller carbons. With " Le Carbone " Electrographitic solid carbons the impedance
of the arc between two 11-millirn. carbons is about 17 per cent, higher than that
between two 9-millim. carbons. Observations on the arc between these two 11-millim.
carbons, which was very unsteady and difficult to maintain, lead the author to think
that the observed difference in impedance is not due to the change in size of the
electrodes, but to the material of the two sizes of carbons being different, though
nominally the same. It is also to be noted that the power-factor, 01)2 of the arc
between the 11-millim. carbons, is the only one which has not been found equal to + 1,
to within the limits of experimental error, at a frequency of 100,000 — per second.
To be quite certain that this was not owing to some error the experiment was
repeated, but with practically identical results.

It may be mentioned that these Electrographitic carbons are not ordinary arc
lamp carbons, but were specially made for the experiments by " Le Carbone." They
are supposed to consist of pure graphite, and they are said to be made by expelling
the remaining impurities from carefully prepared carbon by heating it in an electric
furnace until the impurities are volatilised.

Both the resistance and the hack E.M.F. of the arc depend greatly on the make of
carbon, that is to say, on the composition of the electrodes, since it is very improbable
that any two makers' carbons have identical chemical composition. The experiment
of soaking a pair of solid "Conradty Noris " carbons in potassium carbonate, drying,
and re-determining the resistance, shows that the effect of introducing this potassium
salt was to reduce the resistance from 3 '81 ohms to 2 '92 ohms, and to increase the
back E.M.F. from 12 volts to 15 volts for the same arc length and current. A
similar effect is produced by drilling out one of the carbons and inserting a glass rod
as a core, probably due to the introduction of sodium into the vapour column. The
lower resistance and higher back E.M.F. of arcs between cored carbons than of those
between solid carbons is also probably due to a similar cause, namely, the presence of
potassium silicate in the core. In fact, it seems probable that the whole of the
observed differences between solid and cored arcs, and between arcs for which
different makes of carbons are used, not only in resistance and back E.M.F., but also
in all their physical properties, are due to the different amounts of the traces of
foreign substances present in the arc.

The author believes that if it were possible to obtain perfectly pure carbon
electrodes, then the resistance of the arc between them would be very high, so high
that it might be impossible to maintain a true arc between them at all. He is of the

328

ME. W. DUDDELL ON THE RESISTANCE AND

opinion that traces of impurities, such as the vapours of the alkaline earths, are
essential to provide the carriers of the electric charges in the vapour column, so as
to render it conducting and the electric arc as we know it a physical possibility.
Unfortunately it has not, up to the present, been possible to obtain pure carbon
electrodes in order to test this theory. In favour of it is, however, the known fact
that, given an arc of fixed length and current between the best commercial solid
carbons, then any addition to it of such substances as potassium or sodium reduces
the P.D. required to maintain the arc and its resistance and increases its stability.

The difference between the 11-millim. and 'J-millim. Electrographitic carbons
mentioned above is probably caused by the last traces of impurities having been
more completely expelled in the manufacture from the ll-millim. size than from
the 9-millim.

TABLK IV. ^Various Carbon Electrodes.

Arc length :l millims. Added alternating currents O'l ampere

Direct current through arc if!) I amperes. Frequency 100,000 -»- per second.

Nature of Electrodes Varied.

i

Resist-

Make and description

Diameter. ^.
Direct

Resist-

Power-

ance Back

of carbon

P.D.

ance factor

of arc E.M.F.

Remarks.

electrodes used.

arc.

of arc. of arc. x cur- of arc.

+

-

rent.

minims, millims. volts.

ohms. volts. volts.

" Conradty Noris " solid .

11 11 49-8

3-81 0-99-. 37-8 1:2-0

?> i ? i •

11 9 49-8

3-83 0-99,, 38-0 11-8

>! )> 1 •

9 9 50 • 8

3-90 0-99!, 38-6

12-2

" Apostle " solid

11 11 49-3

4-05 1-00 40-1

9-2

n 15 ••

11 9 19-9

4-07 0-99: 40'3

9-6

" Brush " solid . .

11 11 50-6

4-04 0-99! 40-0

10-6

" Le Cai'bono " solid

11 11 50-4

4 -20 1-00 42-2

8-2

" Le Carbono " . . f

11

11 51 -5

4-66 0-91; 46-1*

5-4

Very unsteady arc.

Electrographitic. . <

11

9 51-2

4-45 0-99!

44-1

7-1

Fairly steady arc.

Solid [

9

9 50-1

3-95 1-00

39-1

11-0

Very steady arc.

f Potassium rapidly

"Conradty Noris " solid, ~|

burnt out of car-

soaked in 10 per cent. 1
solution of KiCOj for f

11

11 44-5

2-92

1-0 28-9

, _ . ,, J bons, results are
means of two

36 hours and dried . J

sets of readings

only.

"Conradty Noris "solid, ]
centre of negative 1
drilled out and filled I
up with a glass rod

11

11

33-1

2-08

0-985

20-6

12-5

|" Very unsteady arc,
I results are means
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2'5 mlliims. diameter J

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resistance, described ^

3-50

0-99,,

—

page 317 .... J

f This is an impedance, as the power-factor is not unity, the only one not found to be unity within the
limits of experimental error.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 329

Seat of the Back E.M.F. Search Carbons in the Arc.

The fact that the arc has a back E.M.F. which appears to increase with the amount
of foreign substances present in the vapour column, at once leads to the question
whether this E.M.F. is located at one or the other of the electrodes, or distributed
along the vapour column. In order to obtain an answer to this question, some
experiments were made on a 6-millim. 9 '91 ampere solid arc by introducing a search
carbon, 2 millims. diameter, into the arc, and measuring not only the direct P.D.
between the search carbon and each of the main carbons, but also the impedance to
the high-frequency testing current of that part of the arc between it and each ot
the main carbons.

In the experiments, three different positions of the search carbon were employed,
(1) with its centre 1 millim. from the positive electrode, (2) central in the arc, (3) with
its centre 1 millim. from the negative electrode. The fine point to which the search
carbon burns was always kept, so far as possible, just reaching to the axis of the
main carbons. The results of these experiments are given in Table V.

The introduction of a search carbon into an arc always greatly disturbs the
conditions of the arc, and the present case was no exception. The introduction of
the search carbon increased the direct P.U. arc by 4'0 volts, and the impedance of
the arc lamp by 0'44 ohm. So that the introduction of the search carbon, either by
deflecting the arc and so increasing its length, or by chilling the vapour column,
increases its resistance by an amount which approximately accounts for the observed
increase in P.D. arc. The back E.M.F. of 'the arc, as a whole, was but little affected
by the introduction of the search carbon. This distortion of the arc by the search
carbon probably also accounts for the observation that the measured impedance of
the arc as a whole is not equal to the sum of the impedances of the t\vo parts
comprised between the search carbon and the main electrodes.

Owing to the correct method of apportioning between the two electrodes, the
resistance and self-induction of the loop formed by the carbons, holders, and frame ot
the lamp, being unknown ; and owing to the fact that the measured quantities are
only roughly approximate, due to the disturbing effect of the search carbon, no
attempt was made to apply the small correction to the observations for the self-
induction and resistance of the carbon holders and lamp frame, and the observed
impedances were treated as resistances, and the back E.M.F.'s calculated as usual.
Further, the three arcs which had the same length and current will be considered as
having been identical, though such was not strictly the case.

On these assumptions, consider the resistance between the positive electrode and
the search carbon when the search carbon is 1 millim. from the positive electrode,
and then 5 rnillims. from the positive electrode (i.e., 1 millim. from the negative).
The change in resistance due to this -change of 4 millims. in the position of the
electrode is 1 72 ohms. Taking next the measurements made between the negative

VOL. ccm. — A 2 u

330

MR. W. DUDDELL ON THE RESISTANCE AND

electrode and the search carbon, the difference for the same movement of the search
carbon is 1'64 ohms. The mean of these two results is 1'68 ohms for a movement
of the search carbon of 4 millims. If this distance really represented the length of
the vapour column between the two positions of the search carbon, and if its
resistance is uniform, then its resistance per millim. would be 0'42 ohm. It is
probably less than this, owing to the length of the vapour column between the two
positions being appreciably longer than 4 millims., due to its distorted shape,

TABLE V. — Search Carbon in Arc.

Carbons both 11 millims. diameter. Solid " Conradty Noris."

Arc length 6 millims. Direct current through arc 9 '91 amperes.

Added alternating current O'l ampere. Frequency 100,000 -»- per second.

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Assuming the resistance to average 0'42 ohm per millim., then the resistance of
the G millims. of vapour column is only 2-5 ohms, as against 5 '2, the measured
value for the lamp and arc. There must therefore be a large resistance at or near
the electrodes. Calculating its value from each of the three experiments, the resist-
ance at or near the positive electrode is 172, T36 and 176 ohms respectively, mean
1'61 ohms; and the resistance at or near the negative electrode is I'll, 1'29 and
T15 ohms respectively," mean 1'18 ohms. As .there seems no very good reason to
suppose that the resistance of the vapour column is very much greater near the

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 331

electrodes than elsewhere, the above resistances are probably located at the contact
between the electrodes and the vapour column.

The resistance of the arc as a whole may therefore be considered to consist of three
parts, the resistance of the vapour column and the resistances of the two contacts
between it and each of the electrodes.

The mean value of the back E.M.F. of the arc lamp, with the search carbon in
place in the arc, i.e., 12'3 volts, is practically identical with its value of 127 volts,
obtained without any search carbon in the arc. The back E.M.F. of that part of the
arc between the search carbon and the positive electrode had, for the three different
positions of the search carbons, the values IG'G, 17 '9 and 15 '6 volts respectively,
which may be considered as indicating that this back E.M.F. is independent of the
position of the search carbon to within the limits of accuracy of the present experi-
ments. The back E.M.F. between the search carbon and the negative electrode
calculated in the same way, i.e., by subtracting the product of resistance by direct
current, or ohmic drop in this part of the arc, from the observed direct P.D. between
the search carbon and the negative electrode, is a negative quantity, so that there
exists a forward E.M.F. which helps the flow of the direct current. The value of
this forward E.M.F. was 6'5, G'4 and 5'3 volts respectively, again practically inde-
pendent of the position of the search carbon.

These experiments show that there is in this arc at or near the positive electrode
a back E.M.F. having a mean value of 107 volts, and at or near the negative
electrode -A forward E.M.F. of G'l volts. The fact that the algebraic sum of these
two voltages is not equal to the observed mean back E.M.F. 12'3 volts of the arc, as
a whole, is probably caused by the distortion of the lines of flow of the current
through the arc, produced by the introduction of the search carbon, which renders
the sum of the resistances measured between each of the main electrodes and the
search carbon greater than their total as measured between the main electrodes.

The back E.M.F. of the arc as a whole consists, therefore, of two E.M.F.'s located
at or near the electrodes, the larger E. M.F. situated at or near the positive electrode
opposing the flow of the current, the smaller E.M.F. situated at or near the negative
electrode helping the flow of the current.

The approximate values of the resistance E.M.F. of each part of the arc, and the
drop of volts due to each of these causes, corrections for the self-induction and
resistance of the holders and carbons being omitted as before, are given in Table VI.
The P.D.'s in this table multiplied by the direct current of 9'91 amperes give the
power supplied to each part of the arc.

2 u 2

332

MR. W. DUDDELL ON THE RESISTANCE AND

TABLE VI. — Distribution of the P.D. in a Solid Arc.

Carbons, both 11-millims. "Conradty Noris." Arc length 6 millims. Direct current

through Arc 9 '91 amperes.

Ohmic drop
due to
resistance.

Back E.M.F.

Sum of
back E.M.F.
and ohmic drop.

At or near the + crater

ToltS.

+ 16-0

YoltS.

+ 16-7

volts.

+ 32-7

Vapour column
At or near the crater

+ 25-0
+ 11-7

0
- 6-1

+ 25-0
+ 5-6

Total . . .

+ 52-7

+ 10-6

+ 63-3

Conclusion.

Ever since the arc was first assumed to have a back E.M.F., speculation has been
rife as to its cause and its location in the arc. So long as the value assigned to it
was of the order of 40 volts, and it was supposed to be located only at the crater
surface, there was great difficulty in offering any consistent explanation of it. It
remains, therefore, to consider whether the new facts set forth in this paper render
the matter more susceptible of a satisfactory explanation.

So far as the resistance of the arc is concerned, there seem to be no difficulties, the
known relations between the size and shape of the vapour column, the size of the
craters, the current, and the arc length, explain the observed changes in resistance
when the two latter variables are altered. The magnitude of the resistance of the
vapour column and of the contacts between it and the electrodes are not such as to
offer serious difficulties, nor does the fact that they are altered by the presence of
foreign bodies.

Any explanation of the back E.M.F. of the arc as a whole must, in the light ot
the results given in the last section, account for the existence of two unequal E.M.F.'s
of opposite signs, of which the larger opposes the flow of the current. The observed
back E.M.F. of the arc is the resultant of these two, their values being of the order
of 17 volts and 6 volts respectively in the solid arc. The existence of two E.M.F.'s
of opposite signs, situated at or near the electrodes, considerably simplifies matters,
since any explanation which would account for a back E.M.F. in the direction
carbon-to-vapour would probably also explain a forward E.M.F. vapour-to-carbon.
These E.M.F.'s are probably either due to a polarisation at the electrodes or to
thermo-electric forces. The polarisation E.M.F.'s include those due to chemical
changes and those which have been assumed' to be caused either by the volatilisation

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 333

or the tearing-off of particles from the electrodes. There appears to he very little
evidence in favour of these last two explanations.

The statement sometimes made that, as it requires a certain amount ot work to
be done to convert the solid electrodes into the gaseous state, and as this work is
done by the current, therefore the current must be flowing against a back E.M.F., is
without sufficient foundation. It is true that, in the above case, a P.D. will exist
which opposes the flow of the current, but that this P.D. is a reversible phenomenon,
and therefore an E.M.F., is not necessarily the case. The volatilization of the
electrode is of course reversible, but it requires experimental evidence to prove that
it is accompanied by a corresponding reversible electric phenomenon, so that the
energy, supposed supplied electrically, to cause the volatilisation, tends to be returned,
on condensation, in the form of electric energy. There is no evidence that any
appreciable E.M.F. is produced by the tearing-off of the solid particles from the
electrodes, and HERXFELD'S experiment of attracting these particles out of the arc by
means of an electrostatic field, which he says did not affect the P.D. or current
through the arc, seems to indicate that the back E.M.F. of the arc cannot be due to
this cause.

That more than a small part of the back E.M.F. is due to a polarisation, such as
occurs in an ordinary cell, is difficult to conceive. If such were the case, what is the
nature of the chemical compound produced, and what becomes of it ? It is certain
that practically the whole of the energy supplied to the arc is emitted again as light
and heat, and that there is no considerable portion stored up in the products produced
by the arc. If, therefore, any chemical change takes place at the positive electrode
accompanied by absorption of energy, this energy must be given out again in the arc
or flame, and the reverse chemical change take place. Some of the energy might be
given out at the negative electrode, and account for the forward E.M.F. observed
there. The nature of the substance in which the chemical change must take place,
which change reverses and so forms a cyclic process, is an almost insuperable
difficulty, since the rate at which the energy must be constantly absorbed and given
out again is considerable, and the materials present in which this chemical change
must take place are only carbon, its vapour, and the slight trace of impurities which
the author thinks essential to the existence of the arc.

If the impurity be assumed to be a salt of potassium, there is the possibility that a
carbide of potassium might be formed, and that part of the P.D. might be due to the
arc forming a cell, having carbon and potassium carbide as its electrodes, and the
vapour column as the electrolyte, the products produced by the flow of the current
through the cell being conceived to be destroyed in the flame which surrounds the
arc proper.

That a chemical combination between the carbon and surrounding gas is not the
cause of the back E.M.F., at any rate in the normal silent arcs considered here, is
evident from the fact that the arc is but little affected by the nature of the gas in

334 MR. W. DUDDELL ON THE RESISTANCE AND

which it burns, as has been shown by Professor S. P. THOMPSON and others. Putting
aside the possible combination of the carbon with the slight trace of impurity, which
may account for a. small part of the back E.M.F., the polarisation E.M.F.'s do not
seem able to account satisfactorily for the whole back E.M.F. of the arc, without
making assumptions which are at present unsupported by any satisfactory evidence.

Against the back E.M.F. of the arc being due to thermo-electric forces, it is
generally urged that these forces are usually reckoned in tenths and hundredths of
a volt, and that therefore the order of magnitude of the hack E.M.F., which was
then supposed to be about 40 volts, rendered it highly improbable that it was due to
thermal causes.

There is no doubt that the temperature of the positive crater is very high — said to
be about 3500° C., and it is quite possible that there may exist large temperature
gradients near the electrodes, since the true back E.M.F. to be accounted for is only
of the order of 17 volts at the positive crater. It is therefore worth while
re-considering whether after all the back E.M.F. of the arc may not be due to
thermo-electric forces.

In this connection the experiment of DUBS* is of considerable interest. He took
two carbon plates, about 1 millim. apart, and caused a blow-pipe flame to impinge
across their edges, and found that a small current could be obtained from one plate to
the other through a galvanometer. He considered this result might be analogous
to the back E.M.F. of the arc. A similar experiment has also been described
by OLIVETTI.!

The views of the author that impurities, such as the salts of the metals of the
alkaline earths, are essential to the existence of the arc, have led him to try a
considerable number of experiments on the P.D.'s produced by unequally heating
carbon electrodes, either with the addition of such salts to the electrodes, or to the
flame used as a source of heat. Most of these experiments were carried out by
heating the tips of ordinary arc-lamp carbons held in the hand-feed arc lamp already
described, by causing a "blow-through" or a "mixed" oxy-coal-gas flame to impinge
upon them.

The terminals of the arc lamp were connected to a direct-current voltmeter and
were not joined in any way to a source of current, so that the P.D.'s observed were
not due to leakage from any extraneous source. The choice of the voltmeter presents
some difficulty. At first sight, owing to the high resistance of the flame between the
two tips of the carbons, it might seem that an electrostatic or a very high resistance
voltmeter would be the best. Electrostatic voltmeters were however found unreliable,
owing to the friction of the gases in the jet and against the carbons charging them
up electrostatically. Two voltmeters were therefore used, the one an ordinary
Weston pivot instrument, having a resistance of about 600 ohms, and the second

* ' Beiblatter,' 1889, vol. 13, p. 197.

t ' Electrical Review,' 1892, vol. 31, p. 728.

ELECTROMOTIVE FORCES OP THE ELECTRIC ARC. 335

a reflecting moving-coil instrument of 20 ohms, having a resistance ot about
16,000 ohms in series with it. The resistance of both these instruments is so low
that no deflection due to the frictional charges could be observed on them. By
comparing the P.D.'s obtained when one or the other of the voltmeters was used an
estimate of the resistance of the vapour between the carbon tips could be made.

If the flame was caused to impinge on the two carbon tips so as to heat them
equally, as judged by eye, then no P.D. was observed between them. If the flame
was now moved so as to heat one carbon more than the other, then a P.D. was
observed between the carbons, the hotter being positive to the cooler as indicated by
the voltmeter, that is, in the same direction as the back E.M.F. of the arc, assuming
that the positive crater is the hotter.

The highest P.D. obtained was 1'5 volts when using cored carbons which had
been previously soaked in potassium carbonate solution. This P.D. was not in any
way due to differences in the quality of the two carbons, since by moving the jet so
as to heat either carbon more than the other the P.D. changed sign, the hotter
carbon always being positive. By setting one carbon in front of the other it was
found that the direction of the P.D. was unaffected by whether the stream of flume
gases flowed from the hotter to the cooler carbon or vice versa.

With two pieces of the same solid carbon, about 2 millims. apart, no foreign bodies
being introduced in any way, the highest P.D. obtained was about -O'o volt on the
higher resistance voltmeter, and the resistance of the flame between the carbons was
deduced as about 4000 ohms, so that the E.M.F. between these solid carbons
was only about 0-G2 volt. After soaking this same pair of carbons in potassium
carbonate the P.D. obtained was I/O volt., the resistance of the vapour being only a
few hundred ohms, so that the effect of introducing this potassium salt was to
greatly increase the E.M.F. between the carbons. If these E.M.F.'s are due to
Peltier effects, then it would seem as if the introduction of the potassium greatly
increased the thermo-electric power of the junction.

The above experiment is in agreement with the fact that the back E.M.F. of the
cored arc has been generally found larger than that of the solid arc, as the cored arc
undoubtedly contains more impurities.

By varying the proportions in which the oxygen and coal gas were mixed before
burning, it was found that the highest P.D.'s appeared to correspond with the gases
being burnt in their combining proportions, so as to produce the hottest flame ;
if either gas was considerably in excess, a much lower P.D. was obtained. It is to be
noted, however, that by suppressing the coal gas altogether and allowing the carbon
to burn in the oxygen, P.D.'s up to 1'5 volt could be obtained, as before the hotter
electrode always being positive to the voltmeter. These latter experiments, though
not conclusive, are not in favour of the P.D. being due directly to a combination of
the carbon and either of the gases.

The temperature of the positive crater of the arc is much higher than the highest

336 MR. W. DUDDELL ON THE RESISTANCE AND

temperature obtained by the oxy-coal-gas flame, and the temperature differences and
gradients which may exist in the arc are probably many times greater than those
obtained in any of the above experiments, for it must be remembered that the carbon
at the lower temperature must always be at a bright red heat, or else it does not
appear to make electrical contact with the flame. It does not therefore seem
improbable that the P.D.'s of 1 volt or 1'5 volts obtained by unequally heating the
two carbons may have the same origin as the 10-volts to 18-volts back E.M.F. found
in the arc, especially as they agree both in direction and in the effect of impurities on
them. On this assumption, the probable causes of the back E.M.F. of the arc reduce
themselves to two, viz. :—

(1.) A thermo-electric force at the junction of the carbon and vapour, causing the
major part of the observed back E.M.F.

