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PROCEEDINGS 



OF THE 



CAMBRIDGE PHILOSOPHICAL SOCIETY 

VOLUME XIX 




CAMBRIDGE : PRINTED BY 

J. B. PEACE, M.A., 
AT THE UNIVERSITY PRESS 



PROCEEDINGS 



OF THE 



CAMBEIDGE PHILOSOPHICAL 
SOCIETY 



VOLUME XIX 

30 October 1916—24 November 1919 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 

AND SOLD BY 

DEIGHTON, BELL & CO. AND BOWES & BOWES, CAMBRIDGE 

CAMBRIDGE UNIVERSITY PRESS 
C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.C. 4 

1920 



CONTENTS. 

VOL. XIX. 

PAGE 

A self-recording electrometer for Atmospheric Electricity. By W. A. 

Douglas Rudge, M.A., St John's College. (Eleven figs, in Text) . 1 

On the expression of a number in the form ax^ + by^ + cz^ + du'^. By 
S. Ramanujan, B.A., Trinity College. (Communicated by Mr 
G. H. Hardy) H 

An Axiom in Symbolic Logic. By C. E. Van Horn, M.A. (Communicated 

by Mr G. H. Hardy) 22 

A Reduction in the number of the Primitive Propositions of Logic. By 
J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. 
Hardy) 32 

Bessel functions of equal order and argument. By G. N. Watson, M.A., 

Trinity College 42 

The limits of applicability of the Principle of Stationary Phase. By 

G. N. Watson, M.A., Trinity College 49 

On the Functions of the Mouth-Parts of the Common Pravni. By L. A. 

Borradatlb, M.A., Selwyn College 56 

The Direct Solution of the Quadratic and Cubic Binomial Congruences with 

Prime Moduli. By H. C. Pocklington, M.A., St John's College 57 

On a theorem of Mr G. Polya. By G. H. Hardy, M.A.. Trinity College . 60 

Submergence and glacial climates during the accumulation of the Cambridge- 
shire Pleistocene Deposits. By J. E. Marr, Sc.D., E.R.S., St John's 
College. (Two figs, in Text) 64 

071 the Hydrodynamics of Relativity. By C. E. Weathers urn, M.A. 

(Camb.), D.Sc. (Sydney), Ormond College, Parkville, Melbourne . 72 

0>i the convergence of certain multiple series. By G. H. Hardy, M.A., 

Trinity CoUege 86 

Bessel functions of large order. By G. N. Watson, M.A., Trinity College . 96 

A particular case of a theorem of Dirichlet. By H. Todd, B.A., Pembroke 
College. (Communicated, with a prefatory note, by Mr H. T. J. 
Norton) HI 



VI 



Contents 



On Mr Ramanujari' s Empirical Expansions of Modular Functions. By 
L. J. MoRDELL, Birkbeck College, London. (Communicated by 
Mr G. H. Hardy) . . . 



Proceedings at the Meetings held dm-ing the Session 1916 — 1917 . 

Extensions of Abel's Theorem and its converses. By Dr A. Kienast, Kiis 
nacht, Ziirich, Switzerland. (Communicated by Mr G. H. Hardy 

Sir George Stokes and the concept of uniform convergence. By G. H. Hardy, 
M.A., Trinity College 

Shell-deposits formed by the flood of January, 1918. By Philip Lake, M.A. 
St John's CoUege 

75 the Madreporarian Skeleton an Extraptrotoplasmic Secretion of th 
Polyps? By G. Matthai, M.A., Emmanuel College, Cambridge, 
(Communicated by Professor Stanley Gardiner) . 

On Reactions to Stimidi iii Corals. By G. Matthai, M.A., Emmanuel Col- 
lege, Cambridge. (Communicated by Professor Stanley Gardiner) 

Notes on certain parasites, food, and capture by birds of the Common Earwig 
(Forficula auricularia). By H. H. Brindley, M.A., St John's College 

Reciprocal Relations in the Them-y of Integral Equations. By ]\Iajor P. A. 
MacMahon and H. B. C. Darling 

Fish-freezing. By Professor Stanley Gardiner and Professor Nuttall 

On the branching of the Zygopteridean Leaf, and its relation to the probable 
Pinna-nature of Gp'opteris sinuosa, Goeppert. By B. Sahni, M.A. 
Emmanuel CoUege. (Communicated by Professor Seward) 

The Structure of Tmesipteris Vieillardi Dang. By B. Sahni, M.A. 
Emmanuel CoUege. (Communicated by Professor Seward) , 

On Acmopyle, a Monotypic New Caledonian Podocarp. By B. Sahni, 
M.A., Emmanuel CoUege. (Communicated by Professor Seward) 

Proceedings at the Meetings held during the Session 1917 — 1918 . 

On Certai7i Trigonometrical Series which have a Necessary and Sufficient 
Condition for Uniform Convergence. By A. E. Jolliffe. (Com- 
municated by Mr G. H. Hardy) 

Some Geometrical Interpretations of the Concomitants of Two Quadrics. 
By H. W. TxjRNBULL, M.A. (Communicated by Mr G. H. Hardy) . 

Some properties of p (n), the number of partitions of n. By S. Ramanujan, 
B.A., Trinity CoUege 

Proof of certain identities in combinatory analysis: (1) by Professor L. J. 
Rogers; (2) by S. Ramanujan, B.A., Trinity CoUege. (Communi- 
cated, with a prefatory note, by JVIr G. H. Hardy) .... 



117 
125 

129 

148 

157 

160 
164 
167 

178 

185 

186 
186 

186 

187 

191 
196 
207 

211 



Contents vii 



On Mr Ramanujan'' s congruence properties of p [n). By H. B. C. Darling 
(Communicated by Mr G. H. Habdy) 

On the exponentiation of well-ordered series. By Miss Dorothy Wrinch, 
(Communicated by Mr G. H. Hardy) . . . - . 

The Gauss-Bonnet Theorem for Multiply -Connected Regions of a Surface 
By Eric H. Neville, M.A., Trinity College .... 



217 
219 
234 



On an empirical formula connected with Goldbach's Theorem. By N. M. 
Shah, Trinity College, and B. M. Wilson, Trinity College. (Com- 
municated by Mr G. H. Hakdy) 238 

Note on Messrs Shah and Wilson's paper entitled: 'On an empirical formula 
connected ivith Goldbach's Theorem'. By G. H. Hardy, M.A., Trinity 
College, and J. E. Littlewood, M.A., Trinity College . . . 245 

The distribution of Electric Force between tivo Electrodes, one of which is 
covered ivith Radioactive Matter. By W. J. Harrison, M.A., Fellow 
of Clare College. (One fig. in Text) 255 

The conversion of saw-dust into sugar. By J. E. Purvis, M.A. . . 259 

Bracken as a source of potash. By J. E. Purvis, M.A 261 

The action of electrolytes on the electrical condtictivity of the bacterial cell 
and their effect on the rate of migratioii of these cells in an electric field. 
By C. Shearer, Sc.D., F.R.S., Clare College 263 

The bionomics of Aphis grossulariae Kalt., and Aphis viburni Schr. By 
Maud D. Haviland, Bathurst Student of Newnham College. (Com- 
municated by Mr H. H. Brindley) 266 

Note on an exp)eriment dealing with mutation in bacteria. By L. Don- 
caster, Sc.D., King's College. (Abstract) 269 

Golourimeter Design. By H. Hartridge, M.D., Fellow of King's College, 

Cambridge. (One fig. in Text) 271 

The Natural History of the Island of Rodrigues. By H. J. Snell (Eastern 
Telegraph Company) and W. H. T. Tams. (Communicated by 
Professor Stanley Gardiner) 283 

Preliminary Note on the Life History of Lygocerus {Proctotrypidae), 
hyperparasite of Aphid i us. By Maud D. Haviland, Fellow of 
Newnham College. (Communicated by Mr H. H. Brindley) . . 293 

Note on the solitary wasp, Crabro cephalotes. By Cecil Warburton, 

M.A., Christ's College 296 

Neon Lamps for Stroboscopic Work. By F. W. Aston, M.A., Trinity 
College (D.Sc., Birmingham), Clerk-Maxwell Student of the Uni- 
versity of Cambridge. (One fig. in Text) ..... 300 



viii Contents 

PAGE 

The pressure in a viscous liquid moving throiigh a channel ivith diverging 
boundaries. By W. J. Harrison, M.A., Fellow of Clare College, 
Cambridge. (One fig. in Text) 307 

The Effect of Ions on Ciliary Motion. By J. Gray, M.A., Fellow of 

King's College, Cambridge ........ 313 

A Note on Photosynthesis and Hydrogen Ion Concentration. By J. T. 

Saunders, M.A., Christ's College 315 

The distribution of intensity along the loosiiive ray parabolas of atoms and 
molecules of hydrogen and its possible explanation. By F. W. Aston, 
M.A., Trinity College (D.Sc., Birmingham), Clerk-Maxwell Student 
of the University of Cambridge. (Three figs, in Text) . . . 317 

Gravitation and Light. By Sir Joseph Larmor, St John's College, 

Lucasian Professor 324 

On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex College 345 

The self-oscillations of a Thermionic Valve. By R. Whiddington, M.A., 

St John's College 346 

Proceedings at the Meetings held during the Session 1918 — 1919 . . 347 

Index to the Proceedings with references to the Transactions . . 350 



PEOCEEDINGS 



OF THE 



CAMBRIDGE PHILOSOPHICAL 
SOCIETY 



VOL. XIX. PART I. 

[Michaelmas Term 1916.] 




ODamijrilJgc: 

AT THE UNIVERSITY PRESS 
AND SOLD BY 
DEIGHTON, BELL & CO., LIMITED, 
AND BOWES & BOWES, CAMBEIDGE. 

CAMBBIDGE UNIVEESITY PRESS, 
C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.G. 

1917 

Price Ttuo Shillings and Sixpence Net 

January 1917. 



NOTICE^. 

1. Applications for complete sets of the first Seventeen 
Yolumes (in Parts) of the Transactions should be made to the 
Secretaries of the Society. 

2. Separate copies of certain parts of Volumes i.— xi. of the 
Transactions may be had on application to Messrs BowES & 
Bowes or Messrs Deighton, Bell & Co., Limited, Cambridge. 

3. Other volumes of the Transactions may be obtained at 
the following prices: Vol. xn. £1. 10s. Qd.; Vol. xm. £1. 2s. 6d. 
Vol. XIV. £1. i7s. 6d. ; Vol. xv. £1. 12s. 6d. ; Vol. xvl £1. 10s. Od. 
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No. ^, 2s.; No. 8, Is. 6d; No. 4, Is. 6d.; No. 5, 2s.; No. 6, Is. Qd. 
No. 7, 2s. ; No. 8, 2s. ; No. 9, 2s. 

4. Complete sets of the Proceedings, Volumes i. — xviil., 
may also be obtained on application to the Secretaries of the 
Society. 

5. Letters and Communications for the Society should be 
addressed to one of the Secretaries, " 

Mr G. H. Hardy, Trinity College. [Mathematical.] 

Mr A. Wood, Emmanuel College. [Physical.] 

Mr H. H. Beindley, St John's College. [Biological] 

6. Presents for the Library of the Society should be ad- 
dressed to 

The Philosophical Library, 

Neiu Museums, 

Cambridge. 

7. Authors of papers are informed that the Illustrations and* 
Diagrams are executed as far as possible by photographic "process" 
work, so drawings should be on a large scale and on smooth white 
Bristol board in Indian ink. 

8. Members of the Society are requested to inform the 
Secretaries of any change of address. 



PROCEEDINGS 

OF THE 



A self-recording electrometer for Atmospheric Electricity. By 
W. A. Douglas Rudge, M.A., St John's College. 

[Received 18 October 1915.] 

In the course of the writer's work on the local variations of the 
atmospheric potential gradient, the need was felt for a simple self- 
recording electrometer. Most of those in use are costly and at the 
same time rather elaborate in construction. A new arrangement 
has therefore been devised which has answered the purpose in view, 
and as the apparatus may be useful in other directions a description 
is now given. 

It has been shown* that very considerable variations of the 
normal potential gradient are produced by clouds of dust raised by 
the wind, etc. ; and also by clouds of steam escaping under pressure 
from steam boilers f. These variations are very sudden and do not 
last for a long time, so that an instrument used for recording them 
must be fairly quick acting. After a considerable amount of 
preliminary work, the type of instrument adopted was a modified 
form of the quadrant electrometer, the record being photographed 
upon a piece of bromide paper attached to a revolving cylinder. 
One special use to which the electrometer was to have been applied 
was to find the relation between the potential gradient and the 
altitude of the place of observation, and for this purpose it was 
proposed to construct ten or more instruments, so that a number 
of observations could have been carried out simultaneously. Some 
work of this kind has already been done in South Africa from 
which it appears that the potential gradient near to the ground 
diminishes with the height of the place of observation above sea 
level;]:. In order to get satisfactory results it is necessary for the 

* Proc. Roy. Sor. A, Vol. 90. t Proc. Pflij. Soc. A, Vol. 90. 

J Trans. Poy. Soc. SnntJi Afrirn, Vol. vi, pai-t 5. 

VOL. XIX. PT. I. 1 



2 Mr Ri(d[/e, A self- recording electrometer 

stations to be chosen as far removed as possible from the disturbing 
influence of manufacturing operations, and of railways, and it was 
intended to have taken a set of observations in the neighbourhood 
of the Dead Sea, as in that district, stations for the instruments 
might have been chosen with altitudes varying from 1400 ft below 
sea level to 3000 ft above, and in an open country. As a number 
of instruments were required it was necessary to keep the cost of 
construction low, and this has been achieved in the instrument to 
be described, so that the cost of material is less than ten shillings 
and a moderate amount only of mechanical skill is required in the 
construction. 

The complete apparatus consists of: 

(1) The Electrometer. 

(2) The recording cylinder. 

(3) The illuminating arrangement. 

(4) The charging battery. 

(5) The collecting system. 

(1) The Electrometer. This consists of four curved pieces of 
brass cut from a tube of 3 cm. diameter, and attached to a block of 
ebonite. The alternate pieces were connected together in the usual 
way. Each conductor subtended an angle at the centre of the 
mirror of about 60°, and the adjacent conductors were about 
1 mm. apart. The " needle " was formed from a piece of silvered 
paper, 2*5 x 1"5 cm. carrying a small mirror, or a piece of silvered 
thin "cover" glass could be used for both needle and mirror. A fine 
wire was attached to the needle to support a piece of wire gauze 
which was immersed in a small bottle containing paraffin oil, for 
damping the motion of the needle. The system was suspended by 
a fine phosphor bronze wire by means of which the needle could be 
charged, Fig. 1. The whole was enclosed in a thin wooden case 
having a small window in front, and ebonite plugs to allow of 
connection being made to the quadrants. 

(2) The recording cylinder. This is the most novel feature of 
the instrument, and is constructed from one of those small round 
clocks which may be bought from a shilling upwards. Two sizes 
of clock-case are common, of diameters 10 cm. and 6 cm., and both 
of these sizes have been used. A brass tube is substituted for the 
hour hand at the fi'ont of the clock, and a similar piece of brass 
tubing is attached to the arbor at the back of the clock, which is 
attached to the minute hand and used for setting the clock. These 
two tubes are in the same straight line and furnish a convenient 
axis about which the clock as a whole can rotate. If the tube 
attached to the hour hand is fixed, the clock-case will turn round 
once in twelve hours, whilst if the minute hand is fixed, the clock 



for Atmospheric Electriciti/ 



3 



rotates once in one hour. Two scales of measurement are thus 
possible and both have been employed. No difficulty was found in 
taking twenty-four, or two hour records, for although the records 
overlapjjed it was (|uite easy to distinguish one part from the other. 
A light zinc tube was slipped over the clock-case to give a good 
support for the bromide paper which was wrapped round outside. 
The whole clock was made to balance by fastening small pieces of 
lead to the inside of the case, but during the working a little 




Fig. 2. 
hour hand arbor fixed by the pin P. 



Fig. 1. 
The electrometer. 



irregularity occurs as a consequence of the unwinding of the spring; 
this however is not very great and a number of clocks could be 
made to keep time together. The recording cylinder was enclosed 
in a light tight case with a long narrow slit in front, Fig. 2. 

(3) Illuminating system. As the apparatus was used out of 
doors, a lamp was unsuitable as a means of illumination, so that 
daylight was used and found to be very suitable. The electrometer 
and recording cylinder were placed at the opposite ends of a light 
tight box measuring 20 x 17 x 14 cm. A hole was made in the 

1—2 



j\[r Rudge, A self-recording electrometer 





J L 



I , v- 



\ I 
\ I 



u 



for AtniospJterio Electrlcitij 5 

top of one of the ends of the box, and covered over with a piece 
of silvered glass, upon which a fine vertical scratch — to serve 
as a slit — had been made. A lens, which could slide upon 
a rod inside the box, was employed to project the light upon 
the electrometer mirror, whence after reflection it was returned 
to the same end of the box as the slit, but at a lower level, 
and fell upon the horizontal slit in the case of the recording 
cylinder. By this means a point of light impinged upon the 
bromide paper, and as the latter rotated, traced out the curve 
which appears after developing the paper in the usual manner. 
Fig. 3. 

To Collector 




Fig. 4. E electrometer. R recording apparatus. .S' slit. 

(4) Charging battery. In using the electrometer the opposite 
pairs of quadrants were kept charged to a fixed potential by means 
of a battery of the small Leclanche cell used for " flash " lamps. 
These cells are sold in sets of three and a batch of eight, giving 
about 35 volts, is quite sufficient for atmospheric observations. The 
centre of the battery was earthed. The complete apparatus is 
shown in Fig. 4. 

(5) The Collector. This consisted of a small plate of brass coated 
with a radioactive preparation. The plate was fixed in the centre 
of a very short piece of brass tubing and the open ends of the 



6 Mr Rudge, A self-recordiiKj electi'uineter, etc. 

tubing covered over with wire gauze, so as to prevent loss of radium 
by rubbing, etc.; Avhilst allowing it to take up the potential of the 
air. The collecting plate was supported at the end of an insulated 
wire, and at such a height above the ground as would give a 
deflection suitable to the sensibility of the electrometer. 

Up to the present time the apparatus has been used for the 
purpose of taking records of the variations in the potential 
gradient, due to the presence of clouds of dust raised by traffic on 
the roads, or to the variation caused by the steam escaping from 
passing trains. A number of representative curves are given. 

No. 1. This is a twelve hour record, taken at a station on the 
Gog Magog Hill about four miles fi'om Cambridge, and so far from 
the railway and roadway that traffic had no disturbing influence. 

No. 2 is a simultaneous record taken in Hills Road at a distance 
of less than a quarter of a mile from the railway, so that every 
passing train shows its influence in increasing the positive potential. 

Nos. 3 and 4 are a pair of simultaneous hour records, three 
being taken at Cherryhinton reservoir, and four at about 300 yards 
from the railway. The " peaks " in the latter indicate the passing 
of a train. 

No. 5 is a one hour record taken on Hills Road and shows the 
remarkable influence of the dust raised by passing vehicles. Every 
vehicle, even an ordinary bicycle, if it raises dust, disturbs the 
normal electrification. Nos. 6 and 7 are simultaneous records 
taken at some little distance from the road. Nos. 8 and 9 were 
taken near the "Long" road railway crossing and show the 
influence of passing trains. Nos. 10 and 11 are a pair of simul- 
taneous records, 10 being taken in the Railway yard, and showing 
the effect of passing train and "shunting " operations ; 11 was taken 
about a mile away from the line. 

All the potentials indicated are positive and the records are 
reduced in reproduction, but an equal range of negative potentials 
could be recorded, as only one half of the width of the photo- 
graphic paper was used in the records given. The sensibility of 
the instrument may be changed by varying the number of cells of 
the charging battery. 



220 volts > 



220 voU^^ 



:z; 




^)vi^f^4^ r\/^^(^ 



/V*— *— " 




1 P.M. 2 P.M. 

^0. 5. Variation in positive potential due to the clouds of dust raised by traffic on the roads. 




1 P.M. 



No. 6. Taken simultaneously with 5. 



2 P.M. 



1 P.M. 



No. 7. Taken simultaneously with 5. 



V 

2 P.M. 




1.30 A.M. 10.30 A.M. 

No. 8. Variation in potential due to steam from passing trains. 




2.15 P.M. 3.15 P.M. 

No. 9. Variation in potential due to steam from passing trains. 



W^%#»jiw 





No. 10. Variation in positive potential due to "shunting" of trains. 




No. 11. Taken simultaneously with No. 10, but at a distance of more than a mile from 

the railway. 



Mr Rainanujau, On t/ie ej^pressiun of a nv/inber, etc. 11 

On the expression of a number in the form ax- + by- + cz- + dv-. 
By S. Ramanujan, B.A., Trinity College. (Communicated by 
Mr G. H. Hardy.) 

[Received 19 September 1916 ; read October 30, 1916.] 

1. It is well known that all positive integers can be expressed 
as the sum of four squares. This naturally suggests the question : 
For what positive integral values of a, b, c, d can all positive 
integers be expressed in the form 

ax- + by- + cz- + du" ? (ll) 

I prove in this paper that there are only 55 sets of values of 
a, b, c, d for which this is true. 

The more general problem of finding all sets of values of 
a, b, c, d, for which all integers luith a finite number of exceptions 
can be expressed in the form (II), is much more difficult and 
interesting. I have considered only very special cases of this 
problem, with two variables instead of four ; namely, the cases in 
which (I'l) has one of the special forms 

a{x^ + y'- + z^) + bu- (1-2), 

and a{x'^ +y-)-\-b{z''-\-u?) (1-3). 

These two cases are comparatively easy to discuss. In this 
paper I give the discussion of (1"2) only, reserving that of (1"3) 
for another paper. 

2. Let us begin with the first problem. We can suppose, 
without loss of generality, that 

a^b^c^d (2'1). 

If a > 1, then 1 cannot be expressed in the form (I'l) ; and so 

a = l (2-2). 

If b> 2, then 2 is an exception ; and so 

1<6^2 (2-3). 

We have therefore only to consider the two cases in which (1"1) 
has one or other of the forms i 

X- + y- + cz^ + du-, X- + 2y'- + cz- + dii-. 
In the first case, if c> 3, then 3 is an exception ; and so 

l^c^3 (2-31). 

In the second case, if c > 5, then 5 is an exception ; and so 

2^c$5 (2-32). 

We can now distinguish 7 possible cases. 

(2-41) x"- -\- y- + z'- + du?. 
If rf > 7, 7 is an exception ; and so 

X^d^l (2-411). 

(2-42) X- + y- + 2z- + du'. 



12 



Mr Raniaiiujan, On tlie expreasiuii of a namber 



If c? > 14, 14 is an exception ; and so 

2^cZ^14 (2-421). 

(2-43) x" + y' + 3^2 -1- dii\ 
If c? > 6, 6 is an exception ; and so 

'^^d^Q (2-431). 

(2-44) X' + ly"" + ^z^ + du-. 
li d >*1, 7 is an exception ; and so 

2^d^1 (2-441). 

(2-45) x'' + 2?/- + '2>z' + diC'. 
If (Z > 10, 10 is an exception ; and so 

3^c^^l0 (2-451). 

(2-46) X- -r '±y'' + ^z- + diJb^. 

If c? > 14; 14 is an exception ; and so 

4^c^^l4 (2-461). 

(2-47) x'-^'ly-^-hz^^diC-. 

If c?> 10, 10 is an exception ; and so 

o^cZ^lO : (2-471). 

We have thus eliminated all possible sets of values of a, b, c, d, 
except the following 55 : 



1, 


1 




2 


3, 


5 


1, 


2 




2 


4, 


5 


2, 


2 




2' 


5, 


5 


2, 


2 




li 


1, 


6 


1, 


3 




1, 


2 


6 


2, 


3 




2, 


2' 


6 


2, 


3 




1', 


3, 


6 


3, 


3 




2, 


3, 


6 


*3, 


3 




2 


4, 


6 


1, 


4 




2' 


5, 


6 


2, 


4 




1, 


1, 


7 


2, 


4 




1, 


2, 


7 


3, 


4 




2, 


2, 


7 


3, 


4 




2' 


3, 


7 


4, 


4 




2, 


4, 


7 


1, 


5 




2 


5, 


7 


2, 


5 




1^ 


2, 


S 


2, 


5 




2, 


3, 


8 


3, 


5 











, 2, 


4, 


9 


5, 


, 1, 


2, 


, 2, 


3, 


, 2, 


4, 


, 2, 


5, 


, 1, 


2, 


2, 


3, 


, 2, 


4, 


2 


5, 


, 1, 


9 


, 2, 


4, 


, 1, 


2, 


2 


4, 


, 1, 


2, 


, 2, 


4, 


, 1, 


2, 


, 2, 


4, 



8 

8 

9 

9 

9 

9 

10 

10 

10 

10 

11 

11 

12 

12 

13 

13 

14 

14 



in the for in ax- + by" + cz" 4- da- 13 

Of these 55 forms, the 12 forms 

1, 1, 1, 2 1, 1, 2, 4 1, 2, 4, 8 

1, 1, 2, 2 1, 2, 2, 4 1, 1, 3, 8 

1, 2, 2, 2 1, 2, 4, 4 1, 2, 3, 6 

1, 1, 1, 4 1, 1, 2, 8 1, 2, 5, 10 

have been ah-eady considered by Liouville and Pepin*. 

8. I shall now prove that all integers can be expressed in each 
of the 55 forms. In order to prove this we shall consider the seven 
cases (2*41) — (2*47) of the previous section separately. We shall 
require the following results concerning ternary quadratic arith- 
metical forms. 

The necessary and sufficient condition that a number cannot be 
expressed in the form 

x^-Vy--\-z' (3-1) 

is that it should be of the form 

4^(8ya + 7), (\ = 0, 1,2..., /x = 0, 1,2, ...) (3-11). 

Similarly the necessary and sufficient conditions that a numbei- 
cannot be expressed in the forms 

x'+ tf+2z- : (3-2), 

.T^+ f + '^z' (3-3), 

.r 2 + 2 2/- + 2^'^ ( 3 • 4 ) , 

a;-^ + 23/- + 3^2 (3-5), 

x- -^ "lif + ^z- (3-6), 

^2+ 2y^+ bz- ...(3-7), 

are that it should be of the forms 

4^(16/i + 14) (3-21), 

9M 9/*+ <)) (3-31), 

4^( 8/.+ 7) (3-41), 

4^(16ya + 10) (3-51), 

4^(16/^+14) (3-61), 

25^(25/x+10) or 25^(25/* + 15)t (3-71). 

" There are a large number of short notes by Liouville in vols, v-viii of the 
second series of his journal. See also Pepin, ibid., ser. 4, vol. vi, pp. 1-67. The 
object of the work of Liouville and Pepin is rather different from mine, viz. to 
determine, in a number of special cases, explicit formulae for the number of 
representations, in terms of other arithmetical functions. 

t Results (3-11)— (3-71) may tempt us to suppose that there are similar simple 
results for the form ax- + hy- + cz-, whatever are the values of a, b, c. It appears, 
however, that in most cases there are no such simple results. For instance. 



14 Mr Raiitavujan, On tJie exjyression. of a nninher 

The result concerning ./- + y^ + z- is due to Cauchy : for a proof 
see Landau, Handhuch der LeJtre von der Verteilung der Prim- 
zahlen, p. 550. The other results can be proved in an analogous 
manner. The form x- + y" + ^z"- has been considered by Lebesgue, 
and the form x'^-\-y'^-{-'^Z' by Dirichlet. For references see Bach- 
mann, Zahlentheorie, vol. iv, p. 149. 

4. We proceed to consider the seven cases (2'41) — (2*47). In 
the first case we have to show that any number N can be expressed 
in the form 

N' = x- + y- + z- + du" (4- 1 ), 

d being any integer between 1 and 7 inclusive. 

If JSf is not of the form 4^(8yLt + 7), we can satisfy (4-1) with 
u = 0. We may therefore suppose that iV^= 4^ (S/jl + 7). 

First, sup]3ose that d has one of the values 1, 2, 4, 5, 6. 
Take u = 2\ Then the number 

N-du' = ^^(8fM+7-d) 

is plainly not of the form 4^(8/4 + 7), and is therefore expressible 
in .the form x^ + y^ + z^. 

Next, let d = S. If /i = 0, take u = 2\ Then 

N - dii- = 4^+1. 

the numbers which are not of the form .r- + 2?/- + 10^- are those belonging to one 
or other of the four classes 

25^(8^ + 7), 25^(25^ + 5), 25^ (25/x + 15) , 25^ (25/^+20). 
Here some of the numbers of the first class belong also to one of the next three 
classes. 

Again, the even numbers which are not of the form x'^ + ij- + lOz- ai'e the numbers 

4^(16^ + 6), 
while the odd numbers that are not of that form, viz. 

3, 7, 21, 31, 33, 43, 67, 79, 87, 133, 217, 219, 223, 253, 307, 391, ... 
do not seem to obey any simple law. 

I have succeeded in finding a law in the following six simple cases: 
•«■''+ ?/2 + 4,--, 
X-+ y- + 5z-, 
x^+ y' + 6z', 
x^+ y^ + 8z-, 
x^ + 2y-^ + Qz\ 
.r2 + 2y- + 8^2. 
The numbers which are not of these forms are the numbers 
4^(8^ + 7) or (8^1 + 3), 
4^(8^ + 3), 
9^(9ya + 3), 
4^(16/x+14), (16m + 6), or (4^ + 3), 
4^ (8m +5), 
4^^(8^4-7) or (8^ + 5). 



in the form (ur- + hy" + cz" + du? 15 

If ya^l, take« = 2^+\ Then 

i\r-rf«;^ = 4^(8/^-5). 

In neither of these cases is TV — d}f- of the form 4^ (8/i + 7), 
and therefore in either case it can be expressed in the form 

X- + 2/2 4- z". 

Finally, let d=l. If ^u, is equal to 0, 1, or 2, take « = 2\ 
Then N - d^ir is equal to 0, 2 . 4^+\ or 4^+-. If /a^S, take 
M. = 2^+1. Then 

iV-*/,^=4^(8/x-21). 

Therefore in either case N — du- can be expressed in the form 

a;2 + 2/2 4- Z-. 

Thus in all cases N is expressible in the form (4'1). Similarly 
we can dispose of the remaining cases, with the help of the results 
stated in § 3. Thus in discussing (2-42) we use the theorem that 
every number not of the form (3'21) can be expressed in the form 
(3*2). The proofs differ only in detail, and it is not worth while 
to state them at length. 

5. We have seen that all integers without any exception can 
be expressed in the form 

m. {x^ +'if + z^) + mi'^ (5-1), 

when m = \, li^n^l, 

and m= 2, n = 1. 

We shall now consider the values of m and n for which all 
integers with a finite number of exceptions can be expressed in 
the form (5'1), 

In the first place 7?? must be 1 or 2. For, if m > 2, we can 
choose an integer v so that 

7iu' ^ V (mod 7)1) 
for all values of u. Then 

(nifx + v) — mi^ 
m 

where fi is any positive integer, is not an integer ; and so 7nfj, + v 
can certainly not be expressed in the form (5'1). 

We have therefore only to consider the two cases in which m is 
1 or 2. First let us consider the form 

cc- + 2/- + z- + nit^ (5'2). 

I shall show that, when n has any of the values 

1, 4, 9, 17, 25, 36, 68, 100 (5-21), 



16 Mr lia/uanujan, On tJie expression of a nwniber 

or is of any of the forms 

4yt+2, 4yb + 3, 8^•^-o, 16A; + 12, 32/.^ + 20 ...(5-22), 
then all integers save a finite number, and in fact all integers from 
4?i onwards at any rate, can be expressed in the form (5*2) ; but 
that for the remaining values of n there is an infinity of integers 
which cannot be expressed in the form required. 

In proving the first result we need obviously only consider 
numbers of the form 4*^ (8yu, + 7) greater than n, since otherwise 
we may take w = 0. The numbers of this form less than n are 
plainly among the exceptions. 

6. I shall consider the various cases which may arise in 
order of simplicity. 

(6-1) 7^ = (mod 8). 
There are an infinity of exceptions. For suppose that 
N = Sfju + 1. 
Then the number 

N - nu- = 7 (mod 8) 
cannot be expressed in the form cc- + y- + z'^. 

(6-2) n=2 (mod 4). 

There is only a finite number of exceptions. In proving this 
we may suppose that iV=4^(8/i + 7). Take u=l. Then the 
number 

]SF - oiu^ = 4'" (S/M + 1) - n 
is congruent to 1, 2, 5, or 6 to modulus 8, and so can be expressed 
in the form x^ + y^ + z^. 

Hence the only numbers which camiot be expressed in the 
form (5"2) in this case are the numbers of the form 4^(8/i+ 7) not 
exceeding n. 

(6-3) n=h (mod 8). 

There is only a finite number of exceptions. We may suppose 
again that i\r = 4^ (8/i + 7). First, let X =|= 1 . Take u=\. Then 

N - nu- = 4^ (8/i + 7) - n = 2 or 3 (mod 8). 
If X = 1 we cannot take u = l, since 

N - n = 7 (mod 8) ; 
so we take u = 2. Then 

JSf- nu- = V (8/i + 7) - 4n = 8 (mod 32). 

In either of these cases N — nu^ is of the form cc'^ -\-y'^ + z". 

Hence the only numbers which cannot be expressed in the 
form (5*2) are those of the form 4^ (8/a + 7) not exceeding ??, and 
those of the form 4 (8/ti + 7) lying between n and 4?l 



in the funii aw- + Inf + cz- + da- 1'7 

(6-4) /i= 3 (mod 4). 

There is only a finite number of exceptions. Take 
iV^=4M8/A + 7). 
If X^l, take ti=l. Then 

N — nil- = 1 or 5 (mod 8). 
If X = 0, take n = 2. Then 

N - nu? = 3 (mod 8). 
In either case the proof is completed as before. 

In order to determine precisely which are the exceptional 
numbers, we must consider more particularly the numbei'S between 
n and 4» for which X = 0. For these \i must be 1, and 

N -nu-= (mod 4). 
But the numbers which are multiples of 4 and which cannot be 
expressed in the form .x- + y ' + z' are the numbers 

4''(8i. + 7), (/c = l, 2, 3, ..., v^O,l, 2, 3, ...)• 
The exceptions required are therefore those of the numbers 

n + ¥{^v + 1) (6-41) 

which lie between n and 4» and are of the form 

8;i + 7 (6-42). 

Now in order that (6'41) may be of the form (6'42), k must be 
1 if 11 is of the form 8A- 4- 3 and k may have any of the values 
2, 3, 4, ... if n is of the form 8A;+7. Thus the only numbers 
which cannot be expressed in the form (5'2), in this case, are those 
of the form 4^ (8/i + 7) less than n and those of the form 

?i + 4''(8i/+7), (y-0, 1, 2, 3, ...), 

lying between n and 4?i, where k=\ if n is of the form 8A; + 3, 
and K>\ if ?r is of the form 8A; + 7. 

(6-5) n = 1 (mod 8). 

In this case we have to prove that 

(i) if n ^ 33, there is an infinity of integers which cannot be 

expressed in the form (5"2) ; 
(ii) if n is 1, 9, 17, or 25, there is only a finite number of 
exceptions. 
In order to prove (i) suppose that iV = 7 . 4^. Then obviously 
u cannot be zero. But if u is not zero n^ is always of the form 
4''(8t/+l). Hence 

N - nu^ = 7 . 4^ - ?i . 4" (8v + 1). 

Since n ^33, X must be greater than or equal to k + 2, to ensure 
that the right-hand side shall not be negative. Hence 

N - jui^ = ^^ (Sk + 7), 

VOL. XIX. PT. I. 2 



18 31r Ramanujan, On the ex'pression of a number 

where k = 14 . V-''-- - nv - ^ {n + 7) 

is an integer ; and so N — nu- is not of the form x- +y'^ + z\ 
In order to prove (ii) we may suppose, as usual, that 

N = 4^ (Sfi + 7). 
IfX = 0, take w=l. Then 

iV - nil'' = 8// + 7 - 7? = 6 (mod 8). 
If X^l, take w= 2^-1. Then 

where k = 4< (fji + l) -^{n + 7). 

In either case the proof may be completed as before. Thus the 
only numbers which cannot be expressed in the form (5'2), in 
this case, are those of the form 8/u. + 7 not exceeding n. In 
other words, there is no exception when n = 1 ; 7 is the only 
exception when n = 9; 7 and 15 are the only exceptions when 
n = 17 ; 7, 15 and 23 are the only exceptions when n = 25. 

(6-Q) n = 4 (mod 32). 
By arguments similar to those used in (6'5), Ave can show that 
(i) if w ^ 132, there is an infinity of integers which cannot 

be expressed in the form (5*2) ; 
(ii) if n is equal to 4, 36, 68, or 100, there is only a finite 
number of exceptions, namely the numbers of the 
form 4'^ (8yu, + 7) not exceeding n. 

(6-7) ?i = 20 (mod 32). 

By arguments similar to those used in (6'3), we can show that 
the only numbers which cannot be expressed in the form (5'2) are 
those of the form 4^ (8yLi +7) not exceeding n, and those of the form 
4^(8/A + 7) lying between n and 4n. 

(6-8) n= 12 (mod 16). 

By arguments similar to those used in (6'4), we can show that 
the only numbers which cannot be expressed in the form (5"2) are 
those of the form 4^ (Sfi + 7) less than n, and those of the form 

n + 4>^(8v + 1), (i/ = 0, 1, 2, 3, ...), 
lying between n and 4>i, where /c = 2 if n is of the form 4 (8k + 3) 
and AC > 2 if w is of the form 4 (8A; +7). 

We have thus completed the discussion of the form (5 2), and 
determined the exceptional values of iV precisely whenever they 
are finite in number. 

7. We shall proceed to consider the form 

2 (ic^ +y" + z') + mir .(7-1). 



in the fornt (i.r- + bij'^ -\- cz- + (hr ID 

In the first place n must be odd ; otherwise the odd iiuinbers 
cannot be expressed in this form. Suppose then that n is odd. 
I shall show that all integers save a finite number can be expressed 
in the form (7"1): and that the numbers which cannot be so 
expressed are 

(i) the odd numbers less than n, 

(ii) the numbers of the form 4^^ (16yu. + 14) less than 4n, 
(iii) the numbers of the form n-\- 4^(16ya+ 14) greater than 

n and less than 9w, 
(iv) the numbers of the form 

cn-\-¥{l(w+U\ (i^ = 0, 1, 2, 3, ...), 

greater than 9/i and less than 25w, where c = 1 if 
n = \ (mod 4), c = 9 if w = 3 (mod 4), « = 2 if n^= 1 
(mod 16); and /c > 2 if /(-=9 (mod 16). 

First, let us suppose N even. Then, since n is odd and N is 
even, it is clear that u must be even. Suppose then that 

We have to show that M can be expressed in the form 

x"->r y- -\- Z' + 27?,?'- (7-2). 

Since %i = 2 (mod 4), it follows from (6'2) that all integers except 
those which are less than 2n and of the form 4-^ (8//. + 7) can be 
expressed in the form (7*2). Hence the only even integers which 
cannot be expressed in the form (7'1) are those of the form 
4^(16/* + 14) less than 4n. 

This completes the discussion of the case in which N is even. 
If N is odd the discussion is more difficult. In the first place, 
all odd numbers less than n are plainl37^ among the exceptions. 
Secondly, since n and N are both odd, u must also be odd. We 
can therefore suppose that 

iV = ?i + 2il/, xv" = 1 + 8A, 

where A is an integer of the form |^•(^• + l), so that A may 
assume the values 0, 1, 3, 6, .... And we have to consider 
whether n + 2if can be expressed in the form 

2 0r2 + 2/'^ + ^'^) + w(l +8A), 
or M in the form 

«- + 2/- + ^- + 4nA (7-3). 

If M is not of the form 4^ (8/a + 7), we can take A = 0. If it is 
of this form, and less than 4?i, it is plainly an exception. These 
numbers give rise to the exceptions specified in (iii) of section 7. 
We may therefore suppose that M is of the form 4^ (8^* + 7) and 
greater than 4/?. 

2—2 



20 Mr Ramamtjan, On the eo;pression of a number 

8. In order to complete the discussion, we must consider 
the three cases in which n = 1 (mod 8), n = 5 (mod 8), and 
n = S (mod 4) separately. 

(8-1) /I- 1 (mod 8). 
If X is equal to 0, 1, or 2, take A = 1. Then 
M - 4?iA = 4^ (8/i + 7) - 4?i 
is of one of the forms 

8z; + 3, 4 (Sv + 3), 4 {8v + 6). 

If A. ^ 3 we cannot take A = 1, since if — 4?iA assumes the 
form 4 (8i/ + 7) ; so we take A = 3. Then 

M - 4n A = 4^ (SyLt + 7) - 12n 

is of the form 4 {8v + 5). In either of these cases M — 4nA is of 
the form x^ + y'^ -\- z^. Hence the only values of M, other than 
those already specified, which cannot be expressed in the form 
(7*3). are those of the form 

4«(8i; + 7), (z/ = 0, 1,2, ...,«>2), 

lying between 4?i and 12??. In other words, the only numbers 
greater than 9'n which cannot be expressed in the form (71), in 
this case, are the numbers of the form 

n+4«(8j; + 7), (i^ = 0, 1, 2, ..., /c> 2), 

lying between 9?i and 25?l 

(8-2) ?i = 5 (mod 8). 
If X 4= 2, take A =1. Then 

ili - 4wA = 4^ (8/i + 7) - 4?i 
is of one of the forms 

8i/ + 3, 4 (8z/ + 2), 4 (8z/ + 3). 

If \ = 2, we cannot take A = l, since ilf— 4?iA assumes the 
form 4 (8v + 7) ; so we take A = 3. Then 

M- 4mA = 4^ (8/A + 7) - \%i 

is of the form 4 (8y + 5). In either of these cases M — 4/? A is of 
the form (x? ■\-y'^ A- z^. Hence the only values of M, other than 
those already specified, which cannot be expressed in the form 
(7-3), are those of the form 16 (8^t + 7) lying between 4n and 12?i. 
In other words, the only numbers greater than 2n which cannot 
be expressed in the form (7"1), in this case, are the numbers of the 
form n + 4"^(16/ti + 14) lying between 9?? and 2.5». 



in the form ax~ + by- + oz- -f dti^ 21 

(8-3) n = 3 (mud 4). 
If \ =1=1, take A = 1. Then 

M - 4»,A - 4^ (8ya + 7) - hi 
is of one of the forms 

Si. + 3, 4(4i/+l). 
If \ = 1, take A = 3. Then 

M - 4mA = 4 (8/* + 7) - 12/i 

is of the form 4(4^- + 2). In either of these eases M — 4?iA is of 
the form j/-^ + -tf + z-. 

This completes the proof that there is only a finite number of 
exceptions. In order to determine what they are in this case, we 
have to consider the values of M, between 4?i and 12w, for which 
A = 1 and 

M - 4h A = 4 (8;i + 7 - n) = (mod 16). 

But the numbers which are multiples of 16 and which cannot be 
expressed in the form x- + y^ -\- z- are the numbers 

4''(8z/+7), (/c = 2, 8, 4, ..., y=0, 1, 2, ...)• 

The exceptional values of M required are therefore those of 
the numbers 

47« + 4*^ (8i. + 7) (8-31) 

which lie between 4fi and \2n and are of the form 

4(8/iA + 7) (8-32). 

But in order that (831) may be of the form (8"32), k must be 
2 if n is of the form 8/^' + 3, and k may have any of the values 
3, 4, 5, ... if n is of the form 8A; + 7. It follows that the only 
numbers greater than 9« which cannot be expressed in the form 
(7"1), in this case, are the numbers of the form 

9w + 4« (16i^ + 14), {v = 0, 1, 2, . . .), 

lying between 9w and 25?i, where k=2 if n is of the form 8/v + 3, 
and /c > 2 if ?i is of the form 8^- + 7. 

This completes the proof of the results stated in section 7. 



22 Mr Van Horn, An Axiom in Symbolic Logic. 



An Axiom in Symbolic Logic. By C. E. Van Horn, M.A. 
(Commimicated by Mr Q. H. Hardy.) 

[Received 30 August 1916: read 30 October 1916.] 

Philosophy's task is a search for the primal and fundamental 
elements of the world. Its face is turned in the opposite direction 
to that of science and mathematics. Philosophy hands back to 
them its results, and they as best they can construct systematic 
bodies of doctrine that purport to show us what the world may bo 
on the one hand (science) and what the world might be on the 
other (mathematics). As philosophy advances in the pursuit of its 
task it is continually vacating old ground to science and mathe- 
matics. The history of this change of boundary can be traced in 
the changes in the nomenclature of human knowledge : Natural 
Philosophy has become Physics ; Mental Philosophy has become 
Psychology ; Moral Philosophy is becoming the inductive science 
of Ethics. Thus (paradoxically speaking) philosophy's advance is 
to be marked by the retreat of her boundaries. 

It is interesting to Avatch this retreat in a field occupied b}' 
philosophy from its very beginning, and until recently supposed to 
be its permanent possession. I refer to the field of the foundations 
of mathematics. Here large areas once occupied by philosophy by 
sovereign right of long control are slowly passing into the possession 
of pure mathematics; and by the way both are gainers by the 
transfer*. 

To facilitate the mathematical treatment of these new areas a 
new instrument of investigation had to be invented, namely, Mathe- 
matical, or Symbolic, Logic. This new logic, which is infinitely 
more powerful than the traditional logic, and which embraces all 
that is really self-consistent in the old logic, makes possible a 
precise and easy handling of all the highly abstract and complex 
ideas occurring in the noAv fields. For example, both philosophy 
and the old logic found themselves involved in many a tangle on 
questions concerning classes and relations because neither possessed 
the requisite instruments of analysis. Again, philosophy had 
wandered into a veritable labyrinth of difficulties concerning 
infinity, quantity, continuity, and so on. Here too the secret of 
the trouble lay in the inadequacy of the instruments of analysis 
afforded by the traditional logic. 

* Much valuable light is thrown upon the details of this process in the writings 
of Bertrand Russell, especially in the preface and introductory chapters of the 
Frincipia Mathematica, Vol. i. 1910; and more recently in his Scientific Method in 
Philosophy, 1914. 



Ml' Van Horn, An Axiom in Symbolic Logic 28 

Nuw however the matter is all changed. Philosophy, equipped 
with the latest instruments of mathematical logic, is able to deal 
successfidly with the problems of these fields. In fact so fully have 
these ideas been analysed that at last philosophy as such has 
relinquished these fields to pure mathematics. Even more, the 
whole field of deduction has now become the foundation-branch of 
mathematics and has developed into a precise Calculus of Pro- 
positions. Out of it grow by easy stages the Calculus of Classes 
and the Calculus of Relations, and these in turn grow by equally 
easy stages into all the manifold branches of pure mathematics as 
more commonly known. It is in these and similar ways that 
philosophy and pure mathematics are both gainers by the transfer 
of the fields recently acquired by mathematics from philosophy. 

It is now easy to understand why the axioms of mathematical 
logic (and so of all pure mathematics) lie in the borderland between 
philosophy and mathematics, and are thus the concern of the 
philosopher equally with the mathematician. To depart entirely 
from our figures and adopt others, the rootage of mathematics is in 
philosophy. It is here too that we meet the innovations of mathe- 
matical logic that appear so fantastic to the philosopher trained 
only in the old logic. Its definitions and treatment of some of the 
common terms of language seem so at variance with what the 
traditional logician is familiar with that he often views the new 
logic as the victim of some delusion. It appears however from the 
nature of the case itself that many of those peculiarities, which 
from the view-point of traditional logic would be described as 
abnormal, do not deserve to be so described ; that in fact it is in 
the theories of the traditional logician and philosopher that the 
abnormalities really occur*. 

In order to indicate what seems to me a possible simplification 
of the axiomatic basis of mathematical logic I wish to introduce in 
a new form an idea advocated by Shelfer. Its importance lies in 
the fact that in terms of it Sheffer was able to define the four 
fundamental operations of logic, namely. Negation, Disjunction, 
Implication, and Conjunction or Joint Assertion. It is a familiar 
fact that Kronecker found the use of certain auxiliary quantities 
(let us call them ' parameters ') of great value in his algebraic 
investigations, the chief value lying in the fact that their dis- 
appearance led to desired relations among numbers essential to his 
investigations. It is a precisely similar use of Sheffer's idea that 
I desire to make in the field of the philosophy of logic. In terms 
of it I define, after him, the four fundamental operations of logic. 
Then, unlike him, I work by means of an axiom to eliminate that 
idea from the formulae, and in so doing to arrive at the desired 

* Cf. Russell, Scientific Method in Philosophij, chap. i. 



24 Mr Van Horn, An Axiom in Si/ntholic Logic 

properties and relations of the four fundamental operations. The 
chief excellence of my method seems to reside in the fact that 
proceeding as indicated above I have been able to prove as pro- 
positions of mathematical logic some of the axioms hitherto laid 
down at the basis of this logic. 

In its most satisfactory form the axiomatic basis of mathe- 
matical logic has been stated by Bertrand Russell in the first 
volume of the Principia Mathematical. In *1 of Vol. i., pp. 98-101 , 
of the Principia will be found the primitive propositions required 
for the theory of deduction as applied to elementary propositions. 
I confine myself to these purposely, for it is here that I have 
succeeded, I believe, in simplifying the axiomatic basis of 
mathematical logic. 

Let p and q be any two elementary propositions. The four 
fundamental operations give us (1) ~ p {not-p), (2) pv q (either p 
or q), (3) j9 D q (p implies q), and {4<) p • q (both p and q). After 
Sheffer, I define these four results in terms of a single undefinable 
operation. I will call this undefinable operation Deltation. The 
result of performing this operation upon two elementary propositions 
p and q is symbolized, after Sheffer, 'pAq' (read " j) deltas q'). 
The four fundamental operations of logic can be expressed as 
logical functions of this parameter thus : 

Negation: ~p. = .jjAj9 D£ 

Disjunction : /;vg. = .~^jA~g Df 

Implication : pD q . = .p A <^ q Di 

Conjunction : p • q . = . '^ (p A q) Df. 

These definitions of the four fundamental operations of logic 
as functions of the one undefined parameter, Deltation, are made 
relevant to our discussion by means of the following axiom. 

Axiom. If p and q are of the same truth-value, then ' p A q ' 
is of the opposite truth-value ; but if j) and q are of ojjposite truth - 
values, then ' p A q' is true. 

For convenience of reference it might be well for me to state at 
this point Russell's primitive propositions concerning elementary 
propositions as he enunciates them in *1 of the first volume of 
the Principia. 

*1.1 Anything implied by a true elementary proposition 
is true. Pp|. 

t Whitehead and Russell, Princiina Mathematica, Vol. i. 1910, Vol. ii. 191'2, 
Vol. III. 1913 (Cambridge University Press). 

X Eussell uses the letters "Pp" to stand for " primitive proposition, " as does 
Peano. 



3fr Van Horn, An Axiom in Sipnbolic Lo(jio 25 

*1.H When <^x can be asserted, where x is a real variable, 
and ' (fjxD yfr x ' can be asserted, where x is a real variable/then yjrx 
can be asserted, where x is a real variable. Pp. 

*1.2 h : pvp.D .p Pp. 

*1.3 \- : q.D .pv q Pp. 

* 1.4 [■ : pv q .0 .qv p Pp. 

* 1 .5 \- : py/ (qv r).D .qv {pv r) Pp. 

*1.6 h: .q'^r.D-.pvq.D.pyr Pp. 

*1.7 If j9 is an elementary proposition, ~ p is an elementary 
proposition. Pp. 

*1.71 If p and (/ are elementary propositions 'pvq' is an 
elementary proposition. Pp. 

*1.72 If ^p and i/rj? are elementary prepositional functions 
which take elementary propositions as arguments, ' (f) pv -ylrp' is an 
elementary prepositional function. Pp. 

These are all the primitive propositions that are needed for the 
development of the theory of deducti(jn, as applied to elementary 
propositions, according to Russell's method of treatment. 

It is my purpose to show that by means of my axiom 
Russell's primitive propositions *1.2 to *1.7l can be demon- 
strated. I do this by starting at the very beginning and 
developing the immediate consequences of three of the axioms 
which I lay down as the basis of the theory of deduction as applied 
to elementary propositions. The resulting deductive development 
at length reaches a point where it includes among its theorems 
Mr Russell's seven pi'imitive propositions and two others that can 
take the place of his definitions of Implication and Conjunction. 
Altogether I prove seventeen theorems. Some of these theorems 
occur as propositions in the first volume of the Principia. Al- 
though many more theorems can be proved as simply as the ones 
given, to economize space I shall stop at the point where my 
development of Mathematical Logic includes the nine theorems 
mentioned above. 

I will now state the three axioms used in this paper. The 
first is * 1.1 given above, the last is my axiom as already enunciated. 

Axiom 1. Anything implied by a true elementary proposition 
is true. 

Axiom 2. Ifp and q are elernentary propositions, then " p Aq' 
is an elementary proposition. 

Axiom 3. If p cund q are of the same truth-value, then ' p Aq' 
is of the opposite truth -value ; hut if p and, q are of opposite truth- 
values, then ' p Aq' is true. 



26 Mr Van Hum, An Aadoiii in Symholic Logic 

Theorem 1 

If }) is an elementary proposition, ~ p is an elementary pro- 
position, 

Deni. 

Axiom 2 gives us ' p Ap' elementaiy when j) is elementary ; 
'pAp' is ~ p, by Definition of Negation. Hence the theorem. 

This is a proof of Mr Russell's primitive proposition *1.7 given 
above. 

Theorem 2 

Ifj} and q are elementary ])ropositions, ' pv (j' is an elementary 
proposition. 

Dem. 

By Theorem 1 , if p and q are elementary so also are ~ p and 
~ q. Therefore, by Axiom 2, ' -^ p A ~ (/ ' is elementary ; but this, 
by Definition of Disjunction, is ' pv q'. Hence the theorem. 

This is Mr Russell's primitive proposition *1.7l quoted above. 

Theorem 3 
The propositions p and ~ p are of opposite truth-values. 

Dem. 

Two possibilities can occur : 

1°:^ true. By Axiom S, " p A p' is false; but this by 
Definition of Negation is ^ p; hence in this case p and ~ p are 
opposite in truth-value. 

2° : j9 false. By Axiom 3, 'pAp' is true; but this by 
Definition of Negation is ~ jo ; hence in this case also ^j and f^ p 
are opposite in truth-value. Hence the theorem. 

This theorem states in precise form the information usually 
given in text-books on logic in more or less vague statements that 
are called ' definitions ' of negation. 

Theorem 4 
\-. pDp. 

Dem. 

[Th. 3] h. p and <^ p of opposite 

truth-values (1) 

[(1). Ax. 3] \-. pA r^ p (2) 

[(2). Def. of Implication] h. theorem. 

This is proposition *2.08f of the Principia. 

I 0]}. cit. Vol. I. p. 105. 



Mr Van Horn, An Axiom in Symholio Loyic 27 

Theorem 5 

If 2) is false, ' p A q' is always true. 

Bern. 

Two possibilities can occur : either q true, or q false. In either 
case ' p A q' IB true by Ax. 3. 

Theorem 6 

If q Is false, ' 2J A q' is alivays true. 
Proof similar to that of preceding theorem. 

Theorem 7 
llie jyropositions ' p A q' and ' q A p' Jmve the same triUh-ualue. 

Deni. 

li' p and q are of the same truth- value then, by Ax. o, ' p A q' 
and ' q A p' are both of the opposite truth-value. If p and q are 
of opposite truth-values then, by Ax. 3, ' p A q' and ' q A p' are 
both true. Hence the theorem. 

Theorem 8 
The proposition 

f"^ p A 1^ {(^ q A f^ r) 

is true if any one or more of the propositions p, q, r are true; but 
if all of these propositions are false then the proposition 

~ jj A '^ {^ q A ~ r) 
is false. 

Dem. 

Eight possibilities can occur : 

1° : p, q, r all true. Then (Th. 3) ~ ^j, ~ q, ~ r are all false. 
Hence (Ax. 3) ' ~ 9 A ~ r ' is true. Hence (Th. 3) ~ (~ g- A ~ r) 
is false. Hence (Ax. 3) the proposition '~ jo A ~ (~ </ A ~ /•)' 
is true in this case. 

2^ : jj and q true, but r false. By Th. S, r^ p and ~ q are 
false, while ~ r is true. Hence (Ax. 3) ' ~ r/ A ~ ?■ ' is true. 
Hence (Th. 3) ~ (~ (/ A ~ r) is false. Hence (Ax. 3) the 
proposition is true in this case. In a similar manner in the 
following cases : 

3° : j) true, q false, r true ; 

4° : ]) false, q, r true ; 

o" : p true, q, r false ; 

6° : J) false, q true, r false ; 

7° : p, q false, r true ; 
we have ' ^' j) A ^ (■-- </ A ^^ /•) ' true. 