(2.) A combination of carbon with the impurity present.

Whether the thermo-electric force at the junction of carbon and vapour be due
either to the Peltier effect, or to a high temperature gradient at the contact, i.e., a
Thomson effect, or both, it seems at the present time to afford the most satisfactory
explanation of the back E.M.F. of the arc and the P.D.'s observed when two carbon
electrodes are unequally heated. The main objection is the great difference of
thermo-electric power the components of the junction must have, which, if the
difference of temperature be assumed to be only 1000° C., amounts to about 15 to 20
times that between bismuth and selenium.

In favour of the view that the back E.M.F. is mainly due to thermal causes may
be mentioned the unilateral behaviour of the alternating arcs between a metal ball
and a metal point,* between carbons and metals, t and the observation of CROSS and
SHEPARDJ on the effect of cooling the positive crater of the direct-current arc-
All these experiments indicate that there is some cause existing which enables the
current to flow much more easily up the temperature gradient than down it.

A tentative explanation of the chief causes which oppose the flow of the current
through the arc may be given under four heads : —

(1.) The resistance of the true vapour column, which is probably an electrolytic
conductor, whose conductivity greatly depends on the traces of impurities, such as
potassium and soda, present. Pure carbon vapour, like pure water, has probably a
very high specific resistance.

(2.) A high contact resistance between the electrodes and the vapour column,
which leads to a large generation of heat, and consequently high temperatures and
temperature gradients at these points.

(3.) Possibly a small back E.M.F., due to the electrolytic cell formed by the
electrodes and the vapour column as electrolyte.

* ARCHBOLD and TEEPLE, 'American Journal of Science,' 1891.

t DUDDELL and MARCHANT, 'Journal of the Institution of Electrical Engineers,' 1899, vol. 28, p. 74.

I 'American Academy of Sciences,' 1896, p. 227.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 337

(4.) Considerable thermo-electric forces at the contacts between the electrodes and
the vapour column, whose resultant is the greater part of the observed back E.M.F.
of the arc. The force at the positive opposing the flow of the current, and that at
negative helping it, account for the much greater conversion of electric energy into
heat at the positive electrode.

In the above, it will be noticed that the volatilisation or sublimation of carbon is
not supposed to have any influence on the back E.M.F. except in so far as it limits
the temperature the electrodes can obtain.

The introduction of impurities up to a certain limit into the arc will cause the
conductivity of the vapour column to increase, and for a fixed current its temperature
will probably decrease, so the thermo-forces and the back E.M.F. may be expected to
increase as has been observed (see Table IV.).

There is another possible explanation of the larger back E.M.F. when impurities
are present, viz., the thermo-electric force of the junction carbon-vapour may be
larger the greater the quantity of impurity present. If this is the case, may not the
explanation of the drop in P.D., when the arc hisses, or when oxygen or hydrogen
come in contact with the positive crater, as found by Mrs. AYRTON, be that the <jas
combines with the impurity and reduces the back E.M.F., and not that it combines
with the carbon as suggested by Mrs. AYRTON ?

In conclusion, the author wishes to express his thanks to Professor AYRTON and
Mr. MATHER of the Central Technical College for the very valuable assistance and

O i/

advice they have given him during the course of these investigations ; he also wishes
to thank the many students who have from time to time helped him with the
experiments, and especially Messrs. DEL MARR, LYNN, BROWN, WATSON, and VINKS.

ArPENDIX I.

On the Relation betivcen the P.D. Current and Length of the Arc.

Mrs. AYRTON has pointed out that the relation between the P.D., the current, and
the arc length, may for the solid carbons which she used be accurately expressed by
an equation of the form V = a + /3l + (y + SI)/ A, where V and A are the P.D. and
current respectively, and a, ft, y, and S are constants, between the limits over which
she varied the current and length. It was thought, therefore, that it would be of
interest to determine the values of the constants «, ft, y, S for the solid " Conradty
Noris " carbons used in so many of the experiments in this paper. The two series of
results in Curves III. and V. give P.D. when the length is kept constant and
the current varied ; and also the P.D. when the current is kept constant and the

VOL. ccni. — A, 2 x

338

MR. W. DUDDELL, ON THE KESISTANCE AND

length varied ; and in order to increase the accuracy of the equation some extra
results were obtained in which both current and length have been varied.

On attempting to find the constants a, ft, y, 8 to fit all these values, the first
difficulty encountered was that, assuming the current to be constant and the length
varied, Mrs. AYRTON'S equation required the P.D. to be a linear function of the
length ; a glance at Curve V. shows that this was not even approximately true for
these carbons and the long range of lengths from 1 millim. to 30 millims. used. At
first it was thought that this might in some way be due to the kind of carbons used
in the experiments. The connection between P.D. and length for constant current
was, therefore, determined for the same size and make of carbons used by Mrs.
AYRTON in her experiments* ; the results are given in Curve VII.

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The divergence between the P.D.'s calculated from Mrs. AYRTON'S equation and ihe

* The ' Electrician,' 1895, vol. 35, p. 420.

ELECTROMOTIVE FORCES OF THE ELECTRIC ARC. 339

observed P.D.'s is very marked, the error in the calculated result being no less
than 32 '6 volts for an arc length of 30 millims. At the same time this is not due to
any difference in the nature of the carbons, since the actual experimental values
obtained by Mrs. AYRTON,* and on which her equation is based, fit the Curve VII.
with considerable accuracy. The whole of this large difference between Mrs. AYRTON'S
equation and the experimental results is due to the fact that the equation is based on
the range of arc lengths of from 1 millim. to 7 millims., and that the equation no
longer represents the facts if we extrapolate any considerable amount. Even over
the range of from 1 millim. to 7 millims. the connection between the P.I), and length
for constant current is not accurately a straight line, as can be seen either in
Curve VII. or by a careful examination of Mrs. AYRTON'S experimental results.

If the arc length was kept constant and the current varied, a similar difficulty was
found with very small currents of 1'5 amperes and lower, namely, that the observed
P.D. was less than the calculated P.D., so that Mrs. AYRTON'S equation can only he
considered as approximately representing tlie facts within tJte limits site used, and
must not be applied to very small currents or long arc lengths.

Taking, in the present case, as limits from 1 '5 amperes to 12 amperes, and from
1 millim. to 6 millims., then the results obtained with the arc between 11 millims.
solid " Conradty Noris" carbons can with a very fair approximation be represented
by the equation

V = 39-G 4- 17/ + (15-5 + H'5/)/A,

which is in very close agreement with the equation given by Mrs. AYRTON for solid
"Apostle" carbons, 11 millims. and 9 millims. diameter, for the range 1 millim. to
7 millims., viz. :

V = 38-88 -f 2-077 + (H'GG + 10'54/)/A.

APPENDIX II.

On the Resistance of an Electrolyte.

In measuring the resistance of an electrolyte by the ordinary Koblrausch method,
using alternating or induced currents, it is usually assumed that the influence of
polarisation of the electrode is avoided, and that the frequency of the alternating
current used is unimportant, provided that it is moderately high, say a few hundred
periods per second. If an appreciable polarisation of the electrodes is produced by
the testing current very soon after its application in any given direction, then the

* ' Electrician,' vol. 35, p. 635.

2x2

340 MR. W. DUDDELL ON THE RESISTANCE AND

above -assumptions will not be true, and the resistance as measured at such low
frequencies will differ from that measured at much higher frequencies.

In order to investigate this matter, the arc was replaced by an electrolytic cell, and
its impedance and power-factor tested in a similar way to those of the arc, except
that no direct current was sent through the cell. To make the polarisation effects
important compared with the ohmic drop, the resistance of the cell must be small,
therefore sulphuric acid was chosen as the electrolyte. The cell used was a glass
trough 100 millims. long, 104 millims. deep, and having a mean thickness of
15 '53 millims. The electrodes were two platinum plates 90 millims. apart (not
platinised) which fitted the trough tightly and extended to the bottom.

The mean depth of the electrolyte was 61 '6 millims. So that the liquid whose
resistance was measured was a rectangular parallelepiped, having a length of
9 centims. and a mean cross-section of 9 '5 7 sq. centime.

The temperature was kept constant at 20° C. to within 0'1° C. during the test, by
immersing the trough in a water-bath. The results of the tests of the impedance
and power-factor at different frequencies are given in Curve VIII.

With the above cell, neither the impedance nor the power-factor are independent
of the frequency at ordinary frequencies, and it is evident that the polarisation in
this electrolyte occurs so rapidly that it is not until the frequency is well above
10,000 -»- per second that it is unable to follow the variations of the testing current,
and that the electrolyte behaves like an ordinary resistance. If the resistance of
this cell were tested in the ordinary way, at a frequency of about 100 •-- per second,
the value obtained would be over twice the true resistance of the cell.

The slight drop in impedance observed when the frequency is increased from 50,000
to 100,000 ^ per second is possibly due to the electrostatic capacity of the plates
and liquid in the cell, and to the water-bath surrounding it acting as a shunt to the
cell. Attempts were made to determine experimentally the exact value of this
correction, which indicated that it was less than 1 per cent, at 100,000 — • per
second.

The possible errors due to polarisation, even when using alternating currents, were
thoroughly appreciated by KOHLRAUSCH, and in the ' Leitvermogen der Elektrolyte,'
by KOHLUAUSCH and HOLBORN, means are described for minimising these errors. By
interpolation from their results the conductivity of sulphuric acid of the same
density and at the same temperature as that used in this experiment is
0758 (ohm X centim.)"1, or a specific resistance of 1'319 X 10° c.g.s., which
confirms the value of 1'30 X 10° obtained in the present experiment at a frequency
of 50,000 — per second.

The fact that the power-factor is not unity at low frequencies proves that at
these frequencies the electrolyte does not behave like an ordinary resistance, and
consequently all the instantaneous values of the P.D. between the terminals of the
telephone sometimes used in the Kohlrausch method cannot be made to vanish at

ELECTEOMOTIVE FORCES OF THE ELECTRIC ARC.

341

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342 THE RESISTANCE AND ELECTROMOTIVE FORCES OF THE ELECTRIC ARC

these frequencies, even supposing the other arms of the bridge quite free from self-
inchiction and capacity. This explains the fact that only a minimum in the sound is
obtainable, and not absolute silence.

The conclusion to be drawn from this experiment is that in any case where the
P.D. due to the polarisation of the electrodes cannot be made very small compared
with the ohmic drop along the liquid whose resistance is being measured, and where
the errors due to the polarisation cannot be eliminated by taking two or more tests,
then it must not be assumed without proof that the use, of alternating currents at
ordinary frequencies of a few hundred periods per second eliminates the possibility of
errors due to polarisation. For in the case of sulphuric acid used above, the
polarisation can vary as rapidly as the resistance of the cored arc.

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[ 343 ]

XI. On

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HAKKKK. /).#<•., Fvlluii-
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HABKBB, J. A.— On the High-Temperature Standards of the National 'he National

Physical Laboratory ; an Account of a Comparison of Platinum
Thermometers and Thermojunctions with the Gas Thermometer.

Phil. Trans., A, vol. 203, 1904, pp. 343-384.

Platinum Thermometers, Comparison of, with Gas Thermometer at High
Temperatures.

HAEKEE, J. A. Phil. Trans., A, rol. 203, 1904, pp. 343-384.

Page

Thermojunctions, Comparison of, with Gas Thermometer at High Tem-
peratures.

HABKEE, J. A. Phil. Trans., A, vol. 203, 1904, pp. 343 -384. 345

III. Gas thermo'i ... 347

IV. Ba: 348

348

VI. Fillin 350

VII. I>il:ut 350

VIII. PP-S 351

IX. The ],!;, ... 352

352

XI. 'IV pou.: 353

XII. St-wrkv. 356

XIII. Th'.> it.. .... 357

XIV. Formula for HUM • • • 358

nt 359

361

XVII. Furiuu-e correction . . 362

XVIII. Exploration of furnace 362

XIX. Method of calculation of ga* < 364

SfH-i-imon determinations' »i !'•«•• wl 365

• tcunicy of constant* i>i %*.-> then:..*!* 366

jries II. . . . 368

iwxl of calculating wnaparliHJB eiqmriBMOt . 309

i»f experim«nto and ct«uit*nt« iwwl in calculation 372

377

a of T corresponding to given v.iluea of pi for - = 1 '5. . . . 378

384

13.8.04

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[ 343 ]

XT. On the High-Temperature Standard* of the National Physical Laboratory:

an Account of a ('omparixon of Platinum Thermometers and

Thermoj unction* with the Gas Thermometer.

By J. A. HARKER, D.Sc., Fdlow of Owens College, Manchester, Assistant at the

National Physical Laboratory.

Communicated by R. T. GLAZEBROOK, F.R.S.,from the National

Physical Laboratory.

Received January 20, — Bead February 11, 1904.

CONTENTS.

Page

I. Introduction 314:

II. The gas thermometer — description 345

III. Gas thermometer details scale, reservoirs, kc 347

IV. Barometer and auxiliary measurements 348

V. Preparation of the gases 348

VI. Filling of the reservoir 350

VII. Dilatation of the porcelain reservoir 350

VIII. Pressure coefficient of the reservoirs 351

IX. The platinum thermometers .352

X. The resistance box for platinum thermometry 352

XI. The potentiometer for thermocouple measurements 353

XII. Standard of electromotive force 350

XIII. The thermojunctions .357

XIV. Formula; for thermojunctions .... . 358

XV. Determination of fixed points 0°, 100° and sulphur boiling-point 359

XVI. Electric furnaces. . . .

XVII. Furnace correction 362

XVIII. Exploration of furnace 362

XIX. Method of calculation of gas thermometer experiments

XX. Specimen determinations of ice- and steam-points ....

XXI. Accuracy of constants of gas thermometer

XXII. Summary of calculation of gas thermometer temperatures. Series II. . . . 368

XXIII. Method of calculating comparison experiment

XXIV. Summary of experiments and constants used in calculation . . - 372
XXV. Conclusions • 377

XXVI. Table of values of T corresponding to given values of pt for 8= 1 -5. . . . 378
XXVII. Table to calculate change in T for a given small change in 8 . . .
(369.) 13.8.04

344 DR. J. A. HAEKER ON THE HIGH-TEMPERATURE STANDARDS

I. Introduction.

IN a paper " On the Comparison of Gas and Platinum Thermometers," read before the
Eoyal Society in 1900,* Dr. P. CHAPPUIS and the author described a series of
experiments in which several platinum-resistance thermometers, constructed of wire
of specially high purity, were compared with the gas thermometer at a number of
steady temperatures from below zero to above the boiling-point of sulphur, and in one
set of measurements to just short of 600° C.

The results were such as to substantially confirm the conclusion of CALLENDAR and
GRIFFITHS that the indications of platinum thermometers may be reduced to the
normal scale by the employment of CALLENDAR'S well-known difference formula

where d = the difference between T, the temperature on the normal scale, and
pt = the " platinum " temperature. The constant 8 for pure platinum wires is
approximately 1'5, the three temperatures chosen for its determination being 0°, 100°
and the boiling-point of sulphur.

The paper concludes with the sentence, " until further investigations have been
made as to the relations of the various gas scales at high temperatures and as to the
influence of the initial pressure and the effect of impurities and traces of water vapour
in the gases employed, and until exact determinations have been made up to high
temperatures of the coefficient of expansion of the material used as thermometric
reservoir, we think that for the purposes of high-range thermometry a scale deduced
by the parabolic formula from that of the platinum thermometer will suffice. In the
present state of our knowledge any attempt to improve on such a thermometric scale
would be attended with such uncertainties as would probably render it futile."

Since that time, however, a substantial advance has been made in our knowledge,
direct determinations of the expansion of porcelain up to high temperatures having
been made by different observers, namely, Mr. BEDFORD, t at Cambridge, and Messrs.
HOLBORN and DAY at the Reichsanstalt.J A discussion by Dr. CHAPPUIS of the
results obtained by these observers and their influence on high-range thermometry is
found in the 'Philosophical Magazine,' (5), October, 1900, and February, 1902.

An examination of the difference formula for the platinum thermometer shows that
it can only represent a physical reality over a limited range, the value of pt for a wire
having a 8 of 1*5 reaching a maximum about 1700° pt, a value numerically not far
exceeding such as may safely be attained. It woiild not therefore be surprising if the
formula which actually holds remarkably closely at low ranges should be found to

* 'Phil. Trans.,' A, vol. 194, pp. 37-134.

t BEDFORD, 'Proc. Phys. Soc.,' XVII., Part III., p. 148, and 'Phil. Mag.'

\ HOLBORN and DAY, ' Ann. Phys.,' vol. 6, 1901, p. 136.

OF THE NATIONAL PHYSICAL LABORATORY. 345

give erroneous results at temperatures well below the maximum to which the
materials used in the construction of a platinum thermometer can be subjected
without injury. The investigations dealt with in the present paper have been carried
out at the National Physical Laboratory during the past two years, and consist
mainly of a continuation of the work of CHAPPUIS and the author on the platinum
thermometer, testing up to 1000° C. the validity of the difference formula for two
thermometers made of representative platinum wires of high purity, by comparison of
these instruments with the constant volume gas thermometer. With these instruments
were also compared simultaneously standard thermojunctions, whose electromotive
force at a series of temperatures had been determined with special care at the
Reichsanstalt at Charlottenburg.

o

II. The Gas Thennometer.

The gas thermometer employed for this work is a duplicate of the one used bv
HOLBORN and DAY at the Reichsanstalt. It was obtained from the same maker,
FUESS, of Berlin, and was presented to the laboratorv by Sir ANDREW NOBLE. For
this munificent gift and for the kindly assistance and advice rendered by the
President of the Reichsanstalt, Dr. KoHLRAUSCH, and by Dr. HOLBORN in procuring
for us the gas thermometer, thermocouple, wire and materials for the construction of
electric furnaces, the laboratory is greatly indebted.

The instrument is specially designed for rapid work at high temperatures, and was
arranged so that measurements could be made with any desired initial pressure and
with bulbs of different materials. The principle employed by CHAPPUIS, in the two
gas thermometers at Sevres, oi making all the measurements depend upon the
determination of a single length, though undoubtedly capable of giving by far the
most accurate results, becomes somewhat inconvenient when great changes of pressure
are needed. For this reason, therefore, in the present apparatus the manometer is
arranged so as to measure directly the difference of height between the level of a very
short metal point, to which the mercury in the closed limb A, fig. I, is adjusted, and
the mercury surface in the long tube B, which during the measurements communicates
with the atmosphere by the tap H.*

The tubes A and B communicate by means of cone joints with the lower part of a
closed iron reservoir in the base plate of the apparatus, into which mercury can enter
from the upper reservoir G by means of the long tube C and steel tap D. The fine
adjustment of the height of the mercury to the point in A is made by a steel screw
with capstan-shaped head projecting from the bottom of the apparatus and working
on a thin steel diaphragm let into the bottom of the reservoir.

* In the original form of the apparatus, tube A was joined to the reservoir below by a large three-way
glass tap, through the side tube of which the filling of the gas into the reservoir was made. It was found,
however, that this tap was a source of danger in the measurements, the results of one set of comparisons
YOL. CCIII. — A. 2 Y

346

DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

ft —

C

Joint between platinum capillary
and neck of reservoir.

a, metal cap ; I>, split metal collar ;
C, porcelain capillary ; c, plati-
num capillary, in this case drawn
down to smaller size for lower
4 or 5 centims. ; </, metal washer
soldered to platinum capillary.

Fig- .1.

OF THE NATIONAL PHYSICAL LABORATORY. 347

IT]. The. Scale, <Cr.

The manometer scale, nearly two metres long, is graduated into millimetres and
carries a slider with vernier and fine adjustment screw by which the position of the
mercury column in B could be read directly to '02 millim. Over the part of the
scale used in these experiments no corrections were required.

Accurate setting of the mercury to the point in the closed limb of the manometer
was facilitated by the use of a small short-focus telescope clamped to the stand.
Measurements of the height of this point on the scale, as given by the vernier
readings when the sighting edge of the slider was brought in line with the point,
were taken frequently throughout the work. The transfer of level was effected by a
Quincke microscope fixed on a plate-glass table which could be attached to the gas
thermometer supports. This constant was different in the several sets owing to the
tube A having for different reasons been several times rebuilt.

The Rejwrvoirs and their Attachment, &c.

For some early experiments up to 200° ('. reservoirs of normal Jena glass If/" were
fixed having a long capillary stem attached by fusion. For the later series at high
temperatures reservoirs of Berlin porcelain, similar to those formerly employed by
CHAPPUIS and HARKKH, were kindly ordered for us by Dr. HOLBORNT at the Berlin
factory. They were made in manner described by HOLBORN and DAY in their paper
in ' Wiecl. Ann.,' vol. 68, pp. 8:10, 831.

For attachment of the reservoir to the platinum capillary lending to the manometer
joints made by sealing-wax or cement are too treacherous to be trusted for long
periods, so the joints were in all cases made in the manner shown in fig. 2. The
brass washer soldered on to the platinum capillary is clamped down tight against the
ground end of the porcelain or glass neck of the reservoir by a hollow cap which
screws on to the two halves of a split brass collar, a thin leather washer, on which is
smeared a little marine glue, being interposed to make the joint. The collar is
prevented from sliding upon the neck by fitting closely into a rounded channel cut
into the glass or porcelain by a small emery wheel. This form of joint has given
perfect satisfaction.

being lost through the plug being forced outward by the mercury pressure. Although the possibility of
the recurrence of this accident was avoided by the subsequent addition of a spring frame arrangement
embracing the plug of the tap, it was found extremely difficult to arrange the lubrication to be free
enough to permit of its constant use for shutting oft' communication between the thermometer and the
bulb and at the same time to prevent very slow inward leakage of air into A through the tap when the
bulb was under low pressure. The apparatus was therefore modified by putting into tube A a plain steel
tap and introducing a smaller and perfectly ground three-way tap into the side tube, where it was not
needed after the bulb was once filled. Under these conditions the whole .arrangement could be kept
perfectly gas tight.

2 Y 2

348 DE. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

IV. Barometer and Auxiliary Measurements.

The barometer used was a Fortin with f'-inch tube, made by HICKS and
standardised at Kew. It was compared from time to time with an old Royal
Society's standard and found to have the same correction, —'05 millim.

The temperatures of the mercury columns in the gas thermometer were obtained by
means of two auxiliary thermometers disposed at convenient heights, having their
bulbs immersed in short test tubes filled with mercury. These are called (2) and (3)
in the examples of gas thermometer readings given later. The temperature of that
part of the dead space over the mercury in tube A. is given by a thermometer
inserted into the metal stopper carrying the reference point. This thermometer is
labelled (l). The temperature of the platinum capillary is kept constant by
surrounding it with a water-jacket of rubber tube, through which a circulation of
water from a supply vessel kept in the room is maintained during the experiments,
the temperature of the flow being given by the thermometer (4).

In making the correction of the pressures of the various mercury columns to
latitude 45° and sea level, the latitude of the thermometric laboratory was taken as
51° 26' 47", and the height above sea level of the cistern of the barometer as
10 metres.

The relation

G latitude Bushy House , ,

^ — . — r-"o— was taken as 1 '000561.
G latitude 45

V. Preparation of the Gases.

The nitrogen used in the earliest experiments was obtained from atmospheric air by
the ordinary method of absorbing the carbonic acid by potash and the oxygen by hot
copper, but for the sake of continuity with the work of CHAPPUIS and the author, in
which chemical nitrogen was used in all the high -temperature experiments, the
method of preparation employed by them was used in all the later work. Into a
flask containing 100 grammes of potassium bichromate, dissolved in air-free distilled
water, was introduced gradually by means of a tap-funnel a strong solution containing
100 grammes nitrate of soda and 100 grammes nitrate of ammonia. When gently
heated on a water-bath this mixture gives off a steady stream of nitrogen, which is
passed over hot copper and copper oxide to destroy any oxides of nitrogen it may
contain, and collected over weak boiled potash solution in a gas holder. From this it
passes through solutions of silver nitrate and strong potash, boiled sulphuric acid, and
over solid caustic potash to a larger U-tube containing phosphoric anhydride, where
it is left several days before use, or is collected in larger quantity in the dry gas
holder over mercury. The tightness of all the joints and taps was ultimately made
so perfect that no measurable inward leak of air could be detected by the attached
manometer, even when the whole was left standing under reduced pressure for some
months. A sketch of the gas preparation apparatus is shown in fig. 3.

OF THE NATIONAL PHYSICAL LABORATORY.

350 DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

VI. Filling of the Reservoir.

The experiments of CHAPPUIS and the author, ' Phil. Trans.,' vol. 194, p. 93,
showed how important it is that previous to and during the tilling of the reservoir
with gas the whole should he heated to as high a temperature as possihle — indeed
that with a reservoir of verre dur it was impossihle to obtain an accurate value for a
temperature in the neighbourhood of the boiling-point of sulphur unless it had been
previously heated for a considerable period to a temperature considerably higher than
this. Although very little is known as to the behaviour with change of temperature
and pressure of the films of condensed gas,- which by their surface tension adhere
closely to the walls of a glass or porcelain vessel, and how they influence the
coefficient of expansion of any gas enclosed in it, it seems at any rate advisable to
remove the existing film in any reservoir to be used for high-temperature work as far
as possible by repeated heating and evacuation. Accordingly for each new filling
during this work the procedure was to heat the reservoir very strongly, exhausting it
and the connecting tubes leading to the pump so thoroughly that an electrodeless
vacuum-tube attached by a side tube showed strong green fluorescence both when the
reservoir was hot and the next dav after cooling. This was only attained after much

, n »/

labour by taking special care in the manipulation of all the taps and joints in the
circuit, and by blowing together into one piece the whole of the long canal leading
from the gas thermometer in the main laboratory to the pump and gas preparation
apparatus in a small room on an upper floor. After the apparatus had stood this test
the gas was filled into the reservoir to 100 to 200 millims. pressure at least three
times iu each case before the final filling and pressure adjustment were made, the
bulb being heated meanwhile as strongly as expedient in the particular case.

VII. Dilfttatio-n of the Porcelain Reservoir.

In 1898 and 1899, OHAITUIS and HARKEK, in the absence of precise data as to the
expansion at high temperatures of the reservoirs of Berlin porcelain they employed in
their gas thermometer, were obliged to content themselves with extrapolation of the
expression for the expansion obtained by them from determinations made in the
Benoit-Fizeau dilatometer between the limits 0° and 100°, confirmed by further
experiments made by a weight thermometer method. The values for the expression
they employed seemed at the time to be amply confirmed by the measurements of
HOLBORN and WIEN, made in 1892, which gave for the mean coefficient of linear
expansion between the limits 0° and 600° the value 4400 X 10~9, the value calculated
by CHAPPUIS and HARKER for the same temperature limits being 4484 X 10~9. On
the other hand, the later experiments of HOLBOKN and DAY made on Berlin porcelain
rods heated horizontally in an electric furnace show fairly conclusively that the
expansion of porcelain cannot be represented by any simple function for more than a

OF THE NATIONAL PHYSICAL LABORATORY.

351

few hundred degrees, and that therefore an expression deduced empirically cannot be
applied outside the limits of actual experiment without risk of serious error.

To obtain the most probable value over the whole range for the expansion of Berlin
porcelain, a combination was made of the values obtained directly by two independent
methods over the range 0° to 100° C. by CHAPPUI.S and HARKKU with those obtained
for higher temperatures by HOLBOKX and DAY. As pointed out by CHAPPUIS in the
paper in the 'Philosophical Magazine' alluded to in the introduction, the formula of
HOLBORN and DAY undoubtedly gives too high values below -JoO" (.'. This has just
been confirmed by the experiments of SCHEEL, published in vol. 4 of the ' Wiss. Abb.
der Keichsanstalt,' whose results over the range I 4° to 100° obtained in the Fizeau
dilatometer are in close agreement with those of CHAPPUIS and HAKKEH.

1 have therefore taken for this work the following values of the mean dilatation
between 0° and T°, and from them calculated a table giving the volume at T° of a
reservoir, whose volume at (JJ is unity, for temperatures up to 1.100° C.

T.

0

100
200
300
400
500

ilatation of Berlin 1

'orcelain between 0

and T°.

Dilatation,

T

Dilatation.

x 10-s.

x 10 •-*.

269

000

363

299

700

374

.318

800

385

:i29

900

397

340

1000

408

352

1100

119

whence volume at T° of reservoir whose volume at 0° — I

o

100
1000

1 • 000000
1-000897

1-013834, ,tc.

Vlll. Ptvwtre Coefficient of tin' Res

In the work of CHAPPUIS and HAKKEH account was taken of the change of volume
of the bulb produced by changes of internal pressure. The value of the change of
volume was deduced from observations made by varying the external pressure on the
reservoir at ordinary temperatures. With the thick-walled reservoirs here employed
the whole effect is only very small, a change of 1 metre in the internal pressure
producing a change of about 4o~ooo °*' the whole volume of the reservoir at ordinary
temperatures. The effect of high temperatures in changing the elastic constants of
the porcelain here involved being quite unknown, it was deemed justifiable to omit
this correction altogether for the present high -temperature experiments.

.352 DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

IX. The Platinum Thermometers.

The platinum thermometers chosen for the comparison at high temperatures with
the gas thermometer were one of a group of six constructed in 1902 at the laboratory
for the British Association and described as BA., in the report of the Electrical
Standards Committee for 1903, and the much older one, lettered Ko, which had been
employed by CHAPPUIS and HARKER in their work at Sevres. Since the date of the
Sevres comparisons in 1898 and 1899 this thermometer had been preserved as a
standard and used comparatively seldom.

Its characteristic dimensions are : —

Length from end of porcelain tube to below the wooden collar, 3 5 '5 ceutims.
External diameter of tube, 11 -5 millims.

The head was of the old pattern with terminals, the leads being entirely of
platinum, '020 inch diameter, and the "bulb" wire, '006 inch diameter.

Thermometer BA2 is similar in design to CHAPPUIS' and HARKER'S thermometers
Ks and Ky, illustrated in their paper, the head being arranged to be capable of
standing a vacuum for a considerable time.

Its dimensions were as follows :•—

Length of tube to under head = 43'5 centims.
Internal diameter of porcelain tube = 1 1/5 millims.
Length of mica cross = 40 millims.
Thickness of " bulb" wire = '006 inch,
"lead" „ ='020 „

Both "bulb" and leads were constructed from the stock of wire ot extra high
purity prepared by JOHNSON and MATTHEY for the British Association. Details as
to the construction of these British Association thermometers are given in the
appendix to the Report of the British Association, 1903.

X. The Resistance Box for Platinum Thermometry.

The resistance box used was the one originally described by GRIFFITHS in ' Nature,'
November 14, 1895, pp. 39 to 46, which had subsequently been modified by the
Cambridge Scientific Instrument Company by the substitution of Doulton-ware plug
holders for the usual brass blocks to which the individual resistances were connected.
In this form the box was used by CHREE in his investigations described in ' Proc.
Hoy. Soc.,' vol. 67, pp. 3-58.

During the period covered by the present work the coils were standardised in the
manner described by HARKER and CHA*PPUIS (loc. cit., p. 52), and great care was
taken throughout as to the cleanliness of the plugs and to prevent the accumulation
of the black deposit which has a tendency to form in the plug holes. The box unit
is very exactly '01 ohm, and the nine coil values are arranged on the binary scale

OF THE NATIONAL PHYSICAL LABORATORY.

353

from 5 up to 1280 units, with an additional coil of 100 units for the determination ot
the fundamental interval. The coils are of fine platinum-silver wire wound into
loose hanks not paraffined, so as to follow with as little lag as possible the tempera-
ture of the massive copper tank in which they are placed. The working of the
resistance box has under the conditions which obtained for this work been sufficiently
good to render unlikely the introduction of any systematic errors. The sensitivity of
the platinum thermometer bridge was such that even at the highest ranges a change
of about '01° C. could be detected, though an accuracy of '1° at 1000° C. is probably
about the limit in the absolute measurement of temperature even under favourable
conditions with resistance coils of platinum-silver, such as those employed here, with
a temperature coefficient of -00026 per degree C.

The flexible leads which were never detached from the thermometers during any
series were carefully checked before and after the experiments, and the possible
checks, got by using different combinations of coils in measurement of the same
temperature, always agreed reasonably well.

XI. The Potentiometer for Thermocouple Measurements.

The potentiometer used in these experiments was specially designed and made in
the laboratory for accurate thermocouple work, and was described in detail in a paper
in the ' Phil. Mag.' for July, 1903. It is direct reading, and i.s arranged so that for

ST. | CELL

ai. I ^t

r'h

-o o-o

ACC.

\

REG. RES.

6

/W\A

4O I I I -5 I 1-5

b 6=6 6=6 6=6 6=0 o

Fig. 4. Connections to potentiometer.

ordinary thermoelectric work where the maximum electromotive force to be measured
is of the order of '02 volt the individual settings could be made to about '1 microvolt
if required. A diagrammatic representation of the connexions is shown in fig. 4 and
VOL. ecu r. — A. 2 z

354

DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

a plan of the top of the instrument in fig. 5. It is designed with a view of rendering
possible the use of a short slide-wire of large cross section. The balancing coils, on
which the fall of potential is adjusted to a definite value, are in two rows, the centre

GALV
0,00° °4oO

©0©0©e©e © O © ©

THERMO JUNCTIONS . ST.CELL ACC + RES

O,O O 5 O O s O Oo Oi O2 O3 O4 O5 Os Or O8 Os O

I o o o n

o o o o o

oooooouooooo

;

I O i«O i«O ITU O isO wO isO a

n

iO ioO »O sO TO eO sQ O O 20 'O <O

I 0 O O O

o o o o

u

oooooooooooo

A A

ST. TH. ST. TH

o o <

] ©

J 4

oonooooooooo

© © 1

1 2.

© G f

lO O 9U 8O 7Q 6O SO 4Q 3Q 2Q IO OO

IUOOOOOOOOOOO

~l— 1 ft I — f—

o o i

It ... II -.-:-, = , -v", » 1 1

c

~n J'[K ,

i uJJ>j^~ ~~i -

Fig. 5. Plan of potentiometer.

row of the box consisting of 20 coils of yg of an ohm each. In series with
these is a second row, immediately behind the bridge-wire, consisting of 11 coils
of y^y of an ohm each. By means of an arrangement of thick copper bars
connected with the ends of the slide-wire, which has a total resistance of "02 ohm,
any two adjacent coils of this latter series may be put in parallel with the slide-wire.
The 11 coils of '01, two of which are thus shunted, are therefore always exactly
equivalent to '] ohm.

For ordinary thermoelectric work the fall of potential along these two sets of coils
is adjusted so that each of the back row represents 1000 microvolts, each coil of
•01 ohm being therefore 100 microvolts. The slide-wire, 200 millims. long, is
provided with a divided scale on which Y5W5 Par^ °f its length can be easily
estimated. It will be seen that the slide-wire thus connected acts like a vernier to
the small coils.

The adjustment of the electromotive force is made by a standard Clark or Weston
cell and the auxiliary set of coils in the back row, a feature of the instrument being
that without any external alteration either form of standard may be used at will.
The five coils to the left are permanently connected in series, but are arranged so
that any coil may be cut out of circuit when required. Their values are 100, 40, 1,
•5, and '5 ohms respectively. Those to the right are a set of 10 simple series coils
of '01 ohm each, arranged so that a connexion can be taken from any one of them to

OF THE NATIONAL PHYSICAL LABORATORY. 355

the long copper bar just in front. Suppose we wish to use as a standard a Clark
cell whose electromotive force at the prevailing temperature is T4333 volts. It is
obvious that we may make the fall of potential over the "1 ohm balancing coils have
the desired value of 1000 microvolts by putting into circuit coils 100, 40, 1, -5, and
three of the "01 series, leading from the third hundredth by means of the copper bar
to No. 17 of the balancing set, when altogether we shall have

100
40

1 [> in the back row,
•5

•03 J
17

L7 "I

>• in balancing coils and bridge- wire,

making in all 143'33 ohms.

Should a Weston cell having an electromotive force of 1'OISG volt be substituted
for the Clark, the only alteration necessary would be to short-circuit coils 40, 1 , and
•5, and to move the connector from the third to the sixth of the set of hundredths.
The compensating current is furnished by a small secondary cell, in series with which
is a dial resistance capable of variation up to 200 ohms by steps of '005 ohm.

The four thermojunction circuits provided are connected to a selector switch, by
means of which each successively or any two of them connected in opposition may be
brought into circuit, and the change-over from the standard cell connexion required
in the preliminary adjustment is made at the two-way switch at the front left- hand
corner, by means of which the galvanometer may be put into the circuit desired.

Constructional details are given in the paper referred to. All the coils employed
are of selected manganin, carefully annealed, and all connexions are made by mercury
caps and copper short-circuiting pieces, the only metals employed anywhere in the
parts carrying current being copper and manganin.

The values of those coils in the box which were used in this work as determined at
the conclusion of the comparison experiments are given in the following table. Any
alteration in their relative values which had taken place since the first standardiza-
tion is undoubtedly so small as to be quite negligible compared with other errors in
thermocouple work.*

* It is obvious that in building clown to obtain a convenient standard of thermal electromotive force —
100 or 1000 microvolts — so long as the relative values of the coils employed remain the same, their absolute
value is of no moment.

A matter of great importance, however, is to measure the value of each coil in exactly the same way as
it is used. Accordingly, for this standardization a potential method of measurement was employed in all
pases, the current and potential leads being connected exactly as in actual work.

2 z 2

356

DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

COIL Values in ohms at 17° C.
Coils in main row nominal value '1 each.

No. 1

•100011

,. 2

•100028

,, 3

• 100025

„ 4

•100021

„ 5

•100022

No. 6

•100022

No. 11

•100019

No. 16

•100026

„ 7

•100046

10

J) ±w

•100026

„ 17

• 100025

„ 8

•100026

„ 13

• 100023

„ 18

• 100022

„ 9

•100019

» I*

• 100025

„ 19

• 100029

„ 10

•100022

„ 15 • 100020

„ 20

• 100033

Mean value of the 20 tenth-ohm coils = '100024.

Total ,, ,, slide-wire set of 11 coils shunted by slide-wire = '100043.

Maximum variation in the resistance of slide-wire set for different positions of the
slide-wire connector — '000008.

Value of nominal 100-ohm coil = 100'029.

The auxiliary set of '01 ohm coils for temperature compensation were all found to
be within J-^QQ part of their face value.

XII. Standard of Electromotive Force.

The standards of E.M.F. used with this potentiometer were two similar H-form
cadmium-sulphate cells with saturated solution of the type employed at the
Reichsanstalt, and made up as part of a large batch of similar ones by Mr. F. E. SMITH
in 1902. From Mr. SMITH'S measurements as to the relation of the E.M.F. of these
cells to the standard Clark cells of the laboratory, and from other data, it is practically
certain that the error committed in assuming their E.M.F. to be identical with those
at the Reichsanstalt is not greater than 1 part in 10,000, which corresponds to a
tenth of a degree at 1000° C. with the thermoj unctions employed. For the E.M.F.
of each of these cells, which throughout the work were never found to differ by more
than '0001 volt, the Reichsanstalt official value, namely 1'0186 volt at 20° C., was
assumed.

The following table gives the value of the total resistance used in the potentiometer
at different temperatures to adjust the E.M.F. over each coil of the main row to
exactly 1000 microvolts: —

Temperature . .

5°. 10°.

1

15°.

20°.

25°.

Resistance; .

i
101-90 101-89

I

101-88

101-86

101-84

The table of the values of the coils in the potentiometer shows that the actual
resistance corresponding to a nominal value of 101 '86 ohm is 101 '8 8 9. The relation
of this to the mean value of the main set of tenths, namely '100024 ohm, is well

OF THE NATIONAL PHYSICAL LABORATOKY.

357

within 1 part in 10,000 of its nominal value, and an inspection of the table shows
that the cumulative effect of individual coil errors is very small. No corrections were
therefore applied to any of the potentiometer readings.

Special precautions were taken to avoid the effect of temperature variation both on
the accumulator furnishing the compensating current of '01 ampere and on the
standard cells, both being placed in double-walled boxes surrounded by a thick layer
of cork clippings. Under these conditions the daily temperature range in the boxes
was found to be reduced to about one-fifth of the value outside them, and the
compensating current could generally be kept to within yorWo °f ^s value for an
hour at a time without adjustment.

XIII. The Thermoj unctions.

The thermoj unctions used in this research were composed of pure platinum with
platinum containing 10 per cent, of rhodium, and were all "6 millim. diameter. They
were obtained from HERAUS of Hanau, through Dr. HOLBORN of the Reichsanstalt,
and were compared by him with the standard junctions of the Reichsanstalt at a
number of fixed points.

In a letter to the Director of the Laboratory, Dr. HOLBORN says : " Two elements
were compared at four points, the melting-points of Zn, Sb, Ag, and Cu, and gave
the following results'" :—

Temperature.

Microvolts.

0 C.

Zinc

419-0

340,,

Antimony

630-5

5504

Silver (in graphite). .
Copper (in air) ....

961-5
1065-0

9088

1027;

Before commencing comparisons with the gas thermometer, three independent
determinations of the freezing-point of silver were taken in an electrically heated
crucible furnace with one of these junctions N.P.L. 2 ; in these experiments two
different observers took part, and the three results were :—

9087]

9092 > microvolts.

9082J

The mean of these, 9087, agrees very closely with the datum given above.
Junction N.P.L. 2 was selected for the comparison, while N.P.L. 1 was reserved as a

* In all experiments with thermojunctions here referred to, it is to be understood that the cold
junction is at 0° C,

358

DE. J. A. HAEKER ON THE HIGH-TEMPERATURE STANDAEDS

master standard, and was compared with No. 2 in a specially arranged electric furnace
at temperatures up to 1200° C. before and after the investigation.

The results of these comparisons show conclusively that Junction No. 2 has not
suffered any material alteration during its protracted heatings at high temperatures.*

The E.M.F. of the two junctions, as given by a comparison made at the close of the
work, is shown in the following table :—

1

Microvolts.

Approximate

•

Difference,

Difference in

temperature.

1-2.

•c.

No. 1.

No. 2.

290

2202-3

2202 • 1

+ -2

+ •02

385

3089 • 1

3089-4

•3

- -03

473

3930 -G

3930-3

•3

- -03

•189

4088 • 1

4084-8

+ 3-3

+ •34

628

5474-3

5471-3

+ 3-0

+ •30

795

7254-5

7249-3

+ 5-2

+ •48

859

7937-5

7931-5

+ 6-0

+ -54

1152

11308-5

11303-8

+ 4-7

+ •39

XIV. Formula for Thermojunctions.

From the values of the E.M.F. of N.P.L. 1 and 2, as determined above by
HOLBORN, a formula involving two powers of the temperature was calculated by least
squares to give the relation between E.M.F. and temperature to represent the
Reichsanstalt's scale.

The formula

E, = — 304 + 8'1 G5 t + 0-001663 t2

gives residuals at the four melting-points given above much smaller than their
probable error.

The corresponding formula for HOLBORN and DAY'S own standard junction T2,
using only the values of the temperature obtained from the gas thermometer with
bulb of platinum-indium and employing the revised data for the expansion of this
material at high temperatures,! is

E, = — 310 + 8-048 t + 0-00172 t-.