But in 8" : p, q, r false, we have ~ jj, r^ q, ^ r all true, by 



28 Jllr Van Horn, An Axiont in H[iinholic Logic 

Th. 8. Hence (Ax. 3) '~ </ A '^ ?• " is false, making ~ (~ (/ A ~ /•) 
true (Th. 3). Hence (Ax. 3) in this case the proposition is false. 

Hence the theorem. 

Theorem 9 
The propositions 

' <>•' p A f^ (<^ q A f^ r)', 'r^(/A~(~^:>A~ r) ', 

always have the same truth-valm. 

This follows at once from Th. 8. 

At this point I introduce Mr Russell's definition of Equivalence f 
as it occurs in the Principia. 

Equivalence: p = q. = .pDq.qDp Df. 

Theorem 10 

h. p= <^ (^ p). 

Dem. 

We first prove h. jj D ~ ( ~ p). Two cases arise : 
1°: p true. By Theorem 3, '^ ^ is false, ~ (~p) is true, and 
f^ [f^ {^ py] is false. Hence 

[Ax. 3] h. I? A ~ [~ (~ jj)] (1) 

[(1). Def. Implica.] Kj9D~(~p) (2) 

2° : p false. By Th. 3, r^ p is true, ^^ {<^ p) is false, and 
,^ [r^ ('^i^)] is true. 



[Ax. 3] 


h. 


j5 A ~ [~ {"^ p)\ (3) 


[(1). Implica.] 


V. 


p D (^ (<^ p) (4) 


Hence in all cases we have 








V. 


p'^r^{<^p) (5) 


We now prove 


V. 


~(~_p)Di9. 


[Th. 3] 


V. 


q and ~ g of opposite 
truth-values (6) 


[(6). Ax. 3] 


h. 


r^qAq (7) 


[(7). /] 

[(8). Def. Implica.] 


V. 


'^ {r^ p) A r^ p (8) 


V. 


~(~p)Dp (9) 


[(5). (9). Def. Equiv.] 


V. 


theorem. 



This is proposition *"4.13 of the Principia^. It is the Principle 
of Double Negation, and asserts that any proposition is logically 
equivalent to the denial of its negation. 

t Op. cit. Vol. I. p. 120, *4.0l. 
% Op. cit. Vol. I. p. 122. 



^fr Van Horti, An A.iiom, in Si/)iibolic Logic 29 

Theorem 11 
H: pvp . D .p. 

Dem. 

[Ax. 3] ■ h. ~ jj and ' <^ p A f^ p' of 

opposite truth- values (1) 

[(1). Ax. 3] |-:~_p A ~^9. A . ~_p (2) 

[(2). Def. Disjunc. Implica.] I-. theorem. 

This is Mr Russell's primitive proposition *1.2 given above. 

Theorem 12 

h: q.'^.pwq. 

Dem. 

Two cases need only be treated : 

I'' : q true. Then (Th. 3) ~ q is false. Hence (Th. 6) 
' ~ jj A <^ q ' is true. Hence ~ (~ ^j A ~ q) is false, by Th. 3. 
Hence 

[Ax. 3] h : (/ . A . <^ ( ~ p A ~ (/) ( 1) 

2° : q false. 

[Th. r, |-~(~pA ~g)j F. , . A . ~ ( ~ ,, A ~ ,/) (2) 

[(1). (2). Def. Disjunc. Implica.] h. theorem. 

This is Mr Russell's primitive proposition *l.o given above. 

Theorem 13 
h: pv q .D . qy P' 

Dem. 

[Th. 7] h : ' ~ jt) A ~ g ' and ' ~ g A ~ p ' 

of the same truth- value (1) 

[(1). Th. 3. Ax. 3] h: ~ jj A ~ (/ . A . ~ ( ~ ry A ~ p) (2) 

[(2). Def Disjunc. Implica.] h: theorem. 

This is Mr Russell's primitive proposition *1.4 given above. 

Theorem 14 
V : p y {q y r) ."^ . qy {p M r). 

Dem. 

[Th. 9] I-: '~p A ~(~(/ A ~?-)' a-nd '~(/ A ~(~|) A ~?-)' 
of the same truth-value (1) 

[(1). Th. 3. Ax. 3] h: ~ ;j A ~ (~ (? A ~ /•) 

. A . ~ [~ q A '^ {r^ p A ~ r)] (2) 

[(2). Def. Disjunc. Implica.] h: theorem. 

This is Mr Russell's primitive proposition *1.5 given abt)ve. 



30 Mr Van Horn, An Aj'iunt in Sfjtnbolic Logic 

Theorem 15 



J 



\-:.qDr.D:pvq.D.2i'v r. 

Bern. 

There are three cases to be discussed : 
1" : li p is true, or if r is true, or if both p and r are true, 
q being any elementary proposition. 

[Th. 8] }-: r^l. A.^i^p A '^r) (1) 

[(1). ::ii. Th. 10] h: Z. A .~(-2J A ~?-) (2) 

[(2). ~i^^ ~5j h: ~p A ~(/. A . ~(~2) A ~ r)(3) 

[(3). Th. 3. Th. 6] 

h: q A ^ r . A . <^ [^ p A '^ q . A . <^ (r^ p A '^ r)] (4) 

Taken together with the Definitions of Implication and 
Disjunction, (4) gives the theorem in this case. 

2° : If both p and r are false, but q true. In this case ~ ^j and 
oo r are true by Th. 3. Hence (Ax. S) ' ^ p A ~ r ' is false. The 
proof in this case proceeds as folloAvs : 

[Th. 3] 1-: ~(-2) A ~ r) (5) 

Since q is true, '^ q is false (Th. 3). 

[Th. 6] h. - 19 A ~ 5 (6) 

[(5). (6). Th. 3. Ax. 3] V: ^[^^p A ^q. A.'^i^p A ^r)]{1) 

By Ax. S, ' q A <^ ?' ' is in this case false. 

[(7). Ax. 3] 

h: q A ~r.A.'^[~/jA <^5.A.~(~jjA ~ ?•)] (8) 

As in the previous case this result gives the theorem. 

3° : All three false. Hence ~ p and ~ ?■ true as before. In 
this case ' ^ p A <-^ q' is false by Ax. 3. The proof in this last 
case proceeds thus : 

[Th. 3, as in 2°] h. ~(~pA~?') (9) 

[(9). Ax. 3] h: ~p A ~ g. A . ~(~p A ~ r) (10) 

In this case q and f--' r are of opposite truth- values. 

[Ax. 3] h: f^ A~r (11) 

[(10). Th. 3. (11). Ax. 3] 

h: ^A~?'.A.'^[~pA'^^.A.'^ (~i^ ^ ~ ''')] (12) 

As in the two preceding cases, this result, together with the 
Definitions of Implication and Disjunction, gives the theorem. 

No other cases can arise. Hence the theorem. 

This is Mr Russell's primitive proposition *1.6 given above. 
It asserts that an alternative may be added to both premise and 



Mr Vail Horn, An Axiom in Si/nihohc Loffic ol 

conclusion in any implication without impairing the truth of the 
implication. 

This completes the list of Mr Russell's primitive propositions 
that I proposed for proof by means of my axiom, on the basis of 
the definitions given in this paper of the four fundamental 
operations of logic. 

I now propose to prove two propositions which can take the 
place of his definitions of Implication f and Conjunction j, or Joint 
Assertion. 

Theorem 16 

Dem. 

[Th. 4^^'"'^] h: p A -</.D.;9 A ~g (1) 

[Th. 10] D . ~ (~ p) A ~ ry (2) 

[(2). Def. Implica. Disjunc] h: pD q .D . ^^ pw q (3) 

[(1). Th. 10] h: ~(~j9)A ~(y.D.j)A ~ 7 (4) 

[(4). Def. Implica, Disjunc] h : '^py/q.D.pDq (5) 

[(3). (5). Def. Equiv.] h : theorem. 

Theorem 17 

h: p . q . = . ^ { '^ p W ^ q). 
Dem. 

[Th. 4 *" ^^ ^ '^^ ] h: ~(jo Ary).D.~(p A (/) (1) 

[Th. 10] D.~[~(~y/) A ~(~^)](2) 

[(2). Def. Conjunc. Disjunc] I- : p . q . D . ^^ (^ p v r^ q) {S) 
[(1). Th. 10] h: ~[~(~p)A ~(~5)].D.~(p Afy)(4) 

[(4). Def. Conjunc. Disjunc] h : '^ (^ p v ^ q) . "D . p .q (5) 
[(3). (5). Def. Equiv.] h : theorem. 

With these theorems established the development of the 
Principia Mathematica can proceed as given by its authors. 
All that I have done is to reduce the number of axioms needed 
for that development. 

Baptist College, 
Rangoon, Burma. 

t Op. cit. Vol. I. p. 98, *1.01. + Ibid. p. 116, *3.01. 



32 Mr JSicod, A Reduction in the nimiber 



A Reduction in the number of the Primitive Propositions of 
Logic. By J. G. P. NicOD, Trinity College. (Communicated by 
Mr G. H. Hardy.) 

[^Received and read 80 October 1916.] 

Of the four elementary truth-functions needed in logic, only 
two are taken as indefinables in Principia Mathematica. These 
two have now been defined by Mr Shefferf in terms of a single 
new function p | q, " p stroke q." I propose to make use of Mr 
Sheffer's discovery in order to reduce the number of the primitive 
propositions needed for the logical calculus. 

There are two slightly different forms of the new indefinable, 
for we may treat 2:)\q as meaning the same thing as either 
~jj . ~g, or <^p}/ ^qt- The definition of <^p is the same in 
both cases, namely p \ p, while that of pv q simply changes from 
p/q \p/q with the AND-form into p/p \ qjq with the 07^-form. 

However, the best course is for us to define all the four truth- 
functions directly in terms of the new one. In so doing, we find 
that, while the definition of ~j9 remains the same, and those of 
pv q, p . q simply permute, as we pass from the ^iV^D-form to the 
Oi^-form, the definition of pO q is simpler in the latter form. It 
is p I qjq, as against j;/j) j q \p/p \ q. 

The OJ?-form is therefore to be preferred §. 

Definitions. 

f^p . = . p\p Df. pvq.^.plpiq/q Df. 

pO q . = . p\ qjq Df. p . q . — . p/q I p/q Df 

Remaeks on these Definitions, 

One ought not to aim at retaining before one's mind the 
complex translation into the usual system, "-^pv^q" as the 
"real meaning" of the stroke. For the stroke, in the stroke- 
system, is simpler than either ~ or v, and fi-om it both of them 
arise. We may not be able to think otherwise than in terms of 
the four usual functions ; it will then be more in accordance with 
the nature of the new system to think of the j , not as some fixed 
compound of -^ and v, but as a bare structure, out of which, in 
various ways, ~ and v will grow. 

+ Sheffer, Trans. Amer. Math. Soc. Vol. xiv. pp. 481—488. 
X Sheffer, loc. cit., footnote f, p. 488. 

% p\q thus corresponds to what is termed the Disjunctive relation in Mr W. E. 
Johnson's writincrs. 



|i 



of the Primitive Propositions of Logic 33 

The above definitions give clear expression to the symmetiy 
between OR and AND ; and this, notwithstanding the choice that 
we had to make between an Oi?-forni, and an AND-iorva. This 
is of some interest, because, in general, the very symmetry forces 
upon us an arbitrary choice, which, in turn, quite obscures the 
symmetry. 

I shall use q for q\q whenever convenient. Observe that 
p I q, i.e. pD q, forms a natural symbol | for implication, 

allowing of permutation ~q\ p. We may notice in general that 
the new system brings the four functions into relations far closer 
than those in Mr Russell's system. For instance, in 

p/p\p/p-\.p/p 
the two propositions pv p .D . p and r^pv p coincide. 

Every stroke-formula falls into two parts on the right and left 
of a central stem. It will, therefore, add to clearness to use black 
type instead of dots to indicate the central symbol. Further, 
slanting strokes are covered by straight ones : thus p/q j p/q stands 
for (p\q)\ (pj q). 

The definition of the two primitive notions of the Principia 
in terms of a single new one tends to reduce the number of the 
primitive propositions needed. But how far does this reduction 
actually occur ? Does it extend beyond the obvious substitution 
of " If p and q are elementary propositions, p\q is an elementary 
prop." (Sheffer, p. 488) for *r7 and *1'71, stating the same for 
~ p and py q respectively ? The reduction goes, as we shall 
presently find, very much farther. 

It has first to be said, in order that we may be as precise as 
possible, that the tuhole amount gained in applying the stroke- 
definitions cannot with complete certainty be attributed to them. 
For Mr Russell's system, as it now stands, has not said its last 
word in that matter. 

Incidentally, I found that *1'4, pv q .D . q y p, can be proved 
by means of the other four, with the unimportant change of *1'3, 
q . "^ . pv q into q . "^ . q v p. In "Association," *1*5, writers for r : 

p y {q y p) . 1^ . qv ij) V p). 

The left-hand side, by the help of q ."D . qvp and " Summation," 
will be found to be implied in pv q. The right-hand side, like- 
wise, hy p V p . D r p, and " Summation," will be found to imply 
qvj). The result then follows by using "Syllogism" (obtained 

from " Summation " with the transformation — - f) twice. 

p 

P V p' 
t By - or ^-^ I mean (following Mr Russell) the substitution of p for q or 

p, p' for q, q'. By {e.g.) P~ I mean the result of effecting the substitution in P. 
VOL. XIX. PT. I, 3 



34 Mr Nicod, A Reduction in the number 

Let us, however, take Mr Russell's eight propositions in the 
form given in Principia. It is my object to reduce them to three 
— two non-formal and one formal — by means of the stroke-defi- 
nitions given above. 

It can be shown, as a first stage, that two formal propositions 
are enough, namely : 

(1) p\l)/p. 

(2) p\q/q\s/q\^. 

The first proposition is the form of " Identity " (p D p) in the 
stroke-system. It would, at first sight, appear more natural to 
adopt the order q/s \ p/s in the left-hand side of (2), since 

p\qlq-'^-qls\p/s 

is the syllogistic principle of the stroke-system, giving " Syllogism," 
pD q .D : q D s . D .pDs when s | s is written for s. 

It will however be found that the inverted order, s/q 1 p/s, is 
much more advantageous than the normal syllogistic order, 
q/s \p/s. For, owing to this " twist," Identity and (2) yield 
" Permutation," s/p \ p/s, which now enables us to eliminate the 
twist in (2), and revert to the normal order. From the three 
propositions thus obtained, the rest follow. 

This, by the way, illustrates the following fundamental fact. 
Which form of a given principle is the most general, and contains 
the maximum assertion, is a function of the symbolic system used. 
Thus, for instance, in Mr Russell's system, 

p .D . qwp (a) 
is more general than p .0 . qD p (b) 

since (h) is (a) with <^q for q. In the stroke-system, on the 
contrary, p \ q/q \ p/p, meaning the same thing as (a), is less general 

than p\ q \p/p, whose meaning is that of (b), since it is obtained 
from it by writing q\q for q. 

A further step has to be made in order to be left with only one 
formal primitive proposition. It consists in adapting to better 
advantage the form of the primitive propositions to the properties 
of the stroke-symbolism where implication is concerned. We had 
above 

p'^q . = .p\ q/q Df 

If we look for the meaning of the general form p \ r/q, we find this 
to be oo 29 V ~ (~ r V ~ 5'), i.e. p .D . r .q. We thus come to the 
fundamental property that, in the new system, p"^ q is a case of 
p .D . s . q, whereas in Principia the contrary relation of course 
holds, 



i 



of the Primitive Propositions of Logic 35 

This leads us to substitute p \ r/q for 'p \ q/q in the " left-hand 
sides " of both the non-formal rule of implication and the syllo- 
gistic proposition (2) above. The reform may be further extended 
to the proposition (2) as a whole, which might be given the form 
P ! S/Q instead of P \ Q/Q, with the proviso, if the proposition is to 
remain true, that *S' must be implied in P. Now, for S, write the 
pioposition (1) above, p\p/p ; for (as we at this early stage know 
" unofficially ") a true proposition will be implied by everything. 

We then have the three primitive propositions of the stroke- 
system : 

( I. If p is an elementary proposition, and q is an 
Non- elementary proposition, then p\q is an elementary pro- 
formal 1 position f. 

\ II. If J) [ r/q is true, and p is true, then q is true. 

This is the non-formal rule of implication, *1'1, with the modifi- 
cation just explained. 

Formal III. p j q/r \t\t/t.\. s/q [p/s. 

I shall call II " the Rule," and III " the Prop." 

Remarks on these Primitive Propositions. 

Observe p r/q in II, while p | q/r in III. This alternance will 
prove essential for the working of the calculus. 

In III, I shall use ir for 1 1 t/t, P for p j q/r, Q for s/q \p/s, and 
shall speak of III as P \ tt/Q. 

P I ir/Q, by the Rule, yields the same result as the syllogistic 
proposition (2) above, when the left-hand side P is a truth of 
logic. This restriction of the syllogistic form to its categorical 
use with an asserted premiss is a peculiar character of the first 
proofs to follow, and is of some philosophical interest. 

One feels inclined to think that III merely asserts together 
(1) and (2) above. This view, whatever may be the amount of 
truth it contains, takes AND too much as a matter of course, 
and tends to lose sight of (a) the fact that III, as a structui;^^s 
simpler than (2) alone : for III is (2) with t \ t/t instead of s/q \p/s ; 
and (y8) the very real step from p .q to q, together with the philo- 
sophical difference between two assertions and only one. 

The main steps in the formal deduction are : 

1. Proof of " Identity," t \ t/t. 

2. Passage from P \ ir/Q to the u sual implicative form P [ Q/Q. 

3. Elimination of the twist s/q \p/s in Q, and return to the 
normal order q/s \p/s. 

t This is the proposition shown by Sheffer to imply the analogous propositions 
*1*7 and *1-71 in Principia. 

3—2 



36 Mr Nicod, A Reduction in the numher 

4. Proof of " Association," p \ q/r .D.q [p/s. 

5. Theorems equivalent to the definitions of p . q, p q in 
Principia. 

Proof of Identity, t\t\t. 

As this first proof from a single formal premiss stands in a 
unique position, I shall, without in any way obscuring the precise 
play of the symbols, expound it after a more heuristic order than 
is usually followed. 

We start with the Prop. P | tt | Q, and the Rule enabling us to 
pass from the truth of P to that of Q ; and we have to prove tt. 
This can only be reached through some proposition of the form 
-4 1 5 1 TT, where A is a truth of logic f. The proof will thus consist 
in passing from P | tt | Q to J. 1 5 | tt by some permutative process. 

A simple two-terms permutative law s 1 5' | ^ | 5, we do not yet 
possess. Our Prop, yields only a roundabout three-terms per- 
mutation, slglpjs, subject to the condition of ^jglr being a 
truth of logic f. This, however, is enough for our purpose. 

In the Prop., write t. for p, q, r : 

(a) 7r|7r!Qi, 

Qi being s|^|^|s. Write now tt for p, q; Q^ for r: then by (a) 
and the Rule, 

(b) S ! TT I TT I s. 

From (b), in the same manner. 



(c) u I tt/s I s/tt j u. 

This enables us to pass, by the Rule, from P | tt | Q to 

(d) Q|7r|P. 

In order to complete the proof of tt, we need only find some 
expression which : (a) can be a value for P, i.e. is a case of p\q\ r, 
and (/3) is implied in some truth of logic, say T. For, by T'lP | P, 
the Prop., and the Rule, as above, 

(e) s\P[T\~s. 

In (e), write Q | tt for s: first by (d) and the Rule, then by T 
and the Rule, we obtain T\Q\7r, and so 

(/) 

t This use of the Rule by anticipation, with still undetermined P's and Q's, is 
in truth contrary to the nature of a non-formal rule, which must never be used to 
build up the structure of an argument. It must always be possible to dispense 
with all such ' anticipated ' assertions in the final form of a proof. This will be 
seen to be very easy in the present case. 



of the Primitive Propositions of Logic 37 

Now, Qi I 7r| TT fulfils (a) and {^). For (a) tt being the complex 
expression t\t\t, i s a case of the form q \ r, and (/3) we have, by 
(c) above, tt i tt/Qi \ Qi/tt \ tt, and by (a) tt | tt | Qj. 

To obtain the strictest development of the proof we have only 
to write Qi/tt tt for P and ir ; tt/Qi for T all through the preceding 
argument. 

Permutation, s | p | p I s 
Gives sv p .1) . py s hy ^ , 

Dem. : Prop. - — — — - , Id., and Rule. 
j3 q r 

Tautology, p/p \ p/p \ pjp 

i.e. py p . -p 

Dem.: Id.^, Perm., and Rule. 
P 



Addition, s\p\sls 

Gives s ."D .py s by — . 
Dem. : By Perm, (twice), p \ s/s\sjs \p (a) 



By Prop, ^-y qrs ' ^ (")' ^ W. +, p \ s/s \ s 

By Perm., result. 

Return froivi Generalised Implication P \ tt/Q to P Q/Q. 



Lemma, pjp \ s/j) 

Dem. : By Perm, (twice), s/p \ p/s (a) 

By Prop. -^ — , I- {a), 

-^ ^ p q r s 



u\p\ s/p I It 
Write p/p for ii : by Id. and Perm, (twice), result. 

t \- (a) means the use of the Rule to pass from a to b iu a sjl). 



38 Mr Nicod, A Reduction in the number 



Theorem, P\irlQ\QIQ\P 

Dem. : Prop. -^^^-^^ -^ , r Lemma, reeult. 

p q, r s 

Hence, by Perm., P \ Q/Q, i.e. 

P I 5'/^ I s/q i P/^ (^') 



Syllogism, i? | 5*/^ I q/s \ p/s 

o s s 
Gives _p D (/ . D : fy D s . D . jj D 6' for ^^ — 

Dem. : In this Dem., Permutation is used to correct the 
twisting action of S\ much as handwriting has first to be inverted, 
if it is to be seen right in a mirror. 

By 8' ~ -^ , I" Perm., and Perm., 

•^ p q, r s 

qjs I u I u I sjq {a) 

•^ p q, r s 



qjs I u I sjq I u (b) 
By ^- i^lgA^ ^/glW^ g/HW^ ^ H^', h6, result. 



Association, p \ q/r | q \pjr 

The structure of the proof is this : 

Syll. " Il'' ' • 
p q, r s 

gives _p I g/r . D : q/r | ?' . {p/r. 



We now need only the Lemma q \ q/r | r for our result to 
follow by Syll. twice. 



Lemma, q \ q/p | p 

The proof of this lemma — call it L — is as follows : We prove 
(a) q I LjL, (b) L/L \ q/q. From this, by Syll. and TautoL, the 
result follows. 

Dem. : (a) By Syll. ^ , 
r, s 

p\qlq-:^-q/p\plp (1) 



of the Privative Propositions of Logic 89 

By Ackl, SylL, I- (1), 



q. D:q/p\p/2) 


(2) 


The right side of (2) implies, by Syll., 




plp\p.'^.q/p\p 


(3) 


By Id., Perm., Add.^/^'^' -'^ , 
^ ' ' p, q' 




q.D:p/p\p 


(4) 


By Syll. twice, h (2), h (3), h (4), 




qD : q ."^ . q/p \p, i.e. q 


L/L. 



(b) By lemma to Syll., q/q\s/q; by Perm, and Syll, q/q Iq/s. 
Hence, q/q \ L/L ; by Perm., L/L \ q/q. 
Now, by Syll. : 

L/L 1 q/q .D:q\L/L.D . L/L | L/L. 

By 1-6, h a, and Taut. -, result. We can now complete the proof 
of ' Association.' 



Association, p \ q/r \ q \ p/r 



Dem. : By Syll., /) | q/r . D : q/r \r .\. p/r 
By Syll. twice, h Lemma, result. 

Summation, qDr .D : pvq .D .pv r 

Dem. : By Syll., Assoc, 

q \s .D : p\ q/r . D . p\s (1) 

^ . s/s, q, p/p , 

By (1) — ^-^, result. 

-^ s, r, p 

Theorems Equivalent to the Definitions of p Dq, p . q, 
IN Principia. 

p"^ q • 3 . ^pv q, and reciprocal theorem. 
That is, p I q/q . D .p/p \ q/q. 

Bern. : Taut., and Syll. 

sis D 

Reciprocal theorem by Add. -^ — — , and Syll. 

.9, p 

p\ q ."D . ^p V ~ q, and reciprocal theorem. 
That is, j9 1 5 . D . p/p \ q/q. 



40 Mr Nicod, A Reduction in the nmnher 

Devi. : Taut. SylL; then, Perm., Taut., and SylL, or S'. 

Reciprocal theorem by Add, ^-^ instead of Taut. 

^ . g . D . ~ {"^p V '^q) and reciprocal theorem. 
That is, p .q .D . p/q \ p/q. 

Dem. : Id,, Def, of ~, preceding theorem, and Syll. 
Reciprocal theorem in the same manner. 



Appendix, 

After the substance of this paper had been written, I was 
given the opportunity of seeing Mr Van Horn's very interesting 
and original paper dealing with what is practically the same 
subject, Mr Van Horn recognises clearly the superiority of what 
has been called above the Oi^-form over the j4iVD-form chosen 
in Sheffer's text. This deserves the more notice, as Mr Van 
Horn, I understand, had not Sheffer's article at hand in the time 
he was writing his own paper. His A, as will be seen from the 
definitions he gives, is indistinguishable from |. I was much 
attracted by the harmonious character of Mr Van Horn's third 
Axiom. It seems to me therefore all the more desirable that 
certain objections, which Mr Van Horn's proofs in their present 
form naturally suggest to the reader, should be dealt M'ith, 

(a) It is not quite plain to me whether " of the same truth- 
value " (say S for short), " of opposite truth-values " (say 0), are 
used as indefinables, or as abbreviations. If the former, we have 
no right to go, e.g., from p q, and '^p, to q, etc., without some 
axiom to that effect, connecting and S with A, If, on the 
other hand, S and are abbreviations — as it seems to me they 
are — the two parts of Axiom 3 stand for not less than four 
propositions : 

1, If jj and q, '^{pAq). 

2. If (^p and ~(/, pAq. 

3. If p and ^q, pAq. 

4, If ^p and q, pAq. 

We cannot assert the first two, or the last two, or all four, 
propositions together, because we should then need p . q . D . p, 
p . q . D . q, before we could make any use of such a synthetic 
Axiom, 



of the Primitive P7'opositions of Logic 41 

This uncertainty as to the status of S and is not without its 
effect upoii the proofs. Consider, for instance, Th. 3. In the proof, 
"1°: p true. By Axiom 3, pAp false" will be seen to require p Sp, 
concerning the origin of which, and the relation it has to p D j^ 
(Th. 4), which it indirectly serves to prove, Mr Van Horn says 
nothing. 

(/3) In his extensive use of the Principle of Excluded Middle, 
Mr Van Horn makes no explicit mention of the last steps, that 
lead from pOq, ^^ pDq, to q. These steps would seem to require 
several propositions: (1) those carrying us from ^^pyp to qvq 
— " Summation," plus " Permutation," presumably — and (2) " Tau- 
tology " qv q .D . q. As Mr Van Horn uses the principle of 
Excluded Middle in this particular way in the first formal proof 
given — that of Th. 3 — both the principle itself and the proposi- 
tions required for its use ought, I think, to be deduced immediately 
from Axiom 3 ; and I do not see how this is possible. 



42 Mr Watson, Bessel functions 



Bessel functions of equal order and argument. By G. N. 
Watson, M.A., Trinity College. 

[Received 1 November 1916: read 13 November 1916.] 



A proof of the approximate formula 



Jn{n)' 



TT 2» 3« w« 



(the order and argument of the Bessel function being equal and 
large) was apparently first published by Graf and Gubler*, 
although the formula had been stated by Cauchyf many years 
before. The formula has been discussed more recently by 
Nicholson J and by Lord Rayleigh§, while Debye|| has given a 
complete asymptotic expansion of Jn{n) in descending powers 
of 71 ; this expansion is obtained by the aid of the elaborate and 
powerful machinery which is provided by the mode of contour 
integration known as the "Methode der Sattelpunkteir"(Methode 
du Col, method of steepest descents). 

The earlier writers, just mentioned, employed Bessel's formula 



1 /■'" 
Jn (*') = — I COS (nO — X sin 6) dd, 
ttJo 



valid when n is an integer, and it is by no mea.ns obvious to what 
extent their methods of approximating are valid**. 

As the correctness of the approximation can be established 
without the use of contour integration on the one hand and 
without appealing to physical arguments ff on the other hand, 
it seems to be worth while to write out a formal and rigorous 
proof (based on comparatively elementary reasoning) that, when 
n is large and real, then 

* Einleitung in die Theorie der Bessehcheii Funktionen, i. (1898), pp. 96 — 107. 

+ Comptes Rendus, xxxviii. (1854), p. 993; Oeuvres (1), xii. p. 163. 

J Phil. Mag., August 1908, pp. 273—279. 

§ Phil. Blag., December 1910, pp. 1001—1004. 

II Mathematische Annalen, lxvii. (1909), pp. 535 — 538. 

1[ This method of discussing Je"/W<^(s)(fi; consists in choosing a contour on 
which If{s) is constant, and so Bf{s) falls away from its maximum as rapidly as 
possible (/(s) being monogenic); it is to be traced to a posthumous paper by 
Eiemann, Werke, 1876, p. 405. 

** See § 4 below. 

ft For example Kelvin's "Principle of stationary phase" {Phil. Mag., March 
1887, pp. 252—255 ; Math. Paiyers, iv. pp. 303—306) is really based on the theory 
of interference. See also Stokes, Camh. Phil. Trans, ix. (1850), p. 175, foot-note 
(Math. Papers, ii. p. 341). 



of equal 07'der and argument 



43 






Jnin)=^ 






7r2*3-^ 



2. In order not to restrict ourselves to the case in which n 
is a positive integer, we take the Bessel-Schlafli integral*, namely 



sin IITT 



J II (^) = - cos {nd — X sin 6) dO — 



,-nd -X sinh ^ 



dO, 



(which is valid whether n be an integer or not), and, after writing 
n for x, we integrate by parts. This process gives 



Jn (it) = ^ 



d 



nir j 1 "" cos 

sin mr 



-^ -Tj. {sin n (0 — sin 6)] dd 



+ 



d 



IT J 1 + cosh 9 dd 



|g-«(^ + siuh^), ,^ 



nir 



sinw(^ — sin^) 



1 — cos 6 
1 



+ 



sm nir 



-n{0-^sm}\d)-\ ^ 



1 + cosh^ 



+ 



'^smn(^-sm^)^^^^^^^ 



/, 



mr] Q (1 — cos 6)" 

sinnTT p sinh^ ^-n(d + s.\nhe) ^n 

"^ TT Jo (T+COsh^)^^ ;''^' 

The integrated parts cancel ; and 

^^^^^ -n[6 + ^mh0)^0 ^ f"(l+C0sh6^)«-»(^ + «i"l^^)(/^ 
(1 + cosh^)- Jo 

= lln; 

and so, when ?« is large and real, 

r y . 1 f '' sin ^ sin w<f> , , „ , ^, 

?i7r.lo (1 —cose')* ^ 

where </> has been written in place of 6 — sin 6. It is obvious that 
<l> inci-eases steadily from to tt as ^ increases from to tt. 

When 6 is small, (f)r^^6^ and sin ^ . (1 — cos 0)~^ ^ 80-'^. Hence, 
as ^ -> 0, • 

<^^ sin 8 



(1 — cos 0y Qi ' 
Now write | (60)* sin ^ . (1 - cos 0)~' =/ (</>) ; 



Schlatii, 3Iath. Ann. m. (1871), p. 14«. 



44 Ml' Watson, Bessel functions 

then it is fairly evident* that when ^0 ^ir (i.e. when ^ ^ tt), 
fi{<f>) is bounded and has only a finite number of maxima and 
minima (and therefore it has limited total fluctuation). Con- 



sequently, since f -^jr ■• sim/rc/i/r is convergent, we have;]: 
. 
v 77 r ^ 

Lim n~i ^ (f)"'^ sin (n(f>) . f\ {(f)) d(fi =/i (0) | yjr'" sin yjr d-yjr. 

Therefore, since /i(0)= 1, we have 

n-ijy~fsm(ncf)).f,{cf>)d<f> = ^T(V> + o(l), 

and so Jn(n) = 2~ ^ S~ '"^ ir-^ T (^) n~ ■- + o (n~ ^). 

To obtain the second approximation to Jnin), we obser^ 
that, when 6 is small, 

(6«^)^sin^ (i-i&^+^e^-...){i-^ ^e-^+ ^L^e^- ...f 

Consequently, if </) " ^' { (1-^0^6'^ ~ efj ~ "■^' ^'^^' 

we have /o (0) = 6 ~ ^ ^ 35, Also, as in the case of/i (</>), we assume§ 
for the moment that /2(^) has limited total fluctuation in the 
range (0, tt). The application of Bromwich's theorem is therefore 
permissible, and we deduce that 

Lim ni ["(^ - -^ sin (?i</)) ./, {j>)d<l> = 3^ 2 " ^- T (|)/35, 

M-s-oo J 

* A formal proof will be given in §5a that /j (^) is, in fact, monotonic and 
decreasing (we use the term decreasing to mean non-increasing). 

t Euler's result that / ip''^^~'^ sin \p cl^p — V [m] sin {\mir) , when -\<m<l, is 



well known. 

J Bromwich,I?i/ini<e Series, p. 444, proves that, if f{<p) has limited total Jiuetua- 

f ^ sin Hd) 
tion in the range (0, h), where 6>0, and if U,^— I — - — f((p)d(p, then 



H-^ao «-*oo J W J f 

but his analysis is equally applicable to the more general integral 
V^=n'"' i (p'^-'^ sin {7i4>) . f (^) d(p (-l<m<l), 



and hence 

Lim F„=Lim I i/^^-i sin i// ./(i///?i)(7i/'=/(0) I f-^ sin xj^ df. 
rt^-x M-*-Qo Jo J i) 

% A formal proof will be given in § 5 b that/^ (<p) is monotonic and increasing. 



that is 



of equal order and argument 
["(/)-« sin (7i(/)) . /; (<^) ^0 = 3* 2 - ^ ?i - * r (|)/35 + o {n " '), 



45 



and so 



^ 8 ( f'^ sini/r r'-^ sin^/r ] 



(6f) 

-r 

??7r j 



</> ~ I/2 (</)) sin (7?</)) . f/<^ + (/i-2) 






secon 
r such 

sin-^lr 



Now, by the second niean-vahie theorem, there exists a number a 
exceeding nir such that 



liTT "^^ 



dyjr 



1 

(nirf 



sin^p• d/\jr I < 2(mr) •', 



and so we have at once that, when n is large and real, 
^» (n) = ^T—^ ^ + (n - -^ ), 

which is the result to be established. T(3 obtain a closer approxi- 
mation by these methods would necessitate some very tedious 
integrations by parts. 

3. We next consider the approximate formula for Jn (n). It 
is immediately deduced from the Bessel-Schlafli integral that 



j:/(7i)=- sin (9 . sin 71 (^ - sin (9) . c?^ 



TT Jo 

Now we get, on integrating by parts, 

rsinhde-^^^ + ^'^'^'^^dd 

Jo 



~ nJo 1 



sinh 6 d 



^-n(d + sinhd)^^0 



1 

nj 



+ cosh 6 ' dd 



^r e-''Ud = 0{n-% 
2njQ 



46 Mr Watsoji, Bessel functions 

Hence /„' (n) = - I , -p^ sin ii6 cl(b + («~-), 

TT / 1 — cos ^ ^ 

where ^, as previously, stands for 6 — sin 6. 

Now, if fs ((}>) = ^^ sin ^ . (1 - cos e)-\ then/: (0) = 2^ 3 ' ^ and 
/^{(fi) has limited total fluctuation* in the range (0, tt). 
Hence, applying Bromwich's theorem we have 

o f'^ sin n(6 . , ,, , , , ,., f'" sin ilr - , ,,, 

J (p'' Jo T^-- 

and so J",/ (n) = „ ^^^ + o (?i " *) + (n-'), ^ 

TTIV- 

when n is large and real ; and this is equivalent to the result 
stated in § 1. The approximation could be carried one stage 
further (as in § 2), but it seems hardly necessary to give the 
analysis. 

4. As an example of the necessity for the caution which has 
to be taken in approximating to integrals with rapidly oscillating 
integrands, it may be remarked that some of the earlier writers 
mentioned in § 1 assumed that when x and n are large and nearly 
equal [in fact, when \x — n\ = o (n^)], then Airy's integral 

An («?) = -[ cos [nd - cc {6 - ^6')} cW 

is an approximation to Bessel's integral for J^ (^). This assumption 
is correct, and it happens that the first tivo terms in the asym- 
ptotic expansions of An (;») and Jn {oc) are the same. 

But Airy's integral for An {not) is not an approximation! to 
Jn (no) when a is fixed and < a < 1, while n — > x . 

To establish this statement we use Carlini's formula:}: 

Jn (na) ~ -_ 

{1 + ^/(l - a2)}» . (1 _ a')i V(27rw) 

(valid when < a < 1), and after observing that we ma^- write 

An (yia) = - I — ) / cos IW (mw + lu^)] dw, 
7T\naJ J Q '■^ ^ n ' 

* A formal proof will be given in § 5 c that /I, [cp) is monotonic and decreasing. 

t For example, the arguments given in the P/u7. Mag., August 1908, p. 274, 
to justify the approximation seem to me to be as applicable to the second case as 
to the first. 

X A translation of Carlini's memoir (published at Milan, 1817) was given bv 
Jaeobi, Astr. Nach. xxx. (1850); Qes. IVerl^e, vii. pp. 189—245. See p. 240 for 
the formula quoted. 



of equal order and argument 47 



2?t(l-a) /37r\^ ( nOL\\: ^ 



where m — ( — J , w 

IT \naj 

X 

we use Stokes' asymptotic formula* 

cos IItt (mw + 10^)] dw ~ 2 " ^ (3m) " * exp | - ir (|w)^}, 

.'0 " 

valid for large values of m. 
This process gives 

exp { - in 2^ a ~ ^ (1 - g)^ | 

{2a(l-a)}V(27rn) 



^" ("«)'- r^ ., Mi ./ 



Hence ./nOi«) ^ /JgLVe^xM 

where 

X («)= V(l - «•-■) + log « - log {1 + V(l - «"^)l +i«"'(2 - 2a)i 
Since % (i) = -02047, 

a rough approximation to .7io„o (500)/J.iooo (500) is (f )* e-"'*l 

5. We now prove the monotonic properties (valid for $ ^ ^ tt) 
stated in §§ 2, 3 : 

(A) To prove that /^ (</>) = i (6<^)* sin ^ . (1 -cos ^)-- is a 
decreasing function, we have 

d ... .,.,.., _ (3 + 2cos^)<^^^,(^) ' 

re ^^^' ^^^/^ ^- - — (i-cos^)3 — ' 

where g, (6) = [5 sin 0(1- cos ^)/(9 + G cos 9)]- 6 + sin 6, 

so that 

(y/ ((9) = - 6 (1 - cos ey/{9 + 6 cos 6)' ^ 0, and ^r (0) = 0. 

We now see that gi{0)^0, and so f/{<ji)^0, which is the 
result stated. 

(B) To prove that 

/,{<!>) =^-^ [(8/6*) - {(^^ sin 6/(1 - cos Of]] 

is an increasing function, we first prove two subsidiary theorems, 
namely : 

B (i). If c = cos 0, s^ sin 0, then the function 
g, (0) = (85 + 163c + 84c- + 18c^) (/> - is (1 - c) (149 + 157c + 44c0 
is not positive. 

* Math. Papers, ii. p. 343. The result may also easily be derived from 
NicholBon's expression of Airy's integral in terms of Bessel functions of order ± J, 
Phil. Mag., July 1909, pp. 6—17, 



48 ilf?' Watson, Bessel functions etc. 

B (ii). The function 

g,{d) = 2s (7 + 3c) (/)-^ - 3 (3 + 2c) (1 _ c)^ <^ + f s (1 - c? 
is not positive. 

To prove B (i) we observe that 

^ [g, (6)1(85 + 163c + 84c-^ + ISc^} 

= - s- (1 - cy (644 -I- 416c + 60c-)/(85 + 163c + 84c- + 18c--)- 

The denominator may be written in the form 
I873 + 3O72 + 497 - 12 
where 7 = 1 + c, and so the denominator changes sign once on 



i 



when < ^ < TT, say a,t 6 = ^. Hence 

g, (^)/(85 + 163c + 84c^ + 18c0 

decreases from to — czd and then from + 00 to tt as ^ increases 
from to yS and then from /3 to tt. .Hence gi(6) cannot be 
positive. 

To prove B (ii) we observe that 

^ [15^5 (0)1 [s (7 + 3c)|] = (1 - c) g, (0)/{2 (1 + c) (7 + dcf} ^ 0, 

by B (i) ; and so g^ (6) ^ g^ (0) = 0, as was to be proved. 
To prove the main theorem, we have 

where g (6) = {(3 + 2c) f ^ - s (1 - c) (f)^ (1 - c)-\ 

Now g' (6) = - ^a (0) f " (1 - c)- ^ 0, 

so that g(0)^g (0) = 64/6i 

and so // (^) ^ 0, as was to be proved. 

(C) To prove that f(<j>) = (f>i sin 6 . (1 -cos (9)-i is a de- 
creasing function, we have 

^^ = i<^-ni-cos^)-^3(^), 
where g^ (0) = sin ^ (1 - cos e)-S(d- sin 6). 

Since g/ ((9) = - 2 (1 - cos ey we may use the arguments of (A) 
to prove the truth of theorem (C). 



Mr Watson, The limits of applicability etc. 49 



The limits of applicability of the Principle of Stationary 
Phase. By G. N. Watson, M.A., Trinity College. 

[Received 22 November 1916.] 

1. The method of approximating to the value of the integral 

». = ^r- cos [m {oD — tf(m)}] dm, 

Ztt .' 

where x and t are large, by considering the contribution to the 
integral of the range of values of m in the immediate vicinity 
of the stationary values of m {.r — tf(m)], is due to Kelvin*, though 
the germ of the idea may be traced in a paper published nearly 
forty years earlier by Stokes f. 

Kelvin's result is that, if m{x — tf(m)] has a minimum when 
m — [x> 0, then, as ^ — > oo , 

u ~ i^-nt) - * {- ,xf" (/.) - 2/-' (fx)] - * cos [t,x\f' (/x) + iTr} ; 

and this result has imjDortant applications in connexion with 
various problems of mathematical physics J. 

Kelvin, in his analysis of this interesting asymptotic formula, 
takes for granted, on physical groimds, the validity of a certain 
passage to the limit. This process requires justification from the 
purely mathematical point of view ; and the necessary justification 
is afforded by a convergence theorem due to Bromwich§. This 
theorem plays the same part in dealing with integrals as an 
analogous theorem, due to Tannery |j, plays in connexion with 
series. 

The special form of Bromwich's theorem, which is required in 
the rigorous investigation of Kelvin's theorem, may be enunciated 
as follows : 

If f{x) be a function of x with limited total fluctuation in the 
range x ^ 0, and if 7 be a function of n such that ny —^ 00 as 
n —^ 00 , then, if — 1 <m<l, 

* Phil. Mag., March 1887, pp. 252—255 {Math, and Physical Papers, iv. 
pp. 303—306). 

t Camh. Phil. Trans, ix. (1851), p. 175 (Math, and Physical Papers, 11. p. 341). 

t See Macdonald, Phil. Trans. 210 a. (1910), pp. 134—145. 

§ Bromwich, Theory of Infinite Series, p. 444. In the special case 7H = 0, which 
is explicitly considered by Bromwich, the result is important in the investigation 
of Fourier series by the method of Dirichlet. The theorem given by Bromwich on 
p. 443 is equally applicable to the more general case. 

II Fonctions d'une variable, p. 183. 

VOL. XIX. PT. I, 4 



50 3Ir Watson, The limits of applicahility 



i 



-co 

x'"^-^f{x) sin nxdx ->/(+ 0) t"'-' sin tdt 

.' 

=/(+ O)r(w) sin I m-TT. 



[// 0<m< 1, the sines may be replaced throughout by cosines ; 
and, if ny—^ a as n-^ oo , where a is finite, the infinity in the upper 
limit of the integral must be replaced by o-.] 

As the formal analytical proof of a theorem* slightly more 
general than Kelvin's theorem is quite simple, and as sufficient 
general restrictions to be satisfied by the function' /(?n) are 
apparent in the course of the investigation, it seems to be worth 
while to place the theorem on record. It is applicable to all 
kinds of stationary points, whereas Kelvin considered only cases 
of true maxima or minima of the simplest type. 

2. The main theorem which will be proved in this paper is 
as follows f: 

Let a, /3 be any numbers {infinity not excluded), possibly depending 
on the variable n, such that the real function bt — tf(t) has only one 
stationary value in the range a^t^ ^, at t = /m, b being independent 
of n. Let the first r differential coefficients with regard to t o/i| 
bt — tf(t), be continuousX in a range of values of t of which t = fxis^^ 
an interior point, it being supposed that the last of them is the lowest 
which does not vanish at t= /j,, so that r ^ 2. 

Let F (t) be a real function, continuous when a<t < /3, except 
possibly at t = fi, and let 

Urn F{t).{t- ixY = A, Lim F (t) . {tju - 1)"^ := A„ 
where A, Ay^ are not zero ; for brevity, let (1 — A,)/?- = m.. 
Then, if the function 

F(t).\bt-tf{t)-,ji'^f{,,)Y-^-.\b-tf'{t)-f{t)\-^ 
has limited total fiuctuation^ in the range a^i^/9, and if 

I nb^ - n^f(,8) - n/ii'f (/.) | , | nba - noif (a) - ntif (^) j 
both tend to infinity luith n, the approximate value of the integral 

/ s ^ [ F{t) cos [bnt - ntf(t)} dt, 

* For the connexion between this theorem and a problem, due to Riemann 
(Werke, p. 260), which has been discussed by Fej^r {Comptes Rendus, November 
30, 1908, and a memoir published by the Academy of Budapest in 1909) and by 
Hardy (Quarterly Journal, xliv. 1913, pp. 1—40 and 242 — 263), see §4 below. 

t It is convenient to modify Kelvin's notation. 

X It is necessary iox f{t) to have a continuous first differential coefficient when 

§ If the fluctuation depends on n, it must be a bounded function of n as 7i-».x . 



of the Principle of Stationary Phase 51 

wlien n is large, is 
r- 1) !(r !)'"-! r(m)[^ cos {nfx,^ f'(fi)+ ^em-n-] + A, cos {nfj:'f'{fM) + 1 7?m7r}] 

provided that < 1 — X < r ; wAere e = ± 1 according as bt — tf{t) 

increasinq , . , , ^ ,. , 

%s an , . f unction when t > a, ana ?? = + 1 accoratnq as the 

decreasing n • - 

same function is . . when t < a. When n—^cc hti only 

increasing 

such values that cos [nfM-f {/u.)} is always zero, A, may lie in the 

extended range —r<l — X< r. And, finally, F{t) and bt — tf(t) 

inay be infinite at t= a, ^, provided only that the integral converges 

for all sufficiently large values of n. 

3. For brevity, write tf (t) ^ (f) (t). Then ^ is given by the 
equation 

b-<f>'(,M) = 0, 

so that, when t — fi, is sufficiently small, 

bt - tf(t) - t(}>' (/jl) -(f){t) 

= {/.f (/.) - </) (/.)} -(t- fxyr' {t')/r !, 

where, by Taylor's theorem, t' lies between fx, and t. 
Now define a new variable yjr by the equation 

bt-tf{t) = fi<l>'{ix)-c},ifi,) + f, 

and let 7, F be the values of yfr corresponding to t = a, t— /3. 
Noticing that /x^' (/j,) — (f> (fi) = fi^f (/n). we have 

J. cos {n/M'f (fi)} r^„,,, , u 

sin{n/jb^f' (a)} T^et/^x • , 7^ 
Zir J a 

i/r being a monotonic function of t when a ^t ^ /x and also when 
fi^t^^. 

Now e, 7; have been so chosen that e-yjr and rj^jr are positive 
when t > /M and ^ < //- respectively ; hence, when ^ —>//, + 0, we have 

df 

ef '^ {t- fiY </)<'■' {fi) 1 ^ r !. 



52 Mr Watson^ The limits of applicahility 

It follows that 



^^^^^A'^^'"'""'"'"'-^^^' 



as i — > /Lt + 0, where 



^^ -(r-l)!(r!)^ 



</><'•)(;.) 11 (/,<'•) (/.)!}--' 



Since j wF ] — > oo with h, by hypothesis, we deduce from Bromwich's 
theorem that 



cos 



r p(-)o (If reoo 

^<*> sin «+ St '^ ~ "" '"' .1 . <^'>"'"' -n ^'^- 
Writing %e s &>, we get 

re CO rcc 

I (%^)™~^ COS %<^% = e I &>'"~^ cos ctx^o) = eF (??i) cos -| 

and similarly 

(%e)'"~^ sin %c^% = F (m) sin ^7«7r. 



7?2-7r, 



In like manner, when t-^ /jl- 0, 
dt 



and so, since 1 717 | — > oo with n, we have 

/, ^ (*) sm "^ 4 ^'^ ~ (-)•■ -'" ^■^i „ (^'''•""' sin ^<'^- 

Collecting our results, we see that the first approximation to 
/ is 

/ ~ [{AKe + (— )'■ AiKr]} cos ^iutt cos {nfM-f (fM)] 
- [AK + (-)'• A^K] sin |m7r sin {^i/^y (/i)}] 
= A [cos (w^-/' (/u.) + ^emir] + J.i cos {ufx^f (fi) + ■|?;??i7r}] 

(r - 1) ! (r !)'»-! F (m) 
^ 27rw'^{|<^<'-'(;u)j}'" ' 

and this is the result stated. 

The formula fails to be effective in the neighbourhood of those 
values of n for which the expression in [ ] vanishes, as the error 
in the approximation then becomes comparable with the approxi- 
mation obtained. 

[It is evident that if the cosine in the integral defining / may 
be replaced by a sine, then the cosines in the approximation are 
replaced by sines.] 



of the Principle of Stationary Phase 53 

Cases of practical importance are those in which A = Ai and 
t = fj> is a, true minimum or maximum of bt — tf{t), so that e and r) 
are both + 1 or both — 1. The formula then is 

A cos {n/M'f'ifM) ± ^mTr} . (r - 1) ! (r !)'»-i T(m) 
~ 7r?i"* { I (^e-) (fi) I 1*'^ • 

If nV or ny tend to finite limits, the gamma functions have to 
be replaced by incomplete gamma functions ; and if one or other 
tends to zero, we modify the approximation by writing zero for 
A or A^ respectively in the general formula. 

The general result reduces to Kelvin's formula when r = 2, 
X = 0, 711= h, and e = ?; = 1, provided that (with Kelvin's notation) 
x/t is constant. In that case, a sufficient condition for the validity 
of the formula is that 

^ [{{ma^/t) - mf{m) - i^-f (;.)}*]-^ 

should have limited total fluctuation when m ^ 0. 

If X were a function of t, Bromwich's general theorem {loc. cit., 
p. 443) would have to be used, and the enunciation of sufficient 
conditions (even in their simplest form) for the validity of the 
formula, would be exceedingly laborious. The reason for this is 
that (with the notation employed in this paper) -^ and F (t) dt/dyfr 
would both be functions of n. 

4. The problem of Riemann (see § 1 above) essentially consists 
in obtaining an approximation for integrals of the type 



/■"" / ,x cos sin nt. 



when n is large and a-' (t)—>oo as t—> 0. 

These integrals are expressible by integrals of the type 



t-'pit)^"^^ {nt+ (7(t)]dt, 
Jo sm ! ' 



so that the problem is, at first sight, very similar to that discussed 
in % 2—3. 

There is however an essential difference, namely that, in the 
problem we have discussed, ntf{t) owes its large rate of increase 
(which balances the rate of increase of nbt at the stationary point) 
to the large factor n, whereas, in the problem attacked by Fejer 
and Hardy, the function a (t) owes its large rate of increase to the 
infinity of a (t) at ^ = 0. In our problem fj, is fixed, whereas in 
the other problem the stationary point of nt — (T{t) tends to zero 
as ?i — * 00 . It seems to be this difterence which accounts for the 
somewhat elaborate investigation given by Hardy and which 



54 Mr Watson, The limits of applicahility 

makes the theorems of Fejer and Hardy rather deeper than the 
theorem of §§ 2 — 3. 

It should be pointed out that there is one integral which can 
be regarded as coming under either head, namely*, 

/•oo 

/ X sm . 

I X " (nx + ax~n dx, 

j cos ^ ^ ' 

where n is large, a, \, and r are positive and X and r are chosen so 
that the integral converges. [For the sine-integi-al, the conditions 
for convergence are < X < r + 1.] As the integral stands it is 
of the type discussed by Fejdr and Hardy, with a variable 
stationary point where x''+^ = arjn. But if we make the sub- 
stitution 

and then write v for n''''<»'+i', it becomes 



j,(A-i)/r[ t-^^^^'^lvit + at-'y^dt, 

Jo cos ^ ^ ^•' 



which is of the type discussed in this paper, having a fixed 
stationary point where t = {ray/^''+^K The reader will have no 
difficulty in deducing the approximate formula by either method. 

5. As an example of the apparent inapplicability of the 
methods of this paper consider the integral of Bessel for Jn(x) 

when n and x are both large and a; — ?i, is (n^). 
The integral is 

1 f'^ 
Jn (x) =- I COS (n6 — X sin 0) dd, 

and the stationary point is given by cos = nlx; let the root 

of this equation be = /j,, and let x = n + an^ where a > ; when 
n is large we have 

In considering | cos (n0 - x sin 0) d0, we write 

X = n0 — xsin0 — (n/j, — x sin /i,), 
and the last integral is expressible by integrals of the type 
»(tan^-M) cos d0 , 
sin ^ dx 



f 

Jo 



* I am indebted to Mr Hardy for suggesting that the integral in which a- (t) = llt 
can be reduced to an integral of Kelvin's type. 



of the Principle of Stationary Phase 55 

Now tq—'^~ ^ co^ 6 o^x{6 — ix) sin ix, 

when 6 f^ fM and -^r^^oo^iw fx .{6 — ^y. 
Hence, as ;)^; — > 0, 

Lde -1 

and 



/, 



sin '^ c^x 

\ /-n'tauM-M) (• . 1 rf^") _1C0S 



/•n'tanM-M)( . i rft'l _1C0S , 

Now, as ?i -> CO , ?i (tan /x — ^) -> i (2a)^ and so the limiting 
range of integration is of finite length, 

1 d9 
Moreover, \/{2x sin fi) . ^^ -^ > — l as ?i — > go ^vhen % is ^■ero, 

1 df) 
that is, when d ^ fi. But, when 6 -> 0, the limit of \/{2x sin A*) • %- ^ 

is 

— {2 sin fi (sin /jl — /xcos At)|"^/(1 — cos /a), 

a^icZ, a.9 n—>cc, the limit of this is not — 1 hit — 2 \/(^) ; and so 
we cannot infer that 

•«(tunM-M)f . if^6l) _icos , f' _icos , 

|V(2.-sm;.).X-^^~j-% -^i^^X^^X-j^^X \sin^^^' 

where b is Lim •?? (tan yu, — /x). 

The evaluation of the approximate formula for Jn{^) in the 
circumstances under consideration consequently seems to require 
more elaborate analysis than is afforded by the methods contained 
in this paper. 



56 Mr Borradaile, On the Functions of the Mouth-Parts etc. 

On the Functions of the Mouth-Parts of the Common Prawn. 
By L. A. Borradaile, M.A., Selwyn College. 

[Read 30 October 1916.] 

The food is seized by either pair of chelipeds, or by the 
third maxillipeds, and is usually placed by them within the 
grasp of the second maxillipeds, though sometimes it is passed 
directly to deeper-lying structures. The second maxillipeds are 
the most important of the food-grasping organs. They have three 
principal movements; in one, the broad flaps in which they end 
open downwards like a pair of doors, and with their stout fringes 
gather up the food ; in another, they rotate in the horizontal plane 
to and from the middle line of the body, and thus narrow or widen 
the gap through which the food passes; in the third, the bent distal 
part of the limb tends to straighten, so as to brush forward any 
object which lies between them. Frequently these movements are 
combined. Owing to the facts that the second maxillij)eds cover 
the mouth-parts anterior to them, and that if they be removed 
feeding is not properly performed and usually not attempted, it is 
difficult to trace the food beyond them, but the following seems to 
be its fate. If it be small in bulk, or finely divided, or very soft, 
it is passed to the maxillules, by whose strong, fringed laciniae it is 
swept forwards, and probably caused to enter through the slit 
between the paragnatha, into the chamber which is guarded by 
the upper and lower lips. If it be tough or in large masses, the 
second maxillipeds and maxillules brush it forwards towards the 
incisor processes of the mandibles. The action of the latter is, by 
rotating in a vertical plane, to tuck the food into the gap between 
the paragnatha and the labrum. If the mass be large, pieces are 
torn off it by this action. Finally, to enter the gullet, the food 
must pass between the molar processes and be pounded by them. 