The following table gives side by side the E.M.F. of HOLBORN and DAY'S
junction T2, and of our own junctions at temperatures above 300° : — •

* At the conclusion of the second set of comparisons the metallic lustre of some inches of the platinum
wire was decidedly impaired, due probably to the natural disintegration of the material, but this did not
appear to be accompanied by the smallest change in the E.M.F. of the junction.

t HOLBORN and DAY, ' WIED. Annalen,' 1900, vol. 2, p. 520. The change from the original formula
for the expansion of the bulb involves a correction of the scale amounting to 4° at 1000°.

OF THE NATIONAL PHYSICAL LABORATORY.

359

T.

NPL 1 and 2.

HOLBORN and DAY'S T2.

Difference
NPL - T2.

300

2295

2260

35

933

925

400

3228

3185

43

966

960

500

4194

4145

49

1000

994

600

5194

5139

55

1032

1029

700

6226

6168

58

1066

1063

800

7292

7231

61

1099

1097

900

8391

8328

63

1133

1132

1000

9524

9460

64

1166

1166

1100

10690

10626

64

It will be observed that both the formulae just quoted do not apply at lower
temperatures, being nearly 40° C. in error at 0°, and that therefore extrapolation
downwards even over a narrow range is not permissible. The error of the formula
for NPL 2 was determined to be 4° at 200°.

XV. Determination of the Fixed Points 0° and 100° and Sulphur Boiling-point.

The determinations of the fixed points 0° and 100° for the gas and platinum
thermometers were made in baths specially built for eacli kind of instrument. The
ice-points were taken in glass vessels of a capacity of about G and '3 litres respec-
tively, consisting each of an inverted glass bell-jar with draining arrangement below,
and surrounded by a thick packing of cork clippings. Very little melting of ice took
place even in 12 hours, the upper surface of the ice -being protected by a thick felt
covering wrapped round the stem of the instrument.

The block-ice previous to use was always well washed and finely divided by an ice
plane, and was repeatedly tested for dissolved impurity, the method adopted being to
ascertain the amount of chlorine present by addition of silver nitrate to the drainings.
It was found satisfactory, except on one occasion.

The steam-point apparatus for the gas thermometer was of the usual type with
concentric tubes, but was arranged to be easily changed from the vertical to the
horizontal position by suitable couplings of wide compo tubing, connecting it to the
boiler. The steam- and zero-points for any series of comparisons were always taken
with the reservoir in the position in which it was used in that series, and in all cases
the amount of stem emergent was made as nearly as possible the same as in the
comparisons. The steam bath as arranged for the horizontal position is shown in

360

DE. J. A. HARKEK ON THE HIGH-TEMPERATURE STANDARDS

fig. 6. The bulb of the gas thermometer, resting on a small piece of cork, occupies
the centre of the inner tube, through which steam brought direct from the boiler by
a well protected wide tube is circulated. The steam issuing from the outer jacket is

Fig. 6. Steam bath.

condensed and returned to the boiler as shown. The excess of the steam pressure
over that of the atmosphere, which in these experiments was seldom over 1 millim.
of water, is indicated by a small graduated water gauge.

The sulphur-points of the platinum thermometers were taken in the usual manner
in the well-known Callendar form of boiling-point apparatus. The only departure
from previous custom being that for the glass boiling tube was substituted one of
thin weldless steel,* which is more durable and can be heated up quickly without
being removed from its asbestos cover. Careful comparisons of this form with the
older glass apparatus showed no systematic discrepancy.

For the sulphur boiling-point CALLENDAR'S old value 444 '53° C. at normal pressure

* An ordinary iron tube such as gas or steam pipe cannot be used for this purpose, as owing to
conduction from the flame of the burner upwards there is a tendency for the sulphur vapour to become
superheated.

OF THE NATIONAL PHYSICAL LABORATORY.

was taken. The sulphur-points were always taken
on days when the pressure was not far removed from
760 millims. to eliminate the uncertainty as to the co-
efficient to use when reducing to normal pressure.

XVI. Electric Furnaces.

Two different electric furnaces were employed in
this work. Their dimensions were similar, hut they
differed in that in the first the heating-wire was
wound uniformly, and in the second an approximation
to a logarithmic spiral was made at each end, the
turns being gradually crowded, so that the cooling-
effect of the ends was in a great measure compensated
by the additional heat supply. Both furnaces were
wound with wire of pure nickel about 1 '6 millims.
diameter. The heating current was supplied from a
special battery of 56 accumulators reserved for this
purpose, which was divided into four groups of 14
cells, capable of being coupled in series or parallel,
as desired. A set of large well-ventilated manganin
resistances, formed of two No. 9 wires in parallel, and
capable of carrying 100 amperes without undue heat-
ing, was arranged so that the external resistance of
the circuit could be altered by steps of '025 ohm up
to 3 '2 ohms, thus enabling any desired amount of
energy to be put into the furnace at will. The con-
struction of the furnace and the disposition of the
different instruments within it is shown in fig. 7.
The nickel heating-wire is wound upon the inner tube
of unglazed biscuit porcelain, and in order to prevent
the turns becoming short-circuited when hot, the
whole of the wound portion is covered with a thin
layer of " purimachos " which is baked on at a
moderate heat. The leading-iii wires are doubled
or trebled in all cases. The bulb of the gas thermo-
meter is supported on a small bridge of fire-clay
resting on the furnace bottom, and the standard
platinum thermometer and thermoj unction are
arranged as shown, great care being taken that
neither the wires of the junction nor the thin porcelain

VOL. com. — A. 3 A

361

362 DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

capillary tubes used to cover them shall anywhere touch the furnace wall. As
previous experience with gas and platinum thermometers, whose walls were of
porcelain, had shown how very much more slowly the transfer of heat toolc place
through this material than through metal or glass, even when surrounded by a stirred
liquid, it was judged preferable to make a comparatively small number of high-
temperature experiments, in which great constancy of temperature was attained for
some time previous to and during the observations, rather than to attempt to obtain
mean values from more extended series under less perfect conditions. With this
object, a Callendar recorder was connected to a second platinum thermometer placed
in the furnace, and, during the adjustment of the temperature and the comparisons,
records from this instrument were taken on an open scale. The use of the recorder
greatly shortened the time necessary for the establishment of a steady temperature
by guiding the observer as to the manipulation of the resistances in the heating
circuit. In addition to the large set of resistances in the heating circuit, a set of coils
of •()! ohm each was placed close to the recorder, and it was found that a change of
one step on this set made all the difference between a steady state and a gradual rise
or fall in the temperature of the furnace when equilibrium had been nearly established.
When desired, it was quite easy to keep the furnace temperature constant to about a
fifth of a degree for half-an-hour at a time, at temperatures as high as 1000° C., but
in most of the experiments the temperature was intentionally allowed to rise very
slowly. Without these precautions, comparisons between instruments of such widely
differing ''lag" as a bare thermojunction and a gas thermometer with porcelain
reservoir, whose walls were 2 millims. in thickness, would have undoubtedly been
liable to serious error.

XVII. Furnace Correction.

In order to investigate the distribution of temperature throughout the space filled
by the gas thermometer bulb, which was about 130 millims. in length, a pair of
thermojunctions, quite independent of the standard, were arranged so as to measure
the temperature difference between the centre of the furnace and points further out,
and so obtain a correction to be applied to the readings of the gas thermometer, to
reduce its indications to what it would have registered had the whole of the bulb
been at the same temperature as the middle point.

XVIII. Exploration of Furnace.

For this purpose a thin wire junction of platinum with platinum iridium was chosen
on account of its great sensitiveness at high temperatures. This was made up to
work differentially, and was composed of a piece of platinum iridium between two
pieces of platinum, thus forming two junctions, which were both placed in the hot
space, entering the furnace from opposite ends. The wires were stiffened by threading

OF THE NATIONAL PHYSICAL LABORATORY,

363

them through thin porcelain capillary tubes, and the two junctions of the platinum
with the copper forming the rest of the circuit were placed together in ice.

The E.M.F. given by this element for a difference of 1° between its two hot
junctions is given in the table below.* This is obtained by direct comparison over
the whole range of a simple junction made up of wires from the same reels, which had
been similarly treated.

Temperature in °C. Difference for 1° in microvolts.

500

16-4

600

16-6

700

16-8

800

16-9

900

17-1

1000

17-2

One of the junctions was carefully placed and kept at the middle point of the gas
thermometer bulb, and the second was arranged so that it could be pushed backward
and forward into positions 2, 4, G, and 8 centims. to the right and left of this point,
observations being made of the differential E.M.F. produced in each of these
positions.

From a number of such observations made after the different comparisons and
spaced over the interval 400° C. to ]()()0° C., curves were constructed showing the
distribution of temperature over this space for each of the furnaces used, and by
measuring these curves the difference between the average temperature of the bulb
and that of the central point, where the standard thermojunction was placed, could
easily be found.

For the compensated furnace the mechanical centre was found not to quite coincide
with the position of highest temperature at the higher ranges, though over the lower
part of the scale the curves were practically symmetrical.

The corrections obtained by this method from the different explorations were
plotted as ordinates against furnace temperatures as abscissae, and from the mean
curve thus obtained the following values were deduced as the mean corrections at
different temperatures :—

* In practice, when the junctions were placed as close together in the furnace as possible without actual
contact, owing to the inevitable small secondary effects arising from unsymmetric halting of the junction
wires, the E.M.F. round the circuit was not always zero. When, however, the junctions were so supported
as to nowhere touch the furnace wall, the total effect was always quite small and was allowed for in
each case.

3 A 2

364

DR. J. A. MARKER ON THE HIGH-TEMPERATURE STANDARDS

Furnace temperature.

500
600
700
800
900
1000

Corrections to gas thermometer

reading for compensated furnace

experiments.

+ 1-7
+ 1-5
+ 1-3
+ 1-1
+ -8
+ -0

From this table the numbers in the column headed " furnace correction" in
Series II. and III. have been deduced by interpolation. For the uncompensated
furnace used in the earlier experiments the correction is uncertain to about 0'5° C.,
and the results are, therefore, only given to this accuracy in the final column
representing the " corrected gas thermometer temperature."

XIX. Method of Calculation.

The observations taken with the gas thermometer were calculated according to the
usual formula.

Let H0 = pressure at 0°,
H= „ „ T°,

and let tl and t\ be temperatures of "dead space" with bulb at 0° and at
temperature T°.

Let ' = relation of the volume of the whole " dead space " to the volume of the

bulb at 0°,
« = coefficient of expansion of the gas at constant volume between 0° and

100°,

3/3 = the mean cubical coefficient of expansion of the porcelain bulb between
0° and T° ;

then

Hn

therefore T =

H

v

For a given filling H0 ( 1 + /I- - - ) is a constant if the zero-point remains

V0 1 + a.tll

constant.

The values of a obtained from the steam-point determinations were calculated from
the same formula, inserting the value of T obtained from the Regnault-Broch table
for the boiling-point of water under the prevailing pressure, as given in the example.

OF THE NATIONAL PHYSICAL LABORATORY.

865

XX. Series II.
ICE-POINT; 3. April 16, 1903.

Observer, J. A. H.

Time.

Scale
reading.

Dead
space.*

(1-)

Mercury

(2.)

columns.
(3.)

Barometer.

Barometer
temperature.

h. m.

2 20

159-86

13-8

14-2

14-0

768-73

15-0

23

159-98

13-8

14-2

14-0

•62

15-0

26

160-00

13-9

14-3

14-05

•58

15-2

30

160-00

14-0

14-3

14-15

•60

15-2

159-96

13-9

14-25

14-05

768-63

15-1

Index corr. =

747-18

•o

•00

•00

Lat. corr. + -43

•1

Temp. corr. =
Lat. corr. =

H0 =

Temp. corr. = 1-88
Scale corr. = -05

587-22
1-35
+ 0-32

13-9

14-25

14-05

15-0

Corr. bar. 767-13

Merc. col. "1

f 1*
temp. J

f

15°

586-19
767 13

180-94

It will be observed that during the 10 minutes covered by the observations the
barometric pressure fell rather more than O'l millim. while the scale reading rose a
corresponding amount.

* On this occasion the variations in the temperature of the jacket water were so abnormally great
during the preliminary period before the experiment, that it was decided to dispense with the circulation
in the space round the platinum capillary and take the whole dead space as being at the temperature of
the part below the stopper in che closed limb of the manometer, given by thermometer No. 1.

366

DR. J. A. BARKER ON TBE HIGH-TEMPERATURE STANDARDS

STEAM-POINT; 3. April 16, 1903.

Observer, J. A. H.

Dead
Scale space.
hme- reading.
(1.)

Mercury columns.

Barometer.

Barometer
temperature.

(2.)

(3.)

li. m.

3 15 225-58 14-2

14-55

14-30

768-60

15-5

22 '63 14-2

14-65

14-35

•62

15-5

30 -67 14-35

14-70

14-50

•60

15-5

32 -58 14-45

14-70

14-50

•55

15-5

225-61 14-3

14-65

14-4

768-59

15-5

747-18 -0

•o

•o

Lat. corr. = + -43

_ . j

Temp. corr. = 1-94
Scale corr. = -05

521-57 14-3
Temp. corr. -= 1-23 .

T of pfirr 4- 9Q

14-65

14-4
•5°

15-4

Corr bar 7fi7-OS

Col. temp. 1

14

taken as J
520-63

767-03

millini. millims.

Corr. bar 767-03
246 '40 Water manometer on steam bath = + 1 + -07

Total pressure on steam . . . 767 "10

Boiling-point of water at 767 • 10 millims. = 100-26°.

a = -003669L

XXI. Accuracy of Gas Thermometer Constant Determinations.

As an example of the kind of accuracy attained in the determination of the
constants of the gas thermometer, the individual values of the zero- and steam-points
taken before the comparisons in Series TI. is given along with the readings taken in
one complete determination of each.

For calculation of experiments of this kind, where no gradual systematic drift is
expected, instead of utilising for calculating the individual values of the a. the single
determinations of the ice-point, the mean of the ice-points is taken for this purpose.

The difference between the value of a. found before and after the comparisons is
within the limits' of experimental error. It will be seen from the table appended,
which gives in an abridged form the calculation of the gas thermometer temperatures
in the 13 experiments of Series II., that the difference has, however, been treated as
real, and assumed to vary with the time. A more serious change has, however, taken

OF THE NATIONAL PHYSICAL LABORATORY.

367

place in the quantity of gas present in the bulb. The change in this case has also
been assumed as proportional to the time, since before the first experiment was made
the whole furnace was maintained at a high temperature for some time, and there is
no sufficient evidence that the change occurred otherwise than regularly. So far as
could be found by testing on the pump after the conclusion of the; series no leak
could be discovered, even when the bulb was at a full red heat,

CONSTANTS for Gas Thermometer. Series II.

I. Before the comparisons :—

Ice-points.

H0.

Dead space temperature.

(1)
(2)

(3)

180-90
•93
•94

12-6
13-1
13-9

Mean ice-point before "1
comparison. . - j

180-923

Steam-points.

H1W.

Dead space temperature.

a.

(1)

(2)
(3)

246-24
•26
••23

1.-5-95
13-75
14-3

•003GG9.J
•003671!
•00:3669!

Mean value of a before comparisons =

•003670,,

II. After comparisons : —

(4)

Mean ice-point after "I
comparisons. . = j

Ice-points.

H0.

• 182-67

182-65

182-66

Dead space temperature.

17-4

17-8

H

IOO-

Steam-points.

Dead space temperature.

(5)

248-64
248-63

16-3
16-6

• 00367 15
•003670S

• 00367 12

368

DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

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OP THE NATIONAL PHYSICAL LABORATORY. 369

XXIII. Method of calculating Comparison Experiment.

As an example of the method employed in making and calculating out an
experiment, the results of the observations made in Experiment No. 13 of Series II.
are given in full. During this experiment the steadiness of the temperature of the
furnace was perhaps a little above the average, but the room temperature and that
of the resistance-boxes and of the various mercury columns of the gas thermometer
and barometer were rising more rapidly than usual. According to observations taken
on the thermojunctions and platinum thermometer, the steady state had been
maintained for about 10 minutes before the first recorded reading on the gas
thermometer was taken. As only two observers were available, one of these took
alternate observations on the gas thermometer and thermojunctions, while the second
took the platinum thermometer readings.

•*• O

By graphic interpolation the mean thermojunction readings, corresponding to the
times at which the other instruments were read, were obtained and are given in the
example. ""

The readings of the gas thermometer are made as independent of one another as
possible by lowering the mercury each time before a setting, raising it again slowly so
as to make a new meniscus.

The calculation to the accuracy here necessary of the air temperature corresponding
to definite platinum temperatures is somewhat laborious, if the formula

T = MO + 50) - AA5T + 50Y - 10'°6°>
\ o / \ 8 / S

has to be applied for each observation. The graphic methods used by HEYCOCK
and NEVILLE cannot easily be made sufficiently accurate. Since the value of 8 for pure
platinum wire has been found to be 1'5, varying from one specimen to another within
very narrow limits, the most suitable method of effecting this conversion was found
to be to construct a table giving, for a sufficient number of points, the value of T for
given values of pt when S = 1'5. A second table gives the correction to apply to
the value of T thus obtained, if the S differs from the standard value by a small
amount.

* The letters AB, BA refer to the position of the reversing switches leading to the potentiometer.
Owing to small Thomson and Peltier effects, there was generally a small difference between the two
positions. In this case it is rather above the average. There was no difficulty in setting to '1 microvolt,
so the readings are given to this figure, though it is not considered as having any significance in temperature
measurement.

VOL. CCIII. — A. 3 B

370 DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

GAS Thermometer Readings. Expt. 13, May 1, 1903. Qiserver j A H

Time.

Scale.

Thermometers.

Barometer.

Barometer
temperature.

Dead
space
below
point.

(1-)

Columns.

Water
circula-
tion.

(4.)

(2.)

(3.)

]]. m. s.

550
9 0
14 0
17 30

810-86
816-96
817-00
817-00

14-7
•8
•95
15-0

14-90
•95

15-00
•05

14-75
•8
•9

•95

15-2
•2
•1
•1

749-85
•90
•95
750-00

14-0

•o
•o
•o

Index corr. =

816-95
-747-18

14-85
•00

14-95
•00

14-85
•00

15-15

•40

749-92
Scale corr. = - '05
Lat. corr. — \- '42
Temp. corr. 1 • 70

14-0
•1

+ G9-77
+ -04
•17

14-85

14-95

14-85

14-75

13-9

Lat, corr. =
Temp. corr. =

H =

Corr. bar. =748 -59

+ 69-64
748-59

Mean temperature of dead spac

W1 + 1 • v-1-^ = 184'37

\ V 1 + ati/
a. = -00367
Tgas = 1005-2°.
where T taken from junctio

e = 14-80 = /i'.
5. -*• - -0100.

818-23

V

lo.
(1 + 3£T) = 1-01230,

n = 1004-3°.

A second approximation, employing 1005'2° in the dilatation term, makes the final
gas thermometer temperature 1005'0° C.

THERMOJUNCTION Readings. Expt. 13.

Observer, J. A. H.

Time.

Microvolts.

Cell resistance and
temperature.

Series resistance.

h. m. s.

500

9570-5 AB

101-89

103-036

2 0

75-8BA

T = 13-7

. —

7 30

76-1 BA

—

—

8 15

73-0 AB

—

—

11 30

73-5 AB

—

103-036

13 0

72-0 AB

—

—

16 0

71-2 AB

—

—

16 30

75-2 BA

—

—

20 40

71-2 AB

. — .

21 0

75-4 BA

—

103-036

OF THE NATIONAL PHYSICAL LABORATORY.

371

From these observations plotted, the following were deduced as the thermojunction
readings simultaneous with the gas thermometer readings :

Time.

Mean microvolts.

h.

in.

B.

5

5

0 9573

•8

9

0 9574

•8

14

0 9573

•9

17

30 9573

•2

Mean = 9573

•9

From table of E.M.F. of junctions the value of the E.M.F. for 1000° C.
and the difference for 10'' = 11 51 microvolts.

Whence the temperature corresponding to 9573'9 = 1004*3°

T (thermojunction) = 1004'3.

= 9524-0,

PLATINUM Thermometer Headings. Expt. 13.

Observer, W. H.

Time.

Coils.

Bridge wire.

Box temperature. Centre = -0-29.

h. m. s.

550
9 0
14 0
17 30

ABEFGI

-2-530
-2-660

-2-686
-2-784

14-32
•38
•44
•48

1129-454

2-665

14-40

R.

R - RO. i>t.

T(8=

1

Difference for
•009 in 8.

1125-44

868-79

868-70

1005-18

- -82

Summary . —

T(S= 1-491) = 1004-37°.

o

T gas (found) = 1005'G

Furnace correction = + O'O

T gas corrected = IOOS'0

T thermojunction = 1004'4

T platinum thermometer =. 1004 '37
3 B 2

372 DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

XXIV. First Set of Comparisons. October, 1902.

millims.
336-41
336-25

Mean ice-point of gas thermometer before observations

after

Mean value of a before = '003675,
after = "003676.

f = '0083.

vn

Platinum thermometer BA2 :—

R0 = 2-57453 ohms before commencing,

= 2 '57 46 5 ,, after seventh experiment,
= 2-57433 ,, at conclusion,

F.I. = 1 '00034 before commencing,
= 1 '00020 at end,

S = 1'510 mean of all observations.

•

FIRST Set of Comparisons. Furnace wound uniformly. BA3 and Gas Thermometer.

No-, of
experiment.

ft.

T (5 = 1 -50)
from
Table I.

T (5= 1-491)
from
Table II.

Gas

thermometer
rending.

Compensation
correction.

Corrected gas
thermometer
reading.

1

482-54

514-66 514-47

510-2

+ 5

515

2

604-96

660-48

660-15

655-2

+ 4

659

3

606-48

662-35 662-02

659-1

+ 4

663

4

610-02

666-69 666-36

661-7

+ 4

665|

5

459-21

487-66 487-49 483 -4

+ 5

488J

6

749-07

843-03 842-46 839-7

+ 3

842£

7

750-24

844-56 843-99

840-7

+ 3

843A

8

481-53

513-36 513-17

509-5

+ 5

5Uf

9

394-96

414-52

414-40

411-2

+ 5J

416£

i

RESULTS of Comparison arranged in Order of Ascending Temperature.

T from platinum.

T from gas
corrected.

Difference.
T gas - T platinum.

9

414-4

416J

+ 2

5

487-5

488£

+ 1

8

513-2

514$

+ 1

1

514-5

515

+ -5

2

660-1

659

-1

3

662-0

663

+ 1

4

666-4

665£

-1

6

842-5

842£

+ 0

7

844-0

843£

- -5

OF THE NATIONAL PHYSICAL LABORATORY. 373

Second Set of Comparisons. April and May, 1903.

millims.

Mean ice-point of gas thermometer before comparisons . . . . 180'923
„ ,, „ after „ .... 182'66

Mean value of a from determinations before comparisons = '003670,
„ „ „ after „ = '003671.

v = -01(V

vo

In this case there is a change in the zero-point greater than that observed in the
first set and in the opposite direction. In allowing for this, the change was supposed
to have been proportional to the time.

Platinum thermometer BA2 :—

R0 = 2 '572 51 ohms before commencing comparisons,
= 2'57198 „ at conclusion of ,,

F.I. = 1 '00008 before commencing ,,

= 1 '00000 at conclusion of ,,

8 = 1*491 mean of large number of observations extending over six months.

The platinum thermometer constants of this set are expressed in terms of a slightly
different unit from that employed in first set six months earlier.

374

])R, J. A HAKKER ON THE HIGH-TEMPERATURE STANDARDS

CD

OH

g

O

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CD
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CD

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OF THE NATIONAL PHYSICAL LABORATORY. 375

Third Set of Comparisons. May and June, 1903.

millims.

Mean ice-point of gas thermometer before comparison = 176 "28

after „ = 176'19

Mean value of a from determinations before = '003660

after == '003661

Platinum thermometer K2 :—

R0 = 2'62864 ohms before commencing comparisons,
= 2'62705 ,, at conclusion of „

F.I. = 1 '007 8 9 before commencing ,,

= 1 '008 19 at conclusion of ,,

8 = I -5 10 mean value over a long period.

376

DR. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS

<u

CO

— <

CD

PH

a

o
U

cc

§

.25

'£

cS
PH

I

CO

£

'£
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—

a

H

c<j

PM

5?

CD
I

a
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a
K

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o cs M fl £

fcSgls

S S e
S fc.S

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i— i(Mt-O5i— i

OF THE NATIONAL PHYSICAL LABORATORY. 377

XXV. Conclusions.
These three sets of experiments show that : —

(1.) The readings of platinum thermometers BA2 and K2, which may be taken
as representative samples of pure platinum, when reduced to the air-scale by
CALLENDAR'S formula, employing his value for the boiling-point of sulphur, are in
reasonably close agreement with the results given by the constant-volume nitrogen
thermometer employing chemical nitrogen under low initial pressure and using the
revised values for the dilatation of porcelain. The divergence of the two scales only
exceeds the probable error at the higher part of the range.

(2.) The platinum thermometers, and the thermojunctions representing the
temperature scale of the Reichsanstalt, based on measurements made with the gas
thermometer with bulb of platinum iridium, agree very closely, the thermojunction
giving apparently a slightly higher value throughout the range covered by the
experiments.

As the result of these comparisons is to justify the application of CALLKNUAR'S
parabolic formula up to 1000° C., the tables previously alluded to for deducing the
value of T from pt over the whole range for which platinum thermometers are useful,
— 200° to + 1100°, are given. From — 200° to + 200° the values are given for
each degree,* and from 200° to 1100° for each 10°.

For the figures given in Table I, in their final form, calculated by a method
which renders unlikely the accumulation of errors greater than 1 unit in the last
figure given, I am indebted to Mr. F. J. SELBY.

T have also to acknowledge my indebtedness to Mr. W. HUGO, who assisted during
the comparisons by taking the observations with the platinum thermometers, and
especially to the Director of the Laboratory, Dr. GLAZKBROOK, for his continued
interest and advice on many points.

* In the paper as printed this table has been shortened by only giving the value of pt for intervals of
1° over the range - 50° to + 150°.

VOL. CCIII.- -A,

378

DE. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

XXVI. TABLE I.

ft.

T.

Difference
for 1 ° pt.

Proportional
parts.

pt.

T.

Difference
for 1° pt.

Proportional
parts.

-200

-191-62

- 39

- 38-208

•934

•974

-190 -182-28

- 38

- 37-234

•936

•975

-180

-172-92

- 37

- 36-259

•939

•975

^

-170 -163-53

- 36

- 35-284

•941

•975

-160 -154-12

- 35

34-309

•943

•975

-150 -144-69

- 34

- 33-334

•946

•976

-140 -135-23

- 33 - 32-358

•949

•976

-130 -125-74

- 32 - 31-382

•951

•976

-120 -116-23

- 31

- 30-406

•954

•977

-110 -106-69

- 30 • 29-429

•956

•977

-100 - 97-13

- 29 - 28-452

•959

•977

- 90 - 87-54

- 28 - 27-475

•962

•978

- 80 - 77-92

- 27 - 26-497

•964

•978

70

- 68-28

.

- 26 - 25-519

•967

•978

- 60

- 58-61

- 25

- 24-541

•970

•978

- 50

48-91

- 24 - 23-563

•971

•979

- 49

47-937

970 - 23 - 22-584

•972

•979

. 40

AR -Qftf;

oo oi -firtF;

T:O ~f\j \j \>*j

•972

•979

- 47 45-993

•1

•097

- 21 - 20-626

•972

•979

- 46 - 45-021

•2

•194

- 20 • 19-647

•973

•980

45 44-048

•3

•291

19 • 18-667

980

•973

•980

U 49 .f)7P>

. A

. QQQ

1Q 1*7 .CO*?

TO V 1 rj

•973

4

GOO

— lo

1 1 VJU 1

•980

43

- 42-102

•5

•485

17

16-707

•1 -098

•973

•980

42

41-129

• fi

•582

16

15-727

•2 -196

•973

•981

41

40-156

•7

•679

15

14-746

•3 -294

•974

•981

40

- 39-182

•8

•776

14

13-765

•4 -392

•974

•981

- 39

- 38-208

•9

•873

13

12-784

•5 -490

•974

•982

OF THE NATIONAL PHYSICAL LABORATORY.

379

TABLE I. — continued.

pt.

T.

Difference
for 1 ° pt.

Proportional
parts.

pt.

T.

Difference
for 1° pt.

Proportional
parts.

13

12-784

+ 13

+ 12-832

•982

•989

- 12

11-802

•6

•588

+ 14

+ 13-821

•982

•989

11

10-820

•7

•686

+ 15

+ 14-811

•982

•989

10

9-838

•8

•784

+ 16

+ 15-800

•983

•990

_ 9

8-855

•9

•882

+ 17

•f 16-790

990

•983

1 O

•991

8

7-872

•983

+ 18

+ 17-781

•990

- 7

6-889

+ 19

+ 18-771

•1 -099

•983

•991

- 6

5-906

+ 20

+ 19-762

•2 -198

•984

•991

- 5

4-922

+ 21

+ 20-753

•3 -297

•984

•992

4

3-938

+ 22

+ 21-745

•4 -396

•984

•992

3

2-954

+ 23

+ 22-737

•5 -495

•984

•992

_ 2

1-970

+ 24

+ 23-729

•6 -594

•985

•992

_ i

0-985

+ 25

+ 24-721

•7 -693

•985

•993

+ 0

+ o-ooo

+ 26

+ 25-714

•8 -792

•985

•993

+ 1

+ 0-985

+ 27

+ 26-707

•9 -891

•986

•993

+ 2

+ 1-971

+ 28

+ 27-700

•986

•993

+ 3

+ 2-957

+ 29

+ 28-693

•986

•994

+ 4

+ 3-943

+ 30

+ 29-687

•987

•994

+ 5

+ 4-930

+ 31

+ 30 '681

•987

•994

+ 6

+ 5-917

+ 32

+ 31-675

•987

•995

+ 7

+ 6-904

+ 33

+ 32-670

•987

•995

+ 8

+ 7-891

+ 34

+ 33-665

•988

•995

+ 9

+ 8-879

+ 35

+ 34-660

•988

•996

•

+ 10

+ 9-867

+ 36

+ 35-656

•988

•996

+ 11

+ 10-855

+ 37

+ 36-652

•988

•996

+ 12

+ 11-843

+ 38

+ 37-648

•988

•996

+ 13

+ 12-832

•989

+ 39

+ 38-644

•997

3 c 2

380

DR. J. A. MARKER ON THE HIGH-TEMPERATURE STANDARDS

TABLE I. — continued.

pt.

T.

Difference
f or 1 ° pt.

Proportional
parts.

pt.

T.

Difference
for 1 ° pt.

Proportional
parts.

+ 39

+ 38-664

+ 65

+ 64-657

•997

1-005

+ 40

+ 39-641

+ 66

+ 65-662

•997

1-005

+41 + 40-638

+ 67

+ 66-667

•997

1-005

~^»

+ 42 + 41-635

+ 68

+ 67-672

•998

|

1-005

+43 + 42-633

+ 69

+ 68-677

•998

1-006

+44 + 43-631

+ 70 + 69-683

•998

1-006

+45 + 44-629

+ 71

+ 70-689

•999

1-007

+46 + 45-628

+ 72

+ 71-696

•999

1-007

+ 47 + 46-627

1000 + 73

+ 72-703

•999

1-007

+ 48 + 47-626

+ 74 + 73-710

•999

1-007

+ 49 + 48-625

•1 -100 + 75 + 74-717

1-000 1-008

+50' + 49-625

•0

•200 +76 + 75-725

1-000

1-008

+51 + 50-625

•3

•300

+77 + 76-733

1

1-000

1-008

+52 + 51-625

•4

•400

+ 78 + 77-741

1-001

1-008

+53 + 52-625

•5

•500

+ 79 + 78-749

1-001

1-009

+54 + 53-627

•6

•600

+80 + 79-758

1-001

1-009

+55 + 54-628

•7

•700

+81 + 80-767

1010

I

1-002

1-009

+56 + 55-630

•8

•800

-u 82 + 81 -77fi •

1-002

ouv/

~ O^ T^ O J. 1 t \j

1-010

+57 + 56-632

•9

•900 +83 + 82-786

•1 -101

1-002

1-010

+58 + 57-634

+ 84

+ 83-796

•2 -202

1-002

1-011

+ 59

+ 58-636

+ 85

+ 84-807

•3 -303

1-003

1-010

+ 60

+ 59-639

+ 86

+ 85-817

•4 -404

1-003

1-011

+ 61

+ 60-642

+ 87

+ 86-828

•5 -505

*

1-003

1-011

+ 62

+ 61-645

+ 88

+ 87-839

•6 -606

1-004

1-012

+ 63

+ 62-649

+ 89

+ 88-851

•7 -707

1-004

1-012

+ 64

+ 63-653

+90 + 89-863 -8 -808

1-004

1-012

+ 65

+ 64-657

+91 + 90-875

•9 -909

1-005

1-013

OF THE NATIONAL PHYSICAL LABORATORY.

381

TABLE I. — continued.

pt.

T.

Difference
for 1 ° pt.

Proportional
parts.

Pt.

T.

Difference Proportional
for 1" pt. parts.

+ 91

+ 90-875

+ 117

+ 117-304

1-013

1-021

+ 92

+ 91-888

+ 118

+ 118-325

•6 -612

1-013

1-021

+ 93

+ 92-901

+ 119 +119-346

•7 -714

1-013

1-022

+ 94

+ 93-914

+ 120 + 120-36S

•8 -816

1-014

1-021

+ 95

+ 94-928

+ 121

+ 121- 38',)

•9 -918

1-014

1 • 022

+ 96

+ 95-942

+ 122

+ 122-411

1-014

1-023

+ 97

+ 96-956

+ 123

+ 123-434

1-014

1-023

+ 98

+ 97-970

+ 124

+ 124-457

1-015

1-023

+ 99

+ 98-985

+ 1 25

+ 124-480

1-015

1-023

+ 100

+ 100-000

+ 126

+ 126-503

1-015

1-023

+ 101

+ 101-015

+ 127

+ 127-526

1-016

1 • 02 1

+ 10-2 +102-031

+ 1 28

+ 128-550

1-016

1-025

+ 103 +103-047

+ 129

+ 129-575

1-016

1-024

+ 104

+ 104-063

+ 130

+ 130-59!)

1-017

1 • 025

+ 105

+ 105-080

+ 131

+ 131-624

1-017

1-026

+ 106

+ 106-097

+ 132

+ 132-650

1-017

1 • 025

+ 107

+ 107-114

+ 133

+ 133-675

1-018

1-026

+ 108

+ 108-132

+ 134

+ 134-701

1-018

1-026

+ 109

+ 109-150

+ 135

+ 135-727

1-018

1-027

+ 110

+ 110-168

+ 136

+ 136-754

1-018

1-027

+ 111

+ 111-186

1020 +137 +137-781

1-019

1-027

+ 112 +112-205

+ 138 +138-808

1030

1-019

1 • 028

+ 113 +113-224

•1 -102

+ 139 + 139'836

1-020

1-027

+ 114 +114-244

•2 -204 +140 +140-863

• 1 -103

1-020

1 -028

+ 115 +115-264

• 3 "306

+ 141

+ 141-891

•2 -206

1 -020

1 -029

+ 116

+ 116-284

•4 -408

+ 142

+ 142-920

• 3 • 309

1 -020

1-029

+ 117

+ 117-304

•5 -510 +143

+ 143-949

•4 412

1-021

i|

1 -029

1

382

DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS

TABLE I. — continued.

ft.

T.

Difference
for 1° pt.

Proportional
parts.

pt.

T.

Difference
for 1° pt.

Proportional
parts.

+ 143

+ 143-949

+ 340

+ 353-44

1-029

1-102

+ 144

+ 144-978

•5

•515

+ 350

+ 364-46

1-030

1-106

+ 145

+ 146-008

•6

•618

+ 360

+ 375-52

1-029

1-110

~-

+ 146

+ 147-037 -7

•721

+ 370

+ 386-62

1-030

1-114

+ 147

+ 148-067 -8

•824

+ 380

+ 397-77

1-031

1-118

+ 148

+ 149-098 -9

•927

+ 390

+ 408-95

1-031

1-123

+ 149

+ 150-129

+ 400

+ 420-18

1-031

1-127

+ 150

+ 151-16

+ 410

+ 431-45

1-033

1-132

+ 100

+ 101-49

+ 420

+ 442-77

1-036

1-136

+ 170

+ 171-85

+ 430

+ 454-13

1-040

1-140

+ 180

+ 182-25

+ 440

+ 465-53

1-043

1-140

+ 190 ' +192-68

+ 450 +476-97

1-040

1-149

+ 200 +203-14

+ 460 +488-46

1-051

1-154

+ 210 +213-65

+ 470 +500-00

1-053

1-158

+ 220 +224-18

+ 480 +511-58

1-057

1-163

+ 230 +234-75

+ 490 +523-21

1-060

1-168

+ 240 + 245 • 35

+ 500 +534-89

1-064

1-173

+ 250 ' +255-99

+ 510 +546-62

1-068

1-178

+ 260

+ 266-67

+ 520 +558-40

1-071

1-182

+ 270

+ 277-38

+ 530

+ 570-22

1-075

1-188

+ 280 +288-13

+ 540

+ 582-10

1-079

1-193

+ 290

+ 298-92

+ 550

+ 594-03

1-083

1-198

+ 300

+ 309-75

+ 560 +606-00

1-086

1-203

+ 310

+ 320-61

+ 570

+ 618-03

1-090

1-208

+ 320

+ 331-51

+ 580

+ 630-11

1-094

1-213

+ 330

+ 342-46

+ 590

+ 642-24

1-098

1-219

+ 340

+ 353-44

+ 600

+ 654-43

1-102

1-224

OF THE NATIONAL PHYSICAL LABORATORY.

383

TABLE I. — continued.

ft.

T.

Difference

for 1 ° pt.

Proportional
parts.

pt.

rp Difference | Proportional
for 1° pt. parts.

+ 600

+ 654-43

+ 850

+ 979-11

1-224

1-390

+ 610

+ 666-67

+ 860

+ 993-01

1 • 230

1-399

+ 620

+ 678-97

+ 870 +1007-00

1-235

1-407

+ 630

+ 691-32

+ 880 +1021-07

1-241

1-416

+ 640

+ 703-73

+ 890 +1035-23

1-247

i|

1-424

+ 650

+ 716-20

1 + 900

+ 1049-47

1-253

1-433

+ 660

+ 728-73

+ 910

+ 1063-80

1-259

1-441

+ 670

+ 741-32

+ 920

+ 1078-21

1-264

1 -450

+ 680

+ 753-96

+ 930

+ 1092-71

1-271

1-460

+ 690

+ 766-67

+ 9-40

+ 1107-31

1-277

1 • 4G(J

+ 700

+ 779-44

+ 950 +1122-00

1-283

1 -479

+ 710

+ 792-27

+ 960

+ 1136-79

1-290

1-490

+ 720

+ 805-17

+ 970

+ 1151-69

1-296

1-499

+ 730

+ 818-13

+ 980

+ 1166-68

1-303

1 -508

+ 740

+ 831-16

+ 990

+ 1181-76

1-310

I "Ol'J

+ 750

+ 844-26

+ 1000 +1196-95

1-316

i oza

+ 760

+ 857-42

+ 1010

+ 1212-24

1C 1 1

1-323

•541

+ 770

+ 870-65
1-330

+ 1020

+ 1227-65
1-552

+ 780

+ 883-95

+ 1030

+ 1243-17

IK I ' Q

1-337

ODD

+ 790

+ 897-32

1-344

+ 1040

+ 1258-80
1-575

+ 800

+ 910-76

1-352

+ 1050

+ 1274-55
1-587

+ 810

+ 924-28

1-359

+ 1060

+ 1290-42
1-599

+ 820

+ 937-87

1-367

+ 1070

+ 1306-41
1-611

+ 830

+ 951-54

+ 1080

+ 1322-52

1 -fi9/t

1-374

i t > .. i

+ 840

+ 965-28

+ 1090

+ 1338-76
1 -fiS7

1-383

1 \)O I

+ 850

+ 979-11

+ 1100

+ 1355-13

1-390

384

DE. J. A. BARKER ON THE HIGH-TEMPERATURE STANDARDS, ETC.

XXVII. TABLE IT. — To Calculate, the Cliange in T for a Given Small

Change in 8.

T.

Change in T for change
of + -01 in 5.

T.

Change in T for change
of + -01 in S.

-200

+ -0600

+ 250

+ -0375

-180

+ -0504

+ 300

+ -0600

-160

+ -0416

+ 350

+ -0875

-140

+ -0336

+ 400

+ -1200

-120

+ -0264

+ 450

+ -1575

-100

+ -0200

+ 500

+ -200

- 80

+ -0144

+ 550

+ -247

- 60

+ -0096

+ 600

+ -300

40

+ -0056

+ 650

+ -357

- 20

+ -0024

+ 700

+ -420

0

+ • 0000

+ 750

+ -487

+20 - -0016

+ 800

+ -560

+ 40 - -0024

+ 850

+ -637

+ 60

- -0024

+ 900

+ -720

+ 80

- -0016

+ 950

+ -807

+ 100

- -0000

+ 1000

+ -900

+ 120 + -0024

+ 1050

+ -997

+ 140

+ -0056

+ 1100 +1-100

+ 160 +-0096

+ 1150

+ 1-207

+ 180 +-0144

+ 1200 +1-320

+ 200 + -0200

+ 1250 +1-437

[ 385 ]

XII

Films.

INDEX SLIP.
Commi'nicat.ed i

ambridge.

UAKNETT, J. ('. M. — Colours in Metal Glasses and in Metallic Films.

Phil. Trans., A, vol. 208, 1904, pp. 885-420.

" Allotroptc" Silver — Constitution, Origin of Colour of.

GARNBTT, ,T. C. M. Phil. Trans., A, TO!. 203, 1904. pp. 385-420.

fnt roiiucttoit.

Coloration of Glass by Reduced Metals ; Action of Radium.

GABHETT, J. C. M. Phil. Trans., A, rol. 203, 1904, pp. 385-420.

r> •-».,«. (V)f )f**\| IkU •

Coloured Metallic Films — Cause of Change on Annealing.

&ABXBTT, J. G. M. Phil. Trans., A, TO!. 208, 1904, pp. 3K5-420.

Nascent Crystals of Metal, Spherical Form.

GA'BITBTT, J. C. M. Phil. Trans., A, vol. 203, 1904, pp. 385-420.

proportion may have any vulur. from ,

In Part I. ti>.' ^hsr-i". .--.fii.i-: of SIKI-KN.

operties of a medium
) two Parts : the first
woportion of volume
al films, iu which tliis

' beyond the limit of
>ed. It is shown that
d which, when their
endeavoured to show
-length of light in the
I ruby glass, and that
IBS exhibited by gold
irticles or to excessive
ler. It is also shown
glass the ruby colour

microscopic vision (• Ann. dri Pin
the particles soe.ti in ;i gold rutiv <:''•*>.-:
uetera are kvvi t!i:m O'l^i. ai-.1 arciirat'N
that the presence of many of tlx^o u>.ii
a* \rill account, for ali the optical v

Clarities in colour and in [»••!.
-*s are due to excessive distance betw>-t»ii <
of such ]Mi'*:ii!<?5;, the latter, howescr, involving list
i'n irom radium is capable of producing in g-

dopted enables us to
form in a glass what
colon; • ss will be in its " regular" mtnte.

ur on heating, of the

Proc.,' vol. 72, p. 226),

Professor R. W. WOOD

ng that they can be

,-jsed of minute metal spheres of

.