The mandibular palps, maxillae, and first maxillipeds appear 
to play parts of little importance in regard to the food. The 
palps are present and absent in closely related genera, and appear 
to be disappearing in the higher Carides. The same is true of the 
lobes of the maxillae, which are in constant regular motion to and 
from the middle line, and probably serve to restrain the action of 
the scaphognathite. The large laciniae of the first maxilliped 
may have as their function the covering of the maxillae and 
protecting them from the food. The labrum undergoes active 
movements, whose function is probably to aid in keeping the 
food under the action of the mandibles. The exopodites of the 
maxillipeds set up a strong current forwards from the mouth. 
No doubt this aids in carrying away the exhausted water from 
the gill chamber and the excreta from the tubercles of the green 
glands. Into the same current particles which have been taken 
as food are from time to time rejected by the forward kicking 
of the second maxillipeds. 



CONTENTS. 



PAGE 

A self-recording electrometer for Atmospheric Electricity. By W. A. 
Douglas Ritdgb, M.A., St John's College . . . 

On the expression of a number in the form ax^-\-hy^-^cz--\-dv?. By 
S. Ramanujan, B.A., Trinity College. (Communicated by Mr 
G. H. Hardy) . . . . . . . . . • H j 

An Axiom in Symbolic Logic. By C. E. Van Hokn, M.A. (Com- 
municated by Mr G. H. Hardy) 22 

A Reduction in the number of the Primitive Propositions of Logic. By 
J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. 
Hardy) . 32 " 

Bessel fuiictions of eqiial order and argument. By G. N. Watson, M.A., 

Trinity College . . . 42 ' 

The limits of applicability of the Principle of Stationary Phase. By 

G. N. Watson, M.A., Trinity College . . . . . . 49 

On the Fionctions of the Mouth-Parts of the Common Prawn. By L. A. 

BoRRADAiLE, M.A., Selwyn College . . . . . . .56 



PROCEEDINGS 



OF THE 



CAMBRIDGE PHILOSOPHICAL 
SOCIETY 



VOL. XIX. PARTS IL, III. 

[Lent and Easter Terms 1917.] 




AT THE UNIVERSITY PRESS 
AND SOLD BY 
DEIGHTON, BELL & CO., LIMITED, 
AND BOWES & BOWES, CAMBRIDGE. 

CAMBRIDGE UNIVERSITY PRESS, 
C. F. CLAY, MANAGER, FETTER LANE, LONDON, E.C. 4 

1917 
Frice Two Shillings and Sixpence Net 

October 1917. 



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PROCEEDINGS 

OF THE 



The Direct Solution of the Quadratic and Cubic Binomial 
Congruences with Prime Moduli. By H. C. PocKLiNGTON, M.A., 
St John's College. 

[Received 22 January 1917: read 5 February 1917.] 

1. The solution of congruences by exclusion methods, 
although easy enough when the modulus is moderately large, 
becomes impracticable for large moduli because the labour varies 
as the modulus or its square root. In a direct method the labour 
varies roughly as the cube of the number of digits in the modulus, 
and so remains moderate for large moduli. The object of this 
paper is to develop the direct method. We take or = a, mod. p, 
first, discussing the cases where p = 4<m + 3 and p = Sin + 5 in 
§ 2 and that where p = 8ni + 1 in § 3. We next take ar^ = a and 
discuss the cases where p = Sm + 2, /> = 9m + 4 and p=i9m + 7 
in § 4 and that where p = dm + 1 in § 5. 

2. Throughout the paper we suppose the modulus to be 
p where p is prime*. If p is of the form 4wi + 3 the solution of 
^•- = a is X = ± a'^'^\ If p is of the form 8m + 5 the solution is 
^ = ± a'"+i provided that a-'^+^ = 1. But if not, a'^"*+i = - 1, and 
as 2 is a non-residue 4-'"+' b - 1 ; so that (4a)-»*+^ = 1 and we have 
2/ = + (4a)'"+^ as the .solution of y- = 4o. Hence 

x=±y;2 or x=±{p+ y)/2 

is the solution of x-=a. These values of x can be calculated 
without serious difficulty by repeated squaring (followed by division 
by the modulus to find the remainder) and multiplication of the 
numbers so found (again followed by division). 

* Hence if it is composite we must factorize it and solve the congrnence for 
each of the different prime factors. 



VOL. XIX. PARTS II., III. 



58 Mr Pocklington, The Direct Solution of the Quadratic 

3. Put D=—a, so that we have to solve cc^+ D = where D 
is positive or negative but not divisible by p. Let t^ and Mj be so 
chosen* that t^ — Du-^- = N is a quadratic non-residue of p, and 
let 

tn = [{t, + U, ^Dr + (t, - U, sjDY]l% 

These numbers are clearly integral. Also 

by use of* which (at first with m = n) we can find the remainders 
of tn and Un to our modulus without serious difficulty even when 
n is large. We also have tn — Dun = N'^. 

Supposing that p is of the form 4m + 1, we have D a quadratic 
residue of 'p, and tp = t^P = ^i, Up = u-pD^P~'^''i- = u^ ; and now 

ti = tp^^t-^ + Dup_^Ui^, tij = tp_iUi + t-^Up_^ 

give on solution tp_^ = 1, Up_-^ = 0. Let p-l = 2r. Then 

= l/^_i = 2trUr 

shows that either t,. or u,. is divisible by p. If it is Ur we put 
r = 2s and proceed similarly. We cannot have every u divisible 
by p, for u^ is not. We cannot be stopped by having u,n = with 
m odd, for we always have 4,,^ - Dum'' = N''\ and this would then 
give ^,„' congruent to a non-residue. But if m is even we can 
.proceed further. Hence when we are stopped we must have 
t,n = 0. This gives - Bii,,^ = N''\ and as - D is a residue m must 
be even. Putting m = 2n we have = ^^ = tn' + Bun', so that the 
solution of a^ + D = is got by solving the linear congruence 

UnX =z + tn- 

In applying the method, if n is the largest odd number con- 
tained m p-l we first work to get the suffixes n, and then the 
suffixes 2n, 4<n, 8n, etc. Thus in the case of cc'' + 2 = 0, mod. 41, 
we see that ^j = 3, Ut, = 1 is suitable, and we find t. = 11, u., = 6 ; 
t, = 29, u,= 9; t,= 23, u,= 15; t^o = 36, u,o = 34 ;' t.^ = 0. ' The 
solution of 34^ = 36 is a; =30; and so the two solutions of 
*•- + 2 = are .x=± 30, mod. 41. 

^4. If p is of the form Sm + 2 the only solution of x' = a is 
a; = l/a'«. If p is of the form 9m + 4> one solution is cc = !/«'"■ and 
if of the form 9m + 7 one is .x = a^+\ The other solutions are 
got frojn this by multiplying by (- 1 + 6)12 and (- 1 - e)/2, where 
t/- + 3 = 0, a congruence which we have shown how to solve. 

* We have to do this by trial, using the Law of Quadratic Eeciprocity, which 
" •! ?f i'" *^J® method. But as for each vakie of n half tire vahies of / are 
.suitable, there should be no ditlficulty in finding one. 



and Cubic Binomial Congruences with Prime Moduli 59 

5. Let 8 be the arithmetical cube root of a, which we 
assume* not to be a cube. Findf ti, Ui, Vi such that the norm 
N = t{^ + au^^ + a'-Vi" — SatiU^i\ of the algebraic number 

is a cubic non-residue of p. We see that as a is a cubic residue 
of p we have U^' = ti-\- UiS+ i\8'^, so that if 

U^'-^ = tp-i + Up-i 8 + %_i 8- 

we have iip^i = Vp_i = 0. Now taking ?7'" where m is in turn 
(p— l)/3, (p — l)/9, etc. we see that we cannot always have 
u,n = v„i = 0. Let f/""^ be the last of this series for which this 
happens. Then m is divisible by 3, for otherwise the norm of [7"\ 
which reduces to t^, would be congruent to the non-residue iY'". 
Putting m = on we have 

tsn = tn + <Kln + a-^n + QatnUnVn, 
= Usn = 3 (tn'Un + ^^tnVn' + «",rWn), 
= V.,n = 3 {taUn' + tuVn + aUnVn'')- 

The last two give tn{av^i"' — Vn)= 0; so that if tn is not divisible 
by p we have w = Un/v,,. as one solution of n^ = a, for as Un and v,i 
are not both divisible by p this shows that neither is. They also 
give Vn (ait^n — tn) = 0, and so w = tnjihi is a solution. Eliminating 
a from the same two congruences we see that the ratio A, of the 
two xs, satisfies V + X + 1 = 0, so that they are distinct. The 
third solution follows immediately. 

If however tn is divisible by p the two congruences show that 
either Un or Vn is divisible by p. We now have rtM,i,^=iA'" or 
a-Vn = -Y". In either case n must be divisible by 3 as before, and 
we have as one solution x = N^'jun or x = aVnjN'' respectively, 
where r = n/S. 

* Simply because of the way in wliich for the sake of shortness we are stating 
the method. 

t This again must be done by trial. In order to use the Law of Cubic 
Beciprocity we must express p in the form ii' + xiv + v'^, which requires the solution 
of ^2 + 3 = 0. 



5—2 



60 M7' Hardy, On a theorem of Mr G. Polya 

On a theorem of Mr G. Polya. By G. H. Hardy, M.A., 
Trinity College. 

[Received and read 5 February 1917.] 

1. Mr G. P(51ya has recently discovered a number of very 
beautiful theorems concerning Taylor's series with integral co- 
efficients and ' ganzwertige ganze Funktionen '. The latter 
functions are integral functions which assume integral values for 
all integral (or for all positive integral) values of the independent 
variable. One of the most remarkable of these theorems is the 
following*: • 

Suppose thatg{x) is an integral function, and M{r) the maodmum 
of I g {x) I for \x\^r. Suppose further that 

g(0),g{l),g{2),... 

are integers, and that 

lim 2-''^JrM{l') = (1). 



?'-»-00 



Then g {x) is a polynomial. 

Mr Polya observes that, if it were possible to get rid of the 
factor \/r from the equation (1), the theorem could be enunciated 
in a notably more pregnant form, viz. : 

Among all transcendental integral functions, which assume 
integral values for all positive integral values of the variable, that 
of least increase^ is the function 2^. 

Mr Polya states, however, that he has not been able to effect 
this generalisation. And my object in writing this note is to 
show that the generalisation desired may be obtained by a slight 
modification of Mr P(51ya's own argument, and without the 
addition of any essentially new idea to those which he employs. 

2. Mr Polyaij: reduces the proof of the theorem to a proof 
that the integral 

T( \— ^' r g(x)dx 

^'^^~2^ij x(x-l)(x-2)...{x-n)' 

extended over the circle \x\ = r - 2n, tends to zero when ?i — » oo , 

*■ G. Polya, ' Uber ganzwertige ganze Funktionen ', Rendiconti del Circolo 
Matematico di Palermo, vol. 40, 1915, pp. 1—16. See also 'Uber Potenzreihen 
mit ganzzahligen Koeffizienten ', Mathemathche Annalen, vol. 77, 1916, pp. 497— 
513, where Mr Polya refers to a third memoir (' Arithmetische Eigenschaften der 
Eeihenentwicklungen rationaler Funktionen', Journal flir Mathematik) which. 1 
have not been able to consult, 

t Croissance, Wachstum. 

X Loc. cit., p. 7. 






Mr Hardy, On a theorem of Mr 0. Polya 



61 



This he proves by observing that the modulus of J^^ does not 

exceed 

n\ M{r) ^ V{n + \)V{n) 

(r-l)(r-2)...(r-w) r(2n) ^ '' 

and by an application of Stirling's Theorem. In order to com- 
plete the proof in this manner it is necessary to assume the 
condition (1). 

If however we suppose only that 

lim2-'-ilf(r) = (2), 

or J'f(r) = o(2'-) (2'), 

the proof may be completed as follows. We have 

where ^= 2ne'^. Now 

\x — s\ = V(4?r — 4?2.s cos 6 + 8^)'^ 2n — s cos 
for 1 <s <.n, so that 



de 

.){a)-2)...{a;-n)\\ ' 



U{x- s) 

1 

if cos ^ > 0, and 



> n (2n - s cos 6) = (cos ^f 11 (2« sec O-s) 



n (x - s) 
1 



^ n (2n - s cos ^) = I cos ^ i" n (27* | sec ^ | + s) 
1 1 



if cos ^ < 0. Hence 

where Kn = nl2^''j 



Jn = {Kn) + (X„), 

''' T( 2na-n) 
r {2n(r) 



a'^dO, 



i„ = „,2»|'V„_£(2^iV,.«rf«, 



and a = sec 0. 

A straightforward application of Stirling's Theorem shows that 

uniformly in 6, where 

^ = <t> (0) = {2a - 1) log (2o- - 1) - 2a- log 2o- + log o" + 2 log 2, 
^ = ^ (^) = 2(T log 2o- - (2o- + 1 ) log (2a + 1) + log o" + 2 log 2. 



62 



Mr Hardy, On a theorem of Mr 0. Polya 



= 2 log 2cr - 2 log (2o- + 1) + ^ = ^ - 2 log (l + ^M > 0. 



3. When 6 increases from towards -|-7r, or decreases towards 
— hir, (T increases from 1 towards oo . Also 

^^ = 21og(2c.-l)-21og2^ + ^=2 1og(l-^^)+-^<0, 

da- 

Thus <I> steadily decreases and ^ steadily increases. Moreover 

^ (0) = 0, ^ (0) = 4 log 2 - 3 log 3 ; 

and it is easily verified that both <J» and ^ tend to the limit 

log 2 - 1 
when 6 tends to ^tt. 

We thus obtain, in the first place, 

i,.=of-'.<--vnj:;y(,^)</«}=o(i). 

Secondly, we observe that, if S is any positive number, we have 

^{d)<^{h) = -7^<0 

for h^e^^ir, -^ir^eK-h. 

Hence we may replace the limits in Kn by - 8 and h, the re- 
mainder of the integral being of the form 

4. All that remains, then, is to prove that 
/„ = ,U2»f n^^)w(* = 0(l); 

j_5 1 (2/lcr) 

The function ^{6) may now be expanded in powers of ^. We 
find without difficulty that 

where A = log 2 - ^ > 0. 

It follows that 



and we have 



h = 0bn\ ^e-^»^HO(»^*)^^ 
= 0U/n r e--^^''^'de\^0{l). 



Mr Hardy, On a theorem of Mr G. Polya 63 

The proof of the theorem conjectured by Mr Polya is thus 
completed. 

5. Mr Polya has also proved an analogous theorem concerning 
integral functions which assume integral values for all integral 
values of x, viz.: 

If ...,g{-2), g{-l), g{0\ g{\), g{2)... 

are integers, and 

lim (^^rVrilf(r)=0 (3), 

then g {a;) is a polynomial. 

His proof applies, as it stands, to odd functions only, its appli- 
cation to a completely general function demanding the more 
stringent condition 



lim 



r-*-co 



^4r^l 'rm(r) = (3'). 



He states that it is possible to replace the index | by ^ in all 
cases, but that, as he has not been able to reduce the condition to 

lim (^^^y*'ilf(r) = (3"), 

he has not thought it worth while to publish the details of his 
work. 

A modification of Mr P<51ya's argument, in every way similar 
to that which I have made in the proof of his first theorem, 
enables us to replace (3) by (3") when g{x) is odd. The same 
modification in his unpublished argument would, I presume, be 
equally effective in general. 

That the number 

3 + ^/5 



cannot be replaced by any larger number, and so really is the 
number which ought to occur in any theorem of this character, 
is shown by Mr Polya by the example of the function 

which assumes integral values for all integral values of .r. 



64 Dr Marr, Submergence and cjlacial climates (luring the 



Submergence and glacial climates daring the accumulation of 
the Cambridgeshire Pleistocene Dejwsits. By J. E. Marr, Sc.D., 
F.R.S, St John's College. 

[Read 5 February 1917.] 

A. Introductory. 

The sequence of events during palaeolithic times is still a 
subject surrounded by much uncertainty. The area of the Great 
Ouse Basin is one in which considerable light has already been 
thrown on vexed questions, and as the examination of the area is 
carried out in greater detail, important results will be obtained, 
for in this area we get evidence of the relationship of the palaeo- 
lithic deposits to those which were formed during a period of 
submergence and re-emergence, and also to accumulations which 
give evidence of the occurrence of more than one cold period. 

The general distribution of the palaeolithic deposits of the 
district around Cambridge, and their main characters, have long 
been known, and an account of the deposits, with references to 
the previous literature, is given in the Geological Survey Memoir 
The Geology of the Neighbourhood of Gambridqe, published in 
1881. 

Since that memoir appeared, further light has been thrown 
on the deposits, especially by Professor Hughes, who has given his 
latest views in a paper entitled Tlie Gravels of East Anglia 
(Cambridge University Press, 1916). 

I have devoted much attention to this subject during the last 
six years and hope to describe my detailed results elsewhere. 
The present paper is concerned with a discussion of the main 
problems involved, in hopes that it may direct the attention of 
workers to the importance of further observations, for the deposits 
with which we are concerned are only exposed temporarily during 
the_ working of gravel-pits and the digging of foundations and 
drains, and it is desirable that all temporary excavations should 
be carefully studied, and the objects obtained rendered available 
for study by deposit in Museums, for isolated specimens in private 
collections are usually mere objects of curiosity devoid of scientific 
value. 

B. Submergence and its effects. The actual sequence of deposits. 

In the fenland and on its borders we meet with marine deposits 
above sea-level, which have long been known around March and 
Narborough. They occur above and below fen-level at March 



accumulation of the Gamhridyeshire Pleistocene Deposits G5 

and undoubted marine deposits containing sea-shells are found 
to a height of at least 50 feet above sea-level in the Nar Valley, 
and deposits up to 80 feet above sea-level have been claimed as 
marine. Unfortunately no exposure of these Nar Valley beds 
has been seen for a very long time, and their exact upward limit 
is a matter which must remain unsettled until new excavations 
are made. It is held, with good reason, that the beds of March 
and the Nar Valley are geologically contemporaneous in the sense 
that they belong to the same period of sea-invasion, which was 
subsequent to the accumulation of the chalky Boulder Clay ; and 
as there is good evidence that much of the fenland was low-lying 




Nar Level 



Fig. 1. 

AB. Slope of ground before marine gravels were deposited. 
CD. ,, „ after ,, ,, ,. 

a. Tract of marine gravels. 

b. ,, interdigitating marine and fluviatile gravels. 

c. ,, fluviatile deltaic deposits. 

d. ,, erosion in valley towards its head, during period of deposit of 

a, b, c. 
1, 2, 3. Order of formation of deposits in tracts c and d respectively. (1 is oldest.) 
Vertical scale greatly exaggerated. 

ground after this boulder-clay was formed, it would appear prob- 
able that the March gravels are earlier than those of the Nar 
Valley, and therefore that a gradual silting up of a bay of the 
sea took place, until the sediments reached a height of at least 
50 feet above present sea-level. 

During this period of silting the rivers Ouse, Cam and others 
would build delta-deposits along the lower parts of their courses, 
with interdigitation of marine and fluviatile deposits in an inter- 
mediate belt of ground as shewn in figure 1. In this delta- 
material, the chronological sequence of deposit would be from 
below upward, as shewn by 1, 2 and 3 in the belt c. The upper 
waters of the rivers would still be eroding, and the sequence 
would be from above downwards (see figs, in tract d). 

After submergence had ceased, it would be replaced by re- 
emergence, as shewn by the erosion of the rivers to their present 



66 Dr Marr, Submergence and glacial climates during the 

levels, and new deposits 4, 5, . . . (not shewn in the diagram) would 
be banked against or laid down upon those formed during the 
period of subsidence and general accumulation in tracts c and a. 
It will be seen therefore that relative height of deposits above 
the present river- level is not in itself a necessary indication of 



The geological surveyors gave the following classification of 
the Cam gravels: 

r Lowest Terrace 
Gravels of the Present River System I Intermediate Terrace 

[Highest Terrace 
Gravels of the Ancient River System. 

I shall treat of three of these, leaving out of account the gravels 
of the Intermediate Terrace, which I have not studied extensively 
owing to poor and infrequent exposure of recent years. I shall 
speak of the gravels of the ' Ancient River System ' as the Obser- 
vatory gravels, those of the highest terrace of the ' present river 
system ' as the Barnwell village gravels, and those of the lowest 
terrace as the Barnwell Station gravels. The ages of these 
deposits will ultimately be accurately determined by an exami- 
nation of the fossil evidence, including implements of human 
manufacture. So far, the evidence of this kind points to the 
Barnwell village deposits being of two ages, the older formed 
during the period of delta-growth, the newer during the period 
of re-emergence and erosion. At the end of the period of delta- 
growth, and therefore of an age intermediate between those of 
the supposed two Barnwell village deposits, I would place the 
Observatory gravel, and certain loams, to be referred to later, 
and after all of these, the Barnwell Station gravel marking the 
culmination of the period of re-erosion, for there is evidence of a 
later period of sinking and deposit after this was formed. This 
succession is represented in Fig. 2, which shews a section across 
the Cam valley at Cambridge, before the edges of the valley sides 
had been destroyed leaving the Observatory gravels as a ridge 
with lower ground on either side. 

In the figure the terms Upper, Middle and Lower Palaeolithic 
indicate the ages of the various gravels as inferred by me from 
the palaeontological evidence. I am using the term Middle Palaeo- 
lithic in the sense in which it was used by Prof. Sollas in the 
first edition of Ancient Hunters as equivalent to Mousterian. 
I believe therefore that the older Barnwell village gravel is pre- 
Mousterian, that of the Observatory (in part at any rate) Mou- 
sterian, and the newer Barnwell village gravel and that of Barnwell 
Station post-Mousterian, the former being of earlier date than 
the latter. 

Mr Jukes-Browne, in an essay on the Post Tertiary Deposits of 



■I 



accumulation of the Camhrldgeslnre Pleistocene Deposits 67 

Cambridgeshire, advocated a change in the direction of the rivers 
near Cambridge between the formation of the Observatory gravels, 
and those which he regarded as belonging to the 'present river 
system.' That such a change occurred is admitted, but the evi- 
dence points to all the deposits save those of the Barnwell Station 
terrace having been formed before the river diversion occurred. 

I may now pass on to consider briefly the palaeontological 
evidence in favour of the order of age indicated above, leaving 
details for a future paper. 

In the pits of Barnwell village, and of the Milton Road near 
Chesterton, loams are sometimes exposed at the base of the over- 
lying gravels. These loams contain Corhiculaflaviinalis, and with 
it are associated Unio Wtoralis, Belgrandia margiiiata, and Hip- 
popotamus, On the continent this is recognised as an early 




Fig. 2. 

Section acroiss Cam N. of Cambridge, with higher valley-slopes restored. 
The figures shew the suggested order of formation of the deposits. Cross- 
hatching represents modern alluvium of Cam. 

5. Barnwell Station gravels (Upper Palaeolithic 2). 

4. Newer Barnwell village gravels (Upper Palaeolithic 1). 

3. Loams of Huntingdon Koad area. 

2. Observatory gravels (Middle Palaeolithic). 

1. Older Barnwell village gravel and loam (Lower Palaeolithic). 

Z= Buried channel. 

Vertical scale greatly exaggerated. 

palaeolithic fauna of Chellean or pre-Chellean date, and there 
seems to be no evidence of the reappearance of this fauna at a 
later date. 

In the Geological Magazine for 1878 (p. 400) Mr A. F. Griffith 
described the occurrence of a palaeolithic implement from one of 
the Barnwell pits. A cast of this is in the Sedgwick Museum, 
and it appears to be of Chellean type. 

Further afield, the occurrence of similar implements at or near 
fen-level in Swaffham and Soham fens, and at West Row near 
Mildenhall, and at Shrub Hill near Feltwell, indicates that rivers 
had excavated their channels to fen-level in those times. 

There are patches of gravel between the higher Chesterton 
terrace which corresponds to the Barnwell village terrace and the 



68 Dr Marr, Suhmerc/ence and glacial climates dnrmg the 

Observatory level, but no sections are now seen in them, so we 
may pass on to the Observatory deposits. In these shells and 
mammalian bones are very rare, though the former have been 
found in concretions, indicating that they once lay in the gravels, 
but have since been dissolved. Implements are relatively abun- 
dant, and I have found a large number during recent years. 
Many of them are of Chellean type, others probably Acheulean, 
but there are a large number of Mousterian type, some having 
the facetted platform which, as shewn by M. Commont, came into 
use in Northern France in Mousterian times. It may be noted 
that the implements of Mousterian type are patinated differently 
to and in a less degree than those of Chellean type, and I regard 
the two series as of distinct ages. Either the deposits, which are 
thick and varied in character, are of two dates, or implements 
of different ages lying upon the surface were washed into the 
deposits contemporaneously. This can only be settled by finding 
a number in situ, a work of great difficulty, but the evidence is 
in favour of the latter view. 

I may note that when a valley is being deepened implements 
of one age only are likely to lie in abundance near the spot where 
the gravels were accumulating, but when there is general aggra- 
dation, the highest deposits of the delta-growth are likely to 
receive washings of implements of various ages which have been 
lying together, at or near the surface. In any case the age of the 
newest gravel of a terrace will be determined by the implements 
of latest date. 

Lying on this gravel in channels are reddish sandy loams, 
which must have spread over the gravel, but have since been 
destroyed by erosion except where so preserved. There is also 
a deposit of somewhat similar loam but of a lighter colour flanking 
the gravel at a lower level on either side. It is rarely exposed, 
and only in shallow sections, but I believe it may be of the same 
general date as that lying on the gravel. 

No relics have been found in it, though two implements of 
possible Upper Palaeolithic date were found on the loam when ' 
draining the Christ's Cricket Grouod, but they may well have ■ 
been surface finds. Many other surface finds, some of apparent ' 
palaeolithic type, are found on this loam belt, and will be referred i 
to later. | 

Those gravels of the terraces of Barnwell village age, which | 
I would refer to a date later than that of the Gorhicula gravels, I 
are now exposed in a pit near the Milton Road and in another | 
on the Newmarket Road near Elfleda House, 2\ miles from j 
Cambridge. These contain a fauna differing from the Gorhicida \ 
fauna, and including the mammoth, woolly rhinoceros, horse and 
red deer, the horse being abundant. 



accumulation of the Cambridgeshire Pleistocene Deposits 69 

Implements are scarce, but in both pits I have found some 
suggestion of upper palaeolithic forms, and in each pit a water- 
worn pot-boiler has been discovered. 

In the Barnwell Station pit the common mammal is the rein- 
deer, associated with the mammoth, tichorhine rhinoceros and 
horse. In the Geological Magazine for 1916 (p. 339), Miss E. W, 
Gardner and I recorded the occurrence of an arctic flora in this 
deposit, with abundance of leaves of Betula nana. A long pre- 
liminary list of the other plants which indicate arctic conditions 
was made by the late Mr Clement Reid, F.R.S., but has not yet 
been published. A few worked flints of undeterminable date 
have been found, but the fauna indicates the late palaeolithic 
period, and the late date of these deposits seems to be shewn by 
the fact that whereas all the others are apparently connected 
with the old drainage line extending from Cambridge to Somers- 
ham, these are almost certainly parallel 'to the present course of 
the Cam : they appear indeed to be the upper portion of the 
deposits filling an old buried channel of the Cam, evidence for the 
occurrence of which is borne out by certain observations made by 
Prof Hughes in the paper to which reference has been given. 



C. Climatic Changes. 

There is much difference of opinion as regards the occurrence 
of alternating glacial and interglacial periods in Pleistocene times, 
and it would seem that some light is thrown upon this question 
by the Cambridgeshire deposits and those of adjoining counties. 

I take the prevalent view that the implement-bearing deposits 
from the beginning of Chellean times post-date the period of the 
Chalky Boulder Clay, though others hold a different view, but as 
the local evidence bearing upon this question has already been 
recorded I need not enlarge upon this point. 

If the succession as outlined above be correct the following 
climatic changes seem to have occurred after the cold period 
marked by the accumulation of the Boulder Clay : 

(a) A warm period during the formation of the Corbicula- 
bearing strata. Arguments in favour of this are well known. 

(6) A cold period during the accumulation of the Observatory 
gravels(?) and the newer loams. No evidence of this has been 
advanced in this area, and a few remarks are necessary. 

The fauna of the Observatory gravels tells us nothing, and 
the loams have hitherto furnished no organic remains, but a 
widespread development of loam marks the Mousterian period, 
and N.W. Europe is believed to have been subjected to a cold 
climate during part of the period. 



70 Dr Marr, Submergence and glacial climates during the 

The sections recently seen near Cambridge tell us little, but 
a brickpit in stratified loam with 'race' nodules similar to those 
found in the Cambridge sections has long been worked near the 
railway between Longstanton and Swavesey. It contains boulders, 
and is actually mapped as boulder-clay. A somewhat similar 
loam with boulders at High Lodge near Mildenhall has long been 
known for its implements of Mousterian type. These deposits 
are at an elevation just below that of the highest palaeolithic 
gravels, as are those of Cambridge. 

Further afield there is the very significant section at Hoxne, 
described in detail in a paper drawn up by the late Clement Reid, 
F.R.S.. and published in the Report of the British Association for 
1896. ' 

At that locality we have a stratigraphical sequence. Above 
the boulder-clay lies an aquatic deposit marked by a temperate 
fauna. It is succeeded by loams with an arctic flora, and above 
that are loams with palaeolithic implements. They have been 
usually regarded as Acheulean, but there is one specimen in the 
Sedgwick Museum which is of a distinct Mousterian type. Taking 
these facts into consideration, a period of cold climate in this 
country in Mousterian times seems probable. In any case, the 
evidence points to a difference of date of the arctic plant-beds of 
Hoxne and Barnwell Station. 

(c) The fauna of the beds of the Barnwell village terrace 
claimed here as of newer date than those containing CoriicM^ajj 
suggests an amelioration of the climate, but in the absence of a '' 
well preserved flora, this is doubtful. 

(d) The Barnwell Station flora, as before observed, is distinctly 
arctic, and when this flora lived here, we can hardly suppose that 
our higher hills escaped glaciation. The same remark may be 
made of the Hoxne flora. 

This series of changes would accord with the classification of 
the beds on the continent thus : 

European Continent Cambridgeshire 

Pleistocene 

Wiirm glaciattoxi Barnwell Station beds. 

Waiiii period Newer Barnwell village deposits. 

Riss glaciation Observatory gravels and loams. 

Warm period Corhicula gravels. 

Mindel glaciation Chalky Boulder Clay. 

Pliocene 
Warm period Cromer 'Forest' series. 

Giinz glaciation Chillesford beds. 

I merely put this forward tentatively, claiming however that 
we have in Cambridge proofs of two if not three Pleistocene cold 
periods. 



accumulation of the Cambridgeshire Pleistocene Deposits 71 

D. Surface Implements. 

Implements of all ages from earlier palaeolithic to recent 
times are found lying together on the surftice. Some no doubt 
have got there from the erosion of deposits which contained them, 
others belong to the surfece. My object is to insist on their 
careful collection, with exact records of their localities, even to 
the particular position in a field where thej^ lay. 

If they can be shewn to be limited to heights above those of 
a particular deposit, they may yield valuable information as to 
geological changes. 

Two areas in which surface implements are abundant are 
found very near Cambridge, one on the tract between Castle End 
and Girton on either side of the Huntingdon Road, on the ground 
occupied by the Observatory gravels and loams, the other a little 
south of Fen Ditton, between the railway and the river, and at 
no great height above the latter. They have not been yet 
sufficiently studied to enable one to draw definite conclusions, but 
the former group does not seem to occur below the level of the 
Barnwell village terrace, which suggests that the river may have 
eroded its valley below that level to its present position since 
those implements were made. The other set marks the position 
of a site on a terrace, which is I believe the terrace of the 
Barnwell Station deposits, and would indicate the formation of 
that terrace before this set of implements was manufactured. 

As the above is merely a preliminary account of these deposits, 
I have not burdened it with references, nor have I acknowledged 
the many friends who have helped in the collection of implements 
and other objects. 

The bulk of the implements on which my conclusions are 
based were collected by myself, and the rest by friends chiefly 
under my supervision, and in no case has any implement been 
purchased from workmen, so that the collection, which will be 
deposited in the Sedgwick Museum, is of value, inasmuch as each 
implement is known to have been obtained from the locality 
assigned to it. 



72 Mr Weatherburn, On the Hydrodynamics of Relativity 



On the Hydrodynamics of Relativity. By C. E. Weather- 
burn, M.A. (Camb.), D.Sc. (Sydney), Ormond College, Parkville, 
Melbourne. 

{Received 15 December 1916 : read 5 February 1917.] 

I. The Equations of Motion. 

I 1. Relativistic equations for the adiabatic motion of a 
frictionless fluid have been found by Lamia* and Lauef in the 
form 

dt^ ■' dx^ ■^ dy^ dz^ ' y dx 

^ 7\ 7) f) ^ 7) P 

— (kv) + 11;^ (kv) + V ;^- (kv) + iv X- (kv) + - ^ = Fy...(l); 

where m, y, w are the components of velocity at the point {x, y, z) 
relative to a definite system of reference 8 ; X, Y, Z those of the 
impressed force per unit of normal rest-mass ; and 

ry= ^ (2), 

Vc^ — {u^ •\-v^ + id^) 

c being the constant velocity of light. The significance of the 
symbols P and k is as follows. 

Since the motion is adiabatic the rest-mass of an element of 
fluid is determined by one variable only, say the pressure p. 

If we choose some definite pressure p^ as the normal or 
standard pressure, the element has a definite constant normal 
rest-mass hm^. If the element occupies a volume hV relative to 
the system of reference 8, the density k relative to that system is ^| 
defined by 

, _8mo 

* Ann. der Physik, Vol. 37, p. 772 (1912). 

•|- Das Relativitdtsprinzip, § 36 (2nd ed. 1913). For a more general discussion of 
the mechanics of deformable bodies from the standpoint of Relativity, cf. Herglotz, 
Ann. der Physik, Vol. 36, p. 493 (1911); also a paper by Igndtowsky, FIn/s. Zeit., 
Vol. 12, p. 441 (1911). 



I 



Mr Weathei'burii, On the Hydrodynamics of Relativity 73 

Using a dash t(j refer in every case to the rest-systeui tS", we 
have for the rest-density 

37»o _ hm^ _ k 
~BV'.~y8V y ^"^• 

The function F is defined by the integral 

i'=rt (4), 

•' Pa f^ 

and in terms of this function k is given by 

-^=^(1 + ^) <'''■ 

For the rest-system *S" the quantity y has the value unity, 
while K becomes 

k'=1+-^ (5'). 

The constancy of normal i-est-mass leads, as in the classical 
theory, to an equation of continuity 

¥ + fl.<'-«> + a^<''"> + 3i*^^''> = ° <">■ 

§ 2. Using F and v for the force and velocity vectors, we 
may write the equations of motion more conveniently 

|(«v) + v.V(/cv)-[--VP=F (7). 

ot y 

Then because the gradient of the scalar product of two vectors is 
given by 

V (a • b) = b • Va + a • Vb -f- b x curl a + a x curl b, 
the second term of (7) is equivalent to 

- V (k-v") - V X curl (kv), 
Ik 

while, in virtue of (5'), VP = c'-'^/c'. Hence the equation may be 
exjjressed in the form 

r) 1 

^^ («v) -I- „ V {k-y- -I- cV'-) - V X curl («v) = F. 
ot 'Ik 

But again the second term is equal to 
1 



VOL. XIX. PARTS II., m. 6 



2.^ 



74 ifr Weatherburn, On the Hydrodynamics of Relativity 
and the equation of motion takes the ver}^ convenient form 

^(a:v) + c-V/c + 2w X v= F (8), 

where we have written 

2w = curl (kv). 

In cases where the impressed force F admits a potential, so 
that F = — V F, our equation reduces to 

^(«v) + V(t;-/c+F) + 2wxv = (8'). 

§ 3. Glehsch's transformation^. The equation of motion may 
be expressed in terms of functions analogous to those of Clebsch 
if we write 

KV = V(f) + A.V/X (9), 

(ji, \, fi being three independent functions of x, ?/, z and t. Taking 
the curl of both members we find immediately that 

2w = VXx V;^ ^0). 

The function w = i curl (kv) plays the same part in the present 
analysis as |- curl v in classical hydrodynamics. It will therefore, by 
analogy, be called the vorticity ; and a line whose direction at any 
point IS the direction of w at that point, a vertex line. Since 
by (10) w is perpendicular to both VX and V/x it is clear that the 
vortex lines are the intersections of the surfaces 

A, = const., /J, = const. 

Using then dots to denote partial differentiation with respect 
to t, and assuming the existence of a force potential, we may write 
(8') as -^ 

- V ( F+ c-a:) = V(j) + X/i) + XV/i - ^Vx 

+ (v . VX) VyLi - (V . V/x) VX 

which may be neatly expressed in the form 

§v,-|vx + Vi.= o (u), 

where the function H is given by the equation 

If= (f) + \jiL + V+C"K (12). 

_ * ^f; Basset, Treatise on Hydrodytiamics, Vol. 1, p. 28 ; also Silberstcin, Vectorial 
Meciianics, p. 146. 



Mr Weatherburn, On the Hydro(hjnamics of Relativity 75 

On scalar multiplication of (11) by w, it follows in virtue 
of (10) that 

w.V// = 0, 

showing- that H is constant along a vortex line. It can also be 
shown that H is independent of x, y, z and is therefore a function 
of i only. For taking the curl of (11) we deduce 

On scalar multiplication by Vx it follows, by (10), that 
and similarly that 



w.Vl'^fl^O, 



From these we deduce as in the old theory* that 

^ = ^ = ...(13). 

dt dt ^ 

Thus the first two terms disappeai- from (11), which becomes 
simply VH = 0, showing that H is constant in space and is 
therefore a function of t only ; or 

(j) + \/l + V + c^fc = H (t) (14). 

From (13) it is clear that the surfaces \ = const, and /x = const., 
and therefore also the vortex lines which are their lines of inter- 
section, are always composed of the same particles of fluid. 

§ 4. Steady motion. When the motion is steady partial 
derivatives with respect to t are zero. If then the impressed 
force is derivable from a potential V, (8') becomes 

2v X w = V ( K + c-k) (15), 

and the equation of continuity 

div(/.v) = (16). 

If we multiply (15) scalarly by v the first member vanishes, 
showing that 

vV( l^ + c-/^) = 0. 

Thus the function V + c-k is constant along a line of flow. 
Similarly scalar multiplication of (15) by w gives 

w.V(F+c-^/c) = 0, 
* Cf. Biis-et, lor. cit. p. 29. 



76 Mr Weatherhurn, On the Hijdrodynamics of Relativity 

and therefore V + C'k is constant also along a vortex line. This 
is a particular case of the more general theorem, proved in the 
preceding section, that H is constant along a vortex line. Thus 
the surface 

V + c'-V = const, 
is composed of a double system of vortex lines and lines of flow. 

II. Irrotational Motion. 

§ 5. When the vorticity i curl («v) is zero the motion will be 
termed irrotational or non-vortical, being analogous to the motion 
of that name in the older theory. In this case «v can be expressed 
as the gradient of a scalar function 0,, which may be called the 
velocity potential : i.e. 

'cv = V(f} : (17). 

The lines of flow are orthogonal to the surfaces of equal velocity 
potential. 

The equation of motion can always be integrated when a force 
and a velocity potential exist. For (8') then becomes 
V (<^ + c'k +V) = 0. 

The function in brackets is therefore constant throughout the 
liquid, and will be a function of t only; i.e. 

4> + c'K+V=f(t) (18). 

This is the required integral of the equation of motion. An 
arbitrary function of t may, however, be incorporated in the 
velocity potential cf), and this equation then written without loss 
of generality 

(f) + c-K+ F=0 (18'y 

When the irrotational motion is steady (c'^k + V) is constant 
throughout the liquid, and is also invariable in time. In the 
preceding section, where w was not assumed to be zero this 
function was only proved constant along vortex lines and lines of 
flow. 

The equation of continuity (6), or as it may be written 

dk , -. 

^ + ^-divv = 0, 

may be expressed in terms of 0, if we write kvJk for v, and expand 
the divergence of the quotient. The equation then becomes 

|logA- + v(l).Vc/, + lv^0=.O (19). 



Mr Weatherhurn, On the Hydrodynamics of Relativity 77 

This form is not so short as in the ordinary theory, nor can we 
obtain Laplace's equation, as there, by assuming the Huid incom- 
pressible, for such an assumption is inconsistent with the theory 
of relativity*. 

§ 6. Steadily rotating fluid. Suppose that the Huid is in 
a state of steady rotation about the ^^-axis, and that the angular 
velocity of rotation O is a function of the distance r from that 
axis. We shall now determine what must be the form of this 
function in order that a velocity potential may existf- If i, J: k 
are unit vectors in the directions of the coordinate axes 

V = rn. 
For irrotational motion this velocity must satisfy the equation 

curl {kv) = 0, 

that is IkH + r V ('<:^) = 0, 

dr 

the integral of which is 

/tfir- = const. = fjb, 

say, so that /cH — ^ (A). 

The velocity potential <^ is then given by 

dd) . 1 d<h it 

dr r do r 

showing that (f) = fid + const (B), 

which is an example of a cyclic velocity potential. The integral 
of the equation of motion is by (18') 

c"k + V=0 (C). 

But K involves v" and therefore O, which is itself expressed in 
terms of k by (A). This equation however gives 

K' K-C^ 

whence 12- = -„ , „ „ ,„ , 

r-{/jb' + r''c-K-) 

* Cf. § 10 below. It will be shown, howeyer, in § 11 that V-</) = is the 
equation of continuity for the steady irrotational motion of a Huid of minimum 
compressibility. 

t Cf. Lamb, Hydrodynamics, § 28 (1st ed.). 



78 Mr Weatherhurn, On the Hydrodjjnarnics of Relatiinty 

K being given by (5'). On substitution of this vabio in (C) the 
integral of the equation of motion, viz. 

becomes V + ~ \l uC^ + r-c-K.'- = (D). 

§ 7. FloLV and circulation. We define the flow from a point P 
to another Q, along a path of which ds denotes an element, as the 
quantity 

[Q 

kV • ds. 
J p 

Whenever a velocity potential exists this is equal to <^y — (j)^. The 
circulation round a closed curve is the line integral 



/= /cv.f/s (20) 



taken round that closed curve. This, by Stokes' theorem, is equal 
to the surface integral 



7 = 1" curl («v).nrf;S' (20') 



taken over any surface drawn in the region and bounded by the 
closed curve. When the motion is irrotational the integrand is 
zero, and the circulation round the closed curve vanishes. It 
follows that, for a simply-connected region, the velocity potential 
is single- valued. 



III. Vortex Motion. 

I 8. When the vorticity w is not zero the motion will be 
called vortical or vortex motion. A vortex tube is one bounded bj 
vortex lines. Considering the portion of a vortex tube betAveei 
any two cross sections, we find as usual on equating the volume 
and surface integrals 

= div curl (/cv) dr = I 2w • nd,S, 

that the moment of the vortex tube 1 vj'XidS, Avhere the inte 

gration is extended over the cross section, is the same for all 
sections. And hence, as in the classical theory, the vortex lines 
either form closed lines, or else end in the surface of the fluid. 



Mr Weatherbarn, On the Hydrodijnaniics of Relativitij 79 



I shall now show that, on the assumption of a force potential, 
Kelvin's theorem* of the constancy of the circulation in a closed 
filament moving with the fluid is true in the present case also. 
Consider a closed filament consisting always of the same particles, 
and let ds be a vector element of its length and ds the correspond- 
ing scalar. Then the circulation round it is 



/ = kv •ds. 



The time rate of change of this is 

dl 

dt ' 



|(.v).r/s+.v.(;jj 



f/s 



VF--VP + /cv.(f/s.V)v 

7 



--F+/CV. _- 

OS OS 



ds 



ds 



"lv" 



.(21). 



Now the last integral is 



ds 



c- c 

, V/c — 



(c"- 



^. die K 8y^ , , 

''hs'-^.Ts^^'- 



T 27 Vc2 - V- 

On substitution of this value in (21) that equation reduces to 



dr 

dt 



9«: 



9^ 9 / n' 



96' ds ds 



ds. 



Hence, since the path of integration is closed and k, V, and kv^ 
are single-valued functions, the integral vanishes, showing that 



f- 



.(22). 



Thus the circulation does not alter with the tiipe. 

Corollary. If / is zero at any instant it will remain zero. In 
particular, if the motion is irrotational at any instant it will remain 
so, provided that the impressed forces have a potential. 

§ 9. Helmholtz's theoremsf. That these theorems are true in 
the present theory also follows without difficulty from the form (8') 
of the equation of motion. For taking the curl of both members 
we have 



dw 

'dt 



+ curl (w X v) = 0. 



* Of. Silberstein, loc. cit. p. 161, for the proof of the ordinary theorem. 
t Ibid. Y>p. 163 — 65. 



80 Mr Weatherburn, On the Hydrodynamics of Relativity 

Expanding the second term and using the equation of con- 
tinuity, we find 

rfw T^ dk _ 

dt k dt ~ ' 

which, after division by k, may be written 

d fVT\ w ^ 

s(l)=I-^^ ■; (23). 

Differentiation with respect to t gives 

cZ- /w\ fd Mv\ -. w ;'d _ 

If then w vanishes at any instant it follows from (23) that the 
first derivative of w/Z; also vanishes, and from the next equation 
likewise the second derivative at that instant. Similarly all the 
derivatives with respect to t vanish at that instant, and the 
quantity -wjk remains permanently zero, so that the motion con- 
tinues irrotationa.l. 

Further, the moment of a vortex filament does not vary with the 
time. For if ds is an element of such a filament moving with the 
fluid 

ds = wds/w, 

and -J- (ds) = ds* Vv = — w • Vv, 

at w 

so that (23) is equivalent to 

d /wX w d , -, ^ 

<sUJ = arf(('^'> (24). 

Now if jj, is the moment of the filament, dm^ the constant 
normal rest-mass of the element considered, and a the cross- 
sectional area 

fx = aw, dniQ = kads, 
,1 , w ds 
^°"'''* k = ^d^. (25)- 

Substituting this value in (24), and remembering that dm^ is 
constant, we have 

|(/.^s) = ;x|(rfs), 

and therefore -^ = 0, 

dt 

showing that the moment of the filament remains constant. 



3h^ Weatherhurn, On the Hydrodynamics of Relativity 81 

It has been proved already that a mrtex filament consists 
alvmys of the same particles of fluid, though this can also be now 
deduced from (24) and (25), using the invariability of yu.. 

IV. Fluid of Minimum Compressibility*. 

§ 10. According to the theory of Relativity no velocity can 
exceed that of light. Hence there is no such thing as an incom- 
pressible fluid ; for such a fluid would admit a wave propagation 
with infinite velocity. A fluid of minimum compressibility is one 
in which a wave can attain a velocity equal to that of light ; and 
for such a fluid the quantity k is directly proportional to the 
densityf 

K = k/kJ, K^k'jk; (27), 

where A:,,' is a constant representing the normal rest-density, i.e. the 
rest-density corresponding to the normal pressure p^. 

For a fluid of minimum compressibility the equations of motion, 
energy and continuity may by (27) be expressed in terms of the 
velocity v and the rest-density k'. The equation of motion, viz. 

becomes on substitution 

, dv dk c'-„,, J ,„ 
^' -ZtT + V ,^ + - V^' = k,'F. 
dt at y 

Dividing by 7 and using the equation of continuity to transform 
the second term, we have at once 

k' (^ - V div v) + (c^ - v-^) Vk' = A-o'F/7 (28), 

which is the equation of motion in the required form. 

Multiplying this equation scalarly by v, and transfortning 
V • V/t', we obtain 

,,/lrfv^ , ,. \ ,, ,, fdk' dk'\ /co'F.v 

.11.' .1 //. a/^J _ ^,2\ 

Now 



dk' 


_d fk Vc- - 


-"') 




dt 


~ dt\ c 






_ dk Vc^ - V-' 


k 


dv^' 




dt c 


2c "Jc^ - v" 


dt 




= — k' div V - 


1 y-k' dv- 

2 c- dt ' 





* Latnla, loc. cit. p. 788 ; Laue, loc. cit. § 37. f Laue, loc. cit. p. 241. 



82 Mr Weatherburn, On the Hydrodijnaribics of Relativity 

in virtue of the equation of continuit}-. On substitution of this 
value in the last equation it becomes simply 

7/1- C" V" U/C Wn __ //-»/,N 

//divv+ ^ ~- = --^F.v (29), 

c- ot C^J 

which is the energy equation in terms of k' and v. These equations 
(28) and (29) are identical with those found otherwise by Lamia* 
and Lauef. The equation of continuity is as before 

^" + ^divv-0 (30), 

which takes the required form if k is replaced by jk'. 

§ 11. Steady irrotational motion. In virtue of (27) the 
equation of continuity may also be written 

— + div(/cv) = (81), 

Ob 

and therefore when the motion is irrotational 

| + ^^^ = (31'). 

If it is also steady the iirst term is zero, and we have (as in the 
older theory for the case of an incompressible fluid) 

^^(/) = (31"). 

Thus for steady irrotational motion of a fluid of niinimimi com- 
pressibility the velocity jjotential satisfies Laplace s equation. 

It follows immediately that for such a fluid, filling a simply- 
connected region within a hollow shell, which is fixed relative to 
some system of reference S, steady irrotational motion relative to 
that system is impossible. For by Green's theorem 

.K'V'dT = I {^(py^dr = — (f)/<:v • ndS — I (f)V-(f)dT. 

Now the last integral vanishes by the equation of continuity. 
So also does the last but one : for v • n is zero, being the normal 
velocity at the surface of the fluid. Hence 



/ 



fc-v'-dr — 0, 
showing that v must vanish identically throughout the fluid. 

* Loc. cit. p. 792. 
• t Loc. cit. p. 244. 



Mr WeatJierbuni, On the Hydrodij navvies of Relativity 83 

In the present case* the integral of the equation of motion 
found in § 5, viz. 

takes the form 

c^/, + /,;'F=0, 

or, in terms of the rest-densit}^ //, 

cH-' + Vk\;\/c--v' = 0. 

§12. l^^teady motion in two dimensions. Supposing the fluid 
of minimum compressibility, let its steady motion be parallel to 
one plane — the plane xy. Introduce a function y^ satisfying the 
relations 

dy\ 



dyjr 
KV = ^ 

on; 



•(32), 



u, V being, as in § 1, the components of velocity parallel to the 
axes of a; and y resjjectively. Such a function -v/r exists, the 
equation of continuity 

div («v) = 

being satisfied identically. The function yfr is proportional to the 
flux of matter across a line AP drawn from a fixed point A to the 
variable point P (x, y). For owing to an infinitesimal displacement 
Sx of F the increment in the flux of matter is 

kvSx = ku'icvSx = k,' ~ Sx. 

ox 

Thus if ^ denote the flux 

-,^ o.i: = /in : - dx. 

ox ox 

Similarly Jy ^^ ^ ^'" Jy ^^' 

showing that '^P = kj-yjr, 

as stated. The part played by this function yjr is exactly similar 
to that of the stream function in the two-dimensional motion of a 
liquid in the classical theory. The present function also is a true 
stream function. Its value is independent of the path chosen from 
-4 to P provided the region is simply-connected. For, if ^4PP and 

* Lamia considers only the case of free motion (F= const.) ; loc. cit. p. 71*5. 



84 Mr Weatherbarn, On the Hydrodyncmdcs of Relativity 

ACP are two different paths, the flux across the complete boundary 
ACPBA is 



h-v . ndfi — I div (7bV) dr = 0, 



as is also obvious because the motion is steady. The lines 
■y^r = const, are the actual stream lines : for if P moves subject 
to this condition there is no flux across the path traced out by 
that point. 

The above is true whether the motion is irrotational or vortical. 
The vorticity w is equal to 






,(34). 



and therefore for irrotational motion -^ must satisfy Laplace's 
equation 

V^f = (33). 

If this relation is satisfied there is a velocity potential (f), and (32) 
may then be expressed in the form 

• dcf) _ d-\fr I 

dx dy I 

dy dx ) 

These are identical with the relations subsisting between the 
stream function and the velocity potential in the classical theory 
of the two-dimensional irrotational motion of a liquid. They are 
the conditions that (jy + iyfr should be a function of the complex 
variable x + iy. The theory of such functions may then be used 
as in the theory referred to*, to give various possible forms of 
stream lines and lines of equal velocity potential. 

§ 13. Source, sink and doublet. Similarly the irrotational 
motion of a fluid of minimum compressibility defined by the 
velocity potential 

/-^■l («^>' 

where r is the distance from a fixed point 0, corresponds to the 
assumption of a continual creation of matter at the point 0, 
of amount 4<7rm per unit time. For 



so that ^•v = — . 

r^ 

* Cf. Lamb, loc. cit. chap. iv. 



i 



Mr Weatherbarn, On the Hydrodynamics of Relativity 85 

The velocity is therefore radial from 0, and kv is inversely 
proportional to ?'^. The flow of matter per unit time across the 
surface of a sphere of radius r is ^irm, equal to the rate of creation 
of matter at 0. Such a motion is then that due to a source 
of strength m at the point 0. If the negative sign in (o5) were 
replaced by a positive one, we should have the motion due to 
a sink at of strength ni. And finally the velocity potential 
representing a doublet at of moment M and with its axis along 
the unit vector n is 



*=""-(;) 



86 Mr Hardy, On the convergence 



On the convergence of certain multiple series. By G. H. Hardy, 
M.A., Trinity College. 

[Received 15 May 1917.] 

1. In a paper published in 1903 in the Proceedings of the 
London Mathematical Society*, and bearing the same title as this 
one, I proved a theorem concerning the convergence of multiple 
series, of the type 

which is given (with an improvement in the conditions) on p. 89 
of Dr Bromwich's Theory of infinite series. This theorem is one 
of a class of some importance ; and I propose now to state and 
prove the leading theorems of this class in a form more systematic 
and general than has been given to them before. I shall begin by 
recapitulating, with certain changes of form, some known theorems 
concerning simply infinite series ; and I shall then obtain the 
corresponding theorems for double series in a form as closely 
analogous as possible. The generalisation from double series to 
multiple series of any order may well be left to the reader. • 

Simply infinite series, 

2. I shall say that a function a.,„,, real or complex, of a positive 
integral variable m is of hounded variation if 

■^ i (^m ~ f'-m+l I 
1 

is convergent. It is plain that this condition involves the existence 
of a = lim a^n- 

Theorem 1. The necessai^y and sufficient condition that a^n 
shoidd be of bounded variation is that its real and imaghuiry ptarts 
should be of bounded variation. 

This follows at once from the inequalities 

I ^m ^m+l I ^ j Cini (^m+l \ ; j Pm Pm+l i ^ ! ^^ in ^^9n+l |> 
I (^m C^7n+i I ^ I ^m ^m+i \ "i' [ Pm Pm+i \ > 

where a,,,, = «,„ + i^,„. 

* Ser. 2, vol. 1, pp. 124 — 128. See also 'Note in addition to a former paper on 
conditionally convergent multiple series', ibid., vol, 2, 1904, pp. 190 — 191. 



of certain multiple series 87 

Theorem 2. The necessary and sujjicient cunditiuu that a real 
function a,„ shoidd be of bounded variation is that it should he of 
the form A^n—AJ, where A.^. and A J are 'positive and decrease 
steadily as m increases. 

The sufficiency of the condition follows at once from the 
•inequality 

I Clm ~ ftjn+l ! ^ \A,n, — -4,„ + i) + {Ajn — A ,„,+]). 

In order to prove that it is necessary, let us suppose that a,„ is 
of bounded variation, and let us write 

P„i = ! flm - (I'm+l I (an, " ««<+! > 0), p,„ = (a,,,, - a,„+, < 0), 

Pm'^ ! "m - (Im+l \ (('in " «/»+! < 0), j)„/ = (a,;, - d ,„^.^ > 0), 

B = ^' i) 7? ' = S « ' 

-'J??; — —' /'n> -"m -^ J'n • 

II! VI 

Then B,,, and i?„/ are positive and decrease steadily as m in- 
creases ; and 

B,n - BJ = 2 ((f„ - a„,+i) = ff,,, - a. 

in 

We may therefore take A,„,= B„,, + G and ^j,/= 5,,,' + C, where 
C and C" are suitably chosen constants. 

Theorem 3. // a,,; is of bounded variation, and Su,,,, is con- 
vergent, then 2rt,„M„,, is convei^gent. 

Theorem 1 shews that it is enough to prove this theorem 
when a„, is real. Theorem 2 shews that it is enough to prove it 
when «„, is positive and steadily decreasing. In this form the 
theorem is classical*. 

Lemma a. If 2c,„ is a divergent series of positive terms, we 
can find a, sequence of positive numbers e,„, tending steadily to the 
limit zero, such that 2e,„c,„ is divei'gent. 

Lemma /3. If 2c,„, is a divergent sei'ies of positive terms, we 
can find a sequence of integers m^ such that the series Sc,„' , where 
Cm' = if m = nil and c,,' = c,,, otherwise, is divergent. 

Lemma a is due to Abelf. Lemma (3 is quite trivial, and the 
proof may be left to the reader. 

* See Bromwieb, Infinite Series, p. 48. Theorem 3 is given by Dedekind in bia 
editions of Diiicblet's Vorlesungcn iiber Zahlcntheorie : see e.g. p. 255 of the tbird 
edition. Tbe central idea of all sucb tbeorems is of course Abel's. Tbe line of 
argument followed bere is due substantially to Hadamard, 'Deux theoremes d'Abel 
sur la convergence des series'. Acta Matheinatica, vol. 27, 1903, pp. 177 — 184. 

t 'Sur les series', (Euvves, vol, 2, pp. iy7-~20o. 



88 ilf?' Hardy, On the convergeyice 

Theorem 4. If ^a,f,u,n ■'*' convergent whenever 'lii,,, is cuii- 
vergent, then, a,„, is of hounded variation. 

This theorem is due to Hadamard*. We have to shew that, 
if" X I ^m — f'?n+i j is divergent, v^„ can be so chosen that -n,n is 
convergent and 1a,nUr,n is not. By Lemma a, we can choose a 
sequence of positive and steadily decreasing numbers e„, so that 
e„, — * and Sc,„, where 

is divergent. By Lemma /3, we can then choose the sequence m,- 
so that %c„j' is divergent. We take 

u^ = U,, tL,n = L\n - ?7,„_, (m > 1), 

where Um,. = 0, 

and U,n = em 

if m^mi, the last expression being interpreted as meaning e„, 
if a,„ = a^+i • We have then 

mi mi—1 mi-1 

1 1 1 

which tends to infinity with i. Thus l.amUm is not convergent, 
while ^Um. converges to zero. 

We may call a,,;, a convergence factor if 2o„,m,„ is convergent 
whenever %Um. is so. Theorems 3 and 4 may then be combined 
concisely in 

Theorem 5. The necessary and suffi,cient condition that a^n 
shoidd he a convergence factor is that it should he of hounded 
variation. 

Double series. 

3. The convergence of a double series, in Pringsheim's sense •]-, 
does not necessarily involve the convergence of any of its rows or 
columns |. In this paper I shall confine my attention to con- 
vergent series whose rows and columns are convergent separately : 
in this case I shall say that the series is regularly conve7'gent. 
A regularly convergent double series is also convergent when 
summed by rows or by columns, and its sum by rows or by colunms 
is equal to its sum as a double series §. 

Similarly I shall say that a^.^n tends regidarly to a limit if 

lim a,„,, n = «n , lim « „,,, „ = «,„ , 

* I.e. supra. t Bromwich, Infinite Series, p. 72. 

+ Bromwich, ibid., p. 74. § Bromwich, ibid., p. 75. 






of certain multiple seines 89 

and the double limit 

lim a^n., n= CL, 

all exist. In this case 0^ a.nd «„ tend to a when m and n tend to 
infinity. 

Lemma 7. // SSum^n is regularly convergent, to the sum s, and 

m n 
1 1 

then, given any positive number e, we can find co so that 

I *m,M S I < 6 

if either m or n is greater than co. 

We may suppose s = without loss of generality. Since the 
double limit exists, we can choose w^ so that | 6',„,^„ \< e if ni and n 
are both greater than ^i. When &)i is fixed we can choose o)., and 
«»3 so that the inequality is satisfied for 1 ^m ^(o^, n> Wn and for 
m > &)3, 1 •^ n ^Wj. We can then take to to be the greatest of <wi, 
(On, and 0)3. , . 

Lemma S. In the same circumstances, we can choose w so that 



pq 
m n 



< e 



if p^m, q^ n, and either m or n is greater than w. 
This follows at once from Lemma 7 and the identity 
1^ _ , 

m n 

4. I shall say that a^n^n is of bounded variation in (m, 71) if 

(1) a^n^n is, for every fixed value of m or n, of bounded 
variation in ?i or m, 

(2) the series 

is convergent. And I shall say that ar,i^n is a convergence factor if 
SSa„i_,j,Mm_,i is regularly convergent whenever SSw,„,,„ is regularly 
convergent. My main object is to prove the analogue of Theorem 5 
for double series, i.e. to establish the equivalence of these two 
notions. 

VOL. XIX. PARTS II., III. 7 



90 Mr Hardy, On the convergence 

It will be convenient to write 

The condition that a^n,n should be of bounded variation is then 
that the series i | A„, a,„,,,i|, S|A„a^^,,j|, and iS ] A„i^,i«,„,„ j 
should all be convergent. It is clear that these conditions in- 
volve the regular convergence of a^^n to a limit a. 

Theorem 6. If the condition (2) is satisfied, and a„,,i and «!_„ 
are of bounded variation in ni and n respectively, then a,n,n is 
of bounded variation in (m, n). 

n — l 

For A^a^,„ = A^a^_i- 1, A^_^«.^,^, 

v = l 
m—\ m— 1 ni-\n-\ 

S |A^a^,«!^ 2 |A^a^,i|+ S S |A^_^a^,J, 

/u. = l /u. = l n = l 1^ = 1 

so that 2 I A^,a^,M j 

is convergent. 

Theorem 7. If a^n,n is of bounded variation in {m, n), then 
a,n= lim am.^n, ««= lim cv,n 

are of bounded variation in m and n respectively. 



For a^= ai 1,— % A 



^t=i 



/xfi^Ai, »»' 



W-1 71-1 ao 11-1 

S I «;, — a^+i| ^ 2 I A^tti,^ j + 2 2) I Aj^,^a^,„|, 

and so 2 ] a^ — a^+j j 

is convergent. 

Theorem 8. The necessary and sufiicient condition that a,n,n 
should be of bounded variation is that its real and imaginary parts 
should be of bounded variation. 

This follows from Theorem 1 and the inequalities 

I ^m,n(^'m,n \ ^ j ^m, ?i^m,9i I "r | ^in,n Hm.,n |j 
where am,n = '^m„n + i^m,n- 



of certain multi'ple series 91 

Theorem 9. The necessary and sufficient condition that a real 
function a.m^n should he of bounded va,riation is that it should he of 
the fo7'm Am,n — A\n,n, whcrc 

and A\n^n satisfies similar conditions. 

Suppose first that a-m^n is of tlie form indicated. It is plain 
that the series 

^^7n-^m,n> ■^^7i-^7n,nt ■"-^^m, ?i-^?n., n> 
m n 

and the corresponding series formed from ^',„„>i, are all convergent. 
Further we have 

and similar inequalities for A„o,„_,i and \n,n(^m,n- Hence «,„,„ is 
of bounded variation. 

Next suppose that A,n,n is of bounded variation, and let 

Pm,n = I ^m,n(^7n,n \ \^7n,7i^m>7i ^ ")> Pm, 7i =" " \^rn, 7i^7n,7i < ^)) 
P 7)1,71 ~ I ^m,7iO-7n,7i I ( ^m, Ji C^m, ?i ?$ ^/i P m,7i ~ ^ \^m,n(^7n,7i > ^)- 

Suppose also that 

C» 00 00 00 

^-'711, Jl — -^ — JJlJL, VI ^^ 771, 71 — -^ — ' Z' M, I' • 

771 ji m rt 

Then it is plain that 

^7n-Drn,7i^ ^> ^n^in, 7i-^^i ^7n,7i-t^7n,n^^^ 

and that B'^^n satisfies similar conditions. 
Also 

00 00 

J^7n, n — -O 7JJ , jj = -^ .i ^fi, V Ojh, V ^^ ^m, ?!. ^m ^7i "r '^j 
m w 

(^771,71 ^^ -ttnijTi -O 7ji, n + ftjn ~r ttjj ft. 

But, by Theorems 7 and 2, we have 

where 0^., CJ, D,,, and D,/ are positive and steadily decreasing 
functions. Thus 

^7n, >i ^^^ ■^7n,7i -^ on, 7i > 

where 

Am,7i — Bm^n "H ^m + Dn + Jif, A ,n,7i— B rn.^n + ^m + -L'oi + -^ > 

£^ and E' being suitably chosen constants; and it is clear that 
Am,7i and A',n,7i will satisfy the conditions of the theorem if 
E and E are sufficiently large. 

7—2 



92 



Mr Hardy, On the convergence 



Theorem 10. // a.m.,n *'* of bounded variation, and 2St/,„,,j, is 
regularly convergent, then SSa,„,,„M,„,„ is regidarly convergent. 

In virtue of Theorem 8, it is enough to prove this when o.,n,n 
is real. In virtue of Theorem 9, it is enough to prove it when 

am.,n > 0, A„,a,rt,,i ^ 0, Ana,„,„ > 0, ^m.,na.m,n > 0. 

In the first place, by Theorem 3, every row and column of the 
series S2a,„^„iA,«,,i is convergent. 

In the second place, we have 

m n m n m n 



p-l («. q q-l P V pg 

+ 2 A^ a^, g 2 2 %i, j + , 2 A^ a^^ ^ 22 Uij + a^^ ^ 2 2 Uij * . 

m in n ii m n in n 



It follows that, if _p ^ m, q^n,we have 



p q 



<a 



m,n-'^'m,,n> 



2^2^ a^l y ^fji, V 

mn 

where Hm^n is the upper bound of 

1 '^ " i 
'^'%Ui,j (/-t ^ m, n ^ v). 

\ mn I 

Now 

p q mn f P 1 p n m q^ 

22 a^, ^iV,"- 22 «M,^ '?''/",>'= 2 2 + 22+22 

1 1 ' 11 \in+l n+1 m+1 1 1 w+l 



II 



and so 

p q 

21 
1 1 



1 1 ' ' 11 

where h^^n is the upper bound of 

22«t,j 



for all values of k, I, /x, and i/ such that fi'^k, v^l, and Tom 
or l>n. 

* See pp. 124—125 of my paper quoted in § 1, where the general form of this 
identity, for multiple series of any order, is given. Similar transformations of 
double series were given independently by M. Krause, ' Uber Mittelwertsatze im 
Gebiete der Doppelsummen und Doppelintegrale', Leipziger Berichte, vol. 55, 1903, 
pp. 240 — 263. See also Bromwich, 'Various extensions of Abel's Lemma', Proc. 
London Math. Soc, ser. 2, vol. 6, 1907, pp. 58—76, where further interesting 
applications of the identity are made, 



of certain multiple series ' 93 

Hence, by Lemma 8, we can choose co so that 



p q m n 

11 11 



if m and n are greater than eo. Thus the double series is con- 
vergent, and, since its rows and cokimns are convergent, it is 
regularly convergent. 

When a^n,^n and its various differences are positive, this theorem 
is nearly the same as that referred to in § 1. It is related to the 
latter theorem, in fact, as what Dr Bromwich calls 'Abel's test' for 
ordinary convergence is related to 'Dirichlet's test'.* The more 
direct generalisation is as follows. 

Theorem 11. If a,n^n is of bounded variation and tends 
regularly to zero, and 

m n 

2^ 2^ U/j,^ „ 

1 1 

is bounded, then S'Zam,n%n,7i ^■^ regularly convergent. 

The proof is similar to that of Theorem 10, and I need hardly 
write it out at length. The theorem shews, for example, that 
the series 



2S 



cos (md + n^) 
(a + mco + nw'f ' 



where 6 and <^ are real, w'jui is positive or complex, and the real 
part of s is positive, is regularly convergent except for certain 
special values of 6, </>, and a; or again that the series 

^^ cos (md + n(f)) 



(arn^ + 2bmn + cn'-y ' 

* Theorem 10 itself does uot seem to have been enunciated before, even in the 
specialised form. The nearest theorem which I have been able to find is one 
given by G. N. Moore, ' On convergence factors in double series and the double 
Fourier's series', Tia)is. Amer. Math. >Soc., Vol. 14, 1913, pp. 73 — 101. Moore's 
theorem (a particular case of a theorem concerning Cesaro summability) is as 
follows : if 

(1) 23«„j, ,j is convergent as a double series in Priugsheim's sense, 

VI n 

(2) |2S«^^J<:7i, 

(3) ^m, » ^ 0, 

00 00 

(4) lim 2 |«„,,,J =0, lim S|a„^,„|=0, 
m-^cc »=1 w-^-a)m = l 

(5) 22 I M^, „ a,„, „ I 
is convergent, then 22a„j „ Um,n *® convergent. 



I 



94 Mr Hardy, On the convergence H 

11 
where 6, (f), a, h, and c are real, and a, ac - ¥, and the real part of s 
are positive, is regularly convergent except for certain special 
values of 9 and 0. In either of these series, of course, the cosine \ 
may be replaced by a sine. 

In order to prove the converse of Theorem 10 we require two 
lemmas analogous to Lemmas a and yS. 

Lemma e. If SSc^.n is a divergent series of positive terms, 
we catt find €„i^n so that (1) e^^^n decreases luhen ni or n increases, 
(2) €m,n tends regularly to zero, and (3) the series Sl,e^n,nGm,n is 
divergent. 

(1) Suppose first that at least one row or column of the 
original series, say the vth row 2c„i, ^, is divergent. By Lemma a, 
we can choose a steadily decreasing sequence 97^, with limit zero, 
so that XvmCm,p is divergent. We take 

€vi, n = Vm (n ^ v), €,n, n = {u > v), 

and it is plain that the conditions of the lemma are satisfied 

(2) Suppose that every row and column is convergent; 
and let 

(m) (w) 

Then 'S,j„^ is divergent. We choose a steadily decreasing sequence 
r]m so that 'S^Tjmjvi is divergent. Then SSc'„,,,„, where 

is divergent; and so Xjn, where 

yn ^ ^ Vm Cm, n j 
(m) 

is divergent. We now choose a steadily decreasing sequence ^„, 
with limit zero, so that ^^nln is divergent. It is clear that, if 
we write 

// y 

(^ m,n — Vmbn^^m.jn — ^?w, n C^/i,, ^ , 

all the conditions of the lemma will be satisfied. 

Lemma ^. If SSc^.w is a divergent series of positive terms, we 
can choose a sequence of pairs of integers (nii, Ui), tending to infinity ! 
luith i, so that the series X%c\n,n, where c'm,n = if 771 = nit, n^ni 
or m^nii, n = ni, and c,n,n = c,,,,,,,, otherwise, is divergent. 

The modification to be made in the series is effected by 
drawing perpendiculars on to the axes from the points {mi, n^), 
and annulling all terms which correspond to points on these 
perpendiculars. Let a^ denote the sum of the terms whose 



of certain multiple series 95 

representative points lie on the perpendiculars from (m, m) on 
to the axes. Then So"„i is divergent. Applying Lemma to this 
series we obtain the construction required, nii being in fact 
always equal to Ui. 

Theorem 12. If S2a,„.^,i,i<„,,,j, is regularly convergent ivhen- 
ever SSif.„i.,,i is regularly convergent, then a,n,n "^^ ^f hounded 
variation. 

In the first place it follows from Theorem 4 that ■u,„,^n is, for 
every value of n (or m), of bounded variation in m (or n). It 
remains only to shew that SS | A,„^„a,„,_,i | is convergent. 

Suppose, on the contrary, that it is divergent. By Lemma e, 
we can choose a sequence of positive numbers €,„,„, tending 
regularly to zero, so that 2Sc,rt,n, where 

is divergent. We can then modify this series as in Lemma ^ 
without destroying its divergence. 

Now let 

m n 
U —^l.u 
1 1 

and suppose that 

if m = mi, n ^ ni or m ^ mi, n = ni, and that otherwise 



^ in n, — 






this last formula being interpreted as meaning e^yi^n if 

These equations define u,n,^n uniquely for all values of m and n, 
and it is plain that U'm,n tends regularly to zero, so that 2S^/,„,,t is 
regularly convergent. On the other hand 

^ -^ '■hn,n ^'■m,n "i ■^ '^m,n ^i')n,n ^ m,n — -^ ■^ ^ w,n> 

11 11 11 

which tends to infinity with i. Thus ^'%a„r,nUm,n is not convergent. 
This proves Theorem 12. Combining it with Theorem 10 we 
obtain the analogue of Theorem 5, viz. 

Theorem 13. The necessary and sufficient condition that am^n 
should be a convergence factor is tJiat it should be of bounded 
variation. 



96 Mr Watson, Bessel functions of large order 



* 



Bessel functions of large order. By G. N. Watson, M.A., 
Trinity College. 

\^Received 14 June 1917.] 

1. When the order of a Bessel function is large, the asymp- 
totic expansion of the function assumes various forms depending 
on the values of the ratio of the argument to the order of the 
function. The dominant terms of the asymptotic expansions are 
given by the formulae : 

(i) When n is large, x is fixed and ^x< 1, then 
Jn {nx) ~ (27rw)"* (1 - a^y^x"" [1 + V(l - a?)]"' exp [n V(l - x% 
(ii) When n is large, x is fixed and x>l, then 

Jn{nx)<^{^'jrny^{x'^- l)~^cos [?i V(*'^ - l)-n^e(r^x- lir]. 

_ 2. 
(iii) When n is large and € = 0{n ^), then 

J^{n + ne)^T(i)/{7r2is^n^]. 

The corresponding complete asymptotic expansions, valid for 
general complex values of n and x, have been given by Debye*. 
Accounts of the history of the approximate formulae are to be 
found in Debye's memoirs and also in two papers f which I have 
published recently. 

It is evident that there are transition stages between the 
domains of validity of the three formulae quoted ; and not much 
is known about the behaviour of J„ (jix) in these transition stages. 
Consequently I propose to establish approximate formulae (involv- 
ing Bessel functions of orders'] + ^) which exhibit the behaviour 
of the Bessel function right through the transition stages. These 
formulae are more exact forms of some approximations which 
Nicholson § obtained some years ago without estimating the 
margin of error or the precise ranges in which the results were 
valid. 

* Math. Ann., lxvii. (1909), pp. 535—558. Milnchen. Sitzungsberichte [5], 1910. 

t Proceedings, xis. (1916), pp. 42—48. Proc. London Math. Soc. (2), xvi. (1917), 
pp. 150—174. 

J These functions have been tabulated by Dinnik, Archiv der Math, und Phys., 
XVIII. (1911), p. 337. 

§ Phil. Mag., Feb. 1910, pp. 228—249. 



Mr Watson, Bessel functions of large order 97 

The approximations which I shall obtain are derived by 
shewing that certain integrals of Airy's type* are effective 
approximations to the integrals which occur in Debye's analysis. 
It will be assumed that the reader is familiar with Debye's 
memoirs, although it seems desirable to modify the notation to 
a considerable extent. The formula for Jn (nx), when x^l, is 
of importance in connexion with the maxima of the Bessel 
function f. 

The two formulae which will be obtained in this paper are as 
follows : 

(I) When a ^ 0, 

Jn ( n sech a) ~ 27r~^ 3 " ^ tanh a 

X exp [n (tanh ol + ^ tanlr' a — a)} . if ^ {^n tanh* a), 

where the error is less than 37i"^ exp [?i(tanh a — a)|, and K,,,{2) 
denotes the Bessel function of Basset's type (see § 6). 

(II) When ^ /3 ^ \7r, 

Jn {>i sec /3) ~ ^ tan /3 cos [n (tan ^ — ^ tan^ yS — /3)} 
X [J_ I (-3-*^ tan=^ ^) + /, (i/i tan* /3)] 

+ 3" ^ tan yS sin [n (tan ^- ^ tan* j3 - /3)} 
X [/_ , (iw tan* /S) - J^ O tan* y8)], 

where the error is less than 24/?i. 

Part I. The value of Jn(n.v) ivhen O^^-^l. 
2. We take Sommerfeld's integral 



The stationary points of x sinh w — w, qua function of w, are 
given by cosh w — l/x ; accordingly we replace x by sech a, where 
a ^ ; and then, putting w^ot + t, we have 

jiTTl J 00 -ni 

+ n (sinh t — t)] dt. 

The exponent has a stationary point at ^ = 0, and the method 
of steepest descents provides us with the contour whose equation is 

/ {tanh a (cosh t-\)-\- (sinh t-t)] = 0. 

* These integrals have been expressed in terms of Bessel functions by Nicbolsou, 
Phil. Mag., July 1909, pp. 6—17, and by Hardy, Quarterly Journal, xli. (1910), 
pp. 226—240. 

+ Proc. London Math. Soc. (2), xvi. (1917), p. 169. 



98 Mr' Watson, Bessel functions of lar-ge oi'der 

The portion of this curve which is suitable for our purposes 
consists of an arc* on the right of the imaginary axis in the ^-plane 
with its vertex at the origin and with the lines I {t)= ±tt as 
asymptotes. 

If we write t = u + iv, where tt, v are real, the equation of the 
curve becomes 

cosh (a + u) = V cosec v cosh a. 

We shall put 

tanh a (cosh t - 1) + (sihh t — t) = — r, 

so that as t traverses the contour t diminishes from + x to 0, and 
then increases to + oo ; and therefore 

Jn (^'^) = o-^ e«(t^'ii^*-«) I j + \ e""^ (dt/dr) dr ; 

in the first integral v ^0 and in the second integral v ^ 0. 
Now define T by the equation j- 

1^2 tanh a + ^T' = -T. 

A contour in the T-plane on which t is real is a semi-hyperbola 
touching the imaginary axis at the origin and going off to infinity 
in directions inclined ± ^tt to the real axis. If we write 

T=U+iV, 

where U, V are real, the equation of the hyperbola becomes 

Utanha + ^U'' = iV-\ 

Taking the semi-hyperbola as the T-contour, we shall shew 
that an approximation to 

foD+iri /• 00 exp ( Jtt?') 

e-'^-'dt is e-''^dT. 

J CO - Tii ■' 00 exp ( - JttO 

It is easy to see that the difference of these integrals is 
D idt dT) , 



{] ^ Jo ] (dr dr] 



and so the problem before us is reduced to the determination of 
an upper bound for \d{t— T)/dT |. 

* This curve is derived from the curve shewn in fig. 4 (p. 541) of Debye's first! 
paper by turning it through a right angle and talcing the origin at the vertex. The! 
degenerate case when a is zero is shewn in fig. 5. \ 

t Since T = ^t^ ta,nh a + lt^ + (t*) when | 1 1 is small, the curve in the T-plaue 
closely resembles the curve in the i-plane near the origin ; and, the parts of the 
curves near the origin being the most important when n is large, we are obviously 
able to anticipate that the integrals under consideration are approximately equal. 



Mr Watson, Bessel functions of large order 99 

3. We shall now shew that, whenever t ^ and when, corre- 
sponding t(^ any given value of t, we choose V to have the same 
sign as v, we have the inequality 

\d{t- T)}dT I ^ Stt. 

Since, corresponding to any value of t, the two values of t are 
conjugate complex numbers (and similarly for T), it is evidently 
sufficient to prove this inequality when v and V are both positive. 

On comparing the values for r in terms of t and T, we perceive 
that 

' ( 7' - {H^ + tanh a + 1 (T' + Tt + f')} 



tanh a (cosh ^ - 1 - ^t-) + (sinh t-t- ^t-'). 



Also 
d(t- T)jdT = { jTtanh a + ^T''}~' - {sinh t tanh a + (cosh t - 1)}-^ 
^ t-T 

T {sinh t tanh a + (cosh ^ — 1 )] 

^^(^-r ) + (sin h t - i) ta nh ajl^(cosh t-l-^f) 
^ TXta'nh a + iT) {sinh « tanli^+ (coshT-nf)} " 
Now 

I sinh t tanh a + (cosh t—l)\ 

— sech a v/[(cosh u — cos v) {cosh (2a + w) — cos vW ; 
and since 

{cosh (2a + (t) — cos v] — cosh- a (cosh ii — cos v) 

= sinh- a (cosh m -t- cos y) + sinh 2a sinh a 

and 

{cosh (2a + u) — cos v] — sinh- a (cosh a + cos y) 

= cosh^ a (cosh <( — cos v) + sinh 2a sinh a 

we 6'ee that \ sinh ^ tanh a + (cosh ^ — 1) | exceeds both 

cosh ^^ — cos V = I cosh i — 1 1 

and also tanh a \/(cosh- u — cos- y) = tanh a | sinh ^ j . 

We now divide the range of integration into two parts, namely 
T ^ 1 and :^ T ^ 1. 

4. Consider first what happens when t^ 1. 
If I T| ^ 1, we have (on the T'-contour) 

T = I IT- tanh a + 1^=* | < ^ + i < 1. 



100 Mr Watson, Bessel functions of large order 

Also, if I ^ I ^ 1, we have (on the ^-contour) 

T = I (cosh ^ - 1) tanh a + (sinh t-t)\^ S \t i '"/''* ! ^ e - 2 < 1. 

Hence, when r^l, we must have both \T\^1 and also \t\^l. 
But, when | T| ^ 1, since U^O, we have 

I (dr/dT) I = I Ttanh a + ^-T- 1 ^ ] tanh a + ^T\>^. 

Also, when | i | ^ 1, we have u or v (or both) greater than l/\/2, 
and we always have v less than vr. 
Hence, by the result of § 3, 

I (dr/dt) I = I sinh t tanh a + (cosh t — 1)\^2 (sinh'^ ^u + sin^ ^y), 

and this exceeds the smaller of 

2sinhni/V8), 2sin^(l/v'S). 
Consequently 

i (dt/dr) I ^ 1 cosec^ (l/\/8) = 4-14 < 27r - 2. 
Therefore, when t ^ 1, we have \d(t — T)/dT \ < ^ir. 
We shall make use of this inequality in § 6. 
5. Consider next what happens when ^ t ^ 1. 
If I T| ^ 2, we have (on the I^-contour) 
T = I i-r^ tanh a + iTM ^ 4 ! 1 tanh a + ^T\> l\ Tl^^. 

Also noting that u and v increase together, when v "^ ^tt, we 
have (on the ^-contour) 

T = u + tanh a — cos v sech a sinh (a + it) 
^ M + tanh a 
^ tanh G — a + log {-^-TT cosh a + \/(l7r" cosh" a — 1)}, 

on expressing u in terms of v and noting that v cosec v exceeds -^tt. 
This function of a increases with a and so it exceeds 

log{j7r + V(i7r^-l)} >1- 
Hence, tuhen r ^ 1, we ??iwsi have both \T\^2 and also v^^tt. 
Further, when v^I-tt, we have 

cosh u ^ sech a cosh (a + it) = v cosec w ^ I-tt < cosh I'l, 
so that \t\" -^ ^TT' -h (I'ly < 4, a7id therefore \t\<2. 

That is to say, when r^ 1, neither \ T\ nor j t j exceeds 2. 

Also, for all values of t, 

du/dv = {1 —V cot v)/i\/(v" — sin^ v sech^ a) 
^ (1 — ?; cot «;)/«; ^ ^-v, 
and so u'^^v'^. 



Mr Watson, Bessel functions of large order 101 

Further, when v ^ ^tt, we have* 1 — v cot v ^ ^J{v- — sin- v), and 
so dujdv ^ 1 , i.e. v ^ w, whence at once we have v \/2 '^\t\. 
Next, when | ^ | $; 2, we have 

\^m\xt\^\t\[l - ^\t\^ - -,^^,\t\* - ...] 

In like manner we may prove that, when | ^ | ^ 2, 

I cosh i - 1 I ^ 4 I ^ J-/18 ^l\t\\ I cosh t-\-\t'\^\t IV20, 
i sinh t-t\^^\t IV24, | sinh t-t-^,f\^\t I7IO8. 

We are now in a position to obtain an upper bound for | 7^- ^ |. 
It is first evident that 

\\{T+t) tanh a + 1 (^' + ^« + t') I 

^ / {i ( 2^ + tanh « + 1 ( r^ + r^ + 01 
^ I V tanh OL-^ ^uv 
^ |-w(tanha + i-y-) 

^|i|(tanha + yVl^l')/\/8 
^U|2(itanha + T'H|^I)/V8. 
Hence, by the result stated in § 3, 

\l-t\^d> \t\ -ytanha+ 1^1/18 ^^^^^l^l ^i|«l. 

To obtain a stronger inequality, we write the equation of § 8 
in the modified form 

{T-t){T + t) {1 tanh a + ^{T-\-t)] 

= -{T- tfl24> + tanh a (cosh t-l-^f-) + (sinh t-t- ^t^). 

The expression on the right does not numerically exceed 

I ^ 17192 + tanh a.\t \'/20 + \ t IVIOS ^ tanh a . 1 1\'/20 + | ^1748 ; 

and since \{T + t) [^tixnh.a + ^(T + t)]\ exceeds both -H^ltanha 
and also ^|^1", we see that 

\T-t\^{j^^+i)\t\'=i\tm5. 

If we now f mother restrict t so that | ^ | ^ 1, the last inequality- 
gives 

\{-(T- i)V24 + tanh a (cosh t-1- ^f-) + (sinh t-t- |^«)} | 

^ 4-^ I « 1 7(24 . 150 + tanh a.\t |720 + | ^ J7108 
^tanha. 1^1720 + 1^17104, 

* Since -j~ {siii'^j' (1 - v cot v)^ - sin-i; {v- - sin- c) j- = -- 2 sin 2v . (v^ - sin^ u) < 
when < u < Itt. 



102 Mr Watson, Bessel functions of large order 

and this inequality, combined with the modified form of the 
equation of § 3, gives 

I T- ^1^(1/10 + 1/13) jip'^i^lVS. 



Hence, when |^!^1, \T-t\^^\t\, and so \T\^^\t\; and 
therefore, when U | ^ 1, \T\ exceeds the value which it has when 
1^1 = 1, and so, a fortiori, | T\ ^ |. 

It now follows that, when both r and \t\ do not exceed 1, 
we have 



d^_dT ^ i 1 1 

dr dr 



II ^ I . I (sinh t tanh a + (cosh i — 1)} | 

1 1 [yiO + 5 tanh a \ t |V24 + 1 1 |V20 
"^ (8 1^ IV25) I {sinh t tanh a + (cosh t-l)]\ 
= (23 i ^ 1732 + 125 tanh a 1 1 1/192} 
-^ I {sinh t tanh a + (cosh t— 1)} j . 

Since the denominator exceeds both 1 1 ^ | lanh a and i | ^ |", we 
see that 

\d{t- Tydr I ^ (23/8) + (125/128) < 27r. 

If T ^ 1 and 1 ^ j ^ I ^ 2 we use the second expression of § 3 for 
d{t— T)ldr. Replacing \T—t\ in the numerator by 4 | ^1^15 
and \T\ in the denominator by |, we get in a similar manner 

\d{t- T)ldr \^[\t IV3 + 125 tanh a . | i |7192 + 11 1 ^ | V60| 
-^ j {sinh t tanh a + (cosh t — 1 )} | 
^ 4 {I ^ i/3 + 1 1 I ^ I V60}/13 + 125 i « I-/128 
<37r. 

6. It is obvious, from the results of §§ 4, 5, that, luhenever 
T ^ 0, we have 

\d{t- T)ldT I < Stt ; 

and from this result we have 



27r 



r] „ idt dT\ ,1 ., /•" „ , ^, 



The evaluation of I e""**^(ZT presents no special points 

J CO exp( — -J-irt) 

of interest ; the simplest procedure is to modify the contour into 
two rays, starting from th^ point at which T = — tanh a and 
making angles + -J^tt with the real axis. 



Mr Watson, Bessel functions of large order 103 

If we write T = — i&xih. a ■\- ^e*>' on the respective rays, the 
integral becomes 

/•OO 

e'-'^*' exp (i?i tanh-' a) A exp { - i?ip - ^n^e^""' tanh^ a] d^ 

JO 

- g-i'^' exp (1??, tanh' a) . exp { - i?^p - i^j^^-i'^'' tanh- aj d^. 

These are integrals of Airy's type ; on expanding 

exp (— |n^e* ■•'^* tanh- a) 

in powers of tanh a and integrating term-by-term — a procedure 
which is easily justified — we get on reduction 

Itti tanh a . exp (i?i tanh^ a ) . [/_ j^ (^n tanh' a) — /j^ (J-?i tanh" a)], 

where, in accordance with the ordinary notation, 

On introducing Basset's function K^Az), defined as 
^ TT cot niTT [/_^ (z) - /,„ (z)], 

we obtain the final formula 

2 
t/^ (n sech a) = ~ [tanh a exp {?? (tanh a + i^ tanh' a - a)] 

TT V " '^ 

X Kx Qn tanh' a)] -|- SdiU-^ exp {w (tanh a — a)], 
where | ^i j < 1. 

When w is large the ratio of the error term to the dominant 

term is of order n~^\/tanha, n~^, vT^, according as n tanh'' a is 
large, finite or small. 

The formulae (i) and (iii) of § 1 agree with this result when a 
is finite and when n tanh' a is small, respectively. 

Part II. The value of Jn(nx) when x^l. 

7. It is convenient to regard Hankel's solutions of Bessel's 
equation, iT^'^' and Hn^^^ as fundamental. The ordinary solutions 
are expressed in terms of these functions by the equations 

Jn (nx) = i {^,,w (nx) + HJ'^ {nw)], 
The integral formulae of Sommerfeld's type are 

1 roo+ni 

Hn^'^ (nx) = -—. e« (xsinhw-tv) dyj^ 

TT'l J _oo 

2 roo—iri 



104 Mr Watson, Bessel functions of large order 

The stationary points of xi^m\iw — iv, qua function of w, are 
given by cosh ?<; = !/«. As <\lx^l, we put x = sec/3 where 
^ /3 < I^TT ; and two stationary points are given by w= ± ^i. 

Now it has been shewn by Debye that a branch of the curve * 
I{x sinh lu - w) — I {x sinh ijS — i/3) 

is a suitable contour for HJ^\ and the reflexion of this contour in 
the real axisf is a suitable contour for HJ-K 

On making a change of variable by writing w = t + i^, we 
have 

1 rQO+7r?-(/3 

where i tan /3 (cosh t — 1) + sinh t — t = — T. 

If we put t = 'u + iv, where u, v are real, the equation of the 
contour is 

cosh w = (sin ^ + v cos /3) cosec (v + /3), 

and, on the contour, 

r = u — sec /S sinh u cos (v + /3). 

When V is given, cosh u is given and the sign of u is ambiguous ; 
we take u to have the same sign as v, in order that the contour may 
be of the requisite type. 

Next define T by the equation 

^TH tan /3 + ^T' = - r. 

We write T^U+iV, where U, V are real; a contour in the 
T- plane on which t is positive is that branch of the cubic :|:, whose 
equation is 

(t/^ _ T/2) tan ^ + 1 F(3[7- - V) = 0, 
which passes from — cc — i tan ^ through the origin to x exp (^Tri). 

Taking this curve as the contour, we shall shew that an 
approximation to 

r<x+Tri—ip rco exp (Jn-i) 

e-''^dt is e-'^^dT. 

.'-cxj-i/S J -00— i tan/3 

* This curve is derived from the curve shewu in fig. 2 (p. 540) of Debye's first 
paper by turning it through a right angle and taking the origin at tlie node. The 
reader will observe that tlie character of the contour has changed with the passage 
of X through the value unity. 

t Since iJ^*^), HJ") are conjugate complex numbers when n and x are real, it 
will be sufficient to confine our attention to HJ^^. 

X Of course t is real on the whole cubic ; as T traverses the specified portion of 
it, T decreases from + oo to and then increases to + oo . 



Mr Watson, Bessel functions of large order 105 

8. Before proceeding further, we shall shew that the slopes of 
the contours in the t-plane and in the T-plane never * exceed s/S. 
If we write 

(sin ^ + V cos /3) cosec (v + /3)^ ^|r (v), 

we have dv ^sinhu^ ^ [{^(^)}2 _ i]l ^ 

du '\Jr' (v) ~ ■\lr'(v) 

Now 

yjr' (v) = cosec (/3 + v) {cos yS — cot (^ + v) (sin l3 + v cos /8)}, 

and so -yjr' (v) is positive when /3 + v is an obtuse angle. When 
^/3 + v^^TT, however, we find that 

cos /3 tan (/3 + v) — (sin 13 -\- v cos /3) 

is an increasing function which vanishes with v. Hence yjr' (v) 
has the same sign as v (and therefore the same sign as u), and 
consequently 

dv ^ [{ylr(v)\'-lf 
du I yjr' (v) I 

It is therefore necessary to prove that 

i.e. that x (^) = '^ li"' W' " if (^)}' +1^0. 

Now % (0) = 0, and it is consequently sufficient to shew that 
X {^) h'^s the same sign as v. Since 

X{v) = 2y{r\v){Sf"(v)-ylr{v)} 

and yfr' {v) has the same sign as v, it is sufficient to prove that 

Srfr" (v) - f (v) ^ 0. 

Since yjr (v) sin (v + /8) reduces to a linear function of v, its 
second derivate vanishes, and so the inequality to be proved 
reduces to 

f {v) - 3-f' (v) cot (v + yS) ^ 0, 
i.e. to 

(sin ^+v cos /3) {1 + 3 cof^ {v + /3)} - 3 cos ^ cot (v + /3)^0. 

But 

sin yS + V cos yS - 3 cos ^ cot {v + /3)/{l + 3 cot^ {v + /3)} 

has the positive derivate 4cos yS (1 + 3 cot"(w + /3)}~^ and is 
positive when v = — ^; hence it is positive throughout the range 
— ^^v-^TT-^. And this is the result which had to be proved. 

* In the limiting case when ^ = 0, the ^-contour has slope ^3 immediately on 
the right of theorigin, and the T-contour consists of the rays arg r = 0, arg T — ^tt ; 
so there is no better inequality of the form stated. 

VOL. XIX. PARTS II., III. 8 



106 Mr Watson, Bessel functions of large order 

In like manner, we find that 

dV _ (tan;e + F)^(tan/8 + ^7)^ 

^dU~ tan'/S + Ftan/S + iV- ' 

and it may be proved by quite simple algebra that the square of 
this last fraction does not exceed 3. 

From the results just proved it follows on integration that 

\v\^\u\^/S, |F|^|i7!V3, 
and hence 

h'!^iUI> \v\^^\t\^3, \U\^^\T\, |Fi$i|TlV3. 

9. We now return to the integrals of § 7. As in the corre- 
sponding work of §§ 2 — 3, we have to obtain an upper bound for 
\d{T — t)ldr i ; we shall in fact shew that this function does not 
exceed 127r. 

We notice that formulae corresponding to those given in | 3 
are 

{T-t){hAT+t)i\>?.u^-^^{T^^Tt + 1?)] 

= i tan yQ (cosh t-l-hP) + sinh t-t- ^t\ 
d{t- T)/dT 

= {iT tan /3 + ^T']-^ - {{ sinh nan /3 + (cosh t - 1 )]-^ 

t-T 

" T [i sinh t tan ^ + (cosh ^ - 1)} 

^t{t-T) + i (sinh t - t) tan ^ +_(cosliJ^- 1 - ^i') 
"^ T {% tan y8 + ^ T) (TsmhTtan yS + (coshl -1)} 

Now 

I i sinh t tan /3 + cosh t — 1\ 

= sec 13 V[(cosh u — cos v) {cosh v — cos (2/3 + w)|], 
and since 

(cosh u — cos (2/3 + v)] — cos- B (cosh ii — cos v) 

= sin^ /3 (cosh u + cos v) + sin 2/3 sin v 
^ (1 + cos v) {sin- /S + sin 2/3 tan -^ vj 
^ (1 + cos w) {sin- 13 - sin 2y3 tan ^^] 

we have 

. I i sinh t tan /3 + (cosh ^ — 1) | ^ cosh w — cos « = | cosh t — l\. 
Also 

cosh w - cos (2/3 + ?;) ^ 2 sin= (^8 + -iwi ^ 2 sin- 1/3, 



Mr Watson, Bessel functions of lar^ge order 107 

and so 

I i sinh t tan /3 + (cosh t — l)\^sm^^ sec /S V[2 (cosh it — cos v)} 

^ tan/9 I sinh ^t\. 

That is to say \ i sinh t tan /3 + (cosh i — 1 ) j exceeds botJt, j cosh ^ — 1 j 
and also tan /3 | sinh ht\. 

In order to simplify the subsequent anafysis, it is convenient 
to place a restriction on /S. We shall coiisequently assume in 
futui^e that O^/S^Itt, so that tan^^^l. This restriction is not of 
importance so far as the final result is concerned, because Debye's 
formula, quoted in | 1 (ii), is effective whenever sec/3^1 + S, 
where 8 is any positive constant ; and so it is certainly effective 
when sec /3 ^ \/2. The importance of the analysis in the present 
investigation is due to the fact that it is valid for small values 

of ;8. 

10. Consider what happens when r ^ ^, whether v, V are both 
positive or both negative. 

When I 2'| ^f , we have (on the T-contour) 

T = I iiTHanyS + irT' \ ^\T'\{^ + ^\T\)< h, 
and if I i I < I, we have (on the ^-contour) 
T = I [i tan /3 (cosh ^ — 1) + (sinh t — t)]\ 

00 

$ t |«|'"//?i!^e*-l -1 = 2-12 -1-75 <^. 

Hence, when t ^ |-, we must have both | ^j ^ f and \t\ ^ f . 
But, when | T | ^ | , we have 

I (dr/dT) \ = \T\.\itan/3+iT\^\T\.\^R(T)\^^\T\"-^^\. 
Also (as in § 4) when | ^ | ^ f , we have 
j (dr/dt) \ = \i sinh t tan /3 4- (cosh t— 1)\ 

^ I cosh ^ — 1 I = cosh u — cos v^2 sin- (fjr ^2) = 0-137, 

and so j (dt/dr) | ^ 7-3. 

From these results we see that, when r^^, 

\ d (t -T)/dT\< 15 <57r. 

We shall make use of this inequality in § 12. 

11. Consider next what happens when ^t ^^, whether v, V 
are both positive or both negative. 

When \T\^2, we have (on the IT-contour) 

Also, when \t\ ^2 and v + /3 ^ ^ir, we have u ^ ^tt \/3, and 
then 

T = u — sec /3 sinh u cos (v + /3)^u^ 1. 

H_-2 



108 Mr Watson, Bessel functions of large order 

Next, when | ^ | ^ 2 and ^ <^r + jS -^^Tr, v/e have 
cosh w = (sin + v cos yS) cosec (v + 0) 

^sm/3 + (Itt — /3)cos/3 < ^7r< cosh 1*1, 

since sin yS + (^tt — ^) cos /3 is a decreasing function of y8. 

This gives 2 ^ | i | < \/{(-^7r)- + (1"1)^} < \/3'7, which is impossible; 
so that, when \t\^2, we cannot have /3 ^v + /3 -^ ^tt. 

Lastly, when \t\'^2 and ^ v ^ - /3, we have w :$ 0, and so 

- w ^ V(4 - /S'O ^ V{4 - (iTT)}^ > 1-8, 
and 

T = sec /3 sinh (— u) cos (?; + /3) — (— it) 

^ sinh (- w) - (- u) > ^ ( l-8)» > i . 

Therefore, whenever | ^ | ^ 2, we have r^^. 
Hence, when ^ t ^ |- , we must have both \t\^2 and | T | ^ 2. 
Next we shall shew that R[^t+ T'^/{T + 1)} has the same sign 
as u and U. 

The function under consideration is equal to 

l^u {{U + uf + (F+ vy\ + (U"^-V')(U+u) 

+ 2UV{V + v)]^[{U +uy + (V + vf]. 

Taking U, V, u, v positive for the sake of definiteness, we see 
that the numerator of this fraction exceeds 

lu{U' +V') + u{U-'- F^) = ^u(SU' - V) ^ 0. 

Similarly we can prove that the numerator is negative when 
U, V, u, V are all negative. It follows from this result that 

\R{t+Ty(T + t)}\^^\R{t)\^l\tl 

We ewe now in a position to obtain an upper bound for \1' — t\ 
when 1 1 1 and \ T \ are both less than 2. 
First suppose that | ^ | ^ | . 
Then, from the formula quoted at the beginning of § 9, 

j {T- t) I .\{T+t)\.\ {laan /3 + i^ + iTy{T+t)} | 

= I i tan y8 (cosh t-1- lt~) + (sinh t~t-lt^)\ 

00 

< 2 \t\'"'lm\^D\t\*lllQ. 

1)1 = 4: 

But \T+t\^\t\and 
\{^its.n^ + it + iTy{T+t)}\>i\R{t + ri(T+t)}\^^^^\t\. 

Hence, when | i | ^ l, we have | (T - ^ | ^ 120 | ^ |7119. 

Next, keeping | ^ | ^ | , we take the formula 
i2iT-t)(T+t){iUnl3 + i{T+t)} 

= -^\(T-ty + i tan 13 (cosh ^ - 1 - i t~),+ (sinh t-f-^f) 



3Ir Watson, Besael functions of large order 109 

and observe that 

and also, in view of the fact that, as t varies through positive 
vahies, t + T traces out in the Argand diagram a curve, through 
the origin, whose slope obviously never exceeds V3, the distance 
of all points of this curve from — 4^ tan $ must exceed 2 tan /3. 
Hence | i tan /9 + ^ (T + ^) | ^ ^ tan /3. 

Using these two inequalities, combined with the fact that 
j(T— i) j ^ 120|^|-/119, and the obvious inequalities 

\T+t\^\t\, I cosh t-l-^t'\^\t 1 V23, 
|sinhi-^-i^-|^|^i-Yll9, 

we deduce from the last equation for T— ^ that 

\T-t\^l\t\ {120 1 1 \ll\^Y + 4 1 ^ IV23 + 16 U IV119 < U I'- 

Using now the inequality \T —t\i^\t\^ in place of 

I T- ^1^120 1^17119, 
we get 

\T-t\^l\t\' + ^\t IV23 + 16 I ^ IV119 

< (1/24 + 4/23 + 16/119) \tY ^ ^\t\K 

Using now the inequality \T —t\^\\t\^,yNQ get, in place of the 
last result, 

\T-t\% (1/192 + 4/23 + 16/119) | ^ 1=^ ^ ^ | ^ p'. 

From this result it follows that, when l^j^^, \T —t\% ^\t\, 
and so iri^l^l^l. 

Consequently, from the formula for d{t - T)ldT given at the 
beginning of § 9, we see that, when | ^ | $ |^, 

\dt^_dJ 
dr dr 



i\t\ 



{i|«|.|(cosh«-l)| 



+ 



i (ii I ^ l)N 1* ^i^h ^ ^^^ /^ + (cosh t — 1)| 
Now, when | ^ | < 2, 

1 po«?h /_l|>i|^|2ri_ 4 _ 16 _ 1>1|/|2 

and \smh.t\^\t\[l-l;-^- ...]>^\t\; 

and so, using the results of § 9, we get 

\d(t- T)ldr \ ^ 16/11 + (576/121) [4 (1/6 + 1/23) + 6/5] 
<12, 

when 1^1 ■S^. . 



lio 



Mr Watson, Bessel functions of large order 



Lastly, when |:^|^|^2, we have l^^j^ 11/24, and so, by the 
method of § 10, we get 

\d{t- T)ldr I <S 4 (24/11)^ + \ cosec^ (i ^2) 

<35-3<127r. 

12. It follows from the results of §§ 10, 11 that, for all positive 
values of t, 

\d{t-T)ldT\<12Tr, 

and consequently 



+ 



di 



< 24<7rjn, 



; [dr dr] '^ 

so that 

ir,,« (n sec y8) = A e''^ (t=i»^-^) g-"- dT + 24^6.,/ n, 

TJ"* J-Qo-itan/3 

where | ^al < 1- 

To evaluate this integral, where — r = ^T'^ i tan /8 + ^T'^, we 
take the contour to consist of the two rays arg(T+ itan/8) = 7r, 
^ir ; on writing T= — i tan /3 — ^, —i tan j3 + ^e^"' on the respective 
rays, expanding the integrand in jDowers of ^ and integrating term 
by term we find that 

/■ooexp(47ri) 

e-''"dT 

J -00 — ?!tan/3 

= §771 tan ^ exp (— l^vn tan^ /3) 

X [e" *''' J_ 1 (i?^ tan^* ,8) + e*" /i (iw tan^ ^8)] 
= 3" -7ri tan /5 exp (|■7^^ — |-7w tan^/3) i/^.'^' {^n tan" y8). 
Since J„ {n sec ^) = R [^w"' ('^ sec /3)], 

/_„ (w sec ^) = R [e"'^^' i^,,"' (" sec ^8)] , 
it follows at once that, when ^ /3 ^ ^tt, 
Jn {n sec /9) = 3~^ tan /3 cos {m (tan /3 — ^ tan^ /3 — /3)} . [JL i + t/i] 

+ 3" Han /5 sin {n (tan /3 - i tan^* /Q - /3)} . [/_ .^ - J^J + 24(9/7^, 

J"_,i (?i sec /3) = 3~^ tan /3cos [ji (tt + tan /3 — -^ tan'* y8 — /S)} . [/_ x + Ji] 

+ 3" ^ tan /3 sin {n (tt + tan /3 - i tan=* ^ - /3)} . [J_ j - ^^] + 245'77i, 

where the arguments of the Bessel functions J±x on the right are 
all equal to ^ntan^/3, and | ^ I, \6'\ are both less than 1. It is easy 
to see that, except near the zeros of the dominant terms on the 
right, the ratios of the error terms to the dominant terms are of 
orders Vl^^^^an^), n~'^, ?i~*, according as 7i tan^ /3 is large, finite 
or small. 



Mr Todd, A particular case of a theorem of Dirichlet 111 



A particular case of a theore^n of Dirichlet. By H. Todd, B. A., 
Pembroke College. (Communicated, with a prefatory note, by 
Mr H. T. J. Norton.) 

[Received 14 June 1917.] 

[The following note is an extract from an essay submitted to 
the Smith's Prize Examiners. 

It will, perhaps, be convenient if I preface Mr Todd's argument 
by explaining its relation to the theory of algebraical numbers. 
The principal theorem is a famous one of Dirichlet's on the unities 
of an algebraic corpus or order. It will be remembered that if ^ 
is a root of an irreducible equation of the nth degree, the coefficients 
of which are integers, then, if the coefficient of the nth. power of 
the unknown is 1, ^ is an algebraic integer, and if in addition 
the absolute term is + 1, ^ is a unity ; and further, that if ^ is an 
integer of the ?ith degree, then the order of '^ is the aggregate of 
numbers w of the form 

JUq ~r~ ^1 ^j *T • • • *^"ii 1 'J ) 

where x^... x^-i are rational whole numbers, every member of the 
order of ^ being an integer of the wth or some loAver degree. 
Dirichlet's theorem*, as modified by Dedekind and others, asserts 
that if the irreducible equation satisfied by ^ has r real and 2s 
imaginary roots, then the order of ^ contains r + s — 1 fundamental 
unities, e^, .... e,.+^.._i , which are such that every unity contained in 
the order is expressible in one and only one way as a product 

' ^ r+s-l' 

M'here t; is a root of unity contained in the order and m^, ... , m,.+.,_i 
are rational integers ; and that, conversely, every such product 
is a unity and a member of the order. The simplest cases of 
this theorem are those in which the equation satisfied by "^ is 
(i) a quadratic with two imaginary roots, (ii) a quadratic with two 
real roots, (iii) a cubic with one real and two imaginary roots and 
(iv) a quartic of which all the roots are imaginary. In the first 
case, and in this alone, there are only a finite number of unities in 
the order, and they are all roots of unity ; in the other cases 

* The theorem, when stated completely, has a wider scope, corresponding to a 
wider definition of an ' order ' than is given above : what is there defined is more 
properly called a 'regular order'. A general statement and proofs are given in 
Bachmann, Zahlentheorie, vol. v., eh. 8. 



112 Mr Todd, A 'particular case of 

mentioned there is one and only one fundamental nnit}'^ and in 
cases (ii) and (iii) + 1 are the only roots of unity which the order 
contains. In case (i) the theorem is easy to prove. In case (ii), 
if P + 2bt + c = is the equation satisfied by "^j the unities of the 
order are essentially the same as the solutions of the Pellian 
Equation 

x"" - (b- - c) y^ = ± 1, 

and Dirichlet's results can be deduced from the theory of this 
equation. In other cases the proof of the theorem is much more 
difficult. Mr Todd is concerned with the case in which ^ is the 
cube root of an integer — which comes under the heading (iii) 
above. If ^ = n, the general theorem ass( -rts (a) that the order 
of ^ contains an infinity of unities, (b) that they are all expressible 
in the form 

where 7 is a particular one among them and m is a positive or 
negative whole number, and (c) that every number of this form is 
a unity of the order. Mr Todd's essay contained an elementary 
proof of (6) and (c) ; the proof of (c) does not essentially differ from 
that given in text-books, though this was not known to him at 
the time, but the proof of (6) appears to be new and forms the 
subject of the following note. — H. T. J. N.] 



If ^^ — - n, and T =x + y^ + 2'^^ is a member of the order of ^, 
then 

r^ = nz + x"^ + 2/^2^ 

SO that r satisfies the cubic equation 

\x — t, y, z 

\ nz, x — t, y =0 ; 

I ny, nz, x — t 



*4 



hence it follows that F is a unity of the order if and only if x, y, z 
satisfy the Diophantine equation 

= af^- ny' + n^z^ — Snxyz = ±1 (i). 

It will be the object of this short note to give a simple elemen- 
tary proof of the fact that, if the existence of unities is assumed, 
then every unity of the order of ^ can be expressed in the form 



X, 


y, 


z 


nz, 


X, 


y 


ny> 


nz, 


X 



a theorem of Dirichlet 113 

where V is one particular unity of the order, and m is a positive 
or negative integer or zero. 

In what follows we shall restrict ourselves to the positive sign 
on the right-hand side of equation (i), since the negative sign 
merely replaces {x, y, z) by (— x, — y, — z). Also when x, y, z are 
all positive, we shall refer to {x + 2/'^ -f s^-) as a " unity of positive 
integers ". 

Suppose that 

r = *• + 2/^ + z"^" 
is any unity of the order of '^ : we shall first prove the following 
inequalities, viz. : 

|a--2/^|, JT/^-^^-j, |^^-^-a;!^2/V(3r) (ii). 

For, if we write a-^- x — y*^, 

^ = y^-z^\ 

and 7 = z^- — X, 

we see that the equation satisfied by x, y, z can be thrown into 
the form 

r(a^-h/3-^ + 7^)=2: 

so that we have 

a-^+/3^ + 7' = 2/ri 

and a + /3 + 7 = ]' 

From these two equations, assuming F to be constant, we find 
that the maxima and minima for each of a, /8, 7 are 

± 2/V(3r) ; 

from which the truth of the statement (ii) follows immediately. 

Further, we have the fact that if F = a; + 2/^ + z^" is any unity 
of the order and F> 1, then x, y, z will be positive. 

For, since F> 1, we have the inequalities 

\x-y'^\,\y'^-z'h^\,\z'^^-x\< 2/V3 < US. 

But, r being positive, the only possibilities of negative signs 
occurring amongst x, y, z are either (a) one negative and two 
positive or (6) two negative and one positive ; and in each case 
two of the inequalities given would take the form 

I X, + yU<^ I < 115, 

where \ and fi are positive integers and <^ ^ \/2, which is obviously 
impossible, except in the trivial case of one or more of the quantities 
x, y, z vanishing : it will be seen, on examining the inequalities, 
that the only possibility is x = l, y=0, z = 0, which gives r = l 
and so is excluded. Hence x, y, z must be positive. From this 



114 Mr Todd, A particular case of 

we can easily shew that if there exists an}^ unity in the order 
other than + 1, then there exists a unity of positive integers other 
than + 1 of which any other unity of positive integers is- a positive 
integral power. For suppose that T is any unity of the order 
other than + 1 : then by definition of a unity it follows that the 
three numbers 

-r, i/r, -i/r, 

will be unities of the order also : and of these four it is plain that 
one will be positive and greater than 1, i.e. it will be a unity of 
positive integers. 

Now take any number k> 1; then there will be only a finite 
number of F's for which k >T >1, since for any such F we must 
have kXoO, K>y>0, k> z>0. Hence there must be a unity 
of positive integers which is greater than + 1 and less than any 
other ; let this one be 7. 

Suppose that F is any unity of positive integers which is, if 
possible, not a positive integral power of 7. Then we shall have 
F >,7, so that we can assume that F is intermediate in magnitude 
between 7^ and 7^+S where p is some positive integer. But by 
the last part of Dirichlet's Theorem we know that 

F/7^ - 

is also a unity of the order, i.e. we have found a unity of the order 
which is less than 7 and greater than + 1, which contradicts the 
assumption that 7 was the least unity greater than + 1. Hence 
F must be a positive integral power of 7. Finally Ave have the 
result that, if F is any unity of the order, it can be expressed in 
the form 

where 7 has its previous significance and jo is any positive or 
negative integer or zero. For if F is any unity of the order, other 
than + 1, the numbers 

-F, 1/F, -1/F 

also will be unities, and one of these will be positive and greater 
than 1, and so will be expressible in the form 



where g- is a positive integer. Hence F can be expressed in the 
form 

where p is some positive or negative integer or zero. 