17.8.04

384 DR. J. A. HARKER ON THE HIGH-TEMPERATURE STANDARDS, ETC.
XXVII. TABLE IT. — To Calculate the Change in T/or a Given Small

T.

(

-200

+ -0600

-180

*

-160

., •

-140

-120

-t *\.'2t'4

-100

- 80

-f ' U • 4 4

- 60

40

I" >~jf>

- 20

. .aiiili'i PI

- 0

,,.„. .0£*-888 .qq ,W

+ 20

+ 40

+ 60

,(iS4-<i>!8 .qq ,M

+ 80

+ 100
+ 120

.(**• 588 .qq ,*0

+ 140

+ 160 <™ a»e „„ M

+ 180

+ 200

+ •

+ •

/iug xaani + -1200

f 4 H- •

4-51 + -200

ni hn« va«M) J«feM ni
,808 JOT .A ,.«<unT Mill

,rrs.

+ '487

.io uloioO to (lijjnO .niulli1i1r.no'.) — tBTlif< '' ->ic|(
I ,80S JOT ,A ..«a*iT .lirfl .M .') .1. .ITSHHA«

- •

\nihisU 3o uoiloA ; «i«J-iM hwoubifi vd «Mli.) lo iioi
.Wt JOT .A ,..wiT .lii« .M."'1

rfO lo -

:00

mli'l stf
,80£ ^loT ,A ,.en«T .iiri^ .M tt.TZJO

I wlyi' ~T" 1 * OidU

.nnot laonsriqB ,I«J9M Jo s
.80S lor .A ,.«n«iT .IU? .M .0 .1, ,

fin..]..: )

[ 385 ]

XII. Colours in Metal Glasses and in Metallic Films.

By J. C. MAXWELL GARNETT, B.A., Trinity Collage, Cambridge.

Communicated ly Professor J. L ARMOR, Sec.R.S.

Received April 19,— Read June 2, 1904.

Introduction.

§ 1. THE present paper contains a discussion of some optical properties of a medium
containing minute metal spheres. The discussion is divided into two Parts : the first
Part dealing with colours in metal glasses, in which the proportion of volume
occupied by metal is small ; the second Part dealing with metal films, in which this
proportion may have any value from zero to unity.

In Part I. the observations of SIEDENTOPF and ZSIGMONDY beyond the limit of
microscopic vision (' Ann. der Phys.,' January, 1903) are discussed. It is shown that
the particles seen in a gold ruby glass are particles of gold which, when their
diameters are less than 0'1/x, are accurately spherical. I have endeavoured to show
that the presence of many of these minute spheres to a wave-length of light in the
glass will account for all the optical properties of " regular" gold ruby glass, and that
the irregularities in colour and in polarisation effects sometimes exhibited by gold
glass are due to excessive distance between consecutive gold particles or to excessive
size of such particles, the latter, however, involving the former. It is also shown
that the radiation from radium is capable of producing in gold glass the ruby colour
which is generally produced by re-heating. The method adopted enables us to
predict from a knowledge of the metal present in metallic form in a glass what
colour that glass will be in its " regular" state.

In Part II. the optical properties, and the changes in colour on heating, of the
silver and gold films observed by Mr. G. T. BEILBY (' Roy. Soc. Proc.,' vol. 72, p. 226),
and of the potassium and sodium films deposited on glass by Professor R W. WOOD
('Phil. Mag.,' p. 396, 1902), are discussed, with a view to showing that they can be
accounted for by supposing the films to be composed of minute metal spheres of
varying sizes.

VOL. CCIII. — A 370. 3 D 17.8.04

386 MR. J. C. MAXWELL GARNETT ON

PART I.

§ 2. Consider the incidence of light of wave-length X on a sphere ot metal of
radius a. Suppose the constants of the metal relative to the surrounding medium,
which we may first suppose to be aether, are n, the coefficient of refraction, and K,
the coefficient of absorption. Let us write

N = w(l — IK) .......... (1),

where, as usual, t denotes v — 1.

We shall use the following notation to denote the electric vector :—

Incident light ....... E0 {X0 = exp {ip (t — z/c)} , Y0 = 0, Z0 = 0}.

Transmitted -f reflected light . . Ej {X1; Y1; Z,}.

Here p = 'iTrc/X, c being the velocity of light in vacuo.

HERTZ (' Ausbreitung der electrischeii Kraft,' Leipzig, 1892, p. 150) has shown that
the electric and magnetic forces at any point (x, y, z) due to an oscillating electric
doublet of moment Ae'3" along the axis of x are given by

' E = v - (v-n, o, o) ........ (2),

-, -, 3—5

c\ ozct oy tit]

where

II = A/r.exp [<-p(t — r/c)},

for these expressions satisfy MAXWELL'S equations

,/p 7TT

f = c curl H, . - = - c curl E and div E = div H = 0,

dt dt

and when r is very small compared with the wave-length (X = 2-irc/p) of the emitted
waves the expression for E reduces to

E = V (3n/3x),

which is at any time the electric force which would be electrostatically due to the
doublet if its moment remained constant and equal to its value at that time.

Lord RAYLEIGH ('Phil. Mag.,' XLIV., pp. 28-52, 1897, and 'Collected Papers,'
vol. 4, p. 321) has extended this theorem to the case of a very small sphere. In the
region for which the distance, r, from the centre of a small sphere of radius a excited

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 387

by an electric field E = («"", 0, 0), is small compared with the wave-length, the
electric force due to the sphere is

E . 3 (K-

V -e •

By comparing this with HERTZ'S corresponding result

\ r

for an oscillating doublet of moment e"*, as given above, it appears from (2) and (3)
that the electric and magnetic forces at any point, due to waves emitted by the
sphere, must be given by the equations

E, = V f - (V'n, 0, 0), H, = '-(„, f1? , - f " ) . . (4, 5),

ox c\ ox at oy oil

where now

K — 1 a:i
11 = K + 2 ' r ' eXp ^ ^ ~ r'°^'

Replacing K by N2, where N is the quantity defined by equation (1), we conclude
that when a metal sphere is excited by a periodic electric force E(), it emits the waves
which would be emitted by a Hertzian doublet whicli at time f, was of moment

equal to

2 - 3

N2 - 1 3

~

The same result can be proved directly by adapting the analysis given by
L. LORENZ (' Vidensk. Selsk. Skr.,' Copenhagen, 1890) to the electromagnetic theory.
The problem has also been treated by STOKES (' Camb. Trans.,' vol. 9, p. 1, 1849, and
' Papers,' vol. 4, p. 245, p. 262).

At a great distance from the origin, i.e., when r is great compared with X,
equation (4) reduces to [cf. RAYLEIGH, loc. cit., equation (106)]

7ra- — /\>- X1J x it-\

El= * * eXpN;('"r/C)} ' '

If we transform to spherical co-ordinates X, Y, Z in the respective directions of
increase of r, 0, <f> (fig. 1) we obtain, at a great distance from the origin,

l -r/o)} OO.^OM^

- 2
3 D 2

388

ME. J. C. MAXWELL GARNETT ON

It appears from equations (6) or (7) that such a small sphere, in common with any
other minute system whose moment is proportional to the electric vector of the
incident light, emits light with an intensity proportional to the inverse fourth power,
of the wave-length, provided that N is independent of X. It is this property which,
as Lord RAYLEIGH has shown, accounts for the blue colour of the light received from
the sky.

\

Fig. 1. Fig. 2.

§3. In the ' Annnlen der Physik ' for January, 1903, H. SIEDENTOPF and
R ZSIGMONDY publish some observations on the metal particles in gold ruby glasses.
By their method of illumination they were able to see particles whose dimensions
were of the order of from 4 to 7 //./A, where /A/J. represents 10~6 millim.

The arrangement consisted of a system of lenses following a strongly illuminated
and very narrow slit. The system of lenses, of which the last is a low power
microscopic objective, serves as a condenser and forms a very narrow image of the slit
inside the glass under observation. This image of the slit may not be more than
one or two wave-lengths thick.

The observation is made with a microscope having the tube perpendicular to the
incident light, so that only the light emitted by the metallic particles travels up the
tube. This is the light the electric vector of which has been distinguished by the suffix
unity in the preceding analysis. The image of the slit, which is parallel to Ox in
fig. 2, comes directly under the microscope tube, which is in the direction Oy ; thus
only the particles illuminated at the image of the slit send light up the tube. The
diffraction discs do not pile up on top of one another if the average distance between
two metal particles is greater than the thickness of the image of the slit. In this
case, then, the number of particles per unit area can be counted.

On pp. 1 1 and 1 2 of the paper referred to, SIEDENTOPF and ZSIGMONDY discuss the
appearances in the second focal plane of the microscope when the light incident in
the glass is plane polarised. The figs. 3-6 above are reproduced from their paper.
In fig. 3 the plane of polarisation of the incident light was that of incidence, the
plane of incidence being the plane containing the axis of the microscope and that of

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 389

the incident pencil of light ; in figs. 4 and 5 the plane of polarisation of the incident
light was inclined at 45° to the plane of incidence; while in fig. 6 the two planes
were perpendicular.

In the figs. 3-6 the upper diagram represents the second focal plane of the
microscope when the diameters of the particles of metal in the glass are less than
O-l/i, the small lines being parallel to the planes of polarisation of the emergent
light in various parts of the field, the " emergent light " here meaning the light sent
up the microscope tube by the metal particles in the glass under observation. The

Fig. 3.

Fie. -I,

Fig. 5.

Fig. 6.

lower diagrams in the same figures represent the appearances of a diffraction disc for
the same respective positions of the plane of polarisation of the incident light.

It is to be noticed that the light emitted in any particular direction comes to a
focus at a corresponding point in the second focal plane of the microscope. Conse-
quently a black spot in that plane means that no light is emitted in the corresponding
direction.

If all the particles are spheres sending up no light in some particular direction,
there will thus be a black spot in the second focal plane, as well as in each diffraction
disc, at the point corresponding to that direction.

Suppose now, as in § 2, that the incident light travels in the direction Oz and is
polarised in plane yQz, fig. 1. Instead of conceiving this plane to alter as we consider
the various cases of figs. 3-6, we shall imagine the microscope tube to move in the
plane xOy.

390 MR. J. C. MAXWELL GAENETT ON

Thus in fig. 3 the microscope is along Oy, in fig. 6 along Ox, while in figs. 5 and 4
the tube lies in the intermediate positions, namely, 6 = 90°, (j> = -± 45° respectively.

It will now be shown that the figs. 3-6 are completely accounted for if the particles
are spheres small compared with a wave-length, i.e., appreciably smaller than O'l /u,.

From equations (7) the character of the light emitted by such a sphere in the
direction 0, tf> (fig. 1) is determined by the electric force EL whose composition is :

XL = 0, Y! = B cos 0 cos <f>, Zx = — B sin <j> ..... (8),
where

-D 47r2a3 N2 — 1 , ,. , x,

•-"- f'~r/c}}-

Suppose first that, as in the case corresponding to fig. 3, the microscope tube is
along Oy (fig. 1), the centre of the field then corresponds to 6 = 90°, ^> = 90°.

The fig. 7 represents the direction of E1; as deduced from equation (8), for positions,
the co-ordinates of which are 0, <f>, the centre of the diagram corresponding to

Fig. 7.

0 = (f) = 90°, the axis of y. The same figure will therefore represent the directions
of the electric vector in various parts of the second focal plane of the microscope.

From equation (8) it appears that when either 0 = 90° or <£ = 90°, we shall have
YL=:O, and therefore Ej becomes (0, 0, Zj) and only has a component in the direction <f>.
This is represented by the arrows for positions on the axes in fig. 7.

In the middle of the quadrants the directions of the electric vector are no longer
parallel to the axis of <£ but are tilted as in the figure, being tilted in the same
manner in opposite quadrants.

Now the planes of polarisation are perpendicular to the electric vector, and the
small lines in fig. 3 are perpendicular to the arrows in fig. 7. When, therefore, the
incident light is polarised in the plane of incidence, the appearances are accounted for
if the particles are small spheres.

Next consider the case corresponding to fig. 6, when the microscope tube is
above Ox. The centre of the field is then 0 = 90°, <j> = 0. The arrows represent the
direction of Ej in various parts of the field. All these arrows point nearly towards
the centre. Along the two axes they point accurately towards the centre. There is
no force at the centre, for then both Y, and Zl vanish. Consequently, a black spot
should appear at the centre, if the particles were spheres. Finally, lines perpendicular
to the arrows in fig. 8 are parallel to the lines in fig. 4. Consequently, in this case
also the appearances are explained by supposing the particles to be spheres.

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 391

In this case, namely, when the incident light is polarised perpendicular to the plane
of incidence, it further appears that if an analysing nicol be introduced so as to
polarise the emergent light in the plane of incidence, then the analysing nicol removes
the Yj component of Ej and the vanishing of Z, also, for (f> = 0 causes a dark band
to cross the field over the diffraction disc if there be only one particle sending light
up the tube, the dark band lying along the axis of 6 in fig. 8, i.e., in the plane of
incidence, and this also was observed by SIEDENTOPF and ZSIGMONDY for the particles
in gold glass (loc. cit., p. 12).

The discussion of the cases of figs. 4 and 5 presents no difficulty. The phenomena,
including the correct position of the black spot, are again explained, by means of the
hypothesis that the small particles are spheres.

Thus all the phenomena, observed in the second focal plane of the microscope, due
to particles smaller than 0'1/x, are exactly those which would be produced by spheres
of metal of radius small compared with the length of a wave of light in the glass.

If now the particles were small spheroids, or crystalline in structure, then the
position of the black spots, if indeed any existed, and the positions of the plane of
polarisation of the light emitted from the particles, would depend on the orientation
of the particles. Unless, therefore, the orientation of all the particles were the same,
we should, if many particles were sending light up the tube, get no black spot in the
focal plane, because the black spot, supposing there to be one, due to one particle,
would not coincide with that due to another. And further, even if the orientation
of all the particles were the same, and if every particle alone did send off no light in
some particular direction, so that there were a black spot in the second focal plane,
then, unless the common orientation were such that, for every plane of polarisation
of the incident light, the black spot were in the same plane as if the particles were
spheres, which is an impossibility, spheroidal or crystalline particles could not account
for the effect observed.

These considerations show, therefore, that the small particles in gold ruby glass are
really spheres of gold, so long as their dimensions are considerably smaller than 0'1/x
(10~5 centim.).

This result is of considerable interest in connection with the formation of crystals.
When a metal crystallises out of a vitreous solution, it appears that until the
dimensions have increased beyond a certain limit, the forces of surface tension
overcome the crystallic forces, and the particles of metal are spherical and not
crystalline.*

Mr. G. T. BEILBY has arrived at the same conclusion from microscopic examination

* [Note added Uth May, 1904.— The presence of crystals, whether of silicates or of reduced metal, in
many pottery glazes suggests that minute spheres of the same material as the crystals were present before
the formation of these crystals, and that some may co-exist with the crystals. The colours of the glazes
may therefore be wholly or in part due to the presence of these minute spheres, in the same manner as a
gold ruby glass depends for its colour on the presence of minute spheres of gold.]

392 MR. J. C. MAXWELL GARNETT ON

of the films of metal deposited from solutions (' Proc. Roy. Soc.,' vol. 72, 1903,
p. 223).

In the manufacture of gold and copper ruby glasses and of silver glass, the gold or
copper or silver is mixed with the other ingredients of the glass before the first firing.
If, when the glass is formed in the furnace, the whole be quickly cooled, the glass with
the metal in it is colourless and exactly resembles clear glass. I have had in my
possession several pieces of such clear gold glass, and some of clear silver glass. One
of the former was used in an experiment with the emanation from radium, to be
described later.

In this clear glass the gold or silver is probably in solution in the glass. But
when the glass is re-heated the metal "crystallises" out of solution, or, as we shall
say, is " excreted " from the glass and appears in the small particles observed by
SIEDENTOPF and ZSIGMONDY. These particles of metal, as we shall show, account for
the colour of the glass.

I have seen a piece of copper glass which was allowed to cool down slowly in the
glass pot along with the furnace, taking a week or more in the process. The glass
formed a dark brown, nearly opaque, mass with minute crystals of bright, shining
copper scattered throughout its substance, the crystals being large enough to be easily
distinguishable with the naked eye, while the appearance of the whole mass somewhat
resembled that of the well-known African stone, aventurine.

It is suggested that the second heating, without melting the glass, confers sufficient
freedom on the molecules of the glass to enable the forces of surface tension to exert
themselves in bringing the molecules of the metal, which have been distributed
amongst those of the glass, together into heaps, the phenomenon being similar to that
exhibited when a metal film is heated to 300° or 400° without being melted, when,
as will be described later, the metal forms itself into minute granules, which,
in the light of what we have proved for the particles in gold glass, must be
spheres or spheroids with axes normal to the film. The latter form is possible
for the films of metal, though not for the metal in the gold glass, because a
thin film, as opposed to a piece of glass, is not subjected. to similar conditions in all
directions.

§ 4. We have thus to consider the problem of light traversing a medium containing
many small metal spheres to a wave-length of light.

It has been seen (§ 2) that a small metal sphere produces in all surrounding space
the same effect as would be produced by a Hertzian doublet placed at its centre.
We may therefore imagine the spheres replaced by such electric doublets and thus
avoid considering their finite size.

Let the average (for a large number of doublets) moment of a doublet be, at time t,

Then if there be 3? spheres per unit volume, the polarisation of the medium will be
f (t) = 5J?f (t). If E', due to the regular force E0 together with forces due to the

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 393

neighbouring doublets, be the force causing the polarisation f (t), then we have
proved (§2) that

f /A __ ~3 N — 1 ,,/

' N* + 2

Now by means of the analysis given by H. A. LORKNTZ (' Wied. Ann.,' 9, 1879,
p. 641) and by LARMOR ('Phil. Trans.,' A, 1897, p. 238), and which has been fully
verified in LORENTZ'S own paper and by others, it can be proved that (see § 7 below)

provided the medium under consideration extends throughout a space of dimensions
which in no direction are of an order of magnitude so small as a wave-length of lifht.

o o o

This provision is satisfied except in the case of very thin films. When dealing with
such films in a later portion (§ 7) of this paper we shall return to the consideration of
this point.

From equation (9) we obtain

so that

N- -

CLERK MAXAVELL'S equations written with Hertzian units for this medium, now,
therefore, are

^ = c curl II and -?,H = - c curl E,
at dt

where

e'=(E

_. .

We have therefore proved that a medium consisting of small metal spheres
distributed in vacuo, many to a wave-length of light, is optically equivalent toji
medium of refractive index n' and absorption K' given by N' = n'(l -- IK') = v/e',
where

0-^

We shall throughout use the symbol /* to denote the volume of metal per unit
VOL. com. — A. 3 E

394 MR- J. C. MAXWELL GARNETT ON

volume of the medium (except when p, is evidently used to denote the thousandth

part of a millimetre). Thus JJL = -0- 9Ja3, and equation (10) becomes

O

N2- l
rf/i 2~-~

• <10'>-

If the metal spheres be situated in glass of refractive index v instead of in vacua,
this equation becomes

(11).*

The constants »' and K' of the medium thus depend only on /z, the relative volume
of metal, and not on the radii of the individual spheres. It is clear that the spheres
may now be supposed to be of quite various radii, provided only that there be many
spheres to a wave-length of light in the medium.

We have given the general result which holds for all values of /x, as we shall
require it later. But in the case of metal glasses, by which name we shall describe
glasses- in which a metal is present in metallic form, the value of p. varies from about
10~l for a silver glass down to about 10~(i for a soda glass coloured by radium. The
last equation giving the optical constant N' = n' (\ — i/c') of the metal glass may be
written

2 -

. . (12),

where N is the optical constant of the metal and v the index of refraction of the glass
by itself.

§ 5. Equation (12) may now be written

2m'

., , n- (/c2 — 1) + v- + 2m2/c . _ , / ., 0^
= 3 p.v- • 0 v, ' — v— , - — 3/xv3 (a — 2t/3).

n~ (K~ — 1) — 2v~ + 2un~K

Thus, equating real and imaginary parts, we find, after some reduction,

=

4»V

o =
~

" [n?.(K2 ---1.) .-

f TVT'1' *' "VT"' 2 T

I Note added IQth May, 1904.— This equation may be written ."-„—, -, =/i *~- .}.

L W 2 + 2v2 N2 + 2i''!J

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 395

We have now to see whether by means of these equations (13), and of the values
of n and K for various metals, we shall be able to predict the colour of a glass which
contains a number of small metal spheres, whose linear dimensions and distances
apart are small compared with a wave-length of visible light.-

In the annexed table the refractive index of the glasses has been taken to be
i/= 1-56.

The values of n~ (*2 — 1) and of n~K for the metal are those given by DRUDE
('Physikalische Zeitschrift,' January, 1900), for yellow light (X = '0000589 centim.),
and for red light (X. = '0000630 eeutim.). For the potassium-sodium amalgam,
however, blue and yellow light were used instead of yellow and red.

Now let us suppose that /A, the quantity of metal per unit volume, is very small
If, then, a and {$ represent the numbers in the penultimate and last columns of
Table I. respectively, we have

/.I / f r-i\ ° i o ° '-1 f o no

n" (1 — K ~) = V~ + dfJiVOt., 'II ~K — 6/jLV-p.

Hence

Hence, neglecting higher powers of p,

.......... (14).

Now, suppose that light of wave-length X in racno travels through this composite
medium, whose constants are n' and K. The light in vacuo being given by

X = Aexp {•2wi(l/T-z/\)},
in this medium it is given by

so

that n'«' measures the absorption. In fact, the intensity of the light sinks to
e-.-s (— _i nearly) of its original value in traversing a distance

d= - - - = X • • (15)

2nn'Kr Qirfjiv/B

of the medium.

We have now to apply the formulae to the observations, in order to test the
validity of our analysis as regards the actual phenomena. SIEDENTOPF and ZSIGMONDY
give ('Ann. der Physik,' January, 1903, pp. 33, 34) a table of various gold gLsses
examined by them. This table is reproduced in Table II.