The result obtained can be put into an interesting geometrical 
form as we shall proceed to shew. 



a theorem of Dirichlet 115 

It is evident that any rational point (*•, y, z) in space of three 
dimensions can be regarded as being determined by its affix 
V^x-]ry^ + z^'\ where ^ is the real root of the equation ^^ = n : 
also the affix of any point determines a plane through that point 
and parallel to the asymptotic plane of the surface whose equation 
is i\=(jc? -\- ny"^ + n"z^ — Snxyz = 1 ; such a plane we shall call a 
" r-plane ". 

We shall now prove the following proposition : 

The V-planes of any two consecutive integj^al points on the 
surface A = 1, together with the surface itself, enclose a space of 
constant volume. 

The equation A = 1 can be written in the form 

[x + y^ + 2^-| [{x - y'^y + (2/^ - z^^-'f + {z"^^ - xy^ = 2 ; 

so that the section by the F-plane of the point (^, rj, f) will be 
given by the equations 

x" + 2/-^2 ^ ,^^^.^2 _ ,^^y^ _ ^2^^. _ ^^.y ^ ijY (i) 

and a; + 2/^ + z"^'- = T. 

Evidently the quadric (i) and the surftxce A = 1 are cut in a 
common section by the F-plane of the point (^, 77, ^). It is this 
quadric that we shall now examine. 

If by any rotation of axes it becomes ax- + hy^ + 6'^^= 1, we 
shall have (from the usual properties of invariants) 

a4-6 + c=r(l+ 7i^ + ^-), \ 

ah + bc + ca = ^ P^- (1 + m^ + ^-), I 

abc = ; j 

so that the quadric is evidently a cylinder, and the direction of its 
axis is the line x = y^ = z^". 

Suppose that c = ; then the area of a right section of the 
cylinder will be 

-rr/^/iah) = |^/V3 (1 + w^ + ^^). 

But the angle between the normals to the right section and the 
F-plane is the same as the angle between the two lines 

and x = y/'^ = zl'^^; 

i.e., is cos-i {3^7(1 + n^ + ^-)} : 



116 il/r Todd, A partictdar case of a theorem of Dirichlet 
hence the area of the section made by the F-plane will be 
27r V(l + «^ + ^-)/3nV3 F. 

Now the perpendicular distance between two near F-planes, 
r and r + ST, is 8r/V(l + >i^ + ^-), and so the element of volume 
enclosed by these two planes and the surface A = 1 will be, to the 
first order, 

27r ST 
3nV3'^' 

Integrating this between the limits F = 7^+^ and F = 7^' (i.e. the 
F-planes of any two consecutive integral points), we find that the 
volume of the space enclosed is 27r log 7/3/1^3 ; and since this is 
independent of the integer p, our proposition is proved. 



a 



Mr Mordell, On Mr Ramanujans Empirical Expansions, etc. 117 



On Mr Ravianujan's Empirical Expansions of Modular 
Functions. By L. J. Mordell, Birkbeck College, London. (Com- 
municated by Mr G. H. Hardy.) 

[Received 14 June 1917.] 

In his paper* "On Certain Arithmetical Functions" Mr 
Ramanujan has found empirically some very interesting results 
as to the expansions of functions which are practically modular 
functions. Thus putting 

(^X^{<o„ CO,) = r [(1 - r) (1 - rO (1 - r^) . ..? = S T {n) r-, 

he finds that 

T{mn) = T(m)T{n) (1) 

if m and n are prime to each other ; and also that 

2 ^^ = Ul/(l-T(p)p-^+p^^-) (2), 

n=l "' 

where the product refers to the primes 2, 3, 5, 7 He also gives 

many other results similar to (2). 

My attention was directed to these results by Mr Hardy, and 
I have found that results of this kind are a simple consequence 
of the properties of modular functions. In the case above 

A (&)i, Wa) (r — e'"'^"", (o = coi/coj) 

is the well-known modular invariant (^f dimensions — 12 in co^, 0)2, 
which is unaltered by the substitutions of the homogeneous 
modular group defined by 

ft)/ = aoii + bco.2, (o./ = Cftji + dw2, 

where a, b, c, d are integers satisfying the condition ad —bc = l. 

Theorems such as T {mn) = T {m) T {n) had already been 
investigated by Dr Glaisher f for other functions ; but the 
theorems typified by equation (2) seem to be of a new type, and 
it is very remarkable that they should have been discovered 
empirically. The proof of Mr Ramanujan's formulae is as follows. 

Let f{(Oi, 6)2) be a modular^ form of dimensions ~ k in coj, co,, 
which is a relative invariant of the homogeneous modular gi'oup, 
so that /(fw/, w.^)l f{(o^, 0)2) is a constant independent of «i, (Oo,. 

* Transactions of the Cambridge Philosophical Society, vol. xxii. , no. ix. , 1916. 

t See, for example, his paper " The Arithmetical Functions P (m), Q (in), fi (;n) ", 
Quarterlij Journal of Mathematics, vol. xxxvii., p. 36. 

+ For an elementary introduction to the modular functions, see Hurwitz, 
Mathematische Annalen, vol. 18, p. 520, 



118 Mr Mordell, On M^- Ramanujan's 

Let also p be any prime number; then we may take 

(&)i, pa>^, («! + CDa, p(o<^ . . . (ft)i + ( jO — 1) fUo, J9&)y), ( ptWi , &),.) 

as the reduced substitutions of order j)- Then for many modular 
forms* it is well known that unities ^, fu, ^i, ••■, |>-i can be 
found so that 

is also a relative invariant of the modular group. 

This is also true of the quotient Q = (fi/fioy^Jwo), which is a 
modular function of co. Q is really an automorphic function whose 
fundamental polygon (putting (o = x + Ly)is that part of the upper 
CO plane bounded by the lines x = ±^ and external to the circle 
a;2 + 2/2= 1, but we reckon only half the boundary as belonging to 
the fundamental polygon. The only infinities of Q are given by 
the zeros of /(&)i, 0^2) = 0, and if these zeros are also zeros of the 
numerator of at least the same order as of the denominator, 
it follows that Q has no infinities in the fundamental polygon. 
Hence Q is a constant, so that (f)~Qf((Oi, &>.,). 

Suppose now that 

where ^1 = 1. Then 

becomes (^)" S's' |.^,r^/^e^-'VP 

and in the examples with which we are concerned all the terms 
will vanish, because of the summation in X, except those for which 
s = (mod p), and then the sum will become 



^ pA, 



Hence we have 



Equating coefficients, we find, if s is prime to p, 
pAsp = Qp^Ag. 

* This fact is intimately connected with the transformation equations in the 
theory of the modular functions. We may note that it is often more convenient to 
select the reduced substitutions in different ways. 



Empirical Expansions of Modular Functions 119 

Taking .9=1, pAp = Qp", 

so that -Agp^ AsAp... (3). 

If no restrictions are placed on s we find, by equating coefficients 
of rP', 

^^s+^,Asp.= QAsp. 

From this 

Asp.-ApA,p + ^p''-'A. = (4). 

From equations (3) and (4), we can prove that A^n = A^An 
if m and n are prime to each other. For all we really have to shew 
is that, if 2? is a prime and s is prime to j), then ^,,^a = AifAp\. 
But from equation (4), we have 

Asp\+2 - ApAspK-hi + ^p"-^ Agp\ = 0, 

and Ap\+2 — ApAp\+i + ^j)''-^ A^k = (4a). 

Hence the theorem follows by induction, for if it is true for \ 
and X + 1 it is true for X, + 2. But it is true for X = and for 
X= 1 (equation 3): hence it holds universally. 

We notice also that equation (4a) is a linear difference equation 
of the second order with constant coefficients*. Hence, since 
A,= l, 

1 + ApX + Ap-iX" 4 Ap^x^ + . . . = 1/(1 - ApX + ^p^-Kx-), 
from which, by putting x = l/jf, 

pS p2S p3S '^ ••■ / ^ p' p^^ J' 

Putting for p in succession the primes 2, 3, 5 ..., multiplying 
together the corresponding equations, and remembering that 
A,nn = A^nAn if vi and /; are prime to each other, we have 

« 

where the product refers to the primes 2, 3, 5 — 

The simplest application of these results is given by the 
function 



fa {co„ &).3) = A ^— ft),, (Ooj 



This is obvious if we put fx^= ApK. 



120 My^ Mordell, On Mr Ramanujan's 

where a is a divisor of 12. Its expansion in powers of r involves 

only positive integral powers of r and starts with I — j r. 

/a(&)i, 6)2) is not however an invariant of the modular group. 
We can avoid this difficulty by taking /(co,, eo,) = [A(ft)i, w^)]"'^'^. 
In this case* 

Kp-ni 

^f(pco^,a),) =L(-1) ^ iyA(p&)i, &)2)J , 

provided we exclude p = 2 and p = S. Putting for the moment 



/m \" / "= —4-'! 

we find 



(—:) [iyA(a,„a,,)]- = S^^,ri-^^^ 






CO p-i OKpni ( a \ 2K7ri /■ «■ , „N 1 



2 

K = 

= 22 e"~6 

S = 0/<: = 

But since ^ 4" ^ or 3, p' —1=0 (mod 12). Hence 
2 eP i 12 +*;=o, 

K = 

unless a{\ —p-)/l2 + s = O(mod_p), that is a 4- 12s = (mod jd), and 
is then equal to p. Hence ^ is a power series in r^^^^ (really of the 
form ?-'^/i2(^ + Br+ Gr" ...)), starting with r'<"'+'^^s)i\ip^ where s is 
the smallest positive integer for which a + 12s = (mod p). Now 
the only zeros of /(wi, oa^) = in the fundamental polygon are at 
ft) = iX) or r = 0, and 

/(o)i, «2) = (^y '■'"' (1 + ^^' +^^ ...). 

But putting a = 1 2/6, so that h is an integer, 



a + ] 2s 1 + 6s 1 a 



12p bp "6^12' 

since 1 + 6s = (mod p). 

Hence <^lf{ui^, Wa) is a constant, and equations (8), (4), (5) 
apply to the function 

V a 
We note also that |= (- l)«(^-i)/2. 

* Hurwitz, I.e., p. 572, or Weber, Lelirhuch der Algebra, vol. 3, p. 252 



i 



«/12 



Empirical Expansions of Modular Functions 121 

When p = 2, these theorems hold if a = 4 or 12. For the 
functions ^\f\* are selected as before, and it is clear that the 
argument above applies, as « (1 —p^)l\2 is an integer. 

Lastly, when p = S, these theorems hold if a = 3, 6, 12, and the 
functions ^k/k* are selected as before. 

Hence, altering our notation, we have the following theorems. 
If a is a divisor of 12 and 

p 12 24 36 -j,,^ 00 

r [(l -r") (l-r«) (l -r") .-..J^" = S fa(n)r'\ 

n = \ 

then fa {m)fa (n) =fa (mn) (6), 

if m and n are prime to each other ; and 

^ />- (^) ^ n 1 /(i -^^ ^^^ I ^~ ^K^^''~' ] (7). 



n=i n' / V p' P~ 

The product refers to the primes 2, 3, 5, etc., except that 

p = 2 is excluded except when « = 4, 12, 
and jt) = 3 is excluded except when a = .3, 6, 12. 

We notice that when a — 1, 2, 3, or 6, jt? = 2 is not excluded 
as a factor of say m in (6), as in this case fai'ni) and fa (mn) are 
both zero. Similarly for jo = 3 when a = 1, 2, 4. 

The result (6) is given by Mr Ramanujan wh6n a = 12, as are 
most of the cases of (7). We shall now shew how in many cases 
we can find simple expressions for fa{p). 

If a = 1, it is known that, by a result due to Kulerf, 

r[(l-?-i2)(l-r'^^)...]2 = [ 2 (-If ?• 2 J 

— 00 

(6ot + 1)2+(6w + 1)2 

= SS(-l)'"+"r 2 

= tt(-l)^r^'^^^\ 

where ^ = 3 (m + ?i) + 1, r) = n — m, so that ^, ij take all integer 
values satisfying ^ = 1 (mod 3), ^ + 77 = 1 (mod 2). 

Hence f(p) — 2(— 1)'' if p = ^^ + 9r}^ and we take both ^ and 77 
to be positive. If ^ = — 1 or ±5 (mod 12), f{p) is obviously zero. 
This is Mr Ramanujan's result (118). 

If a = 2, it is known (Klein-Fricke, vol. 2, page 374) that 

r [(1 - r«) (1 - r'^) ...]* = ^2 (- 1)^ ^r^'+^^^+^r,^, 
where f , r) take all integer values satisfying 

1=2 (mod 3), 7; = 1 (mod 2). 

* Hurwitz, I.e., vol. 18. 

t See also Klein-Fricke, Modulfunktionen, vol. 2, p. 374. 

VOL. XIX. PARTS II., III. 9 



122 Mr Mordell, On Mr Ramanujans 

Hence /.(jt>) = 2 i(- 1)^| 

extended to the solutions of p= ^ + S^rj + Srj- for which 

^ = 2 (mod 3), 17 = 1 (mod 2). 

This * can be written as /2 ( p) = ^v, where p = Su^ + v'-, u is positive 
and V = 1 (mod 3). Also f\ {p)=0 if p = - I (mod 3). This is 
Mr Ramanujans result (127). 

If a = 3 we have, from Klein-Fricke, vol. 2, page 377, 

r [(1 - r^) (1 - r«) ...]« = - 12 (p - 7)') rf=+''^ 
where ^ takes all even values and rj all odd values. Hence 

if p = ^^ + 7)"^, I is even, 97 is odd, and both f and ■?; are positive. 
Also /3( j9)= if ^ = 3 (mod 4). This is Mr Ramanujans result 
(123). 

If a = 4, then by Klein-Fricke, vol. 2, page 373, 

r \(l — r^) (1 — r^) . . .? = +2p r^""*"^^*''+^''', 

where ^, rj take all values for which |^ = 2 (mod 3). 
Hence /4 (^) = ^Sp extended to all the solutions of 

^ = p + 3^77 + '67j\ 

where | = 2 (mod 3). Thisf can be written as /4 ( jd) = 2 (v" — 9vu-), 
where p = 3w'^ + ^^ tt is positive, and y = 1 (mod 3). This is 
Mr Ramanujans result (128). 

When a = 6,/6 {n) is known by means of the representations of 
n as a sum of four squares. Mr Ramanujan has overlooked the 
fact that in his result (159) 2c^ is —J\{p)' The theorem 

/b (wO/e in) =/6 (mn), 

is due to Dr Glaisher. 

When a = 12, we have Mr Ramanujan's results given as 
equations (1) and (2) in this paper. 

He also gives results when a = i , |. 

* When ^ is even put ^ = 2v, 7] = u-v, and when | is odd put ^ = du-v,'r) = v-7i. 
Both these cases are admissible, and we find that p=v^ + Bu" and v = l (mod 3). 
Also S {-1)^ ^=2v + 2v - {Su -v) - ( -3u-v) = Gv, where now w is taken as positive. 

t See the last footnote. In addition to the two cases there considered, 7/ even 
is admissible. Put then 7] = 2u, ^— -v -3ii, from which p = v^ + 3ii'^ and v = l 
(mod 3). 



Empirical Expansions of Modular Functions 123 

where [- j and ( ] are symbols of quadratic reciprocity, so 

that (:-^) =(-l)V , (I) = 1 if i, EE ± 1 (mod 12), and (|) = - 1 



if jj = ± 5 (mod 12). If /; = -S, ('^\ = 0. 

These are particular cases of Euler's theorem that 

s-^^^^=ni/(i--^^^) 

if the function y" satisfies the condition 

f{m7i)=f(m)f(n), 

the product refers to any group of primes, and the summation to 
all numbers whose prime factors are included in the group. Thus 

r (1 _ r2J)(l _ r'') ... = i (_i)»y.(««+i)^= v f'}\ ,,.«^ . 

-c»' 1,3, 5... V''^/ 

and r [(1 - ?•») (1 - r^^) ...]'= t (- 1) ^ wr^' = 5 f ^ nr"\ 

1,3,5... 1,3,5... \ n J 

Finally, Mr Raman ujan gives two results, equations (155) and 
(162), of which the first is 

5 -^ = lT9i=^ n 1/(1 - 2c, p-' + (- IV P'-n 

where Cp = u^ - (4w)- and u and y are the positive integers satis- 
fying u^ H- {^v)'=}f: But if JO = 3 (mod 4), Cp is taken to be zero, 
/lo (w) is defined * by 

=^r[(l-r^)(l-r^)(l-r«)...]"/[(l+r)(l-r-^)(l-|-r^)(i-r^)...h 
and this is equal tof 

ii 2(a,- + i2/)^r^'+^'. 

— 00 —00 

The second result is 

I'^^^lT^^W-Sc^p-^+i'^-'X (1^ = 3,5...), 

" The functions /io(?i),/i6 (n) arise in iinding the number of representations of 
n as a sum of 10 and 16 squares respectively and the series 2 S (a; + i?/)-*r^""'"*' is 
well known in this connection. 

t From this, it follows that the result can be also proved as a particular case 
of Euler's product. 



124 Mr Mordell, On Mr Ramanujan's Empirical Expansions, etc. 
where /le (?i) is defined* by 

1 

= r [(1 + r)(l-r2)(l +rO(l -r^) . ..]'/[(! -r"){l-7^){\-r') ...]«. 

Mr Ramanujan overlooks the fact that Cy = ^fu{p)- 
These results can be proved by aid of the principles used in 
finding equations (3) and (4). We should however have to consider 
now invariants of a sub-group of the modular group, and it seems 
hardly worth while to go into details. 

* The functions /lo ('O'/ieC**) arise in finding the number of representations of 
n as a sum of 10 and 16 squares respectively and the series 2 S [x + njY r* "^^^ is 
well known in this connection. 



PROCEEDINGS AT THE MEETINGS HELD DURING 

THE SESSION 1916—1917. 

ANNUAL GENERAL MEETING. 

October 30, 1916. 

In the Comparative Anatomy Lecture Room. 

Professor Newall, President, in the Chair. 

The following were elected Officers for the ensuing year : 

President: 
Dr Marr. 

Vice-Presidents : 

Dr Fenton. 
Prof. Eddington. 
Prof. Newall. 

Treasurer : 
Prof. Hobson. 

tSecretanes : 

Mr A. Wood. 
Mr G. H. Hardy. 
Mr H. H. Brindley. 

Other Members oj the Council : 

Dr Duckworth. 

Dr Crowther. 

Dr Bromwich. 

Dr Doncaster. 

Mr C. G. Lamb. 

Mr J. E. Purvis. 

Dr Shipley. 

Dr Arber. 

Prof. Biffen. 

Mr L. A. Borradaile. 

Mr W. H. Mills. 

Mr F. F. Blackman. 



126 Proceedings at the Meetings. 

The following was elected an Associate of the Society : 
W. Morris Jones, Emmanuel College, 

The following Communications were made : 

1. Methods of investigation in atmospheric electricity. By 
C. T. R. Wilson, M.A., Sidney Sussex College. 

2. On the functions of the mouth parts of the Common Prawn, 
By L. A. BoRRADAiLE, M.A., Selwyn College. 

3. On the growth of Daphne. By J. T. Saunders, M.A., Christ's 
College. 

4. A self-recording electrometer for Atmospheric Electricity. By 
W. A. D. Budge, M.A., St John's College, 

5. An axiom in Symbolic Logic. By C. E. Van Horn. (Com- 
municated by Mr G. H. Hardy.) 

6. On the expression of a number in the form aar -i- hy- + cz^ 4- du". 
By S. Ramanujan, Trinity College. (Communicated by Mr G. H. 
Hardy.) 

7. A reduction in the number of primitive propositions of Logic. 
By J. G. P. NicoD, Trinity College. (Communicated by Mr G. H. 
Hardy.) 



November 13, 1916. 

In the School of Agriculture. ^ 

Dr Marr, President, in the Chair. 

The following were elected Fellows of the Society : 

F. W, Green, M,A,, Jesus College, 
R, I, Lynch, M.A. 

The following was elected an Associate of the Society : 
N. Yamaga, Fitzwilliam Hall, 

The following Communications were made : 

1. The surface law of heat loss in animals. By Professor Wood. 

2. Inheritance of henny plumage in cocks. By Professor Punnett 
and Capt. P. G. Bailey. 



Proceedings at the Meetings. 127 

3. On extra mammary glands and the reabsorption of milk sugar. 
By Dr Marshall and K. J. J. Mackenzie, M.A., Christ's College. 

4. Experimental work on clover sickness. By A. Amos, M.A., 
Downing College. (Communicated by Professor BifFen.) 

5. Bessel's functions of equal order and argument. By G. N. 
Watson, M.A., Trinity College. 



February 5, 1917. 

In the Sedgwick Museum. 

Dr Marr, President, in the Chair. 

The following was elected a Fellow of the Society : 

F. W. H. Oldham, B.A., Trinity College. 

The following Communications were made : 

1. Submergence and glacial climates during the accumulation of 
the Cambridgeshire Pleistocene Deposits. By Dr Marr. 

2. Glacial Phenomena near Bangor, North Wales. By P. Lake, 
M.A., St John's College. 

3. The Cretaceous Faunas of New Zealand. By H. Woods, M.A., 
St John's College. 

4. Exhibition of the Fruit of Chocho Sechium edule : remarkable 
in the Nat. Order Cucurbitaceae, native of the West Indies and culti- 
vated also in Madeira as a vegetable. By R. I. Lynch, M.A. 

5. The limits of applicability of the Principle of Stationary Phase. 
By G. N. Watson, M.A., Trinity College. 

6. The Direct Solution of the Quadratic and Cubic Binomial 
Congruences with Piime Moduli. By H. C. Pocklington, M.A., 
St John's College. 

7. On the Hydrodynamics of Relativity. By C. E. Weather- 
burn, M.A., Trinity j(^ollege. 

8. The Character of the Kinetic Potential in Electromagnetics. 
By R. Hargreaves, M.A., St John's, College. 

9. On the Fifth Book of Euclid's Elements. (Fourth Paper.) By 
Dr M. J. M. Hill. 

10. On a theorem of Mr G. Polya. By G. H. Hardy, M.A. 
Trinity College. 



128 Proceedings at the Meetings. 

February 19, 1917. 
In the Botany School. 

Dr Marr, President, in the Chair. 

The followiug Communications were made : 

1. (1) On an Australian specimen of Clepsydropsis. 

(2) Observations on the Evolution of Branching in the 
Ferns. By B. Sahni, B.A., Emmanuel College. (Communicated by 
Professor Seward.) 

2. On some anatomical characters of coniferous wood and their 
value in classification. By C. P. Dutt, B.A., Queens' College. (Com- 
municated by Professor Seward.) 



CONTENTS. 

PAGE 

The Direct Solution of the Quadratic and Cubic Binomial Congruences 
tuith Prime Moduli. By H. C. Pocklington, M.A., St John's 
College .57 

On a theorem of Mr G. Polya. By G. H. Hardy, M.A., Trinity 

College . 60 

Submergence and glacial climates during the accumulation of the Cam- 
bridgeshire Pleistocene Deposits. By J. E. Maer, Sc.D., F.R.S., 
St John's College . . . . . . . . . .64 

On the Hydrodynamics of Relativity. By G. E. Weatherburn, M.A. 

(Camb.), D.Sc. (Sydney), Ormond College, Parkville, Melbom'ne . 72 

On the convergence of certain multiple series. By G. H. Hardy, M.A., 

Trinity College . . .86 

Bessel functions of large order. By G. N. Watson, M.A., Trinity 

College 96 

A particular case of a theorem of Dirichlet. By H. Todd, B.A., 
Pembroke College. (Communicated, with a prefatory note, by 
Mr H. T. J. Norton) Ill 

Oil Mr Ramanujan's Empirical Expansions of Modular Functions. Bj^ 
L. J. MoRDELL, Birkbeck College, London. (Communicated by 
Mr G. H. Hardy) . . .117 

Proceedings at the Meetings held during the Session 1916 — 1917 . , 125 



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PROCEEDINGS 

OF THE 



Extensioits of Abel's 'Theorem and its converses. By Dr A. 
KiENAST, Kiisnacht, Zurich, Switzerland. (Communicated by 
Mr G. H. Hardy.) 

[Received 26 September 1917.] 
Introduction. 
Abel pruved in 1826 the theorem : 

n 

''If liui !£ (^ ejists and is finite, then 



lim % a^x" = lim ^ a„." 



Let us write 



Sn 


1 


(Ik 


t""' 


_1 


S6V 


n 


n 


' ! 


^(A + 1) 


_1 


i (« 




n 


1 / 



Then H(3lder* proved in 1882 

Theorem 1. If lim t^^ exists and is finite, then 



.(1). 



lim S ai^x" = lim t]^ . 

* Bromwifh, Iiijinite scries, p. 313. 
VOL. XIX. PART IV. 



10 



130 



Dr Kienast, Extensions of 



In 1897 Mr Tauber, and in 1900 Mr Pringshciui, published the 
following converse of Abel's theorem : 

Theorem 2. The two conditions 

lim 2 ««.'■" = I ( finite), 

1 " 

hm - 2 KCi^ = 

n-*'Oo ^^ 1 

are each necessary for the convergence ofX a^, i.e. for the existence of 

n 

lim ^a^ — l; 

n-*-ao 1 

and, taken together, they are sufficient'^. 

In the present paper I replace the means (1) by 



^ (2.) 



1 

« ?l 1 


{n = 2, 3, . 

(«. = X + 2, 

0^ = 1,2,. 
{n = 2, 3, . 


•■) 


Defining r\'^ by 


■) 
•) 



r;'+^^ = V - 7-(^) Ol = X + 1, X + 2, . . . ) 



I prove, in Part I, 

Theorem 3. Tlie tivo conditions 



•(3), 



lim S a^x" = I (finite), 

x-*l 1 
Broniwjch, IntinlU series, p. 251. 



Abel's TJieurcDi and its converses 131 



are each necessary for the existence of 



lim s^^^ = / ; 



(tnd, taken together, they are suMcient. 

This theorem includes the analogue of 1 : 

Theorem 4. If lim s\^ =1 exists and is finite, then 

lim 2 aK*" = I- 

x^\ 1 

It is easy to verify that lim ij,^* = lim 6-J^* if \ = 1 or 2 : for 

higher values of X this relation certainly holds if both limits exist, 
as follows from Theorems 1 and 4. 

In Part II, I propose to extend Theorem 3 to certain other 
mean values; and Part III contains some general remarks about 
the converse of Abel's theorem. 

Part I. 

]. In the researches which follow I have to make use of the 
following theorems. 

n 

Theorem 5. // lim 2 a« = lim s,,. = I (finite), 
and Uk is positive and 

n, 

lim 2 6k = lim t^ = ^ , 

1 " 
then lim ~^b^s^= lim s„ = / (-1). 

This theorem is due to Stolz*. 

Theorem 6. Suppose that b^ is positive and 26« divergent; 
a)id let D be the region defined by 

p<2cosi|^ {\ylr\^\lr^<^'Tr), 

where i — x = pe'''^. Further suppose that 

tb,\x''\/\^b^x''\<G, 

where (f is a finite constant, for all values of x inside the region D. 

* Bromwich, Injinitc series, p. 378. 

10—2 



132 



iJr Kienast, Extensions of 



Finally suppose that a^lbn tends to tlie lint it I luhen n tends to 
infinity. Then 

lim (S a^afjt b^,/f) = I (o), 

tuhen X approaches 1 along any path inside D. 

This theorem is due to Pringsheim*. It is to be supposed 
throughout this paper that, when x tends to 1, its approach to 1 
is along some path inside B. 

Theorem 7. If the radius of convergence of F {x) = '!E a^x" 

is r, then 

lim anX'"' =0 (\x\< r). 

If the radius of convergence of Q(x) = Xa^x" is unity, it will 

I 
remain unchanged if Q(x) be transformed in any of the following 
ways : 

(i) by suppressing a limited number of terms, 

(ii) by multiplying by ./;°", a being an integer, 

(iii) by multiplying by ^ _ ^ , = ^ «'\ 

(iv) by integrating term by term, 

(v) by differentiating a limited number of times. 

Using in succession one or other of these operations, there 
result the following power-series, all with radius of convergence 
unity : 

xF' (x) — S Ka^x", 



\(x)= —A^F'{x)] = lrl'>x, 

X — X I 



F,(xy= ^ 



^ 1 ~l '^ ) 

- Pj (x) dx \ =X r,^ x", 



1 — X 



P.^ (x) dx 



Pg (x) dx 



= ir^^x'^\ 






* 4Qta Mathematica, vol. 28, p. 7. 



Abel's Theorem and its converses 133 

Thus the series S r*^' , ,.r'^"^^~" converges if Lt I < 1 ; the same 
is the case with ^^ 

or i r(\+;\«,-'^^^-l 

J) + A — 1 

Differentiating the last series (X — 2) times, we obtain 

SO^ + 1 )0; + 2) ... (^9 + X - 2)7-<^+;':;,a''^ ; 
which gives 

00 

Theorem 8. // the radius of convergence of P (a-) = 2 a^.r" is 

1 
unity, then for every \x\<\ 

r (A + k) n+A-2 r. 

hm ?■ , , , ./• = 0, 

/l+A — 1 ' 

hm - r ic = 0, 

lim (>i + 1 ) (7? + 2) . . . (n + \ - 2) r^+^^l^ «^" = 0. 

2. The demonstration of Theorem 8 depends on certain 
identities. The formula 

' n " 

leads, by successive summation, to the series of equations 

(6). 



(2) (1) 1 (2) 

n I) j^ /( 



^(A) _ _(A-1I _ J- ^.(A) 



If lim s =1 exists, then, bv Theorem 5, lira s . also exists 

and is equal to /, and therefore one of these identities gives 

r 1 (A+i) r\ 
Inn - ?' = 0. 

Theorem 9. If lim .s^^' = I exists and is finite, then 

n-*-x 

r 1 (A + l( n. 

lim 7" = 0. 

7l-»-X " 

Thus the second condition of Theorem 3 is necessaiy. 



134 



Dr Kievast, E.vtensioiis of 



3. I proceed to prove some other identities. We have 



,(1) 



i KQ 

K 
1 



a« = - r -/ 



U<i>_-,.(i' ! 









,(^) 



,(A-|-1) 



1 K " 



M) 



Jl + 1 M 



.(7); 



and by successive substitution we find 

1 

n ' " 

(3) Q,„.,(3) , O/., 1\^(3) /„, 0\ ,,(3) 



" ' - n—l> n + l n 



+ /2) +1|,.C2)_,,(2) } 



= (n + 1) C - 3m'- , + 3 (.^ - 1) r^' - {n - 2) ,-- ^ 

fl '■ n n-l> 

Writing 



, . (n + X - yu, - 2) r 



(\+k) ^ ^ 

M+A — M — 1 K.K.m 



.(8). 



we can easily verify that 

6r, =tT>.i — (X — l)0. ,, 

\,K,n. \ + l,K,n ^ ■' A,K+l,w 

Moreover 

Developing a^ in this way we obtain, after a finite number p 
of steps, the formula 



cin= S 



A+K = P 



A, K A, K, n \^ 



iUp)_, 



.(p) 



.(10). 



The upper index of all the 7''s is the same throughout this expres- 
sion. For the present purpose it is not necessary to determine the 
coefiicients c^ ^, which are integers. 

In consequence of the definitions of s and r we have 

7>) = (n>vi). 



Abel's Theorem and 'its converses 



135 



But it is not difficult to see that the recurrence formulae (8) and 
(9) still hold, if the number p of steps exceeds the index n. It is 
only necessary to put ?-^" = whenever n > m. The form of the 
relations (7), viz. 

is the cause why the coefficients of the remaining terms are not 
influenced by the fact that some terms disappear. Thus 



'I'' A, K A, K, n 

1 A+K=p \n=\ 



+ S -a;^{7>'_7» I (11). 

n ' » n-V V ^ 

n=p "• 

To evaluate the first of these sums we have 

(A+k) 



(1 - xf s (« + 1 ) . . . {n + X - 2) r;;;;:, ^" 

n = \ 

in 



,»H+1' 



A, (C, 7J 



\,K,Vl + V ' 



say. Each of the i\(\ + l) terms contained in the second sum 
has the form 

K{p + i)(p + 2)...{p + x-2)rl^;;!y+', 

{fi= V, V + 1, ...\; V = 1, 2, ...X; p = m — v + 1). 

Therefore, by Theorem 8, we have, for every j x \ < 1 and any 
finite X, 



limS G,^^^^.r"^(l-a^^{n + l){n+2)...{n+X-2)rl^^;;^^x'\ 



«4-*-» 1 1 

The second sum in (11) gives 



'"- 1 . N , \ U-1 -\ 

vif,.(p)_^,.W }./= S r^' 



(p) 



n + 1 



;i -'^ ^ (p) m 

m «' 



= {\-x) S - - 7>' A'" + S ~ , 7>* X'' + i 7>^ .^'" 

p II + \ " p II (v + 1) " m '« 

(12); 



and again, by Theorem 8, we have, for every , x j < 1 and any 
finite p. 



lim % x' {r^'^ - 9>\} = (1 ^ X) 2 — ^ 7>\*'" + ^—^ 
m-» p *'' " ""' p» + l " ,n(n- 



+ 1) " 



,.(p) _.,."_ 



136 Dr Kienast, Extensions of 

Thus we have established 

Theorem lO. //' l.a^x^ has unity as radius of convergence, 
then 



S ancc" = t c. 

1 A+K=p ' 



00 



M = l 



.(f^) ^," 4. V 1 ,,(P) ,..« 



(13) 

4. Equation (13) has now to be considered when .r->l. To 
the first terms on the right-hand side we apply Theorem 6, which 
gives 

00 

2(n + l)...(n + x-2)r;;)^_^^-'^ ^» 

hm . = lim -Jl±h=l_ , 

T^^— jyi 2 (/i + 1) . . . (h +X- 1)^'* 

Again, by Theorem 6, 

CO 1 -. 

lim (1 - ^;) 2 -4t ^'I^^'" = lim - r^"* : 
and finally Theorem 8 gives 

Theorem 11. If ta^x" has unity as radius of convergence, 
and if 

\im~rl^^ = 0, 

J » 00 1 

then hm 2 a^ x" = lim 2 - j-^"' x" 

x^l 1 0,-^! p «(/?+!) " 

5. Furthermore, equations (7) and (12) lead to 

«'' 1 , , , , m-l n 

'"•"1 1 m—l 1 1 

P w + 1 " p 7z(w+l) ■" ■ ^m ^'^ •'' • 

Putting X = 1, it follows that 

-2 ^ 7^^'"^= 2 L^,>+i» , 1 ,.(p+i) 



Abel's Theorem and its converses 



1:17 



Hence 

Theorem 12. //' Xok-"''" Iki-'^ <inity as f(((Iiiis of convergence, 
and if 

1 






then 



.(p) 



'^^ 1 



r " = lini 2, 



,(p+i) 



I 



Another identity is acquired by developing s\f (;? = p + l^ 
/) + 2, ...) in the form 



P + 1 



P+2 



>)_JP) 1 



j(P-l) 



pVi 



p+2 



1 ^(p-1) 



1 V.*"-^' 



V/i(?W — 1) 



I 



"-1 1 ( \ 



VI (m + 1) 



(p) 



til en 



Theorem 13. If .s"' and r ''' are defined as in (2) and (3) 



S^'^= S 



.(P) 



If lim ,9''^* = I exists and is finite, then, by Theorem 5, 

II ^x 

r ('^+1) 7. 
hm s = I : 

II 

ll->-orj 

and by Theorem 13 

S 1 



.(A+i) 



= /. 



.(^+1) 



Therefore by Abel's theorem 

TV -'- (A + 1) III V ■'- 

inn 2. - "Tvr ,r — z — 7 Ts 

x^i A+i ?H (?/i + 1 ) '" A+i m (m + \) '" 

On the same assumption, Theorem 9 gives lim -■^•^^"^ ■ =0; 
and therefore Theorem 1 1 gives 



= /. 



lim 2 n.^x" — lim 2 



^,(A+l) « 



1X+1«(" + 1) " 



" 7 



1^8 Dr Kienast, Extensions of 

Thus we obtain 

Theorem 4. If Km s\';^ = I exists and is finite, th 



en 



Mni^a^x" = I. 

X-»-l 1 



The first condition of Theorem 3 is therefore necessary. 

6. To demonstrate the rest of the assertion in Theorem 8, it 
follows from the hypothesis lim \/^^'^ = that Theorem 11 is 
applicable. Thus the assumptions are transformed into 



lim 2 - ^rl^+'\-=.i 
lim-7^;^+^'=0. 

71-*- 00 ^i 

From this last equation follows 



I '' 1 <>.^^^ .. 1 



.(A+l) 



lim:^S -^r(^+^'=lim^^^ = 

,,-^00 nx+lK + l « ,,^^ 71 + 1 

Hence Theorem 2 can be applied to the series 2 - x'' • 

11 , • • n{n + \) ' 

and the conclusion is that 

lim t -r~~^, = I. 
Theorems 12 and 13 now yield 

lim 4^^=/, 

with which the proof of Theorem 3 is completed. 

7. The foregoing deductions are valid for X = l, 2,.... For 
X = they still hold, except those in § 6. This case requires the 
proof of the following special case of theorem 2 : 



Theorem 14. // 



limS^-i--r^V = /, 
^^1 1 n(n + 1) « 

lim ~r^'^ = 



I 



Abel's Theorem and its converses 



180 



then 



r'''=l. 



1 n(n+l) 

This proof is actually given by Mr Tanber, and is therefore 
the basis of the theorems of this paper. 



Part II. 

8. Let 6k denote the terms of an infinite sequence of positive 
real numbers, which have the properties 



(1) lim S b^ = lim tn = oo 



.(14), 



(2) 



n 1 



tends to a limit or oscillates between finite limits. Then 
Theorem 16. The two conditions 

QO 

lim S «'««"' = / (finite), 

x^l 1 

1 » 
hm --l^AaA = 0, 

n-*-cc I'll 1 

are each necessary for the convergence of S a«, i.e. for the existence of 

II 
lim %aK= I : 

and, taken together, they are sufficient. 

Abel's theorem states that the first of these conditions is 
necessary. 

If lim Sn = I, then lim a„ = 0, and by Theorem -5 



1 " 

lim - S 6a.?a-i = lim 



ii^X ^'11 



1 n 1 n 

--2&ASA--S6AfO 

'« 1 f»i. 1 



/. 



The identity 



1 " 1 " 

f ft 2 f« 1 



n(^w gives, as a consequence of lim ,<?„ = /, 



1 
Therefore the second condition is necessary too. 



nm ^ uxu\ 



^^^ Dr Kienast, Extensions of 

9. To prove the converse, we require two identities. If 

J^ hi 



n 



we have 

X a.af =f ^ + 2 - |^« - p,_,] x^ 



/ — 

1 ''K + l 



'«+r 



^« 



a' 









w-1 

1 ^K+l 1 t^ ^K + i 

Putting a; = 1, this gives the identity 



.(15). 



_\^t<±i-t^ Pk . p 



L 



t ^f 

''K+i hi 



•(16). 



If we suppose lim ^ = 0, it follows that 

lim^.r'»=0 

for every I .t | < 1 ; and, by Theorem 6, 

lim(l-^)i^,,- = 0. 

Now passing in (15) to the limit (first n -^ oo and then 
a?— » 1 we find that if lim -^'=0, then 

lim 2 o„ a;" = lim 2 ""^^ -^JL.^" n^j) 

Theorem 15 starts from the assumptions 
lim ^ = 0, 

n-^-x I'll 

CO 

lim S Ui^x" = I. 

x^\ 1 

The first assumption shows that (17) is available ; and this equation 
gives, with the second assumption, 

hm z ~ -i-^ a* = /. 

x-^\ 1 t. L ,, 



Abels Theorem and its converses 141 

NoAv Theorem 2 can be applied to the series S --^ — - j-^ m^", 



provided that 



hiu -1,K — , — J = (lo). 



Assuming for a moment that this condition is satisfied, 
Theorem 2 leads to 

lim ^i<±}-zi<l^ = i. 

Il^-X I ix *K+l 

and (16) gives finally 

lim Sn = I, 

proving the theorem, which is the analogue to Theorem 2. 

Condition (18) depends on the 6's as well as on the as; but 
since lim J-^ = 0. it will certainly be fulfilled when 

-Zk -— 

n I Ik 

tends to a limit or oscillates finitely. For, e being given, we can 
choose K so that 

I ^ ^A+, -t^ Pk ^1 "^^ ^ tK+i -tK pK , e V ^ ^A+i - tK 

— ^ A. < - Zi /^ 7 i — A, - . 

n 1 Ik t\+i n \ Ik t^+i " < t^\ 

We may suppose, for example, that 

tn = ri''; \ogn; log log 7i;... 

10. Adding to the notations used hitherto 

1^7 (1) 

/ .1 A A-l 71 ' 

^lbj^, = sf, 

'"n 3 

S Ox ^ = CJn , 

2 ^A-l 

and restricting the choice of the numbers b^ not only as done in 8, 
but further by supposing that the two limits 

lin, L ^1 , lim h+ip^h+^ ^1 (19) 



142 Br Kienufit, Eidenaioiiii of 

shall exist, or at any rate that the functions under the limit sign 
shall oscillate finitely, I procffcd to prove 

Theorem 16. TJie two cunditions 

CO 

lim 2 ««*■" = I (Jinite), 

x^l 1 

M-S-OG hi 2 fA— 1 

are each necessary for the existence of the limit 

lim s^'^ = l: 

n 

mid, taken together, they are sufficient. 

It is not possible to demonstrate this theorem for every set of 
numbers 6^. The following example shows this. 

Mr Riesz has pointed out* that 

1 'i, 1 

hm ^ - Sk 

WH-Qo log n 1 K 

exists and is finite in the case of 



However, Abel's limit 



lim 2 Kr''^~'^^x'^ 

x^\ 1 



does not exist, as the function behaves like 

r(«)(iogi)"' 

when *■— »1. 

].l. The demonstration depends on some identities analogous 
to those employed in the case of the arithmetic means, viz. 





n 


^n 


- 


1 


Pi 


! 


(2) 

H 


11 


- 


1 

tn 


n 
t 

2 


h. 


tK- 



* See G. H. Hardy, ' Slowly oscillating series', Proc. London Math. Soc, ser. 2, 
vol. 8 (1910), p. 310. 



Abel's Theui'em cuid its converses 



143 



which scries of relations might be continued. They show (in con- 
junction with Theorem 5) that lim s' = / whenever lim s — /, 
from which we deduce 



Theorem 17. //" lim s|J' = I exists <incl is finite, then 

lim^ii,^ = 0. 
Thus the second condition of Theorem 16 is necessary. 



12. We have also 



7 Pn—1 

'^n—i 



and thus 



V ,, ...K _ ^ ' h + ^ 'i" ,.K _ ^ ^« + l '/« " ^A-l „.K 

1 1 Ok+1 2 fx f^K 



II -\ 

In. 
1 i«- 



= (1-^) 



V^!^±LZLil,,,K_i^ 



^K + l ^«.+ l 



^, #« - ^«_i ^« - (y«_i 

+ Z ^ — T-?— w'' 

■1 t^ b^ 



+ 



'Jn+i'^ ^n-^ 



^n+\ bn+i 



[ 1 K+2 I t>«+l &«+2 ^J 

+ 1 h ^ ^ i h * • 

"n+1 "n+i 2 f/f Ox 



Now the series 



has a radius of convergence at least as great as 1, since lim -=^ = 

f?t— 1 

and =■ -^ tends to a limit or oscillates finitely. Thus 

n-1 ba •' 

lira p X'' = 



144 Dr Kieiiad, E.dennwns of 

for every x < 1 , arul therefore 

i a.,- = {i-xfi p" ^« + (1 - ^) V ^±LZL^J ^ ^ 

Taking account of the conditions (19), it follows from Theorem 6 
that 

lim(l -xf%p'a:- = 0, 

x^-\ 1 Ok +2 

and Iini (1 - x)t ^tir_^«+> f+1 ,.. = 0, 

X^\ 1 Ok-I-1 0^+2 

SO that lim % a^x" = lim S ^ "^^ ~ ^^ a;'^^^ (20). 

13. Lastly we have the identity 



14 If lim 6-|/^ = ^ exists and is finite, then, by (21), i ^"^' ""^^^ 



1 ^K + l 

converges to the sum I. Therefore by Abel's theorem 



and since (Theorem 17) lim ^ = 0, equation (20) is valid, and 
thus 



lim 1a^x'' = l. 

x^l 1 



We have therefore 



Theorem 18. Let the coefficients h^ be chosen so as to satisfy 
the conditions (19). Then, if lim 6'J/* = I exists and is finite. 



lira 2 a^x" = I. 

x^\ 1 



The hrst condition of Theorem 16 is consequently necessary. 



Abel's Theorem and its converses 145 

15. The proof of the converse begins with equation (20), which 
is valid since lini ~ = 0. Therefore 

X^-l 1 f/C + 1 

This is equivalent to the first condition of Theorem 15. But the 
second is satisfied too, viz. 

lini yit. ^A^i^i ^ n„, 1 [-^^^^ -())+...+ (fy, - r/„_,)] 

«-*-x hi 2 tie M-*-c» f)i 

= lim '^ = 0. 
Thus lim S '^'^^'~'^'' = ^. 



and, by equation (21), 



lim s[y = I, 



which completes the demonstration. 

The conditions (14) and (19) imposed on the numbers b^ are 
not necessary but only sufficient. The conditions necessary and 
sufficient would depend also on the coefficients «« of the power 
series considered, so that for a given series 'S^a^x" a given set 6^ 
)nay be admitted which must be excluded for other series ^CkX". 



Part III. 

16. Theorem 2 is in a sense a perfect converse of Abel's 
theorem, from which all these researches originated. 

Series for which Abel's limit exists may be divided into two 
classes, those which are convergent and those which are divergent, 
series for which the limit does not exist being excluded. 
Theorem 2 shows that the first class consists of those, and those 
only, which satisfy the condition 

lim -i«ct, = (22). 

The second class consists of those, and those only, which do not 
satisfy the condition. 

The condition (22) is satisfied, in particular, if 

lim nau = (23) 

VOL. XIX. PAliT IV. 11 



146 Dr Kienast, Extensions of 

But this condition, unlike (22), is not a necessary condition for 
convergence. 

Recent investigators have generalised the condition (23) in a 
different manner. Thus Mr J. E. Littlewood proved* the theorem : 

" 2 a^ IS convergent, provided lim 2 a^x^" = A and 1 na„ \ < K." 

1 x^l 1 

And still more recently Mr G. H. Hardy and Mr J. E. Littlewood f 
proved 

Theorem 19. If lim 2 a,,x'^ = A , and a„ >--K, then 2 a„ 
x^i n 

converges to the sum A. 

But however interesting in themselves these two theorems and 
their proofs may be they are less perfect than Theorem 2. For 
the conditions j na,, j < K and na,, > - ^ are neither necessary for 
convergence nor is either, together with \imta,x'' = A, necessary, 

nor do they characterise the non-converging series for which Abel's 
limit exists. Their interest is in fact of a quite different character 
from that of Theorem 2. 

It is not difficult to state similar theorems which are open to 
the same objection but which give information in cases where the 
last two theorems fail. 

17. The terms a^ of any sequence can be written in the form 
«« = ^ » where t^ is subject to the same conditions as in Theorem 15. 
This theorem then shows that 

CO ^ 00 ■ T 

" ^ -^ is convergent, provided lim 2 ^- a;" = ^ and lim -2c = " 

1 '''' x^l 1 ^K ,i-*oo tn 1 " 

Now the second condition is certainly satisfied if lim 2 c« tends 

to a limit or oscillates finitely. The only limitation'thus imposed 
upon the order of magnitude of a« is that \c^\<K, i.e. that the 

order of /c | a« | does not exceed that of ^ . Instead of the condition 

T */' ,^- ^ittlewood, 'The converse of Abel's Theorem on power-series', Proc 
London Math. Sac, ser. 2, vol. 9 (1911), p. 438. . -^ '<^c. 

_t G.H^ Hardy and J. E. Littlewood, ' Tauberian theorems concerning, power- 
series and Dirichlet s series whose coeflicients are ijositive ', Proc. London Math '^nr 
ser. 2, vol. 13 (1914) p. 188. See also E. LanJau, I)^.^./..,;"^ ^^'^^^^ 
ciniger neuerer ErgebniHse der Funktionentheorie. (Berlin, 1916) pn 45 etsea ■ ihl 
actual theorem is stated in § 9 and finally proved in § 10 (Die HardyiLittlewoodsche 
Umkehrung des Abelschen Stetigkeitssatzes). 



Abel's Theorem and its converses 147 



««K > — A" of Theorem 19 we have \X t^a,, < K, a condition which 

I 1 
allows Kti^ to tend to infinity in either direction. 

That such cases exist, in which S «« is convergent, is shown by 
the fact that 

t-— (0<e<</)<27r-e) 

is convergent if t^ is any function of k which tends steadily to 
infinity with k. 

18. A similar result can be obtained from another theorem of 
Messrs Hardy and Littlewood, viz. : 

Theorem 20. If f{x) = S a^af is a potver series with positive 
coefficients, luul f{x)^ ^ as x^\, then 

n 

2 a« ~ /I. * 
1 

From this theorem it is possible to deduce Theorem 19 (see 
above) of the same authors. 

Now the hypothesis is equivalent to 

CO 

lim (1 — x) f{x) = lim % (a^ — a^-i) x" = 1, 
and the conclusion is 

lim -^a^= hm - S J 2 (a^ - «a-i) h = 1. 

M-*M '>l 1 M-*.0O ^ 1 ( 1 j 

Thus Theorem 20 is equivalent to 

CO n 

Theorem 21. If limXb^x" = 1, and if the sums «„ = ^ 6« 

x-*l 1 1 

are alt positive, then 

1 'i 
lim - 2 6'k = 1. 

»-*-oo 11 1 

Here again is a condition which, in case the series converges, 
does not prevent the real numbers Kb^ from tending to infinity in 
both directions. 

* G. H. Hardy and J. E. Littlewood, I.e. See also E. Landau, I.e., § 9. 



11 — 2 



148 Mr Hardy, Sir George Stokes and the 



Sir George Stokes and the concept of uniform, convergence. Bv 
G. H. Hardy, M.A., Trinity College. 

[Received 1 Jan. 1918. Read 4 Feb. 1918.] 

1. The discovery of the notion of uniform convergence is 
generally and rightly attributed to Weierstrass, Stokes, and Seidel. 
The idea is present implicitly in Abel's proof of his celebrated 
theorem on the continuity of power series ; but the three mathe- 
maticians mentioned were the first to recognise it explicitly and 
formulate it in general terms*. Their work was quite independent, 
and it would be generally agreed that the debt which mathematics 
owes to each of them is in no way diminished by any anticipation 
on the part of the others. Each, as it happens, has some special 
claim to recognition. Weierstrass's discovery was the earliest, and 
he alone fully realised its far-reaching importance as one of the 
fundamental ideas of analysis. Stokes has the actual priority of 
publication ; and Seidel's work is but a year later and, while 
narrower in its scope than that of Stokes, is even sharper and 
clearer. 

My object in writing this note is to call attention to and, so 
far as I can, explain tw^o puzzling features in the justly famous 
memoir-f- in which Stokes announces his discovery. The memoir 
is remarkable in many respects, containing a general discussion of 
the possible modes of convergence, both of series and of integi'als, 
far in advance of the current ideas of the time. It contains also 
two serious mistakes, mistakes which seem at first sight almost 
inexplicable on the part of a mathematician of so much originality 
and penetration. 

The first mistake is one of omission. It does not seem to have 
occurred to Stokes that his discovery had any bearing whatever on 
the question of term by term integration of an infinite series. The 
same criticism, it is true, may be made of Seidel's paper. But 
Seidel is merely silent on the subject. Stokes, on the other hand, 
quotes the false theorem that a convergent series may always be 
integrated term by term, and refers, apparently with approval, to 
the erroneous proof offered by Cauchy and Moignoj. 

Of this there is, I think, a fairly simple and indeed a double 

* The idea was rediscovered by Cauchy, five or six years after tlie publication of 
the work of Stokes and Seidel. See Pringsheim, ' Grundlageu der allgemeineu 
Funktionenlehre ', Encyld. der Math. Wiss., II A 1, §17, p. 35. 

t ' On the critical values of the sums of periodic series', Trans. Canib. Phil. Soc, 
vol. 8, 1847, pp. 533-583 (Mathematical and physical papers, vol. 1, pp. 236-313). 

X See p. 2-42 of Stokes's memoir (as printed in the collected papers). 



concept of uniform convergence • 149 

explanation. In the first place it must be remembered that Stokes 
was primarily a mathematical physicist. He was also a most acute 
pure mathematician ; but he approached pure mathematics in the 
spirit in which a physicist approaches natural phenomena, not 
looking for difficulties, but trying to explain those which forced 
themselves upon his attention. The difficulties connected with 
continuity and discontinuity are of this character. The theorem 
that a convergent series of continuous functions has necessarily 
a continuous sum is one whose falsity is open and aggressive : 
examples to the contrary obtrude themselves on analyst and 
physicist alike. The falsity of this theorem Stokes therefore 
observed and corrected. The falsity of the corresponding theorem 
concerning integration lies somewhat deeper. It is easy enough, 
when one's attention has been called to it, to see that the proof 
of Cauchy and Moigno is invalid. But there are no particularly 
obvious examples to the contrary : simple and natural examples 
are indeed somewhat difficult to construct*. And Stokes, his 
suspicions never having been excited, seems to have accepted the 
false theorem without examination or reflection. 

This is half the explanation. The second half, I think, lies in 
the distinctions between different modes of uniform convergence 
which I shall consider in a moment. 

Stokes's second mistake is more obvious and striking. He 
proves, quite accurately, that uniform convergence implies con- 
tinuity f. He then enunciates and otfers a proof ;J; of the converse 
theorem, which is false. The error is not one merely of haste or 
inattention. The argument is as explicit and as clearly stated in 
one case as in the other ; and, up to the last sentence, it is perfectly 
correct. He proves that continuity involves something, and then 
states, without further argument, that this something is what he 
has just defined as uniform convergence. It is merely this last 
statement that is false. 

Stokes's mistake seems at first sight so palpable that I was for 
some time quite at a loss to imagine how he could have made it. 
A closer examination of his memoir, and a comparison of his work 
with other work of a very much later date, has made the lapse a 
good deal more intelligible to me ; and my attempts to understand 
it have led me to a number of remarks which, although they 
contain very little that is really novel, are, I think, of some 
historical and intrinsic interest. 

2. There are no less than seven different senses, all important, 
in which a series may be said to be uniformly convergent. 

* See Bromwich, Infinite sfrlea, pp. 110-118; Hardy, ' Notes on some points in 
the integral calculus', XL, Messentjer of Matlieniatica, vol. 44, 1915, pp. 145-149. 
t p. 282. I use ' uniform ' instead of Stokes's ' not infinitely slow '. 
X p. 283. 



150 Afr Hardy, Sir George Stokes and the 

I shall write the series in the form 

00 

S tin (^f^) ; 

1 
and I shall suppose, for simplicity, that every term of the series is 
continuous, and the series convergent, for every x of the interval 
a^x ^b. I shall denote the sum of the series by s (x) ; and I shall 
write 

Sn (OC) = II, (x) + Uo {x)+ ...■\- iln (.«), S (x) = Sn (x) + r,, (cc). 

The fundamental inequality in all my definitions will be of the t3^pe 

\rn(a!)'\^e (A), 

I shall refer to this inequality simply as (A). 

When we define uniform convergence, in one sense or another, 
we have to choose various numbers in a definite logical order, those 
which are chosen later being, in general, functions of those which 
are chosen before. I shall write each number in a form in which 
all the arguments of which it is a function appear explicitly : thus 
no (^, e) is a function of ^ and e, Uo (e) one of e alone. 

It will sometimes happen that one of the later numbers depends 
upon several earlier numbers already connected hy functional rela- 
tions, so that it is really a function of a selection of these numbers 
only. Thus h may have been determined as a function of e ; and 
??o niay have to be determined as a function of ^, e, and h, so that 
it is in reality a function of ^ and e only. I shall express this by 
writing 

«o = ^2o(^, e, S) = no(|^, e); 
and I shall use a similar notation in other cases of the same kind. 
3. The first three senses of uniform convergence are as follows. 
A 1 : Uniform convergence throughout an interval. The 
series is said to he uniformly convergent throughout the interval (a, b) 
if to every positive e corresponds a.n no (e) such that (A) is true^for 
n ^ Wo (e) and a^x^^b. 

This is the ordinary or ' classical ', and most important, sense, 
the sense in which uniform convergence is defined in every treatise 
on the theory of series. 

A 2 : Uniform convergence in the neighbourhood of a 
point. The series is said to be uniformly convergent in the 
neighbourhood of the point ^ of the interval (a, b) if an interval 
(f — 8 (I), ^ -\-B (^))* can be found throughout luhich it is uniformly 
convergent ; that is to say %f a positive 8{^) exists such that (A) 
is true for every positive e, for n ^ n^ (^, S, e) = ??o (?> e), and for 

* A trivial change is of course required in the definition if t = « or ^ = b. The 
same point naturally arises in the later definitions. 



concept of mil form convergence 151 

A3: Uniform convergence at a point. The series is 
said to be uniformly convergent at the point x = f {or for x = ^) 
if to every positive e correspond a positive S (^, e) and an 
»o(|, €, B) = nQ(^, e) such that (A) is true for n ^n^{^, e) and for 
^-S(^,e)^x^^+S(^,e). 

4. Before proceeding further it will be well to make a few 
remarks concerning these definitions and their relations to one 
another. 

The idea of uniform convergence in the neighbourhood of a 
particular point (Definition A 2) is substantially that defined by 
Seidel in 1848*. It is clear, however, that definitions A 1 and 
A 2 were both familiar to Weierstrass as early as 1841 or 1842f. 
It is obvious that a series uniformly convergent throughout an 
interval is uniformly convergent in the neighbourhood of every 
point of the interval. The converse theorem is important and by 
no means obvious, and was first proved by Weierstrass | in a memoir 
published in 1880. This theorem would now be proved by a 
simple application of the ' Heine-Borel Theorem ', and is a par- 
ticular case of a theorem which will be referred to in a moment. 

Definition A3 appears first, in the form in which I state it, in 
a paper of W. H. Young published in 1903§; but the idea is 
present in an earlier paper of Osgood ||. The essential difference 
between definitions A 2 and A 3 is that in the latter S is chosen 
after e and is a function of ^ and e, while in the former it is chosen 
before e and is a function of f alone. In each case n^ is a function 
of two independent variables, ^ and e. It is plain that uniform 
convergence in the neighbourhood of ^ involves uniform conver- 
gence at ^, and at (and indeed in the neighbourhood of) all points 
sufficiently near to ^. But uniform convergence at ^ does not 
involve uniform convergence in the neighbourhood of |. 

It is important, however, to observe that uniforni convergence 
at every point of an interval involves uniform convergence throughout 
tJie interval. This important theorem is proved very simply by 

* ' Note iiber eine Eigensehaft der Reihen, welche discontinuirliche Functionen 
darstellen', Munchener Ahliandlungen, vol. 7, 1848, pp. 381-394. This memoir has 
been reprinted in Ostwald's Klassiker der e.vakten Wisscmchaften, no. IK!. The 
reference tliere given to vol. 5, 1847, is incorrect. 

(■ For detailed references bearing on this and similar historical points, see 
Pringsheim's article already qnoted. 

X See the memoir 'Zur Functionenlehre ' {Ahliandlungen aus der Funktionen- 
lehre, pp. 69-104 (pp. 71-72)). 

§ 'On non-uniform convergence and term-by-term integration of series', Proc. 
London Math. Soc, ser. 2, vol. 1, pp. 89-102. 

II 'Non-uniform convergence and the integration of series', American Journal of 
Math., vol. 19, 1897, pp. 155-190. See Prof. Young's remarks on this point at the 
beginning of his later paper ' On uniform and non-uniform convergence of a series 
of continuous functions and the distinction of right and left ', Proc. London Math. 
Soc, ser. 2, vol. 6, 1907, pp. 29-51. 



152 Mr Hardy, Sir George Stokes and the 

Young, in his paper already quoted, by means of the Heine-Borel 
Theorem * ; and it plainly includes, as a particular case, Weierstrass's 
theorem referred to above. 

5. It seems to me that the definition given by Stokes is not 
any one of A 1 , A 2, A 3 ; and that, if we are to understand him 
rightly, we must consider another parallel group of definitions. 
These definitions differ from those given above in that (A) is 
supposed to be satisfied, not for all sufficiently large values of n, 
but only for an infinity o/ values. 

B 1 : Quasi-uniform convergence throughout an interval. 
The series is said to he quasi-uniformly convergent tlvroughout (a, h) 
if to every positive e and every N corresponds an n^ (e, N) greater than 
N and such that (A) is true for n = n^ (e, N) and a^x^b. 

B 2 : Quasi-uniform convergence in the neighbourhood 
of a point. The series is said to be quasi-uniformly convergent in 
the neighbourhood of f if an interval (^ — 8(f), | + S(f)) can be 
found throughout which it is quasi-uniformly convergent ; i.e., if a 
positive 8(f) exists such that (A) is true for every positive e, every N, an 
«o (f . 8, e, iV) = ?io (f , e. ^) greater than N, and f — 8 (f ) ^ .'c ^ f + 5 (f ). 

B3: Quasi-uniform convergence at a point, llie series 
is said to be quasi-uniformly convergent for iV = ^ if to every positive 
€ and every N correspond a positive S (f, e, N) and an 

no{^,e,8,N) = n,(^,e,N), 

greater than N, such that (A) is true for n = ??o (!> f> N') and for 

Definition B 1 is to be attributed to Dini or to Darboux+. 
Another form of it has been given by Hobson|. As Arzela and 
Hobson§ have pointed out, a series is quasi-uniformly convergent 
throughout an interval if, and only if, it can be made uniformly 
convergent by an appropriate bracketing of its terms. 

Definition B 2 is for us at the moment of peculiar interest, 
for (as I shall show in a moment) it is really this definition that 
is given by Stokes. 

. Definition B 3 is also of great interest, both in itself and in 

* Choose € and determine 5 (^, e) and n^ (|, e), as in definition A3, for every f of 
the interval. Every point of {a, b) is included in an interval {^-d, ^ + o). By the 
Heine-Borel Theorem, every point of (a, b) is included in one or other of a finite 
sub-set of these intervals. If N (e) is the largest of the Hq's corresponding to each of 
the intervals of this finite sub-set, then (A) is true for n^N and a ^ .r ^ 6. 

This is the essence of the proof, though, like all proofs of the same character, it 
requires a somewhat more careful statement if all apj^earance of dfpendence upon 
Zermelo's AusicahUprinzip is to be avoided. 

t See Pringsheim, I. c. 

+ ' On modes of convergence of an infinite series of functions of a real variable', 
Proc. London Math. Sac, ser. 2, vol. 1, 1903, pp. 373-387. Hobson (following Dini) 
uses the expression ' simply uniformly'. 

§ L. c, p. 375. 



concept of uniform convergence 153 

relation to Stokes's memoir. For the necessary and sufficient con- 
dition that s (x) should he continuous for x=^ is that the series 
should be quasi-uniformly convergent for x — ^. This theorem is 
in substance due to Dini*. I give the proof, as it is essential for 
the criticism of Stokes's memoir. 

(1) The condition is siificient. For 

I s {x) - s (I) I ^ { Sn {x) - Sn (|) ' + | r„ {x) j + | r^ (f) |. 

Choose e, N, S (^, e, N), and n = ??o {^, e, N) as in definition B 3. Then 
[ r„ {x) I < e for ^—Z^x^^ + h. Now that n is fixed we can choose 
Si less than 8 and such that [ s,i {x) — «» (^) j < e for ^ — Si ^ a" ^ ^ + Sj. 
And thus 

|s(.«)-s(f)i<3e 

for ^ — §1 ^ .« ^ ^ + Si , so that 5 {x) is continuous for a? = f . 

It is plain that this argument proves, a fortiori, that A 2, A 3, 
and B 2 all furnish sufficient conditions for continuity at a point, 
and A 1 and B 1 sufficient conditions for continuity throughout an 
interval. 

(2) The condition is necessary. For 

I rn {x) \^\S {x) - S (^) I + ! Vn (f ) | + ! S„ {x) - 5„ (|) |. 

Suppose that e and N are given. Then we can choose S (^, e) 
so that \s{x) — s{^)\<e for f — S ^ « ^ ^ + S, and n^ (|, e, iV ) so that 
Vq > N and j r^^ (|) | < e. And, when n^ has thus been fixed, we can 
choose S] (^, e, n^) = Sj (^, e, N) so that Si < S and 1 6'„^ {x) — Sn^ (f ) | < e 
for I — Si ^ .^■ ^ I + Si . Thus | r,i (^) j < 3e for n = no> N and 
^ — Bi^X'^^ + 8i, so that the series is quasi-uniformly convergent 
for x=^. 

6. If a series is uniformly convergent at every point ^ of an 
interval, it is (as we saw in § 4) uniformly convergent throughout 
the interval : definition A 3 (and a fortiori definition A 2) passes 
over, in virtue of the Heine-Borel Theorem, into definition A 1. 
It is important to observe that this relation does not hold between 
B 3 (or B 2) and B 1 : a series quasi-uniformly convergent at 
every point of an interval (or in the neighbourhood of every such 
point) is not necessarily quasi-uniformly convergent throughout 
the interval. We can apply the Heine-Borel Theorem in the 
manner indicated in the first sentences of the footnote * to p. 152 ; 
but the last stage of the argument, in which every one of a finite 
number of difterent integers is replaced by the largest of them, 
fails. What we obtain is the necessary and sufficient condition that 
s {x) shoidd he continuous throughout the interval ; and this is not 

^' Foiulaiiii')iti..., p. 107 ((jerinan translation, GruiuUa(ii'ii...,p\). 143-145). 



154 Mr Hardy, Sir Georr/e Stokes and the 

the condition B 1 but a condition first foi-mulated by Arzela*, 



VIZ. 



C: Quasi-uniform convergence by intervals {convergenza 
uniforme a tratti). ^ The series is said to he quasi- uniformly con- 
vergent by intervals if to every positive e and every N correspond a 
division of (a, h) into a finite number v (e, N) of intervals 8,. (e, N), 
and a corresponding number of numbers n,(e, Nj, all greater than A^, 
and such that (A) is true for ?? = 7?,.(?- = 1, % ...,v) and all values 
of X which belong to 8,.. 

The deduction of Arzela's criterion from B 3, in the manner 
sketched above, was first made by Hobsonf. 

There is one further point which seems worth noticing here, 
although it is not directly connected with Stokes's memoir. Dini J 
proved that if u^ (x) ^ for all values of n and x, and s (x) is con- 
tinuous throughout {a, b), then the series is uniformly convergent 
throughout (a, b). This theorem is now almost intuitive. For it 
is obvious that, for series of positive terms, quasi-uniform conver- 
gence in any one of the senses B 1, B 2, or B 3 involves uniform 
convergence in the corresponding sense A 1, A 2, or A 3. If then 
s {x) is continuous throughout (a, b) it is continuous for every f of 
(a, b) ; and therefore the series is quasi- uniformly convergent for 
every f ; and therefore uniformly convergent for every |; and 
therefore uniformly convergent throughout (a, b). 

7. Let us now consider Stokes's definitions and proofs in the 
light of the preceding discussion. 

It is clear, in the first place, that Stokes has in his mind some 
phenomenon characteristic of a small, hit fixed, neighbourhood of 
a point. 

' Let u^ -{-U.+ ... (66)', he says§, ' be a convergent infinite series 
havmg U for its sum. Let v, + v, ■]-... (Q7) be another infinite 
series of which the general term v.,, is a function of the positive 
variable h and becomes equal to Un when h vanishes. Suppose 
that for a sufiiciently small value of h and all inferior values the 
series (67) is convergent, and has V for its sum. It might at first 
sight be supposed that the limit of V for A=0 was necessarily 
equal to U. This however is not true.... 

' Theorem. The limit of V can never differ from U unless 
the convergency of the series (67) becomes infinitely slow when h 
vanishes. 

* ' Sulle serie di funzioni', Memorie dl Bologna, ser. 5, vol. 8, 1900 up 131-186 
701-744. ' 

t L. c, pp. 380-382. 

J L.c. (German edition), pp. 148-149. See also Bromwich, Infinite series, p 125 
(Ex. 6). ■ '^' 

§ p. 279. 



concept of II inform convergence 155 

' The convergency of the series is here said to become infinitely 
slow when, if n be the number of terms which must be taken in 
order to render the sum of the neglected series numerically less 
than a given quantity e, which may be as small as we please, n 
increases beyond all limit as h decreases beyond all limit. 

'Demonstration. If the convergency do not become in- 
finitely slow it will be possible to find a number n, so great that 
for the value of h tue begin with and for all inferior values greater 
than zero the sum of the neglected terms shall be numerically less 
than e....' 

Stokes's words, and in particular those which I have italicised, 
seem to me to make two things perfectly clear. 

(1) Stokes is considering neither a property of an interval 
(a, b) im Grossen (such as is contemplated in A 1 or B 1), nor a 
property of a single point which (as in A 3 or B 3) need not be 
shared by any neighbouring point, but a property of an interval 
im Kleinen, that is to say a small but fixed interval chosen to in- 
clude a particular point. His definition is therefore one of the 
type of A 2 or B 2. 

Stokes's failure to perceive the bearing of his discovery on 
problems of integration is made much more natural when we 
realise that he is considering throughout a neighbourhood of a 
point and not an interval im Grossen. And this remark applies 
to Seidel as well. 

(2) Stokes is considering an inequality satisfied for a special 
value of n, or at most an infinite sequence of values of oi, and not 
necessarily for all values of n from a certain point onwards. In 
this respect there is a quite sharp distinction between Stokes's 
work and Seidel's. What Stokes defines is (to use the language 
of this note) a mode of quasi-unifo7'ni convergence and not one of 
strictly uniform convergence. 

It seems to me, then, that what Stokes defines is what I have 
called quasi-uniform convergence in the neighbourhood of a, point 
(B2). 

8. If we adopt this view, Stokes's mistake becomes very much 
more intelligible. He proves, quite correctly, that uniform con- 
vergence in his sense implies continuit}^ : his proof, stated quite 
formally and by means of inequalities, is substantially that given 
in 1 5, under (1). He then continues* as follows. 

' Conversely, if (66) is convergent, and if U= Vof, the con- 
vergency of the series (67) cannot become infinitely slow when h 

* p. 282. Tbe italics are mine. 

t Ffl is what Stokes calls 'the value of V for h = 0', by which he means, of 
course, its limit when h tends to 0. 



156 Mr Hardy, Sir George Stokes and uniform, convergence 

vanishes. For if Un, V^ represent the sums of the terms after 
the nth in the series {QQ), (67) respectively, we have 

V^Vn + V,:, U=U^+Un'; 
whence 

v,:=Y-u-{v,,-u,,)^-u,:. 

Now V-U, Yn- Un vanish with h, and Ua vanishes when n 
becomes infinite. Hence for a sufficiently small value of h and 
all inferior values, together with a value of n sufficiently large and 
independent of h, the value of F,/ may be made numerically less 
than ^ any given quantity e however small ; and therefore, by 
definition, the convergency of the series (67) does not become in- 
finitely sloiv when h vanishes.' 

Now this argument is, until we reach the last sentence, perfectly 
accurate, and indeed, if we translate it into inequalities, substantially 
identical with that given in § 5, under (2). Stokes proves, in fact, 
that continuity at | involves quasi-uniform convergence at |. 
Where he falls into error is simply in his final assertion that this 
property is that which he has previously defined, the mistake being 
due to a failure to observe that his intervals of values of h depend 
upon a prior choice of e. In a word, he confuses, momentarily, 
B 2 and B 3. The ordinary view that Stokes defined uniform 
convergence in the same sense as Weierstrass compels us to suppose 
that he confused B 3 with A 1 , or at any rate with A 2 : and this 
is hardly credible. 

I add one final remark. If we could identify Stokes's idea with 
B_3, instead of with B 2, we could acquit him of having made any 
mistake at all, since B 3 really is a necessary and sufiicient con- 
dition for continuity. We could then regard Stokes as having 
anticipated Dini's theorem. This view, however, does not seem to 
me to be tenable. 



• 



Mr Lake, Shell-deposits fornied by the flood of January/, 1918 157 



Shell-deposits formed by the flood of January, 1918. B}- Philip 
Lake, M.A., St John's College. 

[Read 18 February 1918.] 

The heavy snow of the third week in January 1918 was followed 
by a very rapid thaw and a considerable fall of rain, and the Cam, 
in consequence, rose to an exceptional height. In the neighbour- 
hood of Cambridge the floods were the most extensive of recent 
years, the water reaching its highest level on Sunday, Jan. 20. 

The traces of the flood remained visible for several weeks, its 
limits being marked in most places by straws, twigs, silt, etc., with 
a sprinkling of land and fresh-water shells. But below the town, 
near the railway-bridge, the shells were so abundant as to form a 
remarkable deposit, which seems to deserve a special record. It 
was not till the 25th Jan. that I saw it, and the following notes 
are drawn up from the observations made on that day and on two 
or three subsequent visits. 

The deposit lay partly upon the tow-path and partly in the 
shallow ditch on the iimer side of the path, and it extended with 
little interruption from the immediate neighbourhood of the 'Pike 
and Eel ' to a point about 850 yards below the railway-bridge, a 
total distance of approximately 850 yards. Occasional patches 
occurred still farther down, and scattered shells even as far as 
Ditton Corner. Beyond Ditton the tow-path was in several places 
covered with a thick layer of silt, but I saw no more shells until 
within sight of the lock at Baitsbite. 

The deposit was somewhat irregular and it was difficult to form 
an estimate of its average width, but this can hardly have been 
less than a foot, and was probably much more. 

Above the railway-bridge the shells were mixed with silt, 
especially in the ditch on the inner side of the path ; but even 
here the proportion of shells was large, and in places they formed 
the bulk of the deposit. Below the railway-bridge the deposit was 
free from silt and consisted entirely of shells. In the shallow 
hollows formed by the irregularities of the surface, it was often an 
inch or two deep, so that it was possible to scoop up the shells by 
the handful. Owing to its colour it showed conspicuously as light 
streaks upon the slightly darker path. 

By far the greater part of the deposit consisted of Limnaea, 
L. stagnalis and L. peregra being the most abundant species ; but 
other fresh-water shells also occurred and land-snails were by no 



158 Mr Lake, Skell-depu'sits foniied bij the 

means rare. Mr C. E. Gray, of the Sedgwick Museum, went down 
shortly after my first visit, and in a very short time obtained most 
of the following species, but a few names have been added to the 
list from specimens collected subsequently : 

Sphaerium corneum (L.), 
Bithynia tentaculata (L.), 
Vivipara contecta (Millet), 
Valvata piscinalis (Miiller), 
Limnaea stagnalis (L.), 

„ peregra (Mtiller), 

„ auricularia (L.), 
Pkuiorhis corneus (L.), 

„ umbilicatus Miiller, 

„ caiinatits Miiller, 

„ vortex (L.), 

„ contortus (L.), 

Pliysa fontinalis (L.), 
Helix nemoralis L., 
Theba cantiana (Mont.), 
Hygromia striolata (Pfr.), 
Vitrea draparnaldi (Beck), 
„ cellaria (Miiller). 

Even now the list is probably far from complete, and a closer 
examination would no doubt reveal the presence of many other 
forms. 

The last five species are land-shells, and, with the exception of 
Vitrea cellaria, they occurred in Mr Gray's first collection and were 
identified by Mr Hugh Watson. Vitrea draparnaldi does not 
appear to be a native of the county, but is found in and near green- 
houses ; for instance, in the Botanical Gardens. In Mr Gray's first 
collection, which was made below the railway-bridge, it was repre- 
sented only by a single specimen, which we supposed to have come 
from the florist's greenhouses close by. But at a later date he 
found it to occur abundantly at the beginning of the tow-j)ath, 
some five or six hundred yards above the greenhouses. In order 
to make sure that the specimens really belong to this species they 
were sent to Mr Watson, who agreed with the identification. 

Since there were so many specimens of Vitrea draparnaldi at 
the beginning of the tow-path, and so few (at least comparatively) 
below the railway-bridge, it seems clear that they cannot have been 
carried far, for otherwise they would have been more evenly dis- 
tributed. It is most probable indeed that there was a colony of this 
species in the immediate neighbourhood. The nearest greenhouse 
that I have been able to find above the locality where the species 
was so abundant is five or six hundred yards off, and stands well 



I 



flood (if Jcumuvij, 1918 159 

away from the river. The specimens can hardly have come from 
there, and it is more likely that the colony lived out of doors and 
nearer to the river. Nevertheless its progenitors may have been 
'escapes'. The greenhouses below the railway-bridge have now 
been out of use for some time, and the snails that were in them 
must have been forced to seek new quarters. 

Most of the shells, both land and fresh-water, were perfect or 
nearly so, and all of them were empty. Neither Mr Gray nor myself 
found a single specimen with any remains of its former inhabitant. 
The greater number were very fresh in appearance, but some of 
the land-shells had evidently been exposed to the weather for some 
time, and some of the fresh-water shells had lain in the mud long- 
enough to become discoloured or incrusted as if the process of 
fossilization had begun. The specimens of Vitrea draparnaldi, it 
may be noted, were all fresh-looking. 

Apart from the extent of the shelly deposit, its freedom from 
silt below the railway-bridge was perhaps its most important feature, 
for it shows that even a muddy river like the Cam may produce a 
purely calcareous deposit. 

The fact that the shells were all empty indicates that those 
belonging to the river must have lain in its bed for some time; and 
in this connection an observation made by Mr Gray is of interest. 
Some years ago at Bottisham, when dredging operations were going 
on, he noticed that the mud brought up by the dredger was full of 
fresh-water shells. 

During floods the river digs up its bed and, as on the occasion 
here described, it may deposit the shells in one place and the silt 
in another. In the case of an artificially controlled stream like the 
Cam, floods are comparatively rare ; but in an unrestrained river 
we may reasonably expect them to be both more numerous and 
more extensive. It seems quite possible therefore that neither the 
clayey fresh-water limestones of the Wealden nor the purer fresh- 
water limestones of the Purbeck series required lagunary conditions 
for their formation. 



160 Mr Matthui, la the Madveporarian Skeleton 



Is the Madveporarian Skeleton an Extraprotoplasmic Secretion 
of the Polyps ? By G. Matthai, M.A., Emmanuel College, Cam- 
bridge. (Communicated by Professor Stanley Gardiner.) 

[Read 18 February 1918.] 

In 1881 von Heider (5) suggested that the calcareous skeleton 
of the Madreporaria is formed by the deposition of carbonate of lime 
within certain specialised ectodermal cells (calicoblasts*) consti- 
tuting an outer layer, and repeated this conclusion in a subsequent 
paper (6). In 1882 von Koch (8) inferred from embryological obser- 
vations that the skeleton is deposited outside the living tissues, 
i.e. is extraprotoplasmic in origin. In 1896 Ogilvie (9) supported 
von Heider's view and argued that, by repeated calcification of 
"cells" of the calicoblastic layer of ectoderm, successive strata of 
calcareous " scales " are formed, and slightly modified her opinion 
in 1906 (10). Fowler (4) had previously accepted von Koch's view. 
In 1899 Bourne (2), from his studies on the Anthozoan skeleton, 
supported von Koch's conclusions and entirely disagreed with 
von Heider and Ogilvie. He further held that, whilst in Heliopora 
and the Madreporaria the corallum is formed outside the living 
calicoblastic layer, the spicules of the Alcyonaria are formed within 
certain ectodermal cells or scleroblasts which either remain in the 
ectoderm or wander into the mesoglaea (2, p. 506). Following 
von Koch and Bourne, it is noAv generally believed that the 
Madreporarian skeleton is an extraprotoplasmic formation and that 
Alcyonarian spicules are entoplastic products. 

After a ground-down section of an Astrgeid corallite has been 
slowly decalcified on a slide, somewhat homogeneous organic 
remains (distinguishable from algal filaments penetrating the 
skeleton) are left which react to any of the common stains. This 
is clear indication that the calcareous matter has been deposited 
in an organic matrix. Bourne regards this matrix as due to the 
"disintegration of calicoblasts" (2, pp. 520 and 521, fig. 21), 
assuming that the organic basis was not part of the living calico- 
blastic ectoderm. His view is that carbonate of lime is secreted 
by the calicoblastic layer and is passed through its outer border 
(the " limiting membrane ") into the decaying part outside, exactly 
as the Alcyonarian spicule is " from its early origin, separated 
from the protoplasm which elaborated the material necessary for 
its further growth by a layer of some cuticular material" (2, p. 537), 

* Von Heider's original rendering of this word is chalicoblast, of which the first 
half, I am informed, is derived from the Greek x'^^'li which in Eomau characters 
should be spelt clialix. Subsequently, Fowler changed the spelling to calycoblast, 
and in 1888 both this author and Bourne adopted the present form calicob/ast. 



an Extraprotoplasmic Secretion of the Polyps! 161 

viz., the spicule-sheath. At the same time, Bourne contends that 
the spicule is entoplastic in formation whilst the Madreporarian 
coralkim is exoplastic. To be consistent, both the spicule and the 
corallum would have to be regarded as formed either within living 
protoplasm or outside it, but spicules could not be viewed as intra- 
protoplasmic products whilst assuming the extraprotoplasmic origin 
of the corallum. 

Duerden (3) held that the organic basis of the corallum of 
Siderastrea yadians is a " secretion " of the calicoblastic layer of 
ectoderm to which it is closely adherent (pi. 8, fig. 45) and is "a 
homogeneous, mesoglaea-like matrix within which the minute cal- 
careous crystals forming the skeleton are laid down " (p. 34). 
Since he refers to the skeleton as " ectoplastic " in origin (p. 113), 
it is evident that he agi-ees with Bourne in the view that the 
organic matrix was not part of the living tissues when calcareous 
matter began to be deposited in it. But in the account of these 
authors there is no more evidence to show that, in the Madre- 
poraria, the organic ground substance or "colloid matrix " (2, p. 539) 
was non-living at every phase of skeleton formation than that the 
areas of the scleroblasts of the Alcyonaria in which the deposition 
of spicular matter took place had not, at least at the initial stages 
of this process, formed part of the living protoplasm. 

Further if, in the Madreporaria, the calcareous matter were 
deposited outside the living calicoblastic ectoderm, it is difficult 
to understand how the manifold patterns of eoralla so charac- 
teristic of this gi'oup of organisms can have been built up*. But 
if the matrix in which carbonate of lime is laid down is part 
of the living calicoblastic sheet, it follows that the protoplasm 
must regulate the arrangement of the calcareous matter into the 
various skeletal types which, in large measure, maintain their re- 
spective form independent of changes in environmental conditions. 
Similarly, the formation of the various kinds of spicules of the 
Alcyonaria can be adequately explained only if calcareous deposition 
takes place within living protoplasm, and indeed. Bourne has drawn 
attention to the phenomenon that " the spicules of the Alcyonaria 
show a definite and complex crystalline structure, the details of 
which are, indeed, moulded upon and dominated by an equally 
complex organic matrix..." (2, p. 517). 

The intraprotoplasmic origin of spicules in the Alcyonaria might, 
without difficulty, be ascertained since sections can be made with- 
out decalcification, whereas in Heliopora and the Madreporaria 
possessing massive eoralla, satisfactory sections are possible only 
after decalcification, and in this condition the skeleton may appear 

* In explanation of this phenomenon, Bourne suggests that "the general 
arrangement of the fasciculi of crystals is dominated, in some manner of which we 
are ignorant, by the living tissues which clothe the corallum " (2, p. 539). 

VOL. XIX. PART IV. 12 



162 Mr Matthai, Is the Madrejjorarian Skeleton 

as though formed outside the living tissues. A further difficulty 
with regard to the Madreporaria is that, except perhaps at the 
growing points, the skeleton would secondarily lose its intraproto- 
plasmic character and appear to be external to the living tissues by 
having displaced most of the protoplasm in which it was deposited, 
just as the discrete condition of fully developed Alcyonarian 
spicules is due to the increase of calcareous matter at the expense 
of the protoplasm in which it was formed. 

From the above considerations it would appear to be highly 
probable that von Heicler was right in regarding the Madreporarian 
skeleton as formed within the calicoblastic protoplasm. Bourne 
directs much of his criticism to von Heider's suggestion that the 
striae in the calicoblastic layer (i.e., in the processes of attachment) 
are calcareous fibres, but it is not improbable that, in the unde- 
calcified condition, some of these processes of attachment might 
be partially calcified. 

When thin sections of Astrseid coralla are examined under a 
microscope, they frequently appear to consist of calcareous pieces 
united by sutures resembling the " laminae " or " trabecules " of the 
skeleton of Heliopora (1, p. 463, pi. 11, figs. 7 and 8) and the " tra- 
becular parts " of the Madreporarian skeleton as figured by Ogilvie 
(9, p. 124, figs. 13, 19, etc.). Each piece is composed of calcareous 
strands radiating from a dark centre or line which, as Ogilvie sug- 
gested, appears to be the organic remains of the protoplasm in which 
the calcareous needles were laid down. There is some similarity 
between these elements and the spicules of Tuhipora (7, figs. 9 
and 10) which, according to Hickson, are not fused together but 
dovetailed into one another as in the membrane bones of Mammals 
(p. 562). The resemblance is also marked in the case of the scale- 
like spicules of Plumarella (2, figs. 6 and 7) containing dark centres 
from which calcareous fibres or rods radiate. 

It is difficult to gather from Bourne's account what he considers 
to be the unit of skeletal structure in the Alcyonaria. Are 
spicules such units* ? But spicules are not all homologous elements 
since they are formed in protoplasmic areas containing one or more 
nuclei and no limit can be set to their size in the various genera 
(2, pp. 508-517), an extreme case being the scale-like spicules of 
Primnoa and Plumarella, each of which is " formed by several 
cells, or at least by a comparatively large coenocytial investment 
containing many nuclei " (p. 510). Or, is a spicule a calcareous 
piece which behaves like a single crystal when examined under 
crossed Nicols? The same confusion prevails with regard to ske- 
letal units in the Madreporaria — whether they are represented by 
" fibro-crystals " (Bourne), "crystalline sjjhgeroids" (von Koch) or 

* Bourne applies the term spicule to "an entoplastic product of a single cell or 
of a ccenocyte " (2, p. 504). The italics are mine. 



an. Extra protoplasmic Hecretiou of the Polyps ? 163 

" calcareous scales " (Ogilvie). The latter are not calcified calico- 
blastic "cells" as Ogilvie contended since the calicoblastic ectoderm 
is now found to be a multinucleated sheet of protoplasm devoid of 
cell-limits, i.e., a syncytium. 

In fact, there is hardly any evidence to show that the skeleton 
of the Anthozoa is made up of homologous units just as it is highly 
doubtful if their soft parts are composed of uninucleated units or 
cells. The significance of the Anthozoan skeleton would consist in 
its probable formation within syncytial protoplasm according to 
physical laws under the presiding activity of the living protoplasm 
which would direct the complex skeletal architecture. The cal- 
careous deposit further appears to be differentiated into elements 
which remain separate as spicules in most Alcyonarians but are 
united to form a compact skeleton in certain Alcyonarians, e.g., 
Tuhipora, Corallium, Heliopora, and in all the Madreporaria (in 
which the calcareous matter may undergo subsequent rearrange- 
ment). From this point of view, a separate calcareous piece of an 
Alcyonarian might be regarded as a diminutive corallum, and the 
corallum of a Madreporarian as a massive spicule, and finally, the 
formation of the Anthozoan skeleton would be essentially similar 
to the formation of membrane bone in Vertebrates*. 

References. 

1. Bourne, G. C. " On the Structure and Affinities of Heliopora ccerulea, 
Pallas. With some observations on the Sti'ucture of A'euia and Hetero 
xenia." Phil. Trans.., CLXXXVi, p. 455, 1895. 

2. Bourne, G. C. " Studies on the Structure and Formation of the Calca- 
reous Skeleton of the Anthozoa." Quart. Jour. Micr. Sci., xli, p. 499, 1899. 

3. DuERDEN, J. E. "The Coral Siderastrea radians and its Postlarval 
Development." Carnegie Institution, No. 20, Washington, U.S.A., 1904. 

4. Fowler, G. H. " The Anatomy of the Madreporaria : I, Flabelhcm, 
Rhodopsanimia." Quart. Jour. Micr. Sci., xxv, p. 577, 1885; and Stud. 
Owens Coll., I, p. 243, 1886. 

5. Heider, a. R. von. "Die Gattung Cladocora, Ehrb." Sitzb. Akad. Wis- 
sensch. Wien, Lxxxiv, p. 634, 1881. 

6. Heider, A. R. von. " Korallenstudien : Astroides calycidaris, Blainv., 
u. Dendrophyllia ramea, Linn." Arbeit. Zool. Inst. Graz. i, No. 3, p. 153, 
1886 ; and Zeitsch. Wiss. Zool., XLiv, p. 507, 1886. 

7. Hickson, Sydney J. " The Structure and Relationships of Tuhipora.^ 
Quart. Jour. Micr. Sci., xxiir, p. 556, 1883. 

8. Koch, G. von. " Ueber die Entwicklung des Kalkskeletes von Asteroides 
Cali/cularis und dessen morphologischer Bedeutung." Mitth. Stat. Neapel, 
III, p. 284, 1882. 

9. Ogilvie, Maria M. "Microscopic and Systematic Study of Madreporarian 
Types of Corals." Phil. Trans., clxxxvii, p. 83, 1896. 

10. Ogilvie, Maria M. "The Lime-forming Layer of the Madreporarian 
Polyp." Quart. Jour. Micr. Sci., XLix, p. 203, 1906. 

* It is interesting to note that structures analogous to fibrous connective tissue, 
tendon and bone of Vertebrates, occur in the Madreporaria, viz., the middle lamina 
( = mesoglfea), processes of attachment and the calcareous coraUum, a matter which 
will be discussed in a future communication. 

12—2 



164 Mr Matthai, On Reactions 



On Reactions to Stimuli in Corals. By G. Matthai, M.A., 
Emmanuel College, Cambridge. (Communicated by Professor 
Stanley Gardiner.) 

[Read 18 February 1918.] 

The following is a brief record of feeding-experiments made on 
living Astrseid colonies during a short stay at the Carnegie Bio- 
logical Station at Tortugas (July 16 — Aug. 2) and at the Bermuda 
Biological Station on Agar's Island (Aug. 20 — Sep. 14) in the 
summer of 1915, which, though necessarily incomplete as they had 
to be undertaken in the midst of other work, gave some indication 
of the nature of reactions to stimuli in the Madreporaria. In order 
to watch the behaviour of living Corals, colonies of most of the 
recent species recorded from those localities were kept in aquaria 
of running sea- water, viz. : 

Mceandra lahyrinthifo7^mis (Linn.), Moeandra strigosa (Dana), 
McBandra clivosa (Ell. and Sol), Manicina areolata (Linn.), Colpo- 
phyllia gyrosa (Ell. and Sol), Isophyllia dipsacea (Dana), Isophyllia 
fragilis (Dana), Dichocoenia Stokesi, Ed. and H., Easrnilia, aspera 
(Dana), Favia fragum (Esp.), Orhicella cavernosa (Linn.), Orbicella 
annidaris (Ell. and Sol.), Stephanocoenia intersepta (Esp.), Ocidina 
diffusa, Lam., Mycetophyllia lamarckana, Ed. and H., Siderastrcea 
radians (Pallas), Siderastrcea siderea (Ell. and Sol.), Agaricia 
purpurea, Les., Porites astreoides, Lam., Porites furcata, Lam., 
Porites clavaria, Lam., Madracis decactis (Ly.), and Acropova 
muricata (Linn.). 

In Isophyllia dipsacea (Dana), when a particle of meat was 
placed on the oral disc with contracted mouths, the oral lip 
was slowly directed towards the particle and the mouth became 
dilated, to an extent depending on the size of the food-particle. 
The latter was, in the meantime, slowly moved into the oral open- 
ing by ciliary action. To facilitate this event, the periphery of the 
oral disc was drawn over towards the dilated mouth and the disc 
itself was somewhat depressed, thus deepening the peristomial 
cavity. During distention of the mouth, the stomodgeum was everted 
and, consequently, the coelenteric cavity Avith its convolutions of 
mesenteries became exposed.^ After the food-particle had passed 
into the coelenteric cavity, it was caught in the mesenterial coils. 
If the fragment of meat was large, the mouth remained widely open 
till the former had been reduced in size by the digestive action of 
the mesenterial filaments. The stomodtEum was subsequently with- 
drawn and the mouth opening gradually narrowed. But if, before 
this, the oral lip was touched with a glass needle, it did not contract 
as it would do instantaneously if no food-particle had previously 



to Stimuli in Corals 165 

been swallowed. Every mouth that was tested could thus take in 
particles of meat. The touch of the food-particle on the oral disc 
was also a stimulus for the expansion of the tentacles around the 
mouth and of those around the neighbouring oral openings. 

When a particle of meat was placed on the tentacles of a colony 
of Mceandra labyrinthiformis (Linn.), it was slowly passed on to the 
oral disc, but the tentacles did not show any sign of contraction. 
At the same time, the oral disc was depressed and arched over the 
mouth opening till finally its margin closed over the peristome. In 
the meantime, the tentacles were fully distended, the entocoelic 
ones were directed obliquely towards the oral opening, those of 
one side passing between those of the opposite side. The food- 
particle was now hidden from view. After it had passed into the 
ccjelenteric cavity and had presumably undergone partial digestion, 
the periphery of the oral disc gradually moved outwards carrying 
the tentacles with it, thus again exposing the peristomial cavity. 

The principal movements in these two cases are: 

(1) Ciliary movement passing the food-particle into the nearest 
oral aperture. 

(2) The direction of the oral lip towards the food-particle pari 
passu with the dilatation of the mouth. 

(3) The narrowing and deepening of the peristomial cavity, 
which help to roll the food-particle into the oral opening. 

(4) The expansion of the tentacles of the affected oral disc and 
of those of adjacent oral discs. 

(5) The eversion of the stomodeeum and consequent exposure 
of the coelenteric cavity and mesenterial coils. 

(6) The return of the soft parts to their original condition by 
the retraction of the stomodseum into the coelenteric cavity, recoil 
of the oral lip to its normal extent, shortening of the tentacles, 
flattening of the oral disc and withdrawal of its periphery carrying 
the tentacles outwards. 

When a drop of meat-juice was gently placed on a colony of 
Favia frag am (Esp.), the oral apertures in the neighbourhood were 
slowly distended after a short pause. The inner or entocoelic row of 
tentacles was then extended and directed over the oral disc, meeting 
or intercrossing over the mouth as had been noticed in the case of 
Mceandra labi/rinthiformis (Linn.), thus hiding the oral region, 
whilst the exocoelic tentacles were arched outwards. Similar move- 
ments were observed in Mceandra strigosa (Dana). 

When meat-juice was spurted by a pipette on sea- water con- 
taining a colony of Orhicella cavernosa (Linn.), strong contraction 
of the soft parts was set up in the neighbourhood, the polyps en- 
tirely closing up. This was followed by the protrusion of convolutions 
of mesenteries through mouth openings, oral discs and especially 
through edge-zones, combined with secretion of mucus over the 
polyps, the former obviously to paralyse prey and the latter to 



166 Mr Matthai, On Reactions to Stimuli in Corals 

entangle food-particles. Shortly afterwards, the oral apertures were 
widely distended to let in the meat-juice but the process was un- 
accompanied by eversion of stomodsea. Similar events were observed 
in Manicina aj-eolata (Linn.). 

When finely powdered carmine was scattered in sea-water con- 
taining a colony of Manicina areolata (Linn.), it was partly taken 
into the stomoda^a, the oral lips becoming conspicuously stained. 
The carmine was, however, subsequently passed out of the stomodaea, 
showing thereby, that the mouth openings could function as in- 
halent and exhalent apertures. 

When a tentacle of any of the Astraiid colonies was touched 
with a fine glass needle, it was suddenly withdrawn in a manner 
resembling pseudopodial movement and the neighbouring tentacles 
were also retracted. In Porites and Madracis, whose soft parts are 
composed of small polyps, the instantaneous contraction of a polyp 
due to mechanical stimulation caused the contraction of its neigh- 
bours as well. In all these cases, the wave of contraction started 
from a centre, viz., the point of stimulation, but remained local and 
did not spread over the entire colony. 

Series of movements such as the above, made in response to 
chemical and tactile stimuli, are reminiscent of amoeboid or stream- 
ing movement of protoplasm, the soft parts of the colonies appearing 
to serve as the medium for the transmission of stimuli*. If the 
initial stimulus be too strong, the sudden contraction of the soft 
parts, due to the mechanical impact, is followed by slow purposive 
movements. 

The amoeboid character of the movements of the soft parts of 
Astrseid Corals is in conformity with their histological structure 
which, on examination, revealed neither a muscular nor a nervous 
system, although a neuro-muscular apparatus has been supposed 
by most authors to exist in Madreporaria. The so-called muscular 
fibres at the base of the ectoderm and endoderm seem to be of the 
nature of specialised connective tissue fibres, for in both teased 
preparations and in sections of 4/z — 10/i thicknesses these are found 
to be without nuclei and to form part of the middle lamina (= meso- 
glsea) which is itself composed of fine fibres cemented together by 
a homogeneous matrix containing a few scattered nucleated cells. 
Fibrils pass into the middle lamina through the granular stratum 
present at the base of the ectoderm (and less frequently at the base 
of the endoderm), but these fibrils do not show any histological 
differentiation which would justify us in regarding them as belong- 
ing to nerve elements f. 

* Carpenter I'egarded the feeding reactions of Isophyllia as muscular in nature 
and as brought about by the transmission of impulses of a " nervoid character," 
but he had not investigated the histological structure of its soft parts {vide Con- 
tributions Bermuda Biol. Station, No. 20, Cambridge, Mass., U.S.A., p. 149, 1910). 

t For a detailed account of the minute structure of coral polyps vide "The 
Histology of tlie Soft Parts of Astraeid Corals " to be published shorth'. 



Mr Brindley, Notes on certain parasites, food, etc. 167 



Notes on certain parasites, food, and capture hy birds of the 
Common Earwig (Foi-ficula aiiricularia). By H. H. Brindley, M.A., 

St John's College. 

[Read 18 February 1918.] 

(rt) Effects of pa7'asitism. 

In a paper entitled " The effects of Parasitic and other kinds 
of castration in Insects " (Jour. Exper. Zool. viii. Philadelphia, 
1910) Wheeler expresses the opinion (p. 419) that Giard has given 
good reasons for supposing that the dimorphism exhibited by the 
forcipes of male earwigs from the Farn Islands, Northumberland 
(Bateson and Brindley, " On some cases of variation in secondary 
sexual characters statistically examined," Proc. Zool. Soc. Lond. 
1892, p. 585), is due to "differences in the number of gregarines 
they harbour in their alimentary tract." The reference to Giard 
is C.R. Acad. Sci. cxviii. 1894, p. 872, where he writes " J'ai tout 
lieu de croire qu'une interpretation du meme genre (referring to 
the changes evoked in Carcinus by the action of parasites) pent 
s'appliquer pour la distribution des longueurs des pinces des 
Foi'ficules males. II est possible, en effet, d'apres la longueur de 
la pince, de prevoir qu'une Forficule male possede des Gregarines 
et qu'elle en possede une plus ou moins grande quantite." 

In criticism of the above statements Capt. F. A. Potts and 
myself published a letter in Science, Philadelphia, Dec. 9, 1910, 
p. 836, in which we gave reasons for disagreeing with Wheeler's 
conclusion : viz., (i) that in the absence of any further account by 
Giard the above passage could not be taken as direct evidence 
that he had examined the intestine of Forficula for gregarines and 
found a correspondence between their presence and the condition 
of the male forcipes ; (ii) that out of several thousand earwigs 
collected by us on the Farn Islands in 1907 over 50 males of 
different forceps lengths were carefully dissected with the results 
that the gregarine Clepsydrina ovata was found to occur commonly 
in the alimentary canal, that it occurred indifferently and was 
absent indifferently in " low " and " high " males, and that 
no correlation could be traced between the number of parasites 
and the length of its forcipes. Moreover, no difference in the 
development of the testes or other internal sexual organs could 
be detected in low and high males respectively. 

Since the above was written I have (August 1917) examined 
the alimentary canal of 51 earwigs out of a large batch obtained 
at Porthcressa, St Mary's, Isles of Scilly, where the males exhibit 



168 Mr Brindley, Notes on certain parasites, food, and capture 

well-marked dimorphism (Camb. Phil. Soc. Proc. xvii. part 4, 1914, 
p. 831). 

The results summarised are as follows: 



Infection by Clepsydrina ovata. 





Number 
examined 


Not 
infected 


Infected 


Number of 

gregarines 

found 


Average number 

of gregarines in 

the infected 

individuals 


Low males 
High males 
Females 


23 
23 

5 


12 

11 

1 


11 
12 

4 


323 

238 
53 


29 
20 
13 



Thus the evidence so far obtained is that the dimorphism of 
the forcipes in F. auricularia </ is not a result of or influenced by 
gregarine infection — though in view of the well-established effects 
of such parasitism on the secondary sexual characters of another 
arthropod in Geoffrey Smith's case o^ Inachus dorsettensis modified 
by the gregarine Aggregata {Mitt. Zool. Stat. Neap. xvii. 1905, 
p. 406), the absence of positive evidence to the contrary at the 
time Wheeler wrote, but now obtained, certainly afforded ground 
for his support of Giard. 

In this connection I may quote a letter from Geoffrey Smith, 
whose recent death at the battle front brings us into common 
mourning with Oxford zoologists for a friend and colleague. 
Writing to me about 1907 he said, " Have you noticed that Giard 
attributes all cases of High and Low Dimorphism to parasitic 
castration ? I am sure this is not right, but there is no doubt 
that parasitic castration is a much more frequent occurrence than 
is commonly supposed." These words, and a footnote to the same 
effect in his paper " High and Low Dimorphism " (Mitt. Zool. Stat. 
Neap. XVII. 1005, p. 321), are typical of the writer's insight and 
balanced judgment. 

It may be stated that the gregarines in the Porthcressa earwigs 
fell roughly into categories of small, medium, and large, but they 
all seemed to be C. ovata. Rather more than half were small 
individuals, and those of medium size were slightly in excess of the 
large, but the sizes were not recorded in the case of the first few 
earwigs examined. Very large numbers were found in syzygy, 
and such associated individuals were of all three sizes. One 
instance of syzygy of a large with quite a small individual was 
observed. There was no noteworthy difference between the 



hy birds of the Common Earwig (Forficula auriculana) 169 

numbers of gregarines of different sizes or between the proportion 
of free gregarines to those in syzygy in their low and high male 
hosts respectively. 

During our stay on the Scilly Islands in 1912 Capt. Potts and 
myself, in company with Capt. J. T. Saunders, found in St Martin's 
several earwigs parasitised by a gordiid larva {sp. incert.), the coils 
of which, though projecting between the terga of the abdomen, 
seemed to have no effect on the health and activity of their hosts. 
The same apparent absence of deleterious effects was noticed in 
three of the Porthcressa batch of 1917 which were found to be 
similarly infected. In one, a low male, a large gordiid occupied 
most of the body, and no portion of the alimentary canal posterior 
to the crop could be found ; in a high male similarly infested by a 
large gordiid there was very little of the hind gut left ; and an adult 
female contained three or four gordiids of various sizes, the gut in 
this case being intact and apparently healthy. A fourth individual, 
a low male, was not parasitised when examined, but as the gut was 
partially atrophied, it had probably been recently deserted by a 
gordiid. All these infected individuals seemed as active and 
healthy and to possess fat bodies as large as those not infected ; 
the earwig's resistance to such extensive destruction of internal 
organs is very noteworthy. As Clepsydrina ovata inhabits the 
chylific ventricle and hind gut and as the presence of gordiids 
evidently often results in destruction of these portions of the 
alimentary tract, the latter parasite is likely to be exclusive of 
gregarines, and these were absent in all three of the males 
mentioned above (including that with the hind gut intact), while 
only two were found in the female. 

That the presence of parasitic worms has sometimes serious 
effects on the insect's health is suggested by the recent observations 
of Jones recorded in " The European Earwig and its control," 
a report on the invasion of Newport, R.I., in 1911 by Forficida 
auricularia and its subsequent spread ( f/. >§. Dept. Agric. Bidl. 566, 
Washington, June, 1917), from which it appears that 10 per cent, 
of earwigs kept in the laboratory were killed by the infection of a 
worm identified as Filaria locustae, whose average length is given 
as 83 mm. This however is a size exceeding considerably that of 
the gordiids in the Scilly earwigs, which I have called " large " 
when attaining a length of 50 mm. 

In southern Russia Forficula tomis, Kolenati, is parasitised by 
the tachinid fly, Rhacodineura antiqua (Pantel, Bull. Soc. Entom. 
France, No. 8, Paris, 1916, p. 150), but I do not know if it attacks 
the common earwig. The paper quoted mentions the capture of 
the adult fly in Holland and Portugal. 

Lucas {Entom. XXXVII. 1904, p. 213) reports F. auricularia 
(or ? lesnei) attacked by scarlet acarine mites. 



170 Mr Brindley, Notes on certain parasites, food, and capture 

Among fungoid parasites, EntomopMhora forficulae diminishes 
the number of earwigs (Picard, Bidl. Soc. Etude Vulg. Zool. Agric. 
Bordeaux, Jan. — April, 1914, pp. 1, 25, 37, 62). It is possibly this 
species which has caused heavy mortality among the earwigs which 
I have kept in captivity in the Zoological Laboratory during 
recent years. Infection by the above or other fungus is a very 
frequent result of damp in the soil or in the plaster of Paris cells 
bedded with coco fibre which I have employed. The most effective 
preventive of fungus has so far been keeping the earwigs in 
roomy glass dishes lined with virtuall}^ dry sand and supplj^ing 
water only by wetting the vegetable food given. 

(6) Food. 

In " The Wild Fauna and Flora of the Royal Botanic Gardens, 
Kew," 1906 {Kew Bull. Add. Series V), Lucas writes (p. 23) of the 
Common Earwig, " It is an animal feeder. Does it do. as much 
damage as is supposed ? " And Ealand in " Insects and Man," 
1915, p. 266, states "most gardeners would assert that the insect 
is destructive to cultivated plants. Careful observation and 
experiment, however, show that it is carnivorous and that it 
devours caterpillars, snails, slugs, etc.... its habit of hiding in such 
flowers as the sunflower and dahlia have earned it an undeserved 
reputation for evil." 

I find that seven out of nine recent and more or less compre- 
hensive manuals of Economic Entomology do not mention earwigs 
at all, which is fair evidence for considerable doubt as to their 
being harmful insects. Of the two works in which earwigs are 
mentioned one speaks of them as destructive to mangolds, turnips, 
cabbage crops, and plant blossoms, while the other states dahlias 
as attacked, " but nearly all plants suffer." Virtually every fruit 
grower and horticulturist of whom we make enquiry assures us 
that earwigs are most destructive pests, but is the general belief 
thus expressed really well founded ? 

Recent literature leaves the impression that in certain localities 
earwigs may be specially harmful to plants of economic value, 
though an explanation of this capriciousness is wanting. Theobald 
(Rep. on Econ. Zool., South-Eastern Agric. Coll., Wye, April 1914) 
gives hops as attacked by F. auricidaria. Lind and others in a 
summary of the diseases of agricultural plants in 1918 (79 Be- 
retning fra Staiens Forsogsvirksamded i Plantekidtur, no. 30, 
Copenhagen, 1914) state that in one locality in Denmark cauli- 
flowers were completely destroyed by the Common Earwig, which 
seems a very exceptional event. Sch^^iyen in Beretning om skadein- 
sekter og plantesygdommer i land og havchruket 1915 (Report on the 
injurious insects and fungi of the field and the orchard in 1916), 



hy birds of the Common Earivig (Forficula auricnlaria) 171 

Kristiania, 1916, mentions that in many parts of Norway different 
vegetables, cabbage in particular, were extensively damaged by 
F. auricularia. Tullgren, in a report on injurious animals in 
Sweden during 1912 — 1916 (Aleddelande frdn Centrcdanstalten 
for Jorshruksforsok, no. 152 ; Entomologiska Avdelningen, no. 27, 
p. 104), records damage by F. auricidaria to ornamental plants, 
barley, wheat, and cabbage. In the case of the invasion of New- 
port, R.I., by the Common Earwig, Jones {op. cit.) reports that the 
quite young individuals eat tender shoots of clover and grass, and 
possibly grass roots ; while later on shoots of Lima Bean and dahlia 
and blossoms of Sweet William and early roses are attacked, with 
a general preference for the bases of petals and stamens rather 
than for green shoots. Adults are recorded as feeding almost wholly 
on petals and stamens, though clover, grass and terminal buds of 
chrysanthemums and other "fall flowers" are also devoured. Sopp, 
"The Callipers of Earwigs" {Lanes, and dies. Entom. Soc. Proc. 
1904, p. 42), records having seen a female earwig using her forcipes 
to repeatedly pierce damp decaying seaweed on which she was 
apparently feeding. Ltistner {Centralhl. Bakt. Parnsit. u. Infektions- 
krankheiten, XL. nos. 19-21, Jena, April 1914, p. 482) has summa- 
rised the work of over thirty observers of the contents of the crop 
of the Common Earwig. Altogether 162 individuals were thus 
examined, and the conclusion was arrived at that earwigs normally 
feed on dead portions of plants and on fungi such as Gapnodium, 
living leaves and flowers being attacked when circumstances 
favoured the change. Dahlia leaves and petals were very readily 
devoured. How far earwigs are a pest to ripe fruit seems not to 
have been investigated, but it was concluded that as a rule they 
may be regarded as harmless save in special cases. It was admitted 
however that the further the enquiry went the less definite were 
the results. 

In view of the diversity of reports as to the favourite food 
plants of earwigs and the general want of exact information as to 
the damage likely to be done by earwigs in a flower or kitchen 
garden I carried out a small series of observations on the earwigs 
obtained last August from St Mary's, Isles of Scilly, which were 
kept in captivity in the Zoological Laboratory for some weeks, 
primarily for the purpose oj" examining their alimentary canal for 
parasites. These earwigs, several dozen in number, were kept in 
a large glass dish bedded with sand slightly damped occasionally. 
They had no animal food save that afforded by those which died. 
In order to obtain information as to preference for one kind of 
plant above another they were given three different species, taken 
haphazard, at a time for a period of two days or more. 

A summary of the results is as follows : — 

Aug. 20 and 21. Vegetable marrow leaves were ver\' much 



172 ilf?' Brindley, Notes on certain parasites, food, and capture 

eaten ; horse-radish leaves very little touched ; Michaelmas Daisy 
leaves and flowers hardly, if at all, touched. 

Aug. 22 and 23. Beetroot leaves were much eaten, the leaf 
stalks m particular, these being opened out and the pith taken : 
white phlox leaves and flowers, the petals much gnawed and pollen 
grains were found in the gut : dwarf bean leaves, little touched. 

Aug. 24 to 26. Blue Anchusa leaves and flowers, the petals 
were much eaten but the leaves neglected : white rose leaves and 
flowers, petals devoured but leaves untouched: golden rod (Solidago) 
leaves and flowers, leaves nibbled at sides here and there but 
flowers apparently neglected. 

Aug. 27 to 29. Yellow Oenothera flowers and pods, the petals 
were much eaten but the pods remained untouched : white Japanese 
anemone leaves and flowers, petals eaten to some extent, leaves 
neglected : raspberry foliage, the leaves were not nibbled, but the 
earwigs congregated in numbers on their hairy undersides, an 
action much more pronounced than in the case of any of the other 
plants given throughout the observations. 

Aug. 30 and 31. Cabbage leaves were destroyed by the blade 
bemg gnawed down between the veins to the midrib while the 
ends of the veins were shorn off: rhubarb leaves, eaten a good 
dea : scarlet runner leaves, flowers, and pods, apparently quite 

Sept. 1 to 3. Plum fruit unskinned was much attacked- 
potato tuber and rather unripe apple, both unskinned, were not 
touched at all. 

Sept 4 to 10. On the 4th the plum was removed, but the 
apple and potato were not attacked during the seven days. 

Sept 11 to 15. On the 11th the apple was cut across, with 
the result that it was slightly gnawed during the five days : the 
potato remained untouched. 

Sept. 16 to 2:l On the 16th the potato was cut across, which 
was followed by its being very thoroughly attacked, though the 
apple was not entirely deserted. 

Of the 51 earwigs whose alimentary canals were examined for 
gregarine 7 contained spores of Fuccinea graminis (one had as 
many as 180 and another 100), while the food of another individual 
included numerous unidentified enjiomophilous pollen grains 
Both spores and pollen grains appeared to be very slightly if at all" 
digested. It is hoped to extend the observations in the coming 
summer, as those recorded above were limited to only a few of the 
possible food plants and only adult earwigs were kept. It may 
well be that there are differences in the preferences of nymphs 
and adults, and as the former are in the majority till about the 
end of July, it is possible that they may be harmful to certain 
plants m particular, as Jones's observations (o;j>. cit.) suggest. 



hy birds of tJie Covinion Eariuig (Forficula auricularia) 173 

It seems established that a large number of ordinary garden 
species are liable to serious attack by earwigs, and that the latter 
can continue healthy on a purely vegetable diet. But much further 
information of a detailed kind is required befoi'e we can explain 
why in a given locality a particular kind of plant is attacked 
while in another it is neglected. Does it mean that the presence 
or absence of suitable animal food is a factor ? 