3 E 2

396

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398 ME. J. C. MAXWELL GAENETT ON

If Is denote the volume of the gold particles, so that I is given in the 8th or 9th
column of the Table II., according as p., the gold content, is taken from the 6th or
7th column, then the number 91 of gold particles per cubic (10"1 centim.) is given by

We have said that for our analysis to be applicable there must be many spheres to
a wave-length. Since the spheres in glasses A, B, Ca, Cb, Cc (Table II.) can be
separated by a Zeiss -r^th objective, they must be at a distance apart greater than
•2/x., or half a wave-length of violet light. We shall therefore not expect our
analysis to apply to them. It is further apparent that the particles are more widely
separated in Da than Db. If we take for I the mean of the two numbers given, we
have

'$„„'= -019, 91,, = 2-48. 9?, , = 6-24, ' 5R0 = 8'40, «»„ = 6'35,

so that, since 9?,,, 9f,;, and 9?,, are larger than 9tDb and 9?E, the glasses F, G, H
satisfy our condition best.

But here we are presented with a difficulty. The wave-length of the yellow light,
\= -0000589 centim., is in our glass (v = 1'oG) only, X' = '00003775 centim. or
'3775/A. Thus to find the number of gold particles in X'3, we must multiply 91 by
(-3775)3 = '0538. We shall thus, even for glass G, have less than one particle to a
wave-length. < >n the other hand, SIEDKNTOPK warns us (loc. cit., p. 27) that the
linear dimensions of the particles are only to be taken as upper limits and may be
three times too large.

Suppose this is the case, then the number of particles in a yellow wave-length in
the glass is 27 X '0538 (= l-45) times the above numbers, 91, with of course a still
greater value for red light.

On this hypothesis then the glasses F, G, H alone of the series satisfy our
condition. If, therefore, the theory is coiTect, it should explain the colour and other
optical properties of these three glasses as set out in the first five columns of
Table II.

Let us, for instance, consider the colour of glass G. From equation (15) we have
as the distance <l in which the intensity of light of wave-length X is reduced to
1/7 '4 of its original value X/QTTfj.v/3 given by (15).

From the Table 1. we have, supposing v = 1 '5G,

/3 = -5854, for yellow light X = 10~7 '589 centim.
ft = "2501, for red light X = lO"7 "630

The value of yu. found by colorimetry is 72.10"7. Consequently we have

Yellow, d = '48 centim., nearly.
Red, d = 1-19 centims. ,,

Since the latter number is greater than the former, it follows that this glass, and

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 399

indeed all gold glasses for which our condition is satisfied, should he much more red
than yellow. Presumably, therefore, they are still more yellow than green, and more
green than blue.* We should therefore expect the gold glasses F, G, H, which

satisfy our condition, to be red, as in fact they are.

The above values for d are certainly of the right order, but they may be somewhat
too large. If we had taken the value of>, called the Total Gold Content in Table II.,
the corresponding values of d would have been only half those given above.

It is to be remembered that manufacturers, in making gold ruby glass for " flashing"
on to clear glass, use much more gold. A common value for the total gold content is
about :3.10~5.

By means of equation (15) and Table I. we can in this way predict whether a gold,
silver, or copper glass for which v = I '56 will transmit more red or more yellow light,
and whether such a glass containing small spheres of "potassium-sodium" will
transmit more yellow or more blue.

We thus find that when there are several metal spheres to a wave-length

1 Q

Silver glass transmits yellow (/3/A. yellow < /3/A red),
Copper ,, ,, red (/3/A. red < /3/A yellow),

Gold „ „ „ (/3/X „ c/8/X ,. ),

Potassium-sodium glass transmits blue (/3'A blue < /3/A yellow).

Fromjihe values of /3 on Table 1., p. 3%, we see that for a silver glass to absorb as
much red light as a gold glass does yellow, /u, would have to be ^7-3Aj-, or, roughly,
eight times as great for the silver glass as for the gold. And, since the values of /3
for yellow and red light are more nearly equal for silver than for gold, in order to
produce the same coloration there \vould have to be even more than eight times as
much silver (by volume) us gold. I am told that manufacturers put in ten times as
much silver by weight into a silver glass as they put gold into a gold glass.

Again, the very large value of ft for yellow light in a potassium-sodium glass shows
that such a glass would absorb as much yellow light as a gold glass with only ^ of
the amount of metal excreted.

Thus a very slight excretion of the potassium-sodium metal would give a very
strong blue or violet coloration. This probably explains the colouring of soda glass
by radium, the radiation causing the excretion of the metal.

In order to test this hypothesis I asked Mr. F. SODDY, on 9th November, 1903, at
University College, London, to examine whether the emanation from radium was
capable of colouring quartz glass in which there could evidently be no possibility of
the excretion of metal: He stated that he and Professor RAMSAY, had abeady made
this experiment and had found no coloration.
"At my request Mr. SODDY then placed a small piece of -coburless gold glass in a

* See Appendix.

N2 - v2

N2

18

--„ , where \u + «' denotes the

A"

400 MR. J. C. MAXWELL GARNETT ON

tube containing some emanation. Within two days an unmistakable ruby tint
appeared in the glass.*

It seems probable that the violet coloration of soda glass bulbs used in the
production of Rontgen rays may be due to the excretion of metal caused by the 8
rays from the cathode.

The observations of ELSTER and GEITEL, ' Wied. Ann.,' 59, p. 487, 1896, quoted by
J. J. THOMSON, ' Conduction of Electricity through Gases,' p. 496, that salts of the
alkali metals coloured by exposure to cathode rays exhibit photo-electric effects,
suggestive of the presence of traces of the free metal, support this view as to the
cause of the coloration of metal glasses exposed to the radiation from radium.

From equation (6), as modified for the case when the metal sphere is surrounded
by glass of refractive index i>, it appears that the amplitude at any point of the light

emitted from the sphere is proportional to

modulus, + v'n- + '''"'• Using a and 8 as defined in equation (13), we have

W2 2 2

= a~ + 4/32, where a and 8 are to be found from the table on p. 396, where
N2 -4- 2v

v = L'56. Thus at any point the intensity of light emitted by a sphere of radius a
is proportional to (a2 + 4/32)/X4 = I, say. Measuring X in millim./lOOO, the Table 1.
gives the following values of T I—-
Silver. Copper. Gold.

Yellow (X = -589) I, = 27'95 62-11 70-88

Red (\=-630)I., = 38-81 2175 3479.

From these values of I it appears that when white light falls on a small sphere the
light emitted is, for

Silver, more red than yellow, I; > Iy,
Copper ,. yellow ,, red, Iy > I,,
Gold L > Is.

The presumption is that for the two latter the light may be more green than
yellow.

In the table given by SIEDENTOPF and ZSIGMONDY (loc. cit.), of which a copy is
given (Table II., p. 397), it is seen that of the five glasses Cc, E, F, G, H, whose
particles are small compared with a wave-length of light in the glass, the four glasses
Cc, F, G, H contain particles which send out a green cone of light, and the glass E
contains some particles which send out green and some which send out broim.

Thus far -we have confined attention to glasses for which the condition of having

* [Note added \\tli May, 1904.— Sir WILLIAM RAMSAY has lately exposed some clear silver glass and
some soda glass at the same time to the emanation from radium. After a fortnight's exposure the silver
glass had turned a faint yellow and the soda glass a deep blue-violet.]

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 401

many metal particles to a wave-length is satisfied. We have shown that when the
metal is gold such glasses should be pink (cf. column 3 of Table II.) by transmitted
light ; and that the small gold spheres should send up the microscope light whicli is
pre-eminently yellow or green (cf. columns 4 and 5) ; and we have remarked that for
the same reason that explains the polarisation of sky light, such small spheres send
no light directly up the microscope tube when the electric vector of the incident light
is in that direction, so that in this case the cone of light as examined with the low-
power objective will be cut off (cf. column 5), although the large numerical aperture
of the Zeiss -j^-th oil immersion lens will allow some light to go up the tube, but so
as to leave a black spot in the centre of the focal plane of the microscope as shown
in fig. 6.

All these deductions from our analysis are confirmed in every detail by the three
glasses F, G, H (Table II.). And it is these very glasses, of all the glasses in that
table, for which, according to the numbers there given, the particles are both smallest
and closest together.

§ G. Let us now briefly notice the remaining glasses of Table 1 1. For these glasses
the number of metal particles to a wave-length, measured by (gold content) -=- si/.e
of particle, as determined from the Gth and 8th or from the 7th and 9th columns of
that table, is smaller than for the glasses F, G, H, which show the regular pink
colour. For the glasses A to E this number is greatest for the glasses Cc and E, of
which the former and parts of the latter do show the regular pink colour.

Even glasses which do not satisfy the condition of many particles to a wave-length,
and which consequently do not exhibit the "regular" (pink) colour of gold glass,
have many of their properties co-ordinated by the results we have obtained for
regular glasses.

Take, for instance, the glasses A and B (Table II.). Comparison of the gold
content /x with the size of the observed particles shows that tlio.se particles at any
rate are so far apart as not to satisfy our condition. The fact that glass A is
colourless shows that if there are also minute spheres present which escaped
observation, they also lie so far apart as not to be many to a wave-length. On the
other hand the pink colour of glass B suggests the presence of minute unobserved
spheres which are sufficiently close together to satisfy our condition, the absorption
of the glass being proportional to that small part of the gold content (/i) which is
associated with the minute spheres.

In both glasses the large particles reflect much more light than is emitted by the
minute spheres. The colour of this reflected light is the usual yellow-red metallic
reflection from gold. Therefore the colour of the cone of light should be gold-
yellow (i).

When the Nicol is introduced parallel to the plane of incidence, presumably half
the incident light is cut off. Consequently . the large particles send only half the
yellow-red light up the tube that they previously sent. Owing, however, to the fact

VOL. com. — -A. 3 F

402 MR. J. C. MAXWELL GARNETT ON

that the minute spheres send no light directly up the tube when the electric vector
of the incident light is parallel to the microscope tube (Nicol perpendicular to plane of
incidence), less than half the green light from any small spheres will be cut off. The
cone of light will therefore have more green in proportion to the yellow-red than
before the introduction of the Nicol. Therefore the colour of the cone of light will be
more white than before (ii).

When the Nicol is perpendicular to the plane of incidence, the green light from the
small spheres is cut off, so the colour of the cone of light will be more red than
with no Nicol, and therefore the total quantity of light sent up the tube will be
rather lessened (iii). The conclusions (i), (ii), (iii) are in accordance with the
phenomena tabulated in the 3rd, 4th, and 5th columns of Table II.

The glasses Ca, Cb, Cc present no special difficulties. We have seen (§ 3) that
those metal particles in a gold glass whose diameters are less than O'l/i (10~5 centim.)
are spherical, and (§5) that small gold spheres send green light up the microscope
tube. In the above-named glasses the figures in the 7th column of Table II. show
that the particles are so small as to approximate to the spherical form. This is
confirmed by the green cone of light and its approximate extinction when the electric
vector of the incident light is in the direction of the microscope tube.

As here, too, the observed particles are far enough apart to be distinguished under
the microscope, it is necessary to postulate additional minute spheres to explain the
pink colours of these glasses.

In glasses D and E the blue and violet colours of the transmitted light present a
difficulty which I have not yet been enabled completely to surmount.* It is probable
that the particles in this glass are not sufficiently thickly distributed to satisfy the
condition of there being many particles to a wave-length of blue light. When the
incident light is blue, the absorption that we have investigated is therefore not
present. When, however, the incident light is red, there are sufficient particles to a
wave-length for absorption to take place. Thus, although if light of all wave-lengths
were absorbed, the red would be least absorbed ; yet here it is only the larger wave-
lengths that suffer the absorption whose nature we have investigated.

PART II.

§ 7. With a view to examining whether these principles apply to the colour changes
exhibited by translucent films of metal when heated, observed by Mr. G. T. BEILBY
('Roy. Soc. Proc.,' vol. 72, 1903, p. 226) and by Professor R. W. WOOD ('Phil.
Mag.,' vol. 3, 1902, p. 396), we proceed to consider the transmission of light by films of
metal, the metal being in the form of small spheres, many to a wave-length of light
in the film,

* See Appendix added July 8th for explanation of Bine and Violet Colours.

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 403

*

We shall first confine attention to very thin films, defining very thin films to be
such that nd/\' may be treated as small, d being the thickness of the film, and X' the
wave-length of light in the film.

It has already been noticed that equation (9), p. 393, does not hold for very thin films.
That equation is obtained by observing that the average action of its neighbours on
a particle is that due to a medium which is perfectly uniformly polarised in the
neighbourhood of the particle, and whose external boundary is that of the actual
medium, and whose internal boundary is a sphere of radius r(), equal to the smallest
distance between the centres of two particles. POISSON has shown that the effect
of such a uniformly polarised medium is equivalent to that of a surface distribution
over its internal and external boundaries.

The medium actually present here can only be treated as uniformly polarised
throughout the region inside a sphere whose radius, ?•,, is small compared with the
wave-length of light in the medium. When the outer boundary of the medium is
in all directions many wave-lengths distant from the particle under consideration,
the effect of the periodically varying polarisation outside r = rt can be allowed for
by neglecting the Poisson distribution on the outer boundary of the medium.
Consequently, in this case, the effect on any particle of the remaining particles
is that due to a Poisson distribution over the sphere r = ?•„, which leads to
equation (9).

When the external boundary of the medium is, in any direction, at a very small
distance from the average particle, we are not justified in neglecting the Poisson
distribution over that boundary. In the case of a thin film of the medium in the
plane of xy it is, however, clear that when the electric force is parallel to that plane,
there is no Poisson distribution over the surfaces of the film. Consequently the film
has (complex) dielectric constants in the direction of the axes of x and ?/, which are
the same as for the medium in bulk. Omitting the accent in equation (10'), this
constant is given by

The dielectric constant e', parallel to Oz, may be different from e ; if so, the film
behaves optically like a uniaxial crystal whose three (complex) dielectric constants
are e, e, e', the optic axis being normal to the film.

§ 8. Putting v = 1 in equations (12) and (13), we have

where

— . £=- —^-^—2— —*-s- O3')-

3 F 2

404

MR. .1. C. MAXWELL GARNETT ON

We shall henceforward find it convenient to use n and K to denote the constants of
the medium containing the spheres. The constants of the metal itself will therefore
be denoted by nl and /q, and it will appear, as is a priori evident, that the latter are
the values of n and K when //. = 1. Since therefore e = {n(l — t*)}3, equation (1(5)
gives us on substituting from (12')

-•d-O-l-I*.— 1-tf^^y . . . (17),
from which, by equating real and imaginary parts,

»'«=7T r^T, H3H (18)'

H2/e

|\_2_ 3(1

whence

(1 _

(19);

The following table gives the values of a and /8 as found by means of formula (13')
from the constants n and K of the solid metal as given by DBUJJE (loc. cit.) : —

TABLE TIL

Metal.

Colour.

a.

ft-

Gold <

Yellow A = -589

1-4593

•0816

Red X = -630

1-3626

•0446

Silver <

Yellow X = -589

1-2574

•0150

lied X = -630

1-2160

•01277

Blue

3-269

•531

Yellow

2-068

•107

In order to determine the values of /i and K for various values of p., the numerical
values of the functions f, 17, £, where

i

(21)

COLOURS IN METAL GLASSES AND IN METALLIC FILMS.
were calculated for gold and for silver for the following values of p :

/* = •!, p='5, /A ='6, ft = 7, /t = '8, /A = '9, /* =1-0.
Equations (19) and (20) may be written

405

(19'),
4}» ...... (20'),

whence the values of u and «* for gold and for silver were calculated for the above
values of p. The values of n2K thence obtained were checked against those obtained
by means of equation (18), namely,

»i8K = 3£ ........... (18').

In the case of silver with /JL less than '8 it was, however, seen to be better to obtain
HK as the quotient of the value of nzn got from (18'), by the value of u got from (19')
and (20'), owing to the large probable error when HK was determined directly from
(19') and (20').

From equation (13') we find

which are the same as equations (19) and (20) with p. = 1. Consequently, as should
be the case, the medium of spheres is equivalent to the solid metal wherein the
spheres are of such varied sizes that they fill the whole space. Another check on the
tabulated numbers is afforded therefore by a
comparison of the calculated and observed
values of nl2Kl, Mj/q and n}.

I believe that nearly all the numbers here
given for silver and for gold are subject to an ^^
error of less than 1 per cent.

The values of n~K = 3£ and of 7? for the
potassium-sodium amalgam of DKUDE'S table,
' Phys. Zeitschriffc,' January, 1900, are less
carefully calculated.

§ 9. Consider now the incidence of plane

Fig. 10.

polarised light on a plate of this medium. We shall first suppose the plate to be
very thin and therefore optically crystalline.

Suppose the two surfaces of the film are 2 = 0 and z = d, and that zx is the plane
of incidence.

406

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COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 407

First let the incident light be polarised in the plane of incidence, so that the incident
wave is

X = 0, Y = exp [ip {t — (x sin 6 + 2 coa 0)/c}], Z = 0,

« = - cos 6 exp [tp {/. - (a: sin 6 + 2 cos 0)/c}], £ = 0, y = sin 0 exp [tp {...]].
The reflected wave is

X = 0, Y = B exp [tp {/ - (x sin 0 - 2 cos 0)/c}], Z = 0,

a = B cos 0 exp [ip {t - (x sin 0 - 2 cos 0)/c}], /3 = 0, y = B sin 0 exp[tp {...}].
Inside the film, i.e., between 2 = 0 and 2 = d,

X = 0, Y = A' exp [ip {t. — (x sin $ -f 2 cos <£)/Vj]

+ B' exp [ip {f — (x sin <f> — z cos <£)/V}], Z = 0,

«= - -y * {A'exp[tp {< - (xsin^, + 2Cos <£)/¥}] - B'exp[ip {...}]}, ft = 0,

y = sm0{A'exp[...] + B'exp[...J{.
Transmitted wave

X = 0, Y = C exp [ip {^ - (.T sin 0 + z cos 0)/c}], Z = 0,

« = — C cos 0 exp [ip {t — (x sin 0 + 2 cos 0)/c}], /S = 0, y = ( ' sin 0 exp [. . . ].
In these expressions we have

V'2 = c-/t and sin <f»fV = sin 0/c ....... (a).

Since Y and a are continuous at z = 0,

1+B = A' + B' and (1 - B) cos 0 = (A' - B') c/V cos ^ . . (1)).

Since Y and a are continuous at 2 = d, if we replace
exp [i ip rf/V cos ^] by 1 ± ip dfV cos (/>, and exp [— ip d/c cos 0] by 1 — ip djc. cos 0,

we obtain, when the square of 2ir d/\ is neglected, the equations

{ A' + B' - (A' - B') ip d/V cos <j>] = C (I - ip d/c cos 0)1
c cos 4>jV {(A' - B') - (A' + B') tp d/V cos <^} = cos 0C (1 - ^ rf/c cos 0)J

From the last pair of equations we find, neglecting squares of pd/c, that
A'+B' =C

on using (a) ; then eliminating A', B', B from the equations (b) and (d), we finally

have

C = 1 - ITT d/\ . sec 0 . (e — 1).

408 MR. J. C. MAXWELL GARNETT ON

On taking the modulus and substituting for e from (17) we obtain

|C|2= 1 - 47T c//X sec 0 . w2/c ...... . (22).

Secondly, suppose that the incident light is polarised perpendicular to the plane of
incidence, a, /8, y being the magnetic force.
The incident wave is

n = 0, /3 = exp [ij? \t — (x sin 0 + z cos 0)/c}], y = 0.

The transmitted wave is

a = 0, /8 = C exp \ip {t — (x sin 0 + z cos 0)/c}], y = 0.

The velocity V inside the film is connected with the angle of refraction by the
equations

V' - .2 /cos2</" _j_ 5in2<£\ c3fc/

e e' = ee' + sin20 (e' - e)-

The final result after using the two sets of boundary conditions is

/-( wr (^ sec 6 \ .->/>/ , \ • an e' — 1 1

C = 1 — -^ cos20 (t — 1) + snrt' , J-,

whence, using (17), we obtain

r 2/c + tan^6'-n - .; (23)>

X

When the light is normally incident, the crystalline character of the film does not
manifest itself, and we have from (22) or (23)

|C|2= 1 - 4ird/X. n*K ........ (24).

The absorption of directly incident light by a thin film is therefore governed

by nzK.

Owing to the difficulty of knowing whether any particular metal film whose changes
of colour have been observed, but whose thickness has not been recorded, for
example, the films observed by Professor R. W. WOOD or by Mr. G. T. BEILBY
(loc. cit.), is to be regarded as very thin for the purpose of this section, formulae for
thick films will now be found.

We consider here only the case of directly incident plane -polarised light, and
proceed to obtain an equation corresponding to (24), reserving the full discussion of
the behaviour of thick films under oblicpuely incident light till later. Using the axes
shown in fig. 10, suppose that

Incident wave is

E = 0, exp {ip (t — z/c)}, 0 ; H = — exp (ip (t — z/c}, 0, 0.
Reflected wave

E = 0, B exp {ip (t + z/c)}, 0 ; H = B exp {ip (t + z/c)}, 0, 0.

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 409

Wave in film, i.e., between 2 = 0 and z = d,

E = 0, A' exj, { tp (t - z/V) } + B' exp {ip(t + z/V) | , 0,
H = - c/V [A' exp \ip(t- z/V) J - - B' exp \,.p(l + z/V) }], 0, 0.
Transmitted wave

E = 0, (J exp \ip (t - z/c)\, 0 ; H = - 0 exp >ip(t - z/c}} , 0, 0,

where c/V = n (1 — IK).