As regards animal food, there is a considerable amount of 
evidence that earwigs are often carnivorous by choice, very 
possibly they are so usually (cf Rlihl, M.T. Schweiz. Ges. vii. 
1887, p. 310). In respect of eating dead animal matter I have 
found that when kept in captivity they devour the soft parts 
of their fellows who have died even when fresh vegetable food 
is available. In this necrophagous habit they resemble cock- 
roaches. Jones {op. cit.) states that dead flies and dead or dying 
comrades are devoured. Lustner (op. cit.) finds that only dead 
animal matter is taken. This conclusion points to too limited an 
inquiry and want of taking into account the possible presence of 
food plants which were more attractive than available living prey. 
In any case his opinion that earwigs should not be regarded as 
beneficial is traversed by the records of their killing certain insect 
pests of plants. 

Round Island, the northernmost islet of the Scilly group, is 
swarming with earwigs, and they congregate in vast numbers in 
the light-keepers' midden inside the discarded pressed beef tins. 
If, as seems probable, they reached the islet before the lighthouse 
was built a change of diet seems to have occurred, as the indigenous 
vegetation is chiefly Armeria maritima, Cochlearia officinalis and 
Mesembryanthemum edide. There is no turf It is of course 
possible that they seek the potato peelings also thrown into the 
midden and that their numbers inside the discarded tins mean 
that the latter are frequented partly for shelter. If the Round 
Island earwigs have really turned during comparatively recent 
years from a herbivorous to an extensively carnivorous diet, 
Rosevear, another islet of the Scilly group may, in a sense, be a 
converse case. It is the other locality in the Scilly group in which 
(as far as I know) the earwig population is densest. Like Round 
Island, it is very small, but differs from it in being uninhabited. 
But from 1850 to 1858 it was occupied by the builders of the 
Bishop Rock Lighthouse, so is it possible that the abundance of 
earwigs is due to the animal food available in the past ? However 
this may be the present diet of the Rosevear earwigs appears likely 
to be vegetarian in the main, unless the islet harbours some insect 
or other small arthropod suitable for food. The commonest plants 
are Armeria maritima and Lavatera arborea, the latter growing 
luxuriously. But before the abundance of earwigs on Rosevear 



17-i Mr Brindley, Notes on certain jKtrasites, food, and cajytare 

can be discussed adequately something must be known of the con- 
ditions obtaining on Rosevean and Gorregan, its small and only 
immediate neighbours. Of these islets I possess no information 
at present. Also, there are other peculiarities as regards the 
earwigs of Rosevear and Round Island which are beyond the 
scope of the present paper. 

There is no doubt that earwigs sometimes kill and devour 
other insects larger than themselves, though the event is probably 
somewhat exceptional. Chapman ("Notes on Early Stages and 
Life History of the Earwig," Entom. Record, xxix. no. 2, Jan. 
1917) states that "animal food, such as dead insects, seemed always 
acceptable " to earwigs in captivity. Sopp {op. cit. p. 42) regards 
earwigs as probably "omnivorous feeders, largely carnivorous by 
choice, but often phytophagous, frugivorous, or even necrophagous 
of necessity." Whether attack on living animals as prey is 
common I cannot say, I have no observations of my own to 
record ; ^ but it appears that occasionally the forcipes, organs of 
much disputed function, are used for this purpose. Sopp (op. cit.) 
has seen them employed to seize and crush large flies which were 
. subsequently devoured and quotes an instance of a larva similarly 
attacked from the records of another observer. Burr (Entom. 
Record, Sept. 1903) saw a blue-bottle seized by the forcipes of a 
male Labidura riparia kept in captivity. Lucas {Entom. xxxviii. 
1905, p. 267) records a female of this species as using the forcipes 
to capture a cinnabar moth larva, which was afterwards devoured. 
Jones {op. cit.) records that the Newport, R.I., earwigs attack and 
devour " certain sluggish unprotected larvae." 

There are many observations which show that earwigs in some 
localities prey upon small insect larvae, and in certain instances 
they have been recommended as a means of diminishing plant 
pests. Thus the following references, as also others quoted in this 
paper, have appeared in issues of The Review of Applied Entomo- 
logy, 1913—1918. Bernard {Technique des traitements contre les 
Insectes de la Vigne, Paris, 1914) states that they devour the 
pupae of one or more of Clysia amhiguella, Polychrosis botrana, 
and Sparanothis pilleriana {v. also 'Kirkaldy, "^^i^o??*. xxxiii. 
1900, p. 87). Dobrodeev {Mem. Bur. Entom. of Gent Board 
of Land Administration and Agric, Petrograd, XL no. 5, 1915) 
makes a similar report as regards the destruction of the first tAvo 
Tortricidae named above by earwigs. Molz {Zeits. Angeiuandte 
Ghemie, Leipzig, xxvi. nos. 77, 79, 1913, pp. 533, 587) speaks of 
earwigs as natural enemies of the vine moth. Feytaud {Bull. 
Soc. Etude Vidg. Zool. Agric. Bordeaux, xv. nos. 1—8, Jan.— Aug 
1916, pp. 1, 21, 43, 52, 65, 88) states that earwigs destroy the 
eggs and larvae of the coccid vine pests Eidecanium persica and 
(probably) Pulvinaria vitis. Harrison in "An unusual parsnip 



hy birds of the Coiiiiiwn Earwig (Forficula auricuLiria) 175 

pest" {Entomologist, XLVI. Feb. 1913, p. 59) reports them as 
most effective in killing and eating Depressaria heradicwa, the 
"parsnip web-worm." Brittain and Gooderham (Canad. Entorn., 
London, Ont., XLVii. no. 2, Feb. 1916, p. 37) make a similar state- 
ment. 

There is no doubt that our knowledge of the bionomics of 
the earwig is at present very imperfect. As in the case of other 
very common animals far too much has been taken for granted. 
The earwig's nocturnal habit, its tendency to assemble in great 
numbers between two closely apposed surfaces, and its "frightening 
attitude " of flexing its abdomen dorsalwards with opened forcipes 
all tend to give it a reputation for evil which very probably is 
but partially deserved. We all know how the habit of entering 
crevices is responsible for the belief that it gnaws through the 
tympanic membrane with the result of mania or even death. 
Perce-oreille speaks for itself. It seems fairly established that 
its universally bad reputation among gardeners is founded on 
tradition and want of judgment combined with neglect of the 
increasing evidence that its presence is sometimes beneficial by its 
destructiveness to more harmful insects than itself That it eats 
the petals of dahlias and chrysanthemums to some extent is true, 
but as far as my own observations go the outlay of time and 
material devoted to the traditional protection of the flowers by 
inverted flower pots stuffed with straw seems hardly worth while. 
The great attraction which the flowers have for earwigs seems to 
be the closeness and number of their petals, which provide a 
daytime shelter whence nightly excursions for feeding are made. 
Anyone possessing a garden may greatly add to our knowledge of 
favourite foods; observation at night is particularly needed. As 
regards garden varieties of roses the case against earwigs is 
probably more severe. 

(c) Capture by birds. 

During the last decade systematic investigation of the contents 
of the alimentary canal of British wild birds by several observers 
has resulted in most useful information as to which should be 
regarded as harmful and which as neutral or beneficial to agri- 
culture. It is manifest from the laborious and painstaking work 
now at our disposal that many of the reputations, good or evil, 
which certain common birds have in the eyes of farmers and 
gardeners need considerable revision, in some cases even reversal. 

As regards the capture of earwigs by birds, it appears that 
they are not a favourite food when we bear in mind how numerous 
they are sometimes and that they are large enough to be easily 
seized. No doubt their nocturnal habit affords much protection 
from capture. 



176 Mr Brindley, Notes on certain parasites, food, and caj^ture 

Collinge in " The Food of some British Wild Birds " (London, 
1913) reports on the contents of the crop, etc., of 29 of the com- 
monest species, among which only four contained earwigs, and 
these were very few in number. Thus in 404 House Sparrows 
2 earwigs were found, 1 in each of 2 birds; in 721 Rooks 2 ear- 
wigs were found, 1 in each of 2 birds ; in 40 Skylarks 3 earwigs 
were found among 2 birds; in 64 Song Thrushes 7 earwigs were 
found among 2 birds. 

Newstead in " The Food of some British Birds " (Sapp. to Journ. 
of Board of Agric. no. 9, Dec. 1908) records observations on the 
swallowed food of 128 species, the outcome of 871 post-mortem and 
pellet examinations carried out in various years from 1894 to 1908. 
He finds that 10 sj)ecies had eaten earwigs, the numbers of birds 
examined and the numbers of earwigs found being : 1 Whimbrel, 
40 earwigs ; 2 Green Woodpeckers, 24 earwigs ; 2 Starlings, 3 ear- 
wigs; 1 Nuthatch, 3 earwigs; 1 Chaffinch, 1 Great Titmouse, 
1 Redbreast, 1 Song Thrush, 1 Whinchat, 1 Woodcock, 1 earwig 
each. 

Theobald and McGowan in '•' The Food of the Rook, Chaffinch 
and Starling" {Sapix to Journ. of Board of Agric. no. 15, May 
1916) put on record a particularly valuable and interesting series 
of observations, as they examined the food month by month 
during nearly 2^ years, viz., from Jan. 1912 to May 1914, the 
inquiry covering 277 Rooks, 748 Starlings, and 527 Chaffinches. 
An analysis of their results as regards earwigs for the 2^ years is 
as follows: 



3K 

^c^ 





Birds 
examined 


Earwigs 
found 


Average number of 

earwigs taken by 

each bird 


Starling 

Chaffinch 

Rook 


372 

277 
121 


154 
7 
3 


•41 

•025 
•025 


Starling 

Chaffinch 

Rook 


376 

248 
156 


199 
5 
3 


■ -53 

•020 
•019 



I have divided the year into two j)eriods of six months con- 
formably with the seasonal presence or absence of earwigs on the 
surface of the ground. From October to March most male earwigs 



hy birds of the Common Earwig (Forficula auriciilaria) 177 

die and the females are hibernating. In view of this it is curions 
that earwigs should be taken as numerously during this period 
as during the six months when both nymphs and adults can be 
found easily. The numbers recorded for Rook and Chaffinch are 
small, though a large number of birds were examined. The Starling 
is a great insect eater; is it possible that it habitually searches 
for buried insects during the colder months and devours earwigs 
found with the rest ? This action may be true for the other two 
birds also. The figures for all three are certainly curious. 

So we find only 13 species of birds reported as having captured 
earwigs, and most of them as very sparingly. The Starling is not 
recorded by Collinge as an earwig eater. 

The above quoted reports certainly suggest that wild birds 
cannot be relierl upon to diminish earwigs in a garden. Many 
of the most insectivorous are not reported as feeding upon 
earwigs at all. They may be distasteful, and a large number 
together emit a well-defined odour, and the same is true of a 
number preserved in alcohol. Be this as it may, domestic fowls 
always eat them readily, a fact which is noted by Jones {op. cit.) 
in the case of the invasion of Newport, R.I. He also mentions 
that toads will eat them. 

Miss Maud D. Haviland, Hon. Mem. B.O.U., to whom I am 
indebted for assistance with regard to the literature of the subject 
and for kind advice in the preparation of these notes, informs me 
that she has noticed a Redbreast take earwigs in preference to 
earthworms. 



ADDENDA. 

Under (b). 
Mr H. Ling Roth informs me that he has found earwigs very destructive to iris 
pods, with resulting premature fall of seeds, in a garden at Halifax, Yorks. 

Under (c). 
Gurney, in "Ornithological Notes from Norfolk for 1916" {British Birds, x. 1917, 
p. 242j, records that his father in October, 1843, found several earwigs in a Stone 
Curlew. 



VOL. XIX. PART IV. 



13 



178 M(ijo7^ MacMahov and Mr Darling, Reciprocal Relations 



Reciprocal Relations in the Theorij of Integral Equations. By 
Major P. A. MacMahon and H. B. C. Darling. 



[Received 1 February 1918. Read 4 February 1918.] 
1. Let f{oc)K{ajt)dx = ylr,{t) 

J a, 

and f2{cc)/c(a;t)da; = yjr^{t); 

J a., 

then, if we suppose the functions f^,f and k to be such that the 
order of integration is indifferent, we have 

fbi rbo rb, 

/ /i (•^) fa (^t) da; = dy \ f {x)/., {y) k {xyt) dx 

= \\Uy)i^i{yt)dy, 

or, as it may be written, 

/ A(oo)yjr,(xt)dx= f,(x)yfr,(xt)dx (1). 

*i J a.2 

In the Messenger of Mathematics, May 1914, p. 13 Mr Rama- 
nujan has employed this result to deduce a number of ' interesting 
relations between definite integrals. The method is very suggestive 
and appears capable of considerable extension. For example, if 

f{x)K[e{x,t)\dx = ^lr,{t)\ 

[b. \ (2), 

and / fM'c{e{x,t)]dx = ^lrM 

*^^n \j^ (•^) ts [0 {a; 01 dx = ^J, {x) f, {0 (x, t)} dx . . .(3), 

provided that {x, 6 {y, t)]=- d [y, 0{x, t)\ (4). 

The functional equation (4) is satisfied by 

0{^,t) = cl,-^f(x) + cl>(t)\ (5), 

where / and are arbitrary functions ; which is a general form of 
solution and includes among others such solutions as 

H^>t) = c}>-^{f(x).cl>{t)} (6), 

^ ^ \f(^) + cP{t)\ ^'>' 

^{^,t) = cf^~^f(x) + cf,{t)+f(x)cf,(t)] (8). 



in the Tlieory of Integral Equations 



179 



Thus, to derive (7) from (5) let 

f{x) = coth-i [P{oc)], (/> (0 = coth-i [(^1 {t)\ : 

then (5) becomes 

0-' [coth-' [F{x)] + coth-i {(^1 {t) W 

Now let (^"^ (^) = u, 

then ir = <^(m) = coth"' 1^1 («)}, 

whence ^i {u) = coth 0, 

and u = 4>r^ (coth 2) ; 

that is (/)-' (^) = 01-1 (coth ^), 

and therefore (5) reduces to 

_^\ F{x)4>,(t)+l ] 
F{x) + (ji,(t) 



0r 

which is of the form (7 ). 

As an example of the use of (2) and (3) in the determination 
of relations between integrals, let 

/i (•'^) = sin X, /, {x) = cos X, 

and, using the form (6) for 0, let 

0{x, «) = e»'-'o.'^', 

K (x) = X. 

bi = b.2 = a, Ui = «y = 0, 



and 

Then, putting 

we have from (2) yfr^ (t) = sin x . e^iog'' 

J 



dx 



(log t . sin a — cos a) e^^^st 4. 1 

^ l + (logO' ~ 



and 



yfr..it)=\ cos.'r.e-'"'°s'rf.« 

Jo 

(log t . cos a + sin a) e"^^^ — log ^ 



l+(logO-^ 

Substituting these values in (3), and then putting log^ = l/r 
for brevity, we obtain 

'*" [x sin {x — a) + 7' cos {a; — a)} e'**'"' , 






r' + «' 

' -' it' sin X -\- r cos x 



i ■-' X SI 
•'0 



y.2 _j_ ,^2 



f/./ 



= 0: 



13—: 



180 Major MacMalion and Mr Darling, Reciprocal Relations 
so that, provided r is not zero, we have 



X sin {x — a) + r cos {x — a)\ e"'^'^ 



J r- + x^ 



X sm X -{-r cos .r , 

r^ + ic- ^ 

an identity which may be verified by differentiation with respect 
to a. Putting x = r tan ^ and then replacing ^ by a;, (9) becomes 



•*^'^~' «/'• cos(^ + a- r tan .^■) „ tan x 



'^-^"^'^ ' "^ ■_::^-J g a tan x ^^ 

J cos X 



.tan 1 a/;- ^^g (^ _ ^ ^^^ ^A 

= ^ ax (10), 

J cos X 

which admits of ready verification by differentiation with respect 
to a. The identities (9) and (10) hold generally, provided that 
the constants are finite; we have seen that r must not be zero. It 
will be noticed that both (9) and (10) are of the form 

Jo Jo 

where the upper limits of integration involve a. 

2. As another illustration of how the method admits of genera- 
lisation, let 

fAx)'c{0{x,t)]dx = y\r,{t). 
J «, 

rb, 
and f2{x)K{d {x, t)] dx = yjr, (t) : 

J 0.2 

fbi . fb, 

then I /i (x) v/^a {\ (x, t)} dx=\ fo {x) f, {\ (x, t)} dx 

J a, J a, 

when \ {x, t) = 4>^-^ {/(x) + (j>, (t)} 

and e(x,t)=g{f(x) + cl>,it)}, 

f, g, (pi and (f>2 being any functions. It should be observed that A. 
becomes 6 when (f)^ = ^2 and g = 02~^. Other corresponding pairs 
of functions are 

\(^,O = </>rM/(^')-0i(O), 

0(x,t)^g{f(x).cj>,{t)}> 

and M^^0 = <^r^J4^r\^4^|, 

'f(x)cf>,{t) + l] 



e{x,t)=g 



f{x) + (f>,{t) 



so that 



in tJie Theory of Integral Equations 181 

8. A further extension is obtained when the kernel k includes 
more than one parameter t; thus let 

/i (x) K [6 {x, ti , Q} dx = -f, (^1 , t,), 
/„ (x) K [d {x, t„ t^] dx = yfr. (t, , t.^, 

J a.2 

\ fi (!/) « [^ {!/> f^ (^> ii> Q> V {x, ti , t)}] di/ 

J a, 

= -v/tj \/j, {x, ti, t^, V {x, ti, t.^\ 

and 

f Vi (//) « [^ y^ f^ (•'•' ^i> Q, V {x, t, , f,)}] dy 

= \/ro [^ {x, t,, ti), V {X, t,, Q). 

Now consider 

/i (•^-'O -^/^a [/A (*', ^1, t;), V {x, ti, t^)} dx 

= f V"i (^) ( I ' /3 (i/)/^ [0 [y, ti (,*•, ^x, t^}, V {x, t, , t,)}] dy) dx. 

■J tti ^ ■ fl2 ' 

This double integral is equal to 

if ^ {y/, /x (.r, t„ t,), V {x, t„ t.^} = e [x, iM (y, t„ t.;), V (y, t„ L)}. 
Now suppose 

fl (X, t„ «,) = <^,-' [f{x) + (/>! {t,) + (/>! (^2)}. 

^ {x, t„ L) = </)-! \2f(x) -f- (/>! (^0 + </), (t,)} ; 
then 6^ {y, /j.{x, ti, t,), v{x, t^, t,)} 

= </>-! {2/(2/) + 2/ (*■) + (/), (t,) + 01 (t,) + (/), (^0 + 0. (g). 
This is symmetrical in x and y, so that we may write 
/^(^■, ^1, ^2) = 0rM/3(*')+ 03(^1, 4)j, 
/. C^-, t„ t,) = 0,-^ 1/4 (^0 + 04 (^1, ^2)}, 
(a^, t„ t^ = g [f, (x) +f, (x) + 01 {t,) + 0, (QK 
leading to 

5'{/3(^)+/4(i/)+/;(*')+/4(^O+03(^l, ^ + 04(^1, t.^\, 



182 Major MacMahon and Mr Darling, Reciprocal Relations 
which is symmetrical in x and v/ ; and hence it follows that 
I J[(x)ylrn {/m(x, ti, L), v{x. t^, ig)} dx 

■ ' a, 

= I fo,{x)'\^i {/"-(*■> t\, ^2)) v{^> ^1) ^2)) dx. 
As a particular case we may write 

^L {x, t, , L) = cf^r' (./X^O + ^1 (A) + ^1 (^2)}, 

V (X, t, , t.;) = (/),-' {/(*■) + 02 (^1) + C^2 (4)}, 

(x, t„ Q = g {2/(*0 + 4>, (t,) + 4>, (01, 

and again 

^l{x, t„ Q = ct>-' {^,f{x) + (i>(Q + cj)(t,)], 

{x, t„ Q - ct>-' {/(x) + (^0 + (/) (t,)}, 

the case where /x i^ y and each resembles as much as possible. 
It is evident that the case in which the kernel includes any number 
of parameters may be treated in the same manner and presents 
little difficulty. 

4. The method may also be extended to double integrals. 
Thus let 

/i {^> y) K^ [^ (*'. y> ii> 4)1 dxdy = f^ (t,, to), 

J »! J a,' 

/■2 (^-^ y) « [^ (^, 2/. ii, 4)1 dxdy = -f . (^i, 4) ; 

[b, rb,' 

then / /i {cc, y) i/r^ {/x (^■, y, t^, 4), ^ (a-', y, 4, 4)} c^^'f^^/ 

J «, >/ a,' 



6/ 



/2(^, 2/)'fi l/^(«> y. 4, 4)> ^(^S ^» 4, 4)1 dxdy 

if ^ {^r, w, yu, (a;, y, ^j, 4), v {x, y, t^, t^)] 

= [x, y, IX {z, w, ti, t.^, V (z, IV, ti, 4)1- 
If A, B, G, D, E be functional symbols, one solution is 
ix{x, y, t„ O = A-' [B{x, y) + C {t„ t,)] 
v{x, y, t„ t,) = D-^[B{x, y) + E(t„ t,)} 
(x, y, t„ t,) = B (x, y) + kA (t,) + ^D (t,). 



in the Theory of Integral Equations 183 

5. Let US next consider the case of three integral equations 

!'\f\{x)K{dU;t)}dx = f,{t), 
J «, 

/; (a:) K [e {x, t)] dx = ^/r, (t), 



J a,, 

r\f,(x)K{d(x,t)]dx=f,(t). 



We have 

'yA''^)f.{OOr,t)\ir,{d{x,t)}dx\ 

= r Mx)^}r,{e{x, t)}f,{e(x, t)] dx\ (11), 

= fV;cr) ti {^(^'> 01 ir,{0(x, t)} dx ] 

if certain conditions are satisfied. For 

'"' f\{x)y^,[e{x,t)]ylr,[e{x^t)]dx 

= I '' ./; i-'^) f ' /. (z/) '^ [^ (>/> t)\ dy f V; {z) K\e{z, t)\ dzdx, 

and the equalities (11) will hold good if, for example, k (x) = x'^ and 

d[y,d{x,t)\.e[z,e{x,t)] 

is unaltered by the circular substitution {xyz). 

Now suppose that ^ 

e{x,t)^f{x)t-^ (12;; 

then [y, {x, t)] . 6 {z, d (x, t)\ =f(y)f(z) {x, t) 

Hence if k(x) — x'^ the relation (12) satisfies the conditions. The 
generalisation to the equality of n integrals is apparent, and in 
that case 

0(x, t)=f{x)t''-'^ 
is a solution. 
We have also 

/i (.'•) f; 1^ (*•, 0} ^3 {^ (.'<-■, t)] dx 



fb, 
= f, (.'/;) yjfs {\ (x, t)} ^fr, {X (x, t)] dx 

J rta 



184 Major MacMahon and Mr Darlwcf 

if X {x, t) =f{x) t'\ e (x, t) = { f{x)Y'-'t''', 

and in particular if 

X {x, t) =f{x) t^, e {x, t) = [\ {x, t)Y-''. 
A solution may also be obtained when k (x) = [f, in which case 

K[e[y,e{x,t)\'\.K[d{z, 6'(^, 0}] = e^'•^'^^''■'*^^'^^'''^^''•*^^• 
Putting d{x,t)=f{a^ + lt, 
we have 

e [y, d (x, t)] + e{z,e (x, t)] =/(y/) +f(z) + f(x) + ^t, 
which is of the symmetrical form required. 

6. In the cases investigated above the kernels of the several 
integral equations have been functions of the same form. It is, 
however, easy to extend the method to the case where the kernels 
are functions of different form. Thus if 

/i (x) /ci {0 (x, t)\ dx = -v/tj (t), 

bo 
/a (x) K. {6 (x, t)} dx = ylr.2 (t), 

we are led to the condition 

K, [6 [y, \ (x, t)]] = K, [6 [x, \ (y, t)]]. 

Case 1. Let Ki(z)—z, k2{1/z) = z; then the condition becomes 
0{y,X(x,t)].d{x,\(y,t)}^l; 
a solution of which is 

d(x, t)^xWi^\ (A (OK % {0(0, F{^)i 

where \ {x, t) = (jr^ F {x), 

and ;j^ is any function. 

Case 2. Let k^ (z) = z, k.,(-z)^z; then the condition is 
0{y,X(x,t)} + e{x,\{y, 01=0; 
a solution of which is 

d (x, t) = x {F{x)., c/, (01 -x\<^ (0. F{x)\. 
Case 3. Let k^ {z) = z\ k,_ (s) = (1 - ^0'" ; then the condition is 

\0[y,\{x,t)\J^ld{xMy^m=^\ 

a solution of which is 

e {x, t) = x[F{x), </>(0} ^ lixWia^), 4>{i)]y + (%{</> (0, F{x)]yr- 



Prof. Stanley Gardiner and Prof. Nuttall, Fislt-freezing 185 



Fish-freeznuj. By Professor Stanley Gardiner and Professor 
Nuttall. 

[Read 18 February 1918.] 

Fish-freezing commenced in 1888, in connection with Western 
American sahnon. It was started to preserve the excess of fish 
caught during the runs for canning in the shick season. The busi- 
ness proved so profitable that fish began to be distributed all over 
North America and exported to Europe, the chief market in the 
latter being Germany. The fish are, as soon as possible after catch- 
ing, brought to the refrigerator, frozen dry on trays at about 10° F., 
this process taking about 36 hours. The fish then are drawn into 
a room at 20" F., where they are dipped into fresh watei', their sur- 
faces being thus covered with a glaze of ice. They are then packed in 
parchment paper in strong wooden cases and exported to Europe 
by refrigerator cars and cold storage steamers. The process is also 
applied to halibut, haddock, cod, pollack and various flat fish in 
America. It succeeds in preserving the fish for an indefinite period 
of time, but the product breaks up in cooking, tending to become 
rather woolly and loses flavour and aroma. 

To meet this a fresh process has now been developed, freezing 
the fish in brine consisting of about 18 per cent, of salt at a tem- 
perature of 5° to 20" F. The brine is an excellent conductor of 
heat and cold. A large fish freezes thoroughly in three hours, a 
herring in twenty minutes. After freezing, the fish returns to the 
same condition as it was when placed into the brine; there is no 
woolliness, no loss of flavour or aroma. The difference is due to the 
fact that, whereas in dry freezing there is a breaking up of the 
actual muscular fibres, due to the formation of ice crystals, in brine 
freezing the ice crystals are so small that the muscular fibres are 
entirely unaffected and on thawing return to the normal. In neither 
form of freezing is there danger from moulds or putrefaction if the 
fish is stored below 20^ F. 

The authors advocate the creation of a vast store of frozen her- 
rings against time of scarcity, instead of the herrings being pickled 
and exported. The value of fish as food is weight for weight about 
the same as meat, containing the same constituents. If the excess 
of the herring catch were stored in this way, there would be, on 
pre-war figures, a store of herrings in this country to meet the 
necessity for -albuminous food in the British Isles for at least eight 
weeks. 



186 Mr Sahni, On the branching of the Zygopteridean Leaf, etc. 



On the branching of the Zygopteridean Leaf, and its relation to 
the probable Pinna-nature of Gyropteris sinuosa, Goeppert. By 
B. Sahni, M.A., Emmanuel College. (Communicated by Professor 
Seward.) 

[Read 20 May 1918.] 

( 1 ) The supposed quadriseriate " pinnae " of forms like Staurop- 
teris and Metaclepsydi^opsis are tertiary raches, the vascular strands 
of the secondary raches (pinna-trace-bar, Gordon) being completely 
embedded in the cortex of the primary rachis. All Zygopterideae 
therefore have a single row of pinnae on each side of the leaf. 
(2) This revives the suggestion that Gyropteris sinuosa Goepp. is 
a free secondary rachis of a form like Metaclepsydropsis. (3) The 
genus Glepsydropsis should include Ankyropteris because: a. A 
fossil described in 1915 (Mrs Osborn, Brit. Ass. Rep., p. 727) com- 
bines the leaf-trace of Glepsydropsis with the stem of Ankyropteris, 
the leaf-trace in both arising as a closed ring. h. In G. antiqua 
Ung. also the leaf-trace arose similarly, as shown by a section 
figured by Bertrand {Progressus 1912, fig. 21, p. 228) in which a 
row of small tracheides connecting the inner ends of the peripheral 
loops represents those lining the ring before it became clepsydroid 
by median constriction. 



3 



The Structure o/Tmesipteris Vieillardi i)aw^. By B. Sahni, M.A., 
Emmanuel College. (Communicated by Professor Seward.) 

[Read 20 May 1918.] 

The most primitive (least reduced) of the Psilotales. Specifically 
distinct from T. tannensis in (1) erect terrestrial habit, (2) distinct 
vascular supply to scale-leaves, (3) medullary xylem in lower part 
of aerial stem. 

On Acmopyle, a Monotypic New Galedonian Podocarp. By 
B. Sahni, M.A., Emmanuel College. (Commimicated by Professor 
Seward.) 

[Read 20 May 1918.] 

Indistinguishable ivova. Podocar pus in habit, vegetative anatomy, 
drupaceous seed, megaspore-membrane, young embryo, male cone, 
stamen, two-winged pollen and probably male gametophyte. Chief 
differences: (1) seed nearly erect; (2) epimatium nowhere fi"ee from 
integument, even partaking in formation of micropyle; (3) outer 
flesh with a continuous tracheal mantle covering the basal two-thirds 
of the stone. 



PROCEEDINGS AT THE MEETINGS HELD DURING 

THE SESSION 1917—11)18. 

ANNUAL GENERAL MEETING. 
October 29, 1917. 
In the Comparative Anatomy Lecture Room. 

Dr Mark, President, in the Chair. 
The following were elected Officers for the ensuing year : 
President: 
Dr Marr. 

Vice-PresideiUs : 

Prof. Newall. 
Dr Doncaster. 
Mr W. H. Mills. 

Treas'itrer : 
Prof. Hobson, 

tSecretaries : 

Mr A. Wood. 
Mr G. H. Hardy. 
Mr H. H. Brindley. 

Other Members of Council : 

Dr Bromwich. 

Mr C. G. Lamb. 

Mr J. E. Purvis. 

Dr Shipley. 

Dr Arber. 

Prof. Bitfen. 

Mr L. A. Borradaile. 

Mr F. F. Blackman. 

Prof. Sir J. Larmor. 

Prof. Eddington. 

Dr Marshall. 

The following Communications were made to the Society : 

1. On the convergence of certain multiple series. ByG. H.Hardy, 
M.A., Trinity College. 

2. Bessel functions of large order. By G. N. Watson, M.A., 
Trinity College. 



188 Proceedings at the Meetitujis 

3. A particular case of a theorem of Dirichlet. By H. Todd, B.A., 
Pembroke College. (Communicated by Mr H. T. J. Norton.) 

4. On Mr Ramanujan's Empirical Expansions of Modular Functions. 
By L. J. MoRDELL. (Communicated by Mr G. H. Hardy.) 

5. Extensions of Abel's Theorem and its converses. By Dr 
A. KiENAST. (Communicated by Mr G. H. Hardy.) 



November 12, 1917. 
In the Comparative Anatomy Lecture Koom. 

Professor Marr, President, in the Chair. 

The following Communications were made to the Society : 

1. Some experiments on the inheritance of weight in rabbits. By 
Professor Punnett and the late Major P. G. Bailey. 

2. The Inheritance of Tight and Loose Paleae in Avena nuda 
crosses. By A. St Clair Caporn. (Communicated by Professor 
Bitten.) 



February 4, 1918. 
In the Comparative Anatomy Lecture Room. 

Professor Marr, President, in the Chair. 

The following Communications were made to the Society: 

1. On certain integral equations. By Major P. A. MacMahon. 

2. (1) Sir George Stokes and the concept of uniform convergence. 
(2) Note on Mr Ramanujan's Paper entitled : On some definite 

integrals. 

By G. H. Hardy, M.A., Trinity College. 

3. Asymptotic expansions of hypergeometric functions. By G. N. 
Watson, M.A., Trinity College. 

4. (1) On certain trigonometrical sums and their applications in 

the theory of numbers. 
(2) On some definite integrals. 
By S. Ramanujan, B.A., Trinity College. (Communicated by Mr 
G. H. Hardy.) 



Proceedings at the Meetings 189 

February 18, 191S. 
In the Comparative Anatomy Lecture Room. 

Professor Marr, President, in the Chair. 

The following were elected Fellows of the Society : 

E. Lindsay Ince, B.A., Trinity College. 
S, Ramanujan, B.A., Trinity College. 

The following Communications were made to the Society : 

1. Fish-fi-eezing. By Professor Stanley Gardiner and Professor 

NUTTALL. 

2. Shell deposits formed by the flood of January 1918. By 
P. Lake, M.A., St John's College. 

3. (1) Reactions to Stimuli in Corals. 

(2) Is the Madreporarian Skeleton an Extraprotoplasmic Secre- 
tion of the Polyps 1 

By G. Matti^ai, M.A., Emmanuel College. (Communicated by Professor 
Stanley Gardiner.) 

4. Notes on certain parasites, food, and capture by birds of Forficuhi 
cmricnlaria. By H. H. Brindley, M.A., St John's College. 



May 20, 1918. 

In the Botany School. 

Professor Marr, President, in the Chair. 
The following was elected a Fellow of the Society : 

C. Stanley Gibson, Sidney Sussex College. 
The following Communications were made to the Society : 

1. (1) On the branching of the Zygopteridean Leaf, and its relation 

to the probable Pinna-nature of Gyropteris simiosa, 
Goeppert. 

(2) The Structure of Tmesipteris Vieillardi Dang. 

(3) On Acmopyle, a Monotypic New Caledonian Podocarp. 

By B. Sahni, M.A., Emmanuel College. (Communicated by Professor 
Seward.) 

2. Asymptotic Satellites in the problem of three bodies. By 
D. Buchanan. (Communicated by Professor Baker.) 



CONTENTS. 

PAGE 

Extensions of Abel's Theorem and its converses. By Dr A. Kienast, 
Kusnacht, Zurich, Switzerland. (Communicated by Mr G. H. 
Hardy) 129 

Sir Oeorge Stokes and the concept of uniform convergence. By G. H. 

Hardy, M.A., Trinity CoUege 148 

Shell-deposits formed by the flood of Jamtary, 1918. By Philip Lake, 

M.A., St John's College 157 

7s the Madreporarian Skeleton an Extraprotoplasmic Secretion of the 
Polyps? By G. Matthai, M.A., Emmanuel College, Cambridge. 
(Communicated by Professor Stanley Gardiner) . . . .160 

On Reactions to Stimuli in Corals. By G. Matthai, M.A., Emmanuel 
College, Cambridge. (Communicated by Professor Stanley Gar- 
diner) 164 

Notes on certain .parasites, food, and captitre by birds of 'the Common 
Earwig (Forficula auricularia). By H. H. Brindlet, M.A., 
St John's College 167 

Reciprocal Relations in the Theory of Integral Equations. By Major 

P. A. MacMahon and H. B. C. Darling 178 

Fish-freezing. By Professor Stanley Gardiner and Professor Nuttall 185 

On the branching of the Zygopteridean Leaf, and its relation to the pro- 
bable Pinna-nature o/Gyropteris sinuosa, Ooeppert. By B. Sahni, 
M.A., Emmanuel College. (Communicated by Professor Seward) . 186 

The Structure of Tmesipteris Vieillardi Bang. By B. Sahni, M.A., 

Emmanuel College. (Communicated by Professor Seward) . . 186 

On Acmopyle, a Monotypic New Caledonian Podocarp. By B. Sahni, 

M.A., Emmanuel College. (Communicated by Professor Seward) . 186 

Proceedings at the Meetings held during the Session 1917 — 1918 . . 187 



(>.r 



^-.^ 



PROCEEDINGS 

OF THE 

CAMBRIDGE PHILOSOPHICAL 
SOCIETY 



VOL. XIX. PART V. 

[Michaelmas Term 1918 and Lent Term 1919.] 



TfiSTi 



m 



NOV 2 01946 ^J] 

AT THE UNIVERSITY PRESS 

AND SOLD BY 

DEIGHTON, BELL & CO., LIMITED, 

AND BOWES & BOWES, CAMBRIDGE. 

CAMBRIDGE UNIVERSITY PRESS, 

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1919 
Price Three Shillings net 

April 1919. 



NOTICES. 

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2. Separate copies of certain parts of Volumes I.— xi. of the 
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e 



PROCEEDINGS 

OF THE 



On Certain Trigonometrical Series which have a Necessary and 
Suffijcient Condition for Uniform Convergence. By A. E. Jolliffe. 

(Communicated by Mr G. H. Hardy.) 

[Received 1 June 1918; read 28 October 1918.] 

1. The series S^n sin nd, where (a,i) is a sequence decreasing 
steadily to zero, is convergent for all real values of 6, and it has 
been proved by Mr T. W. Chaundy and myself* that the series is 
uniformly convergent throughout any interval if /?a,i-*-0, this con- 
dition being necessary as well as sufficient. 

A generalization of this theorem is as follows : 

If (Xn) is tt sequence increasing steadily to infinity and (an) is 
a sequence decreasing steadily to zero, then the necessary and suffi- 
cient condition that the series Sa„+i(cos A,.,i^ — cosX,i_^i^)/^, which is 
coyiver gent for all real values of 6, shoidd be uniformly convergent, 
throughoid any interval of values of 6, is Xnan^b. 

I shall prove rather more than this, viz. that the condition is 
sufficient for uniform convergence and necessary for continuity. 

When ^ = 0, it is understood that the value assigned to any 
term of the series is its limit as 6 tends to zero, so that for ^ = 
the sum of the series, which I shall denote by Sun, is zero. Since, 
by Abel's lemma, 

i Wn+i + ... + Up\< 2an+i/0, 

it is evident that there is continuity and uniform convergence 
throughout any interval which does not include ^ = 0, so that it 
is only intervals which include ^ = that we have to consider. 

* Proc. London Math. Soc. (2), Vol. 15, p. 214. 
VOL. XIX. PART V. 14 



192 3Ir Jollife, On. Certain Trigonometrical Series which have 

A very trifling modification of the analysis which follows will 
show that, so far as an interval which includes ^ = is concerned, 
the same results hold for the series 

2a„+i (cos \n^ — cos \n+i^) cosec 1)9, 

where h is any fixed number. If either |(X„+i — A,„) or ^(Xn+i + X„) 
is always an integral multiple of some fixed number b, then X„. 
differs by a constant from an integral multiple of 26, and the series 
is periodic with a period Tr/b. In this case the results which are 
true for an interval which includes ^ = are true for any interval. 
The particular series 2a«sinn^ corresponds to b = ^, X,,j=» +^. 

2. Since the sum of the series when ^ = is zero, it follows 
that, for continuity at 6 = 0, the sum of the series, when 6 is 
different from zero, must tend to zero as 6 tends to zero in any 
manner. In particular, the sum when 6 = 7r/2\„, must tend to 
zero, as n tends to infinity. 

When 6 = 7r/2\,j, let m be the integer such that 

Xm-i^ $ TT < \n6- 

It should be noticed that we may have m — l = n, and that 

When m—1 >n, cos \p-i6 — cos \p6 is positive, so long as p is 
not greater than ?7i — 1, and consequently 

0{U^ + Uo+ ... +Um-i) 

> a-n (cos Xi6 — cos \n^) + «w-i (cos X„^ - cos A,„,_i 0). 
Also, by Abel's lemma, 

6 (Um + U,n+i + ... + U,n+g) > "m (cOS X^O - 1) 

for all values of q. 

Hence the sum of the series is greater than 

[an COS \6 - {a,n-i - a„i) cos Xn-iO - am]IO, 

which, since a,„,_i ^ a^ and cos Xm-i ^ is negative, is greater than 

(an cos X^d- a„,)/6 = 2Xn («« - «,»,)/-"■ + bn, 

where bn denotes a„(l —cosXi6)/d and consequentl}^ tends to zero 
as n tends to infinity. 

When m — 1 = ?i, we can divide the series up into 

(U^ + U2+ ... + Un) + (Um + Um+i +...), 

and, noticing that cos X^-i 6 = 0, we see that the sum is greater 
than (an cos Xj^ — am)/ 6, as before. 



a Necessary and Suficiejit Condition /or Uniform Convergence 193 

Hence the sum of the series, when 6 = 7r/2X,i, can in no case 
tend to zero, as n tends to infinity, unless X,i (a,i — a,n) -^ 0. 

If X)i (an — a,n) -* 0, then, given any positive number e, we can 
find V such that X,i (a„ -«,„)< e for n^v. Denote ni by (n, 1) 
and let (/i, 2) be the integer formed from (n, 1) in the same way 
that {ii, 1) is formed from n, and so on. Then 

((n - «n, 1 < e/\n , (/„, i - ttn, o < e/X^, i , , 

for n ^ V, and by addition 

Un < € (1/X,„ + l/Xn,i + ... + lAn,i>) + (hi,p. 

Now X,i_i > 2\,i,, \,i,^2 > 2X„_i, and so on, so that «„ < 2e/\„ + cin,p. 
Also when % is fixed we can choose p so that an^p < e/Xn, and we 
shall have therefore 

Xnttu < Be (n ^ v). 

Hence XnCin^O is a necessary condition that the sum of the 
series should be continuous at ^ = 0, and a fortiori that it should 
be continuous throughout any interval which includes ^ = 0. 

3. To show that this condition is sufficient for uniform con- 
vergence in any interval, and d fortiori for continuity at any point, 
it is sufficient to show that 

I Un+i+ ... +Up\ <AM, 

for all values of 6, where A is some fixed number and M is the 
greatest value of Xt-a^ for r^n + 1. 

Since the value of the series is changed in sign only by changing 
the sign of 6, it is sufficient to consider positive values of 6 only. 
By Abel's lemma 

I Un+i + ... +Up\< 2an+i/d < 2Xn+ian+i/7r, 

if ^ ^ tt/Xji+j. If ^ ^ T^/Xp, every term of Un+i + . . . + Up is positive ; 
and, if u,. is one of these terms, 

Ur ^ M (cos Xr-iO — COS X,.^)/X,.^ 

^ 2if sin l(Xr - X,_i) 6 sin h{X, + X,_,) O/X^d < MO {Xr - X,_i), 
so that Un+\ + . . . + Up < MOXp < ttM. 

If irjXp < 6 < 7r/X„i+i, let Tr/Xq+i < 6 ^ 7r/Xg, and divide 

lln+i + ... ^Up up into Un+i + . . . Uq and Uq+i + ... +Up. 

Then | Un+i + ... + Uq\< ttM, and 

I Uq+i + ... +Up\< 2aq+^l0 < 2aq+iXq^,/7r < 2ilf/7r. 

Therefore j Un+i + ... -\- tip\ < {ir + 2/7r) M. 

U~2 



194 Mr Jolliffe, On Certain Trigonometrical Series which have 

Hence for all values of 6 

I Un+^ + ... + ?i^ i < (tt + 2/7r) M, 

and therefore the condition X,ia„ -*- is sufficient for uniform con-"" 
vergence and a fortiori for continuity in any interval. 



il 



4. If 'Xn tends to infinity more rapidly than n, the series does 
not seem to be capable of any modification. If A,^ = ^?2 + B, where 
A and B are fixed, we obtain practically the series 2a„ sin nO and 
nothing more. But when X^ tends to infinity more slowly than n, 
and with a certain measure of regularity, the theorem can be 
transformed in an interesting manner. We have, in fact, the 
following theorem : 

If \n tends steadily to infinity and \n+i — ^n tends steadily to 
zero, then the necessary and sufficient condition for the uniform 
convergence of 

Zft^i (Xji+i ~ Xji) sm A.,jC7 
is Xndn -* 0. 

As before, I prove rather more, viz. that the condition is suffi- 
cient for uniform convergence and necessary for continuity. 

This theorem will follow at once from the theorem just proved, 
if we can show that the series 

Sa« {(cos \nd - cos Xn+i0)/6 - (Xn+i - Xn) sin XnO] 

is uniformly convergent throughout any interval. Here the con- 
dition Xn+ittn ■-* is equivalent to X^an -^ 0, since Xn+i — X^ -^ 0. 
We can verify immediately that 

cos y — cos X — sin y sin (sc — y) 

= sin^ h{x — y) (cos y — cos x) + ^ sin {x — y) (sin x — sin y). 

It follows by Abel's lemma that, if Xn+i — X^ decreases steadily, 
so that sm{Xn+i — Xn)d and sin ^(Xn+i — Xn) decrease steadily, 
then 

S {cos Xn — cos X,i+i 6 — sin {Xn+^ — Xn) sin Xnd] 



-rt+1 



< 2 sin^ 1 {Xn+i ~Xn)6 + sin {Xn+i - X«) 0- 



Also, given any e, we can choose v so that Xn+i — X^ < e for n ^ v. 
Hence, for n'^v, we have 

p 

2 {cos XnO — cos Xn+i6 — sin (Xn+i — Xn) 6 sin XnO] 

w+1 

< 2e2^2+66'<3e^, 
for any interval of values of 0, if e is sufficiently small. 



a Necessary and Sufficient Condition for Uniform Convergence 195 

It follows also that 

i -& I 

I Z sin (X^+i - \,)6 sin X,^^ < 2 + Se^ < 3, 

|w + l I 

for n ^ V. Now 

(X„+i - Xn) 6 cosec (X.,1+1 - X„) ^ - 1 
decreases steadily to zero, and is less than 

Therefore 



2 6{\n+i - \i) sin Xu^ — 2 sin (X„+i — X„)^ sin XnO 

n+l jj+1 

Hence 



< e'6' 



p 

% {(cos X,i^ - cos \n+i6)ld - (Xn+i - ^/i) sin \nd} 

n+l 

< 3e + e^O < 4e (n ^ i^). 

Hence the series 

2o„ {(cos Xn^ - cos \n+id)l6 - (X„+i - X„) sin X„^} 

is uniformly convergent throughout any interval, and hence the 
result enunciated follows. 

5. If instead of a sequence (X„) we have a function X (x) such 
that, as iT-^oo, X{a;) increases steadily to infinity and \'{a;) de- 
creases steadily to zero, then Xn+i — ^n decreases steadily to zero. 
The series 2 (X'„ — X^+i + X,i), where \'n denotes the value of X' (x) 
when x = n, is convergent and is moreover absolutely convergent, 
since X',^ — \n+i + Xn is positive. Hence, by Weierstrass' M test*, 
the series Sa,i (X'„ — X,i+i + X,i) sin X„^ is uniformly convergent 
throughout every interval. It follows then that ajj,X„ ^ is the 
necessary and sufficient condition that the series Sa^X'„ sinX,j^ 
should be continuous at every point and uniformly convergent 
throughout every interval. 

In particular the series 2a,i?i'~^ sin (n'^), where t is any real 
number not exceeding 1, is continuous at every point and uniformly 
convergent throughout every interval if n^a^-^O, this condition 
being necessary as well as sufficient. 

* Bi'omwich, Infinite series, p. 113. 



196 Mr TurnbuU, Some Geometrical Interpretations 



Some Geometrical Interpretations of the Concomitants of Tivo 
Quadrics. By H. W. Turnbull, M.A. 

(Communicated by Mr G. H. Hardy.) 

[Received 6 July 1918; read 28 October 1918.] 

§ 1. In the Mathematische Annaleii, Vol. LVi, Gordan has given 
a system of 580 invariants for two quaternary quadratics. It appears 
that by carrying out the processes of reduction a little further, the 
irreducible forms can be shewn to number 123 at most. That is to 
say, the system is about as complicated as the ternary system for 
three conies which Ciamberlini* first established. It is therefore 
worth while to give geometrical interpretations to members of the 
system for two quadratics. In the following pages about a hundred 
of them are shewn. The geometrical significance of the residue 
appears to be remote. 

Using the classification introduced by Gordan, the numbers of 
forms of each type J which have not been reduced are shewn in 
the subjoined Table. The rows of the Table give the numbers of 
forms of each particular order in the three sets of coordinates x, 
p, u, which define points, straight lines, and planes respectively. 
Detailed lists of these forms will be found at the heads of the 
paragraphs which deal with separate types. 



References 


Order in x, p, u 


J' 


J2 


Type 

J3 


J' 


J5 


Total 1 

i 
1 


§5 


Invariants 


5 










1 
5 1 


§6 


Covariants 


4 




1 






5 ! 


5) 


Con travari ants 


4 




1 






5 1 


§§ 7-14 


Complexes 


6 


1 


1 


4 


4 


16 ' 


§ 15 


Mixed (1, 0, 1) 


1 




2 






3 


§21 


(1, 0, 3) 






4 






4 1 


5) 


(3, 0, 1) 






4 






4 i 


§22 


(2, 0, 2) 


1 




6 






7 ! 


§§ 17-20 


(0, 1, 2) 




1 


6 


1 


1 


9 ! 


5) 


(2, 1, 0) 




1 


6 


1 


1 


9 ! 


§ 18 


(0, 3, 2) 






1 






1 i 


)) 


(2, 3, 0) 






1 






1 ; 


§23 


(0, 2, 2) 






4 






4 


J5 


(2, 2, 0) 






4 






4 


§ 16 


(1, 1, 1) 




4 


12 






16 ! 


§23 


(1, 2, 1) 






12 


6 


6 


24 1 


» 


(2, 1, 2) 






4 






4 1 


5) 


(3, 0, 3) 






2 






2 


Totals 


21 


7 


71 


12 


12 


123 



Ciamberlini, Giornale di Matematiche, Vol. 




of the Concomitants of Ttuo Quadrics 197 

Notation. 

§ 2. Let Ui, Uo, Ui, u^ be plane coordinates; and let v, w be 
cogredient with u. We may then typify line coordinates by 

Pij = (uv)ij = UiVj - vflij, (i, j = 1, 2, 3, 4) ; 

and X or point coordinates by cc^ = (uvtu)23^ and three similar ex- 
pressions for X2, x-i, ^4, Then the symbolic system of Gordan can 
be exhibited as follows. 

Let the point equations of the quadrics be 

/=<''^' = «/'=..., 
and /" = h^- = bj^ = .... 

Let the line equations be 

u=iApy = {A'py = ..., 

U' = {Bpy = (B'pf=.... 
Let the tangential equations be 

2 = uj = uj = ..., 

X' = M,3" = ?'j3'^ = . . . . 

Then the connections between the symbols are 

A = (ta', B = bb', a = aa'a", ^ = bb'b". 
And all concomitants of the system can be expressed in terms of 

TO P'l'OT'^ 

4, {dd'p), (dd'd%), {dd'd"d"'), 

where d signifies a or b. But the irreducibles can be shewn to be 
composed of the following types, 

(la, b^\ Ucc', b^, (Ap), (Bp), iia, u^, (abp), (Abu), {Ban), (AB), 

a^, ba, (A/3x), (Bax), (a^p), (AB)', F„ F, ; 

where (A^x) — a^a^' — a/a^ = «^(V , 

say, and («/3/>) = UaVp — u^Va = luvp , 

{AB)' = (Ahu)k', 

F, = {(ibp) a^' - (abp) a^ = (Abp^). 

Reciprocation. 

§ 3. By interchanging the symbols (a, a), (b, /S), (u, x) without 
altering vl, jB or ja, we obtain from any given concomitant the 
reciprocal form. Thus the bracket factors (A^x) and (Abu) are 



198 Mr TurnhuU, Some Geometrical Interpretations 

reciprocals. So also would be (ahcu) and (a^ya;), the latter being 
of a type not arising for less than three quadrics. Though the 
process by which Gordan arrived at such symbols as {A0iv) and 
(a/3p) was purely analytic, it is interesting to observe that from 
the geometrical point of view such analytical results were almost 
inevitable. Below will be found several examples of the use of 
this principle of duality. 



The fundamental forms. 

§ 4. A brief investigation would reveal the importance of the 
following forms, to which special symbols are therefore attached. 

Let / denote a^-, f denote hx~, 

S „ u^", %' „ %^ 

n „ {Ap)\ W „ {Bpf, 

k „ {AjBxf, k' „ {Baxf, 

X " {Ahuf, X » {Bauf, 

77^2 „ (abpf, Dia „ (ct^pT, 

and C „ {AB)(Ap)(Bp). 

Some account of these forms may be found in Salmon, Analytical 
Geometry of Three Dimensions (revised by Rogers), Yol. i, Ch. ix. 
There %, %, %', S' are denoted by a, r, r', a' (§ 214) : A^, k' are the 
T, r of § 215 ; n, 7ri2, O' are the ^, ^^, ^' of § 217. 

Invariants. 

§ 5. The irreducible invariants are a^, bj, (ABf, a^^, bj or 
the A, 0, ^, @', A' of Salmon, § 200. In fact, there are no other 
types, for two quadrics of any dimension n, than the n + 1 co- 
efficients of X, in the discriminant of 

The five covariants and contravariants. 

I 6. The covariants (w + 1 in the case of w-ary forms*) are the 
four quadrics /, k, k', f and the quartic J defined by 

apbaaj)x (AB) (A^x) (Bax). 

This is indeed the jacobian of the four quadrics, and represents 
the four planes of the self-conjugate tetrahedron (c£ Salmon, § 233). 

* Of. Turnbull, ' Quadratics in n variables' (pp. 235-238), Camb. Phil. Trans., 
Vol. XXI, No. viii. 



H 



of the Concomitants of Two Quadrics 199 

Correlatively, S, %, x, S' are the four quadrics in u which make 
the system of contra variants together with 

j = a^haUaU^ i-^B) {Ahu) {Ban). 

This latter represents the four vertices of the same tetrahedron. 
In fact, the jacobian of u^, u^, (^Ahuf, (Bau)" is 

(a/3 Ab Ba) Ua,u^ {Abu) {Bau), 

where A = a a", say ; expanding the first bracket this becomes 

iijd^" (bBa) M - d/ b^ {ci"Ba) M + &„d/ {ci"Ba) M, 

where each term represents two, with a, a" permuted, and M is 
short for UaU^{Abu){Bau). But the factor Ua is reducible to Ua' 
(Gordan, il, § 6) ; which means in this case that the symbols u of 
the factors u^,, (Abu) would be bracketed. Hence the product in- 
volving cia is zero. Thus the jacobian is equal to 

baCip'(d"Ba)M, 
= b^a^ {a"Ba')M+ba.h' {a"ab"a) M (if B = b'b") 
= — ba a^ (a'a'B) M (as before) 

A correlative reduction applies to the case of /. 

The complexes. 

§ 7. A complex is a function of ]), or line coordinates, but not 
explicitly of ii or x. There are eight quadratic and eight cubic 
complexes in the system. The quadratics are 

(ApY or n, (Bjyf or TI', (abpf or tt,.,, (a/3p)' or IIi., 
{AB){Ap){Bp) or C, {abp){a^p)a^b„., F;' and Fi. 

Differentiation. 

I 8. Let p be any symbolic product belonging to the whole 

dP 
system; then — - {i=l, 2, 3, 4) would be composed of terms each 

OXi 

with one odd S}?mbol Qj or 6,: left over. Thus the four symbols 

^:— may be considered as the coordinates of a certain plane. For 

dxi ^ ^ 

example the coordinates of the polar plane of a point (x) with 
regard to a^- are («a;«i, axU.,, a^a^, a^ai). Likewise ^ — would give 
a set of point coordinates. 



200 Mr TurnhuU, Some Geometrical Interpretations 

Again, ^ — would give six quantities which would symbolise 

the coordinates of a certain linear complex : and, m some special 
cases, the coordinates of a straight line. For example, 

is a useful way of denoting the six quantities (Aj)) A^ {i,j = 1,2, 3, 4), 
which represent a straight line, since they satisfy the identical re- 
lation existing between the six ^^-coordinates of a straight line. 

Li7ie coordinates. 
§ 9. This identity satisfied by line coordinates (p) is 

^pijPM=0 (1), 

which we denote by co (p) = 0. Symbolically, the condition that 
two lines p and q should intersect is {pq) = 0. If p is the line 
common to two planes u, v, and q is that common to u', v', then 
this condition is (iivuv) = 0. 

If two lines p, q intersect, then Kpij + Xqij represents the co- 
ordinates of any line of the plane p, q passing through the connnon 
point of p, q. Since the line p touches the quadric / if {Ap)- = 0, 
it follows that the line (k, \) touches this quadric if 

k' (Apf + 2k\ (Ap) (Aq) + X2 (^Aqf = 0. 

Hence (Ap)(Aq) vanishes if p intersects the conjugate of q in f; 
for then p and q are harmonic conjugates of the two tangents to / 

r)TT 

in this pencil of lines (k, \). This shews that the coordinates ^::— , 

op 

i.e. (Ap)Aij, are those of the line conjugate to p in the quadric f. 

Analytically it is evident that these coordinates represent a line 

and not a linear complex, since they satisfy the required condition 

(1). In fact 

(Ap) (A'p) (A A') = A {AAy <o (p^. 

But the left member of this equation is the symbolic equivalent of 
substituting (ApJAy forp in (1): which proves the statement. 

Complexes and their polars. 

I 10. Let (Dpy = represent one of the quadratic complexes 
of § 7. Then (Dp) By gives the coordinates of a linear complex 

* Cf. Gordan, ii, § 6. 



of the Concomitants of Two Quadrics 201 

pola7' to (p) in (Dpf. If (p) is a member of the complex (Dp)-, 
the polar is called the tangential linear complex. 