The boundary conditions at z = (/ give

AV-a"'"-/A exp { - t . 27r</H/X} + BV"""'M exp {i . 2*- ^/,/X) = (J exp { - ti>
» (1 -- IK) [A'c<--*''"< * exp { — i. 2ff rfn/X} -- BV"' 'M exp {I-JTT </"/X}]

= (' exp {i-2-ird/K}

It follows from these equations that B' is of the order of A/c~ ''"'""•*; if therefore
ird>iK/\ > 1 we shall be correct within ~2 per cent. (<'"') when we neglect B'. Thus
referring to the Table IV. it appears that if a piece of gold leaf before annealing
be so thick that d > X/1 '5 or (/ > |X, then, so far as yellow and red light are concerned,
TrdnK/\ will be > L for all values of /j.> '5, if we suppose d to vary inversely as p,
the number l-5 being the smallest value of TDIK/IJ. for gold for values of /x from '5 up
to unity.

Eliminating BV"'""* from the last two equations above,

+ "(l "~ l(CH exp {- i

n(l — IK) J

From the boundary condition at ~ = 0 we obtain

Af (I +n(l -«)} = 2.
Eliminating A' from the last two equations

0 ex - l2,rfXl_ n = 4** ......

Taking the moduli, the ratio | C | 2 of the intensity of the transmitted to that of
the incident light is given by

I p I a _ " ~ K~ (,-*« '<<«/* (-26)

- |(1 +n)2+nV}2

It appears that when the thickness exceeds f of the wave-length, the absorption is
governed by HK ; but, to the same order of approximation, by H~K when d is less
than ^-X.

VOL. CCIII. — A. 3 G

410 ME. J. C. MAXWELL GAENETT ON

The (comparatively) small effect of the coefficient

on the colour by transmitted light will be considered later. For the present it is
sufficient to observe that M() becomes small when /A = I/a for any colour, and hence
that the variations of M0 intensify the absorption bands, which will be shown to
occur for gold and silver near /x« = 1.

§ 10. In order to illustrate the discussion of the colours exhibited by films of
metal for various values of p., graphs are given of UK and n~K for gold, for silver, and
for the amalgam, potassium-sodium, the constants of which for ju, = 1 were given by
DRUDE (lo<\ cit.}. The graphs of UK and H~K for gold and for silver when the incident
light is red or yellow are plotted from the vahies given in the accompanying table,
with the help of the additional point corresponding to /xa = 1.

This last point is easy to plot, for we see from equations (18) and (19) that

«2K = and ir (/c — I) = 2 when u = I/a.

4p.ft 4ft

( lonsequently for

p. == I/a, >r(/c- + 1) = 2V//7V + 1,

so that .

r / ' \ •> i

II-K- -- v/V/c2 -f 1 + 1 and HK = j v(8fl) + 1 + 1 [ •

This point is also very near the maximum of H~K, owing to the smallness of /8 in
comparison with «, and is also, in the graph of UK, not far from the maximum, and in
the graph of M0 not far from the minimum. It will be shown that for each of these
reasons there is in general an absorption band in the colour whose a = I//JL.

The graph for H~K when blue light is incident on gold is surmised; i.e., it is
constructed on the supposition that the constants n and K for gold, when /A = 1, are
continuous from red through yellow to blue. The curve for UK for gold under blue
light is made of the same shape as for yellow and red, the value ot HK for p. = a"1
being plotted from the maximum value of ITK assumed in that graph.

The curves for H-K for potassium-sodium are plotted, again with the help of the
points for /A = I/a, the incident light being blue or yellow. The graph of H~K for
red is again surmised.

The graphs of 11 K for potassium-sodium are shown by analogy with those for gold,
the only points plotted being for /j. = I/a, /A = 1. The red curve is constructed from
that for n~K in the same manner as the blue curve for UK for gold was got from the
assumed curve for n~K for gold.

COLOURS IN METAL GLASSES AND IN METALLIC FILMS.

411

£4-0

BELLOW

BLUE ~

.4 -5 -6 '7 8 -9 10

3 G 2

412

ME. J. C. MAXWELL GARNETT ON

•X5M JO

COLOURS IN METAL GLASSES AND IN METALLIC FILMS.

413

9-0

8-0

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414

MR. J. C. MAXWELL GARNETT ON

R
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00

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 415

§11. In a paper on "The Effects of Heat and of Solvents on Thin Films of
Metal," 'Boy. Soc. Proc.,' vol. 72, 1903, p. 226, Mr. G. T. BEILBY gives an account ot
some experiments on the behaviour of gold and silver films when heated to tempera-
tures far below their melting points. He suggests that at such temperatures
sufficient freedom is conferred on the molecules by the heating to enable them to
behave as the molecules of the liquid metal would do, and to arrange themselves
under the influence of surface tension either in films or in drop-like granular
forms.

We have already shown, when dealing with the colours in metal glasses, how the
small particles of metal excrete themselves from the glass into spherical forms.

Mr. BEILBY records that the resistance of silver and gold films increased, on
annealing, from a few (0'2 up to 50) ohms up to many thousands of megohms. This,
of course, strongly supports the theory that the metal breaks up under surface
tension into minute granules. Professor WOOD observed no conductivity in his films
as originally deposited. Mr. BEILBY further states that in one of the gold films there
appeared to be a considerable depth of granules, and Professor WOOD records absence
of conductivity in a film in which granules appeared in contact with and piled upon
top of one another. These observations support our hypothesis its to the structure
of the films, although the granules observed may have been larger than those which
are effective in producing the colour phenomena which we are to investigate.

Let us now see whether our hypothesis as to the structure of the films is in
agreement with the colour effects observed by Mr. BEILBY and Professor WOOD.

First, then, consider a very thin film of gold.

According to the result given in equation (24) the diminution in intensity of
light of wave-length X is, for such a film, iird/K. n~K.

From the graphs of n~« for yellow and for red, it is seen that for the solid metal for
which /A = 1, i.e., before annealing, /r/c/A is less for red than for yellow, and this is
true from p. = I nearly down to p. — "J. Thus, a very thin leaf of gold should not
show the green colour distinctive of gold leaf, but the red colour should predominate
over the yellow. The arbitrary graph for II'-K for blue would, if correct, show that
blue should predominate over either yellow or red.

'The colour of a very thin film of gold leaf would, therefore, be chiefly blue, less
red, and least yellow, i.e., blue-purple, and this is the colour observed by Mr. BEILBY
in the thinnest piece of gold leaf he possessed (loc. cit., p. 227).

It should be noticed that it has been proved that a very thin film will let through
more red than yellow light, and that it, therefore, will not exhibit the green colour of
gold leaf. It has only been stated that it seems probable that it will let through
more blue than either.

We now suppose that when the film is being annealed, surface tension acts and
causes the gold to form into spherical drops, many to a wave-length, but of quite
varying sizes. Thus /ot, the volume of metal in a unit volume of the film,

416 MR J. C. MAXWELL GAKNETT ON

continuously diminishes from unity downwards so long as the metal is kept at a
temperature of about 400°.

Just before \t. has reached '9 the yellow begins to get through better than the red,
but the absorption of both increases rapidly. The value of ??2/c for red becomes
equal to 24 when p, = '734 about. It almost immediately starts to diminish, being
only 15'88 when /j, = '7. There is thus a strong and quite narrow absorption band in
the red for /j. = 734.

Similarly, ?t2/c, when the incident light is yellow, rises to a high value near
fj. = -686, and when p. has that value, ri*K =15 nearly.

Between /x = 7 and /x = 734, red and yellow are absorbed to the same large extent.
It seems probable that blue will not be absorbed so greatly for this value of p.. The
film should therefore probably be blue. Mr. HKILBY finds (loc. cit., p. '228) that a
gold film turns blue or purple (absorption chiefly of yellow) in the earlier stage of
annealing, though, presumably, the films for which this effect was observed fall into
our class of thick films. The turning blue will therefore be again referred to when
we come to consider thick films of gold.

When fj. — "0, the red light is much less diminished in intensity than the yellow,
and probably less than the blue. The film is therefore pink, and remains pink down
to the dimensions of coloured glass for which p. is of the order of 10~5. The thin film
observed by Mr. Bra LBV was rose pink after annealing (p. 'I'll].

The high transparency observed by FARADAY and by Mr. BEILBY corresponds to
the very small values of n~K for values of /i < '5.

Consider next a thick film of gold.

The absorption being now, according to the result given on p. 409, dependent
principally upon the value of HK/\, we see from the table for HK or from the graph
that, for the solid metal, yellow light is less absorbed than red. The colour of thick
gold leaf is, in fact, olive-green by transmitted light. As /u. diminishes the absorption
of both yellow and red increases, the latter more rapidly. Now when /x = 734, there
is a great absorption of red, according to the values of UK, which is intensified, since M0
is for this value of // reduced to '177. The colour should then be more yellow than
red. and probably more blue than either. When /x is < 7, the colour is much more
red than yellow. If our assumed curve for UK for blue is correct, the colour of the
film should be blue between ^ = "85 and 7, purple at 7, and principally red from
Ij. -- -G5 through all the range of values of /JL from gold glass down to /x = 0. (If our
curve for blue is correct, the figure shows that the film is red when the blue curve
crosses the yellow.)

According to Mr. BEILBY, a gold film, originally green, turned blue-purple after
annealing. Gold leaf turned, by annealing, pink with brown-green patches, the
latter, presumably, corresponding to large and the former to small values of /x.

The rise in the absorption as tt begins to diminish from unity was noticed by
Mr. BEILBY (loc. cit., p. 232).

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 417

It has therefore been shown that all the observed colour changes in gold films are
in accordance with the theory and numerical results set forth in this paper.

The points corresponding to /m = 1, referred to on p. 410, which were plotted in for
red and yellow, were

UK = 4-89 for p. = 734 for red,

•riK = 3-80 „ p. = '685 „ yellow.

Let us now consider silver films.

The results for thin films are not of much interest, as probably none of the films-
observed came into this class. We may, however, notice that, according to the
graphs of nzK, the thin film should start by being more yellow than red. There is an
absorption band in the red about p. = '822, for which value of //., TI~K = 71. '4 for red.
There is great absorption in the yellow for p. = 795 when H~K = f>2'9. The nearness
of these vahies of /j. for the maxima of the absorption of yellow and red suggests that
thin films of silver should be blue or else very opaque when /x is about "S. The thin
film should turn more red than yellow for a slightly > -H and remain red down to

t/ C7 »•

small values of /A, at least as far as p. = -1.

Passing to thick films, for which the absorption is measured by UK \. we observe
from the graphs that as p. diminishes from unity the absorption at first increases
rapidly. This may be correlated with the increased conductivity manifested by a
silver film in the early stages of annealing. Shortly before p, = "8 the film becomes
more red than yellow, and although by the time ^ = '(> the absorption has already
become extremely small, the film remains more red than yellow until p. vanishes.

Putting p = I/a, we find the additional points on the graphs of UK

for red .... UK = 8'5\ when ^ = '822.
,, yellow . . .• n-K = 7 "99 .. /J. = 795.

The red colour of silver films for low values of /A is observed in those obtained by
depositing silver on glass in the manner described by Professor WOOD (' Phil. Mag.,'
August, 1903). It is also often seen in fogged photographic plates.

§ 12. We proceed to consider the potassium and sodium films discussed by
Professor R, W. Wood, in the ' Phil. Mag.,' 1902, p. 396, ct s&j. Owing, however,
to the inavailability of the numerical values of the constants for potassium or
sodium for more than one colour when p. = I, the numbers used are those given by
DRUDE (loc. cit.) for "potassium-sodium," for blue and yellow light. Consequently
the same degree of numerical accuracy as for gold and silver has not been aimed at.

The yellow and blue curves for II-K are plotted from the numbers tabulated iu
Table IT., p. 406.

The graph of UK has been constructed to pass through the uutabulated points

Yellow . . . HK = 3-811 for p. = I/a = '484, UK = 2'18 for /* = !..
Blue . . . . «K = 2-225 „ /* = l/« = '306, nic =178 „ p = 1.

VOL. CCIII. — A. 3 H

418 MK. J. G. MAXWELL GARNETT ON

The films made "by Professor WOOD were obtained by the condensation of the
vapour of the evaporated metal on the insides of exhausted glass bulbs. We should
therefore expect the film in its original form to be in drops, which, in accordance
with Part I., § 3, when very small, approximate to the spherical form.

The absence of conductivity in these films supports this view of their structure.
The eii'ect of heating up to and beyond the melting point would be to fuse these
drops into continuous metal, and although surface tension tends to a re-formation into
spheres, it is probable that p. will generally be considerably increased by the fusion.

It appears, from our graphs of n~K and UK, that thin or thick " potassium-sodium "
films should transmit more blue than yellow light, provided that //. > '4, there being a
very strong absorption of yellow for /x = '49 (about) in both cases. It is interesting
to note that Professor WOOD always refers to the yellow absorption bands as
particularly strong. As //, increases, the absorption of yellow relative to blue
increases in both thick and thin films.

If now we introduce our hypothetical curves for H~K and for HK for red light, we
find that for /x < '4 the film should be red. Near the greatest absorption of yellow
(^i = -4i>), red and blue should be equally absorbed and the film be purple. As /t
increases further, red should be most absorbed, and blue least, so the film should be
blue. Thus, in general, the film should turn from purple to blue when heated, as
was the case with most of the films observed. Professor WOOD (loc. cit., p. 407)
further states that the particles which he observed were distinctly closer in the blue
than in the pinkish-purple part, thus again suggesting that a change from purple to
blue accompanies an increase in /JL.

So far, then, as they go, our results are in good accordance with observation.
When, however, numbers can be obtained for n and n/c for potassium and for sodium
for blue, yellow, and red light, it may be possible to state with more certainty that
our explanation of the colours of the films and of the changes in colour, due to
heating is the true one.

Sj l-'i. By considering the oblique incidence of plane-polarised light on thick films of
metal by the method adopted in § 9 in the case of thin films, it can be shown that
equation (26) is replaced by :—

(1.) When the incident light is polarised in the plane of incidence

(2.) When the incident light is polarised perpendicular to the plane of incidence

I n 1 2 _ 1 G (•«'- + v'2) w,, . cos e ., ^

• .((i +u'Y + r'^e

where u, v and u', v' are certain functions of n, K, 6 such that when the angle of
incidence, 9, is zero,

it •=. u' = » and v = v' = UK.

COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 419

It can further be proved that the variations with p. of the coefficients

M.= 11±J?_ Qrl

- 3

. 16

•= {(T+7*')2

are such that a change in (27) from M, to M/ would strengthen the absorption
bands. The complete analysis is somewhat lengthy ; 1 have therefore refrained from
reproducing it here.

This result, however, shows that in general the absorption band should be weaker
when the incident light is polarised in the plane of incidence than when it is
polarised perpendicular to that plane. And tin's effect Professor WOOD observed in
almost every film.

PART 111.

§ 14. Metallic media composed of small spheres of metal, many to a wave-length,
have many interesting properties in addition to those already referred to. The very
vivid colour effects which are exhibited according to the graphs given above for UK
for gold, silver and "potassium-sodium" when light traverses such media, in
consequence of the different absorptions of different colours, suggest enquiry whether
metals in bulk have ever been obtained giving brilliant colours by transmitted and
reflected light, such metals being ordinary metals with /j. less than unity. For
instance, have any of the metals we have discussed been obtained in states in which
the specific gravity was not the normal value for that metal and in which the colour
changed with the specific gravity ?

I hope in the near future to examine CAREY LKA'S \vork in detail with a view to
finding out whether his allotropic silver is a medium of the type we have con-
sidered — silver with //. less than unity. But the first glance at his papers (' American
Journal of Science,' L889) shows the following remarkable correspondence between
the properties he observed and the properties which should, according to our
calculations for yellow and red light, be possessed by silver with /A < 1 : —

(i.) CAREY LEA'S silvers were obtained from solution ; and we have shown that
gold, and therefore, presumably, silver, crystallises out of solutions into
particles which are spherical if they are very small. Our silver (/* < 1) is
composed of minute spheres.

(ii.) CAREY LEA'S silver can be changed by pressure or heating into normal silver.
We should expect ^ to be increased by pressure.

(iii.) The specific gravities of the two principal forms of allotropic silver were
appreciably less than that of normal silver.

{iv.) From our graph of HK for silver we see that red and yellow light are about
equally, and very powerfully, absorbed when ^ = '81. The ratio of the

420 MR. J. C. MAXWELL GAENETT ON COLOURS IN METAL GLASSES, ETC.

specific gravities of CAREY LEA'S gold-coloured silver, C, and normal silver is
given by him to be 8'51/10'62 = '81. This strongly supports the theory
that allotropic silver is of the nature of the media we have discussed.

(v.) CAREY LEA'S silvers were very brittle, but could be toughened by heating.
Further, his gold-coloured silver could be transformed into normal silver by
shaking ; and this transformation could be greatly impeded by packing the
gold-coloured silver in cotton wool. These properties suggest a discontinuous
structure for allotropic silver.

(vi. ) If we might assume an absorption graph of UK for blue light, the fact that
if light is obliquely reflected from a film of " B " silver, then the yellow light
is polarised in the plane of incidence and the blue perpendicular to that plane
can, I think, be explained by our theory : but the proof is not yet complete.

(vii.) The red colour exhibited by all the more dilute forms of the allotropic silver
is in accordance with the fact, exhibited by the graph, that HK is smaller for
red than for yellow light for small values of /x.

j Ai'i'BNDix, added -28th July, 1904. — Using the values of the refractive index
and absorption coefficient of gold for red (C), green (E), and blue light, as given by
RriJKNS (' Wied. Ann.,' 1889), the following values of the quantity /3/X, which
governs the absorption of the gold glass, have been calculated :—

(n. . .
I HK

Golcl^j „

! /8/x . .

The refractive index of the glass has been taken to be T56, as in Table II., from
which the values of /3 for red and for yellow have been copied.

The colours, in the order of the degree in which they are transmitted by gold glass,

therefore are

Red, Yellow, Blue, Green.

The corresponding order for silver as obtained by calculation is

Yellow, Red, Green, Blue.

The order's accord with observations on gold-ruby glasses and silver glasses respectively.
It will be seen that large particles of gold (diameter > O'l /A) in a gold glass would,
by reflecting out the red and yellow light, give the glass a blue colour by transmitted
light, and a brown turbidity by reflected light— as in glasses D of Table I.]

Red (C).

lied (-630).

Yellow (D).

Green (E).

Blue J(F + G).

•38

•31

•37

•53

79

2-91

3'15

'2-3-2

I-8G

1-52

•48

•25

•59

1-07

•46

73

•40

•99

2'03

1-01

[ 421

INDEX

PHILOSOPHICAL TRANSACTIONS,

SKRIKS A, VOL. 203.

A.

Acoustic shadow of sphere (RAYLEIOH), 87.

Arc, electric, resistance and electromotive forces of (BTDDEH,), 305.

Atomic weight and specific heat (TllDEu), 139.

B.

BERKELEY ("Earl of). On some Physical Constants of Saturated Solutions, 189.

C.

CHREE (C.). An Enquiry into the Nature of the Relationship between Sun-spot Frequency and Terrestrial Magnetism, 151.
Colours in metal glasses and films (GARNETT), 385.

D.

DAHWIN (G-. 11.). On the Integrals of the Squares of Ellipsoidal Surface Harmonic Functions, 111.
DUDDBLL (W.). On the Resistance and Electromotive Forces of the Electric Arc, 303.

E.

Earthquakes, elastic theory in relation to phenomena of (LAMB), 1.

Elastic solid, propagation of tremors over surface of (LAMB), 1.

Ellipsoidal harmonics, integrals of squares of surface functions (DAilwiir), 111.

Elliptic integral, the third, and the ellipsotomic problem (GBEEXHILL), 217.

Evolution, mathematical contributions to theory of (PEARSON), 53.

G.

GARNETT (J. C. MAXWELL). Colours in Metal Glasses and in Metallic Films, 385.

Gold-leaf, structure of, and absorption spectrum of gold (MALLET), 43.

GREENHILL (A. G.). The Third Elliptic Integral and the Ellipsotomic Problem, 217.

VOL. CCIII. — A 371. 3 I 8.9.04.

422 INDEX.

H.

HABKER (.1. A.). On the High-temperature Standards of the National Physical Laboratory : an Account of n Comparison of

Platinum Thermometers and Thermojunctions with the Gas Thermometer, 343.
High-temperature standards of the National Physical Laboratory (HAEKBR), 343.

I.

Inheritance, alternative, generalised theory of (PEARSON), ."3.

L.

LAMB (HOHACK). On the Propagation of Tremors over the Surface of an Elastic Solid, 1.

LEGBNDEE'S functions, values from P0 to P.,,, at 5-degree intervals (LODGE), 87.

LODOK (A.). Values of LEOENDRE'S Fnnclions from P,, to P.,, at Intervals of 5 Degrees, 87.

M.

"Magnetism, terrestrial, and sun-spot frequency (CnEEE), 151.

MALLKT (J. W.I. On the Structure of Gold -leaf, and the Absorption Spectrum of Gold. 43.

MENDEIi'8 laws of inheritance, relation to biometric theory (PEARSON), 33.

Metals — specific hents and atomic weights (TILDES), 130.

P.

PEARSON (KARL). Mathematical Contributions to the Theory of Evolution. — XTI. On a Generalised Theory of Alternative
Inheritance, with special Reference to MEXDEI.'S Laws, 53.

R.

HATLETGTT (Lord). On the Acoustic Shadow of a Sphere, 87.

Solutions, physical constants of saturated (BERKELEY), 189.
Specific heats of metals, relation to atomic weights (TlLDEN), 139.
Sun-spot frequency, relation to terrestrial magnetism (CHBEE). 151.

T.

Thermometers, comparison of platinum with gas (HARKEB), 343.

TIIBEJT (AV. A.). The Specific Heats of Metals and the Relation of Specific Heat to Atomic Weight.— Part III., 130.

HABRISON AND SDKS, PRINTERS IN ORDINARY TO HIS MAJESTY, ST. MARTIN'S LANE, LONDON, W.C.

41

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