The complex (Dp)Dij is not usually a special linear complex. 
The preceding case was exceptional. For in that case the quad- 
ratic complex was (Apf = 0, and all the rays touched the quadric/. 

The comptlexes tt,., IIi.,, C, (ahp){oL^p)a^ba. 
§ 11. The principal quadratic complexes which occur are 
'jr,, = (abpy, U,,= ial3pY, G^{AB){Ap){Bp). 

The two former are well known, ttjo being the aggregate of lines 
cutting the quadrics harmonically, and rTi. being the correlative 
complex. The third, G, is the complex of lines whose conjugates, 
in /and /' respectively, intersect. For the conjugate of -p in / is 
{Ap){A) and in /' is {Bp){B). Again, G is satisfied too by the 
singular lines of the complex ttj.,. For if p is a line of {ahp)- = 0, 
its tangent linear complex (§ 10) is {ahp) (abq) = , q representing 
current coordinates : further, jj is a singular line if this tangent 
linear complex is special, i.e. if 

(abp){aba'b')(a'b'p) = 0, 

which reduces to (AB) (Ap) (Bp) = 0. Correlatively C also contains 
the singular lines of the complex ITi.,. 

Again, the singular lines of the complex G belong to the com- 
plex (abp) {a^p) Ufiba. This follows in the same way as in the above 
case. But a more direct interpretation of this last form arises from 
the apolar* condition for two linear complexes ; if the polar linear 
complexes of a line (p) with regard to ttjo and ITis are apolar, then 
(abp) (a^p) cipba vanishes. 

The complexes F{~, Fi. 

§ 12. Besides the original complexes {Apy and {Bpy, and the 
four complexes of § 11, there remain two more quadratics, i'V and 
F^. Just as {abpf is the harmonic complex between / and /', so 
F^^ is the harmonic complex between /' and k, while F^ is that 
between / and k'. To prove this we build up a form (/', ky from 
/' and k, in the same way as (/ f'f, i.e. {ahpY, is built from / and 
/. Then 

{f',ky^{b,^,{ABxrf 

= (bx', 2ft^"^aa;'-— la^a^'axaxY 
= 2ap^ (abpf — 2a^a^' {abp) {abp) 
= [{abp) a/ - {a'bp) a^J = F,^ (| 2). 
* The linear complexes {Dp) — 0, {Ep) — are apolar if {DE) - . 



202 Mr T'urnbull, Some Geometrical Interpretations 

The eight cubic complexes : 

F, (ahp) K (Bp), F, (oL^p) ap (Bp), 

F, (ahp) K (AB) (Ap), F, {oc^p) a^ (AB) (Ap) ; 
and four involving F^. 

§ 13. If aj^, hx, Cx are three quadrics, the lines p cutting them 
in involution are given by the cubic complex 

{hep) (cap) (ahp) = 0. 

Let us denote this complex by the symbol {a^, h^', c^)- Then 
{f,f', k') may be formulated, and we shall have 

(ax% hx^, k') = {{ahp)axb^, {Baxff 

= ((abp).axbx, - 2b:bx"ba"bx + "^^'b^'J 

= - 2 {abp) {ab"p) (bb'p) bjb^' + 2 {abp) (ab'p) (bb'p) b^"\ 

The second term is zero, since b, b' are interchangeable. The first 
term is F^ (abp) ba. (Bp) to a constant coefficient. 

Reciprocally (S, ^',x) represents F.2 (a^p) a^ (Bp) ; and there 
are two like forms involving Fi. 

§ 14. This leaves four complexes such as F2 (abp) ba (AB) (Ap) 
to be interpreted, but the geometrical significance is not at all 
immediate. If however we write (/, /', k') as (Bpf, then the line 
(p) has a polar linear complex 

(npf(Dq) = 0. 

And if q = (Ap)(A), i.e. if q is the conjugate line of p in the 
quadricy, then 

(Dpy(DA)(Ap) = 0. 

This latter form is equivalent to F2 (abp) ba. (AB) (Ap) : and similar 
results follow for the other three forms, as in § 13. 

The mixed concomitants. 

§ 1.5. To denote the order of a form, let (i,j, k) mean that the 
order is i in x, j in p, and k in u. Then there are three linear 
forms (1, 0, 1) and sixteen linear forms (1, 1, 1). 

The three linear forms (1, 0, 1) : 

Up % ax, Uahabx, (AB)(Ab u ) bx'. 

If (v) is the polar plane of a point (x) in /, then (v) = ax(a). 
Hence Usa&av. = is the condition that a conjugate plane of u in/' 



of the Concomitants of Two Quadrics 203 

should be the polar of x in f Similarly for Uahahx- Again, 
{AB){Ahu) h^ vanishes if the polar of x in/' is conjugate to u in ;)^, 
i.e. in {Ahiif = 0. 

The sixteen forms (1, 1, 1) : 

two like ax {Ban) (Bp), two like Ua {Bax) (Bp), 

„ a3^a^{a^p)ua, „ „ ax(abp)baUa, 

„ „ ax{Bau)(AB)(Ap), „ „ u^(Bax){AB)(Ap), 
„ (abp){Abu)(A/3x)ap, „ „ (oL^p)(Abu)(A/3x)ba. 

§ 16. The polar plane of a point (x), with regard to /, meets 
a plane (u) in a straight line whose coordinates are (au) a^. If 
ax(aBu){Bp) = 0, this line cuts the conjugate of p in /'. Let us 
denote this relation by (f^, 11'). The significance of the reciprocal 
of this, viz. (Stt, n'), is obvious. This accounts for four forms since 
either /or/' can be employed. 

Suppose we word this relation differently and say that the 
plane {u) cuts the polar of (x) in/ in a line which lies in the linear 
complex polar of (p) in II' : then a like meaning attached to 
(fx, His) interprets tta;% (^^p) Ua.. So also 

i-u, 'ir^^ = ax{abp)ba.Ua, 
(/,, {AB) {Ap) (Bp)) = a, (aBu) (AB) (Ap), 

with reducible terms, and 

(tu, (AB) (Ap) (Bp)) = lu (Bav) (AB) (Ap), 

while (kx, TTia), (Xu> II 12) denote the remaining two forms of the 
above list. To complete the set of sixteen forms we merely write 
%' for S, k' for k, and so on. 

The polar quadrics (0, 1, 2) and (2, 1, 0). 

I 17. There are nine forms of order (2, 1, 0), any one of which 
represents a quadric associated with a given line (p) ; or, from 
another point of view, represents a linear complex associated with 
a given point (x). The simplest of these is (abp) ajj^. Let this 
denote the polar quadric of the line (p) with regard to the system 
/*+ X/'. It is convenient to use the symbol p (ff) for this relation. 

The equation (abp) aj)x = is the analytical condition required 
when the polar planes of a point (x) with regard to / and /' meet 
in a line which intersects (p). For the coordinates of these polar 
planes of x are denoted by a-^a^, bib^ (* = 1, 2, 3, 4). Hence the 
coordinates of their line of intersection are aj)x(ah)ij; and this 
line cuts (p) if (abp) axbx = 0. 



204 Ml' Turnhull, Some Geometrical Interpretations 



■ 



Forming the invariant of the polar quadric, we obtain an ex- 
pression which reduces directly to {{AB)(A2)){Bp)\^. Hence if j9 
belongs to the complex G, its polar quadric is a cone. 

§ 18. Again, the tangential equation of the polar quadric 
(abp) cij)y; = is formed in the same way as Ua~ is formed from a^-. 
A simple reduction leads to 

{Ap) (Bp) (abp) (aBu) (bAii). 

Likewise the point equation of {ajBp) u^ a^ involves the form 

(Ap) (Bp) (a^p) {AjSx) {Box). 

This interprets the two forms of orders (0, 3, 2) and (2, 3, 0). 

1 19. Again, if we form the polar quadric oi {p) with regard to 
each pair of quadrics /, /', h, k', we obtain the following results : 

p (/, k) equivalent to (Ap) (A /3a;) a^a^, with a like form for p (f, k'), 

p{f,k') „ F,a^(Bax), „ „ p(f',k), 

p (k, k') „ (A^x) {cc/3p) (Bax) [AB). 

If, further, (q) is the conjugate line of (p) in {Bpy,i.e. in/', 
then 

q (/, k) is equivalent to a^a^ (A/3x) (AB) (Bp), 
and q' (/', k') „ b^^b, {Bax) (AB) (Ap). 

All these equivalences are readily verified, but we give a 

special proof for the case of p (k, ¥). In fact, the polar plane of 

dk 
X m. k = 0, i.e. in (A^xy = 0, has coordinates — , which may be 

OXi 

symbolised as (A^x)(A/3)*. So also the coordinates of the polar 
of X in k' are denoted by (Bax) (Ba). Hence the line of inter- 
section of these polars is denoted by (Aj3x)(Bax) [ABa^], which 
is equal to (A ^x) (Bax) (AB) (a^)* ; and the line cuts p if 

(A^x) (Bax) (AB) (aj3p) = 0. 

§ 20. These eight polar quadrics now enumerated, viz. p (f, f), 
p (f, k), ..., q (f, k'), must be supplemented with one more form, 
(a^p) cipbaa^bx, to complete the set of nine forms (2, 1, 0) belonging 
to the irreducible system of two quadrics / and f. The geometrical 
significance of this last form is as follows : the line joining the 
two points, Xi and x^, cuts p ; Xj^ being the pole in / of the plane 
whose pole in /' is x, and x^ being the pole in /' of the plane whose 
pole in / is x. 

* {A^) = a a' -a' a, and the combination of (A^) with (Ba), as a transvectant, 
into [ABa^] is essentially the reduction of Ch. ii, § 15 in the paper of Gordan. 



of the Concomitants of Two Quadrics 205 

Correlatively there are nine forms (0, 1, 2), quadratic in u, 
exactly parallel with the above, of which (a^p) UaU^ is the simplest. 

The four forms (3, 0, 1) and then' correlatives: 

(Abu) (AjSa;) a^axb^, {A^x) (Abu) ba.Ua.u^, 

(AB) (A/3x) (BoLx) a^a.^Ua, (AB) (Abu) (Bau) baiua^, 
and four similar forms interchanging f and f, 

\ 21. If a^', bx", Cx" signify any three quadrics, then (abcu) a^b^Cx 
vanishes when the common point of the polars of (x) in the three 
quadrics lies on the plane (u). Applied to the quadrics ff, k, k' 
taken three at a time, this condition involves the four forms (3, 0, 1) 
indicated above. The correlative condition, applied to each set of 
three from among S, S', %, %', gives rise to the four forms (1, 0, 3). 
For example, if we select /,/', k as the three quadrics, then the 
condition is (Abu) (A^x) a^axbx = 0. 

The polars of (x) in all four quadrics/,/', A:, k' meet in a point 
if (x) lies on any face of the self-conjugate tetrahedron 

(A^x) (BoLx) a^baaxbx = 0. 

The remaining forms of the system. 

I 22. None of the remaining forms appear to have any special 
geometrical importance : but we give a few examples. First, as to 
the forms of order (2, 0, 2), we may exhibit them as follows: 

ax (aBu) (Bax) lu and a similar form, 

(Abu)(A^x)baa^iL„,ax „ „ 

(AB) (Abu) (Bau) axbx and a correlative form, 
and [(AB)J. 

Suppose (q) to denote the common line of the plane (u) and 
the polar of (x) in / and (q) to denote the line joining (x) to the 
pole of (u) in /. Then the condition that q, q should satisfy 
the harmonic relation (Bq) (Bq') = becomes on substitution 

Ux (aBu) (Bax) Ua = 0. 

Thus the first in the above group of forms is interpreted. The 
second form vanishes if two lines (q), (q) satisfy the harmonic 
relation (abq) (abq') = 0, where (q) now denotes the intersection of 
the plane (u) with the polar of (x) in k, while (q^) is the same as 
before. 

Again, the third form of the set vanishes if the lines in which 



206 Mr Turnhull, Some Geometrical Interpretations, etc. 



the polars of {x) in f and /' cut the plane {u) satisfy the har- 
monic relation for the complex C=(AB)(Ap)(Bp). 

Finally the last form [(ABYY, which is equivalent, except for 
reducible terms, to (Abu)bx {Ab'u)hx (§ 2). is involved in the con- 
dition that the line common to (a) and the polar of (x) inf shuuld 
touch /. 

§ 23. Next there are four forms of order (0, 2, 2), such as 
(Ap){Abu) (abp)apUp, (Ap)(AB){Bau){al3p)apUa, and four cor- 
relatives of order (2, 2, 0). All of these have obscure geometrical 
properties, though they present no difficulty to identify. 

After this there are twenty-four forms of order (1, 2, 1). The 
simplest of these is (Abu) b^ (Ap) (Bp), which vanishes Avhen 
u, X, p satisfy the following conditions : if the polar of (x) in /' 
meets (p) at a point {y), and if the polar o{(y) in/' cuts the plane (u) 
in a line (q), then p, q satisfy the harmonic relation {Ap) {Aq) = 0. 
The remainder of these (1, 2, 1) forms are of like nature. 

Beyond this there are four forms (2, 1, 2), and two forms (3, 0, 3), 
none of which present concise geometrical interpretations. 



i 



Mr Ramanujan, Some properties o/p(n) 



207 



Some properties of p (n), the number of partitions of n. By 
S. Ramanujan, B.A., Trinity College. 

[Received 3 October 1918 : read 28 October 1918.] 

§ 1. A recent paper by Mr Hardy and myself* contains a table, 
calculated by Major MacMahon, of the values oip{n), the number 
of um-estricted partitions of n, for all values of n from 1 to 200. 
On studying the numbers in this table I observed a number of 
curious congruence properties, apparently satisfied by p {n). Thus 

(1) |)(4), p(9), 23(14), p(19), ... = 0(mod. 5), 



(2) 


p{^\ 


P(12), 


p{n\ 


i^(26), . 


. = (mod. 7), 


(3) 


P(6), 


P(17X 


P(28), 


i^(39), . 


. = 0(mod. 11), 


(4) 


i^(24), 


yo(49), 


^3(74), 


p{m, . 


.. =0(mod. 25), 


(5) 


p{^% 


P (54), 


^(89), 


i>(124),. 


.. HO(mod. 35), 


(«) 


i>(47), 


i9(96), 


/;(145), 


i>(194), . 


.. = 0(mod. 49), 


(7) 


p(39), 


i^(94), 


^(149), 




= (mod. 55), 


(8) 


pi^n 


p(138). 






= (mod. 77), 


(9) 


^(116), 








B0(mod. 121), 


(10) 


jt^(99),. 








= 0(mod. 125). 



From these data I conjectured the truth of the following 
theorem : 

If h= b'^mV and 24A, = 1 (mod. S), then 

p{\), p(\ + 8), p{X + 28),... = 0(mod.B). 

This theorem is supported by all the available evidence ; but 
I have not yet been able to find a general proof. 

I have, however, found quite simple proofs of the theorems 
expressed by (1) and (2), viz. 

(1) p {57n + 4) = (mod. 5) 

and (2) p (7m + 5) = (mod. 7). 

* G. H. Hardy and S. Ramanujan, 'Asymptotic formulae iu Combinatory 
Analysis', Proc. London Math. Soc, ser. 2, vol. 17, 1918, pp. 75—115 (Table IV, 
pp. 114—115). 



VOL. XIX. PART v. 



15 



208 Mr Ramamijan, Some properties of p (»), 



From these 

(5) p (35m + 19) = (mod. 35) 

follows at once as a corollary. These proofs I give in § 2 and § 3. 
I can also prove 

(4) p {Ton + 24) = (mod. 25) 

and (6) p (49n + 47) s (mod. 49), 

but only in a more recondite way, which I sketch in § 3. 

§2. Proof of {\). We have 

(11) X {(1 - cc) {l-x-){l- x^). ..Y 

= X (1 - 3« + haf -1x^^ ...){\-x-x^^ X' +...) 
= t(- lY+^(2/ji+l)x^+^-'^'i^+'>+i''^^'' + ^\ 

the summation extending from ^ = to fi = x and from v = — oc to 
y = 00 . Now if 

l+i/A(/x+l) + ii^(3i^ + l) = 0(mod. 5) 
then 8 + 4/x (/x + 1) + 4z^(3!^ + 1) s (mod. 5), 

and therefore 

(12) (2ya + l)^ + 2(i; + l)^ = 0(mod. 5). 

But (2/x + 1)^ is congruent to 0, 1, or 4, and 2(v + 1)- to 0, 2, or 3. 
Hence it follows from (12) that '2/x + l and v i-1 are both multiples 
of 5. That is to say, the coefficient of x^^^ in (11) is a multiple of 5. 
Again, all the coefficients in (1 — x)~^ are multiples of 5, except 
those of 1, x^, x^°, ..., which are congruent to 1 : that is to say 

(1 — xf 1 —x" 

1 —x^ 

or -z rr = 1 (mod. 5). 

Thus all the coefficients in 

(1 ~ x') (1 - X'') (1 -x'')... 



1 



{{l-x){l-x"~){l-x') ...Y 
(except the first) are multiples of 5. Hence the coefficient of x^'^ in 
^(l_^..)(l_^ao)... _ (l-a^)(l-x-) ^.^ 

{l-x)(l-x^){l-a^)...~^^ ^^ ""'^'--^ {(l-x){l-x')...Y 
is a multiple of 5. And hence, finally, the coefficient of .r"' in 



(1 -x){l- X') (1 - x^) 
is a multiple of 5 ; which proves (1). 



the number of partitions of n 209 

§ 3. Proof of (2). The proof of (2) is very similar. We have 
(13) a-\(l-x){l-af)il-x^..:f 

= x- (1 - 3a; + 5x' - 7a-« +...)- 
= S (- 1)'^ + " (2//. + 1) (2i/ + 1) a,-+i'^<'^+i> +^v(. + i)^ 
the summation now extending from to oo for both jx and v. If 

2 + l/^(^ + l) + iz/(z; + l) = 0(mod. 7), 

then 16 + 4/A(ya + l) + 4z/(z7 + l) = 0(mod. 7), 

(2/A + l)" + (2i^-l-l)- = 0(mod. 7), 

and 2/x + 1 and 21^ + 1 are both divisible by 7. Thus the coefficient 
of .^■™ in (13) is divisible by 49. 
Again, all the coefficients in 

(l-a;7) (l-a;») (l-a;^!)... 

[{I - x)J\ -^x^) {V-a?) ... Y 

(except the first) are multiples of 7. Hence (arguing as in § 2) we 
see that the coefficient of a'"" in 



(l-^-)(l-^')(l-^')--- 
is a multiple of 7 ; which proves (2). As I have already pointed 
out, (5) is a corollary. 

§ 4. The proofs of (4) and (6) are more intricate, and in order 
to give them I have to consider a much more difficult problem, 
viz. that of expressing 

p {X) + jj (\ + h) X + j) (A, + 2S) X + . . . 

in terms of Theta-functions, in such a manner as to exhibit ex- 
plicitly the common factors of the coefficients, if such common 
factors exist. I shall content myself with sketching the method 
of proof, reserving any detailed discussion of it for another paper. 
It can be shown that 

(14) ^^ " ^'^ ^^ " ^"^ (1 - ^1 . ^. 1 



{l-x''){l-x^){l-x'^) ... ^-'-x' -^X' 

_ |-^ - Zx^ + or (^ -' + 2x^-' ) + ^^(2g-^ - ,rp) + x^{3^-' + xl')+5x^ 
~ ^ -' - 1 1 X - x'^' ' 

where P = ^^ ^ " ^1 ( ^ Z ^(LtI^L • ' 

^ {\-x^)(l-x'){l-x''){l-af)...' 

the indices of the powers of.'?;, in both numerator and denominator 

15—2 



210 Mr Ramanujan, Some properties of p (n) 



of |, forming two arithmetical progi'essions with common difl-erence 
5. It follows that 

(15) (l-oc^) (1 - x^°) (1 - X'') ...{p{4<)+p(9)a; + jj(14).7;-+ . ..} 

5 . 

Again, if in (14) we substitute cox% (o^x% (o"x\ and q}*x% where 
w" = 1, for x^, and multiply the resulting five equations, we obtain 

\ (l-ay^){l-x^^)(l-x^^)... Y^ 1 

^^^ \(l-x){l-x'^)(l-x')... I ^-'-Ux-x'^-^- 
From (15) and (16) we deduce 
(17) ^(4)+_p(9)^'+_p(14)a;-+ ... 

_ {(1 - x'){l - x'^yj l-x^')...}' . 



I 



{(1 -x){l - x')(l-a^) ...Y ' 
from which it appears directly that ^ (5m + 4) is divisible by 5. 
The corresponding formula involving 7 is 

(18) p(5)+p(12)x + p(19)x''+ ... 

{(1-^)(1-^'^)(1-^^)...}^ 

- , ^9^ { a-x')(l-x^^)(l-af^)...Y 
^ {{l-x)(l-x'){l-x'')...Y ' 

Avhich shows that p (7m + 5) is divisible by 7. 
From (16) it follows that 
p (4) X + p (9) x'- + p (14!) x^ + ... 

5{{l-x^){l-x^'>)(l-x'')...Y 

X {l-x'){l-x'">){l-x'')., 



{l-x)(l-x'')(l-af)... {(l-x){l-x^){l-x')...Y' 

As the coefficient of «^'* on the right-hand side is ^ multiple of 5, it 
follows that p {25m + 24) is divisible by 25. 
Similarly 
p(5)x + p (1 2)x'-+p(19)x^+ ... 

7{(1-«0(1-^")(1-^'')---}' 

(l-x')(l-x^')... 



= x(l-3x+5x^- 7a'« + . . .) 



Ki-^)(i-^^)-r 



^^ja-^)(i-^^) 



{(1-^)(1-^-) 



18 ' 



"J 



from which it follows that p {4<9m + 47) is divisible by 49. 

[Another proof of (1) and (2) has been found by Mr H. B. C. Darhng, to whom 
my conjecture had been communicated by Major MacMahon. This proof will also 
be published in these Proceedings. I have since found proofs of (3), (7), and (8).] 



Prof. Rogers & Mr RamanKJan, Proof of certain identities 211 



P roof of certain identities in combinatory analysis: (1) by Prof. 
L. J. Rogers ; (2) by S. Ramanujan, B.A., Trinity College. (Com- 
municated, with a prefatory note, by Mr G. H. Hardy.) 

[Received 3 October 1918 : read 28 October 1918.] 

[The identities in question are those numbered (10) and (11) in 
each of the two following notes, viz. 

q q* q^ 

+ l-l + (I-g)(l-5'=)'^(l-5)(l-f/)(l-9^) + -" 

= 1 (1) 

and 

q^ q^ 512 

+ n^ + (i -q){i- f) + j\-~-^(y-f) (1 - f) + • • • 

= ^ ...(2). 

(1 - r/) (1 - (/) (1 - f) (1 - (/) (1 - q^-^) (1 - f/0 ^ ^ 

On the left-hand side the indices of the powers of q in the 
numerators are n^ and n (n + 1 ), while in each of the products on 
the right hand side the indices of the powers of q form two arith- 
metical progressions with difference 5. 

The formulae were first discovered by Prof Rogers, and are 
contained in a paper published by him in 1894*. In this paper 
they appear as corollaries of a series of general theorems, and, 
possibly for this reason, they seem to have escaped notice, in spite 
of their obvious interest and beauty. They were rediscovered 
nearly 20 years later by Mr Ramanujan, who communicated them 
to me in a letter from India in February 1913. Mr Ramanujan 
had then no proof of the formulae, which he had found by a process 
of induction. I communicated them in turn to Major MacMahon 
and to Prof. O. Perron of Tubingen ; but none of us were able to 
suggest a proof; and they appear, unproved, in Ch. 3, Vol. 2, 1916, 
of Major MacMahon's Combinatory Analysis'^. 

Since 1916 three further proofs have been published, one by 

* L. J. Rogers, ' Second memoir 011 the expansion of certain infinite products', 
Proc. London Math. Soc, ser. 1, vol. 25, 1894, pi?. 318—343 (§ 5, pp. 328—329, 
formulae (1) and (2)). 

t Pp. 33, 35. 



212 Prof. Rogers d- Mr Ramanujan, Proof of certain identities 

Prof. Rogers* and two by Prof. I. Schur of Strassburgf, who appears 
to have rediscovered the formulae once more. 

The proofs which follow are very much simpler than any pub- 
lished hitherto. The first is extracted from a letter written by 
Prof Rogers to Major MacMahon in October 1917 ; the second 
fi-om a letter written by Mr Ramanujan to me in A.pril of this year. 
They are in principle the same, though the details differ :|:. It 
seemed to me most desirable that the simplest and most elegant 
proofs of such very beautiful formulae should be made public with- 
out delay, and I have therefore obtained the consent of the authors 
to their insertion here. 

It should be observed that the transformation of the infinite 
products on the right-hand sides of (1) and (2) into quotients of 
Theta-series, and the expression of the quotient of the series on the 
left-hand sides as a continued fraction, exhibited explicitly in Prof 
Rogers' original paper and in Mr Raman ujan's present note, offer no 
serious difficulty. All the difficulty lies in the expression of these 
series as products, or as quotients of Theta-series. — G. H. H.] 

1. {By L. J. Rogers.) 

Suppose that \q\<l, and let F,,^ denote the convergent series 
(1 - ^»^) - ^«(^«+i-'« (1 - x'"^q"''') C\ 

where 

., _ (1 - ^) (1 - xq) (1 - iC(f) ... (1 - xq'-^) 
^'- (i_5)(i_5-.)(i_23^...(l_5.) -■ 

the general term being 

Then 

V,n - Fm-i = ^*"~' (!-«;)- x"q''+'-'^ {(1 -q) + x'>''~Hf"-' (1 - Ar^)} d 

_,_ ^in^m+z-im |(^1 _ ^^2) ^ ^m-i^im-2 (^l _ ^^2^J (J^^ (• 2^_ 

Suppose now that the symbol 77 is defined by the equation 

vf{«^) =f(xq)^ 
Then ( 1 - f) C, = (1 - x) rj C,-! , (1 - xq^) 0,. = (1 - x) rj C,. 

* L. J. Kogers, ' Ou two theorems of Combinatory Analysis and some allied 
identities', Proc. London Math. Soc, ser. 2, vol. 16, 1917, pp." 315—336 (pp. 815— 
317). 

t I. Schnr, ' Ein Beitrag zur additiven Zahlentheorie uud zur Theorie der 
Kettenbriiche', Berliner Sitzungsberichte, 1917, No. 23, pp. 301—321. 

J I have altered the notation of Mr Eamanujan's letter so as to agree with that 
of Prof. Rogers. 



ill combinatory analysis 213 

Hence, arranging (2) in terms of t^C'i, 776*2, ..., we obtain 
V -V 

' m ' in—\ 
\ — X 

= a.'"^-i {(1 - ^.n-«i+ig»i-"i+i) _ .^H^«+?n (1 _ ^^n-m-H ^jin-sm+i^ r]Ci+ ...} 

= a;'''-'vVn-n,+i (3). 

CO 

If we write v^^ TI (1 — *•(/'■)= F,,^ (4), 

>- = 

then (3) becomes v„, — v,n-i = ^''"^~'^ vVn-m+i (o). 

It should be observed that Fo and Vo vanish identically. 

In particular take n = 2,m = l, a,nd n = 2, ni = 2. We then 
obtain Vi = 7]Vo, V2 — v^ = xrjv^ ; 

and so Vi — 7]Vi = ocqr)'-v^ (6). 

Now let Vi = l + a^x+a2cc-+ (7). 

Then from (5) 

1 + aiX + a-iO? + . . . — (1 + a^xq + a^xif + . . .) 

= a-^ (1 + tti^y/ + aojf-(f + ...) ; 

and so a^=—l~^ (^2 = p. ry^ ^ (8). 

1-q (l-q)(l-q') 

But when x = q, C,- = 1 ; and so 

V, = {l-q)-q^{l-q^) + f^{l-q^)- (!)). 

From (4), (6), (7), and (8) it follows that 

^l-q^(l-q)(l-q'^)^ 

^ a-q)-qH ^- q') + q''('^-q')--- .. ^^ 

(l-5)(l-2'^)(l-f/)... ^ '^- 



Similarly we have 

and, when x — q, 

and V, = {1- q') - q^ (1 - g«) + 5" (1 - f/«) - ... . 



\ ^ X X'Q^ 

77 1-^ (1- 7) (!-(/-) 



l-q (1-g) (!-(/) 



214 Prof. Roger's S Mr Ramanujan, Proof of certain identities 
Thus 

{l-q){\-q'){\-t) ^ 



2. {By S. Ramanujan.) 

Let 
G{x) = l 

+ t^ ^^^^^ ^' ""i \i-q){l-q^){l~f)...{l-qn 

If we write 1 -a;g2'' = 1 - (^^ + 5''(1 -^g"), 

every term in (1) is split up into tAvo parts. Associating the second 
j)art of each term with the first part of the succeeding term, we 
obtain 

1 —xq 



G (a;) = {1- xhf) - x'^q^ ( 1 - x-'q') 



l-q 



+ ^q (1 ^q ) (i_^)(i_^.>) {-)■ 

G(x) 

Now consider II(x) = ., ^ - G (xq) (3). 

\—xq 

Substituting for the first term from (2) and for the second term 
from (1), we obtain 



x^q' 
a;*g" (1 — xq-) 



H {x) = xq - ^3^- {(1 -q) + xc^ (1 - xf)] 



+ 






in combinatory analysis 215 

Associating, as before, the second part of each term with the first 
part of the succeeding term, we obtain 

H {X) = ,rq (1 - ^Y/) jl- ccY' (1 - ^^q') 1 ^ 

'M i^ ^?)(l_^)(l_5.)(l_,/) + ' 
= xg{l — x(f) G {x(f) (4 ). 

II now we write K{x) = ^-^--^^^^^- , 

we obtain, from (3) and (4), 

andso A-(^)=l+fl^^^ (5). 

In particular we have 

1^A_ t^- 1 _ {\-q)0{q) .... 

1+ T + 1 + ... K{\) G{1) ^ " 



or 



I q f _ 1 — g — g* + g' + g^^ — 



(7). 



1 + 1 + 1 + . . . 1 - ^2 _ ^3 ^ ^9 + gll 

This equation may also be written in the form 

1 <!_ f__{i^q){i_::q')iX-j')(i^)i^-_thi: 

1+1+1 + ... {\-f){\-q^){l-q^){l-cf){\-t'-)... 

(8)- 

If we write 

;,,._ G J^) 

^ ' {\-xq){l- xq') (1 - xf) ...' 
then (4) becomes F (x) = F (xq) + xq F {xq"), 
from which it readily follows that 



216 Prof. Rogers d' Mr Ramanujan, Proof of certain identities 



In particular we have 

1 - r/ - rf + (f + ry" - . . 



^l-q^{l-q){l-(f) (i_,^)(l_f/)(l_,/) 



{l-q)il-cf){l-f) 
1 



and 

1 + ^^ + ,g' + ... = {l-q)G(q) 



.(10), 



1_^ (l-g)(l-5^)' ••• (l-5)(l-^-^)(l-r/), 
1 — q — q^ + q' -\- (f^ — ... 

(i-^)(i-9^ya-^Yy^-" 
1 



(l_^.)(l_53)(l_(^7)(l_^8)(l_^l.),.. 



.(11)- 



Mr Darling, On Mr Ramanujan's congruence properties of p (n) 217 



On Mr Ramanujan's congruence properties of p («). By H. B. C. 
Darling. (Communicated by Mr G. H. Hardy.) 

[Received 3 October 1918: read 28 October 1918.] 

1. Proof that p {5ni + 4) = (mod 5). 

Let u ^(l-x){l- aJ") (1 -x')...; 

then by Jacobi's expansion 

a^' = "ST (- 1)" (2» + 1) **"^''+'^ 



n = » 



so that in d'-ur, where d denotes differentiation with respect to x, 
the coefficients are of the form 

i {n - 1) n (n + 1) (n + 2) {2 (n + 3) - 5], 

and therefore 

d^u^= (mod 5) (1). 

Again, in d^u" the coefficients are of the form 

A^{n' + n-4<){n-2)(n-l)n(n + l)(n + 2)\2(u + 4')-7], 

and therefore 

an<3= (mod 7) (2). 

/1\ 1 2 

Now 8- ( - = /ci-u + - (duf ; 

\Uj U' u 

also du^ = Zu'du, and d-ii^ = ^u'^dhi + Qu (du)-. Hence 

a^(-) = -o\9"''' + (r7(9'*')' (3)5 

\uj ?>u^ 9«' 

and thus, by (1), we have 

^' ^ - II ^^'''^^ ^" ^ ~ 27I" ^^'''^' ^"^^^ ^^ ' 

so that 8^ f-] = (mod 5) (4). 

Again if Iju be expanded in powers of x, and the operator 3* 
be applied to the resulting sei'ies, it is evident that the coefficients 
of all powers of x of the forms om, 5m + 1, om + 2 and 5m + 3 will 
be multiplied by a factor divisible by 5 ; but that the coefficients 
of the powers of x of the form 5m + 4 will be multiplied by a factor 
which is not divisible by 5. Hence it follows at once from (4) that 

p {oin + i) = (mod 5). 



2181 3Ir Darling, On Mr Ramanujan's congruence properties of p {n] 

2. Proof that p {7m + 5 ) = (mod 7). 
Differentiating (3), we have 

d' (-] = - ^, dHi' + ^^ dud-u' + A 9 (9"')' ('iiocl 7 ) 

OU SU'^ i)'U' 



= - ~dhi^ + j^^didu^y (mod 7). 
Similarl}^ having regard to (2), 

a* (-] = ^- du^d^u^ + ^a- (du'T- (mod 7), 

d' (^) = ^. d'vPd'w + ~ d' (dtt'f (mod 7) (5), 

^0-^,(^-'y^l,^'('"'y('--^'^ ^6). 

Again a'(-^) = -3'©H-6a.Q = .-.3{..8»Q}; 
SO that, by (5) and (6), 

36 (^] = !_' [43 (x'^d'u'd-u') + 63 {a^«3« (du'f]] (7). 

Now d{dit'')- = 2dHo'dit', 

3^ (8m3)2 = 23^ i(33«.3 + 2 (d-u'^y. 
Thus, by (2), 

3^ (du^Y = Qd'u'^d'-u^ (mod 7) ; 

and therefore, by (7), we see that 



that is, by (2), 

^x 



38 (- j = 3 {w'd'ii'dHi'] (mod 7) ; 



= ai'dhi^dHi^ + 6w^d^ u^'d' u' (mod 7) 
uj 

= d'u^d{£c'd-a') (mod 7) (8). 

But the coefficients in 3 (x'^a-u^) are of the form 

i (n - 1) 9i {n + 1) {n + 2) [2 (n - 3) + 7} {(// - 2) (n + 3) + 14}, 
and are therefore divisible by 7 ; and therefore, by (8), 
3« (-) = (mod 7). 

Hence, by considerations similar to those in the latter part of § 1, 
we see that 

p{7m + 5) = (mod 7). 



Miss Wrinch, On the eicponentiation of well-ordered series 219 



On tlie exponentiation of luell-ordered series. By Miss Dorothy 
Wrinch. (Communicated by Mr G. H. Hardy.) 

[Read 29 October 1918.] 

The problem before us in this paper is the investigation of the 
necessary and sufficient conditions that P'^ should be Dedekindian 
or semi-Dedekindian when P and Q are well ordered series. 

The field of P'^ is the class of Cantor's Belegungen and consists 
of those relations which cover all the members of the field of Q 
with members of the field of P : several members of the field of Q 
may be covered with the same member of the field of P, but every 
member of the field of Q is covered with one member of the field 
of P and one only. In order to prove that P'^ is Dedekindian it is 
necessary to prove that every sub-class of the field of P^ has a lower 
limit or minimum with respect to P^. If there is a last term of 
the series P'^ it is the lower limit of the null class. Unit sub-classes 
have their unique members as minima. It remains, then, to con- 
sider sub-classes with two or more members. 

Now the relation P*'* orders two relations R and *S' by putting R 
before S, if R covers the first Q-term, which is not covered with the 
same P-term by both R and h, with a P-term occurring earlier in 
the P-series than the term with which 8 covers it. Suppose A, is a 
sub-class of the field of P'^ with at least two members. We will call 
Qm'^ the first Q-term which is not covered with the same P-term 

by all \'s; and Tp^\ that subset of X which consists of those members 
of \ which cover Q„/A, with that term, in the class of P-terms with 
which various X's cover Qm'^; which occurs earliest in the P-order. 

Tp'\ will therefore be contained in \ and not identical with it. It 

will be seen that P'^-terms belonging to Tp'\ come earlier in the 
P'^-order than terms of \ not belonging to it. Constructing 

Tp'T'X 

we get a smaller subset of \ : members of this subset occur earlier 
in P''* than other members of X. Continuing this process with 

X, T/X, T/Tp'X, Tp'Tp'iyx,... /*,... V,..., 

we obtain smaller and smaller sub-classes of X: if /a precedes v in 
this order, members of v occur earlier in the P'^-order than members 
of fM which are not members of v. We take the common part of 



220 Miss Wrinch, On the exponentiation of well-ordered series 

all these subsets of \, i.e. the class of relations which belong to all 
the sets 

\, 2 P \, 1 P 1 P \ . . . ', 

and get a subset of X 

p'iT'pW^ 
which, again, consists of members of \ which come earlier in the 
P'^-order than members of A. not belonging to it. Repeating the 
original procedure we get 

Tpy(%h'\ Tp'TpYiT'p)^'\, ..., 

and so obtain a series of sub-classes of A, ordered by the serial 
relation 

A {Tp, \), 

where A is the relation between /j, and v when v is contained in /i 
but not identical with it. And this is a well-ordered relation : 
CQjisequently it will have an end, viz. 

p%Tp^Ay\. 

If this is not null, it consists of a single member, which will be 
the minimum of X in P^. But if it is null we will put 

PQ'X = s'N {a^ . ^,e (Tp^Ayx . iV= (i^/.) r eQm VI- 

Then PQ'X is a relation covering a certain section of the Q-terms 
with P-terms : PQ'\ agrees in the way it covers the Q-spaces with 
each member yu, of ^ {Tp, \) as far as QniV- PQ''^ will therefore 
cover Q-spaces up to z, if there is a yu- which is a member of the 
field of 

A (Tp, \) 

such that z precedes QmV in the Q-order. If no member of the 
field of J. (Tp, X) agrees in the covering of Q-spaces beyond a cer- 
tain member z of the field of Q, PQ'X covers no spaces beyond z 
with P-terms and for this reason is not a member of the field of P'^. 
If P is a P'3-term which agrees with PQ'X in the covering of 
Q-spaces as far as it goes, R precedes all the members of X, in the 
P^-order ; further, any member of the field of P^, following R and 
all relations agreeing with PQ'X as far as it goes, follows at least 
one member of X. Hence, if there were a maximum in the P'^- 
order in the class p of members of the field of P*? which agree with 
PQ'X as far as it goes, this relation would precede all X's and any 
relation following it would follow at least one member of X. If the 
class consists of one term R, it will have a maximum, namely R 
itself: R will then be equal to PQ'X and PQ'X will, therefore, be 
the lower limit of X. But p is a unit class only when PQ'X covers 



Miss Wrinck, On the exponentiation of ivell-ordered series 221 

the ivhole of the Q-terins with P-terras. When PQ'X does nut 
cover the whole of the Q-tenns, but covers Q-terms only up to 
z (say), all p's will agree in their covering of Q-spaces up to z, and 
the remaining Q-spaces will be covered differently by different 
members of p. To get a maximum of the p's with respect to P*^, 
we want a relation S which is a p such that no member of p comes 
later in the P^-order. Now. if P has no last term, every P-term 
is followed by other P-terms. However S covers z and the Q-spaces 
after z, by replacing the term covering any member of the field of 
Q after ^ by a member of the field of P following it in the P-order, 
we obtain a relation T which is a p and follows ;Si in the P'^-order. 
;S' is, consequently, not the maximum of p in the P^-order. Now 
if z in the field of Q is covered by PQ'X, the term innnediately 
following z will also be covered by PQ'X. Therefore, if Q is a finite 
series or an co, PQ'X will always cover the whole of the Q-terms ; 
since, as X has at least two members, it will always cover one Q- 
term. Any X will then have a lower limit or minimum with 
respect to P*^. In such cases, P*? will certainly be Dedekindian 
with the addition of a last term, whether P has a last term itself 
or not. 

But if Nr'Q is greater than o), it is possible to find a subclass 
X of the field of P^ which is such that PQ'X does not cover the 
whole of the field of Q. 

For, let 1 and 2 represent the first and second terms in the P- 
series and let (e.g.) 

i...hi(r)2...h2(aiii 

represent a relation which covers the first ^ Q-terms with 1, sub- 
sequent terms up to (but not including) the ^th term with 2, and 
all remaining terms with 1. Such a relation is clearly a member 
of the field of P'^. Consider the class of I'elations X which cover 
all Q-spaces up to z with 1, and all the Q-spaces following z with 
2, as z is varied from the second Q-term to the ^th, where ^ is an 
ordinal number with no immediate predecessor. We will arrange 
this class of relations in the P^-order. 



1... 1-1(^)2. ..h2(0,22.... {^<0 

11112 h2(f), 22 

11122 f-2(0> 22 

11222 ^-2(0> 22 

12222 ^-2(0, 22 

This class has no minimum in the P'^-'-order, and PQ'X covers all 



222 Miss Wrinch, On the exponentiation of well-ordered series 

the Q-places up to the ^th with 1 and does not cover the subse- 
quent Q-places at all. It is therefore not a member of the field 
of P^. But, as we have seen, every relation which agrees with 
PQ^X as far as it goes, and covers the other Q-places with any P- 
terms whatever, precedes all X's : and any member of the field of 
P^ following this relation, and all relations agreeing with PQ'X as 
far as it goes, follows at least one member of X. Thus, e.g., the 
relation 

ll...hl(^), 2111 

precedes all X's, and any relation following it and all relations 
agreeing Avith PQ'X as far as it goes (as e.g. the relation 

11211... h 1(0, 21211...) 

follows at least one relation belonging to X, e.g. the relation 

11122... \-2{0, 222... 

Thus \ will have a lower limit if and only if there is a maximum 
among the relations covering all places up to the ^th with 1. 
And this is the case when and only when P has a last term u (say). 
For then the relation 

111 \-l{^)uiiu... 

will be the lower limit of X. Thus if Nr'Q is greater than o), it 
will be the case that all existent sub-classes of the field of P'^ will 
have a lower limit or minimum when and only when P has a last 
term. A non-existent subclass (i.e. a subclass with no members) 
will have a lower limit or minimum when and only when P has a 
last term. If Nr'Q is greater than co, P^ is Dedekindian when P 
has a last term, and if P has no last term P^ even with the addition 
of a last term is not Dedekindian. We thus arrive at the following 
conclusions. When P and Q are well-ordered series, (1) P^ is 
Dedekindian when and only when P has a last term ; (2) if Nr'Q 
is greater than co, P*^ with the addition of a last term is Dede- 
kindian if and only if P has a last term ; (3) if P^ is made Dede- 
kindian by the addition of a last term when and only when P has 
a last term, Nr'Q is greater than co. 

These propositions will now be established. 

[The symbols used are those o/Principia Mathematica. Among 
the propositions referred to, those whose nwnhers are greater than 
1 are proved in P.M., ivhile the others are established in the course 
of this paper.'] 

*01. QjX = mmQ'y{s'X'y^eQKjl) Df 

*-02. Tp'X = XnM {M'QJX = mmp's'X'Q^,'X) Df 



Miss Wrinch, On the exponentiation of well-ordered series 223 
*-03. A=\fl{^lQ\.^Ji^\) Df 

*1. I- : P, Q e n . X 6 . D . 5'Cnv'P« = mm'(7^«)'X [*207-l7] 

*11. V:F,QeQ..\el.\C C'F'^ . D T'X = min (P^^)')^ 

Dem. 

h.*l7619. 'D\-:R6a'P^.Dj,.^{RPm) (1) 

l-.(l).*205-18. Dl-.Prop 

*'201. \-:P,Qen.\C C'P'i . E ! T/X . D . f{Tp^Ay\ 

= B'Cnv'A {Tp, X) . A (Tp, X) e O 
Devi. 

[•02] h. TpeRl' A nCh-^1 (1) 

h . (1) . *258-231 . D h . Prop 

*-202. h : E ! Tp'X . D .p'(Tp^Ayx ^eB'Tp [*-201] 

*-203. \-:P,Qen.XC C'P'^ .X^eOwl.D.E! Tp'X 
Dem. 
\-:.XQ G'P'i .R,S€X.X€ a'R . D^^ . R'w = S'x:D .XeO vjI :. 
[Transp] D I- :. Hp . D : P, >Sfe A, . D . g^ . i2^« =|= iS^'a- . iceQ^E :. 

[*250'121] D h :. Hp . D : E ! min,/^ (i-'A,'^ ~ e u 1) :. 

[rOr02] D I- :. Hp . D : E ! Q,„'X . E ! Tp'X 

*-2031. (- : E ! T/X . D . E ! Q,/X [*-02] 

*-204. h : P, Q e O . X C C"P'^ X ~ e D' Tp . D . X e w 1 

[*-2()3 . Transp] 
*-205. f- . Hp r203 . D . p'iTp^Ayx e u 1 [**-202-203-204] 

r211. h . E ! Q,„*\ . D . s'X'QjX ~ e w 1 [*-01] 

*-212. h :. E ! Q,,,'X : D : {s'X)[Q'Q,,,'X el -^C\s: ReX . 

D.P6l->Cls 
Pewi. 

[*176-19] hz.ReC'P'i .D'.zeC'Q.D.R'z^eO (1) 

l-.(l).r02. Dh.Prop 

VOL. XIX. PART V. 16 



224 Miss Wrinch, On the exponentiation of well-ordered series 

r213. \-:S€l->C\s.R(iS.D.S[a'R = R 

Dem. 

V .^u.v . uSv . V € Q.'R . ~ (uRv) . D . g«, v, u' . a 4= w • 
uSv . u'Rv . <^ (uRv) . V € d'R : 
D\-:.R(lS.D:'3^u,v. uSv . v e a'R . -- (uRv) . 

D . g/t, V, n' . u' =j= u . nSv . u'Sv : 
0\-:.RQ.S.Sel->Ch.D:-^ {gu, v . iiSv . v e Q'R . 

~ (itRv)] 
D\-:.R(lS.Sel-^Ch.Di nSv . v e a'R . D„,, . uRv : 
D\-:.R(lS.S€l-^Ch.:>:S\-a'R = R 

*-2131. ^:.'3^l^.aCa's'uT.{s''!^)\-ael^C\s:0:Re^.D.R[a 

= {s''ST)[a = (p''S7)[a 
Dem. 

h . *40-13 . *41-44 . Dhz.Rezy.D: xRy .D.x (sV) xj : 

a'RQa\s'^)'.. 
'^ViaXia'R.Ret;T.':>.R\aQ.{s'zj)\a (1) 

I- *-213 . (1) . D F : a C iVR .Re^. {s'^) pet e 1 -^ Cls . 

:^.R\a = {s'7JT)\a (2) 
h:.g;!tn-.i^ero-.Djj. xRy .yea: 

D : g[*S' . S e-sT . xSy .yea (3) 

I- . (3) . D h : a ! OT . D . {p'^) \a G (s'tsr) \a (4) 

f- . (4) . *-213 . D h :. a: ! w . D : (5^-sr) [^a e 1 -^ Cls . D . 

(i'OT)Pa = (^^i3-)['a (5) 
l-.(2).(5). Dh.Prop 

r214. V :. Hpr20.3 : D : E e \ . D . R[Q'QJX = {s'\)[Q'Q^,'X 

= {p'X) [Q'QJX [*40-13 . *41-44 . r2131] 

r215. h :. Hpr203 :D:Re\- T/X . S e Tp'X . D . 

R [Q'Q^'X = S [Q'Q^'X . {S'QJX) P (R'Q^'X) 
Dem. 

[r02] \-:SeTp'X.D.SeX (1) 

[**-01-02] h : ^ ! Q^,'X . S e T/X .ReX- Tp'X . 

D . S'Q,,,'X = minp 's'X'QjX . R'Q^'X + min's'X'Q^'X (2) 
K (1) . (2) . r2l4 . D h . Prop 



Miss Wrinch, On the expouentiatluit of luell-ordered series 225 

*-216. f- : . Hp *-208 :D:Re\- Tp'\ .SeTi^'X.D. SF'^ R [*-2 15] 

*-217. h : . Hp *-203 : D : /x e ( Tp^A y\ . R eX- /m . S e fi, .D . SP'^'R 

Bern. 

h . *40-23 . D h :. p C (Tp^Ayx .'■^l p : fie p . S e /j, . ReX- fi . 

D . SP'^R :D:Sep'p.R€X-p'p. D^„, . SF'^R (1) 
h . (1) . -r216.*258-241 . D I- . Prop 

r218. h: }ii)*-203. '3^1 p'{Tp*Ayx.D.7Y(Tp^Ayx = mill {P'^yx 
Dem. 

V . *r217-201 . D I- : Hp . D : ^' e \ -p^Tp^Ayx . 

Rep'iTp^Ayx.D.RP^S (1) 
h.(l). Dh.Prop 

*-31. hi.SeX-.D: k=p'{{Tp^Ayx n^/Sf} . D . 

ke(Tp*Ayx.S'^eTp'k 
Dem. 

[*22-43] h :. SeX. D : h = p'{{Tp^AyXn e'S] . D . 

'3_p.pC (Tp^Ayx . g! p . p = [{Tp^Ayx n e'S] . k=p'p 

[*257-125] D h :. A'^eX . D : k=p'{{Tp^Ayx n eSS'J . D . 

a/3 . It/p e {Tp^Ayx . p = {{Tp^Ayx n e'S] . k =p'p 

[*258-211] D h :. 6' e A . D . k = p'{{Tp*Ayx n e'S] . D . 

^^e(Tp*^)'X (1) 

l-.(l).*2o7-125. 

Dl-:.S€X.D.k = p'{{Tp*Ayx n e'^'l . D . 

?p'^^e(^p*^)'x (2) 

[*40-12] D\-:.SeX.D.k^p'{{Tp^AyXne'S].D. 
^ e (T^p*^ yx n e'S .D^.kCr^: 

[Transp] D\-:.SeX.D.k =p'{(Tp^Ayx n e'S] . D . 

16—2 



226 Miss Wrinch, On the exponentiation of vjell-ordered series 

l-.*22-43.*-04.(2). 

D\-:.SeX.D.k =|/{(rp*^)'X n e'S\ : D . 

~ (k C Tp'k) . Tp'k e {Tp^A y\ ( 4) 

l-.(3).(4). D h :. >S'eX . D . k =p'{{Tp^Ay\t^ e'S} . D . 

r^iSeTp'k) (5) 

|-.(1).(5). Dh.Prop 

r32. h : Hp r203 . /. ?p A, . E ! ?p V ■ ::> ■ (Qn/^) Q (<^m» 

Dem. 
[*-212] l-:Hpr203.i?eX.D.^Q(Q^/X)D.i^*5el (1) 

[**-02-203] \- .fiTpX.D.fjiCX (2) 

h . (1) . (2) . O 1- :. Hp -r203 .f^TpX.Sefx. 

:>:zQ(QjX).D.S'zel (3) 

[*-02] f- : Hp *-203 . fjuTpX. S e /j, . 

D . S'Q,,,'X = minp'P [s'X'y - e u 1 } (4) 

1- . (3) . (4) . I- :: Hpr203 .f^TpX.Se/x. 

D :. a/ : (QJX) Qz' : uQz' . D,„ . ^S^(^ e 1 (5) 

K (5) . h : Hp*-203 . /. ?p X . El ?pV • ^ • (Qn/^) Q (Qm'/-) 

r321. h : Hp r203 . ^a (.4 {Tp, X))v.El Tp'v . D . (Q,„V) Q ('Q^'i.) 

Dem. 
[**-02-203] \-ipC (Tp^Ayx .'Rlp.'K^.p'p. 

D.lyl'Q.^Yp'^elyjO (1) 
[*40-12] h'.Xep.D.'p'pCX (2) 

K(l).(2). D\-:.pC{Tp^Ayx.'3_lp.'3_lp'p: 

D-.Xep.O. i'X'Q,,yp'p ~ e 1 u (3) 

h . (3) . -r02 . D\-:.pC {Tp^Ayx .'3,1 p .'Rlp'p . 
*• > 

D : Tp% Xep.D. s'Tp'X'Q,,yx e 1 . i'Tp'X'Q,^'p'pr^e 1 yj : 

D\- :. pC{Tp^Ayx.'3l p .^Ip'p: 

:>:Tp%Xep.D.Q,^'X^Qjp'p (4) 



Miss Wrinch, On the exponentiation of well-ordered series 227 
I- . (1) . rOl . (4) . D f- : pC{Tp^A)'\ . g ! p . g ! jo'p . 

(InP'p ^V !.v'X',y-eO u i; . Q,/\ = miiV.??~e() w 1 (5) 
I- . (o) ■ D h : p C (rp*^)'\ . a ! /9 . a ! jt)'p . 

r,/X,\6p.D.(Q„A)Q(Q,nyp) (H) 
V . (6) . *-82 . *-258-24l . D h . Prop 

*-33. V : Hp *-203 . D . PQ'X e 1 -^ Cls 

[*-04] I- : Hp . t!7 = ^ [gyti . yLt 6 (r/,*4)'\ . 

|-:.Hp(l).D:i¥,iV6t^.Di,,v- 
[r201.*250-113] D h :. Hp (1) . D : M, Ne^. D^^, v ■ 3/^, ^^ - 

IJi [A {Tp,\)]v.v.v \A (Tp, \)] /Lt : 
[**-321-214] D F- :: Hp (1) :. D :. i¥, iYe ^ . D,^, y ■ 

D\-:.Rp(i):D.M,N€^.DAVMCa'N. 
M = N \ QM/ . V . a'N C a'i¥ . N = M\-a'N':. 
D:.Hp(l):D.if,.V6t^. 

D . 2/ e a'lM n a'N . D . 3f'y = N'y (2) 
f-.(2). D.Prop 

*-34. I- : /x e ( '/p*^ )*\ . E ! ?p'X . Hp r203 . 

[*-04] 1- . Hp *-203 . E ! Tp'tx .fx.€( Tp^*A )'X . 

D.(/>V)rW„>CPQ'X (1) 



1 



228 Miss Wrinch, On the exponentiation of luell-oj'dei^ed series 

[*-33] h . Hp . D . PQ'X € 1 -^ Cls 

|-.(l).(2).*-213.Dh.Prop 

r341. h : /x €(Tp*Ayx . Hp *-208 . p'(Tp*Ayx . 

Dem. 
[*-201] !-:Hp.D.'B^Cnv'^(Tp,\) = A: 

D h : Hp . D . E ! ?p V (1) 

|-.(l).r84.DK.Prop 

*-35. f- :. i^ [^ a^PQ'X = PQ'X . Hp *-203 . p'(Tp^AYX = A . 

ReC'P'^::)f>:SeX.Ds-RP'^S 
Dem. 

[**-84-31] h . Hp . <Sf e X . /<: = p^K^P*-^ y^ '^'^\ ■ 

D^.Er'Q'Q^'x=5:r^Qm'x (1) 

[**-34-02] h . Hp . ^ e \ . k=p'\{Tp^Ayx nV'S] . 

D^ . gT . T e fp'k . K^e^'X = T'Q^'X . 

(2''Q„^X)P(,Sf,Q,^X) (2) 

h . (1) . (2) . D I- :. Hp .DpiSeX. D,, ■ PP^>S^ 

*-4. h : : Hp *-203 . p'(Tp^A yx = A.ze a'PQ'X :.D:.S ['Q'z 

= (PQ'X) [~Q'z : (PQ'X'z) P {S'z) .D.'3^U.UeX. UP'^S 
Dem. 

[r04] h-.ze a'PQ'X .:>.'^pi.fMe {Tp^AyX . 2 e'q'Q^'iM : 

[r341] D h : . Hp . D : ayu . ^ = Tp V ■ 3 ■ (PQ'^) [~Q'Qm'i^ 

[r341] D h : Hp . D . a/., f^ . C/^ 6 ?p ^ ■ (i'Q*^) T Q'Qm'Tp^M' 

= U['Q'Q^'Tp'v.zQ^iQ^'v): 

[*176-19] D I- : Hp . D : ,S7 Q'^ = PQ'X [Q'z . 

(PQ'X'z) P (S'z) . D . a ^ . [/ e X . UP'iS 



Miss Wrinch, On the exponentiation of well-ordered sei'ies 229 

*-41. t- : . Hp *-203 . p\Tp^-A y\ = A . i2 e C'P'? . 

7? [ (VPQ'X = PQ'\.D. RP'i V : D : g^S' . ,S' e X . .S'pv V 
Dem. 
[*17G-19] h:Hp.D.-(rra'PQ'\ = PQ'A,): 

[*176-19] D H : Hp . D . 32 , 5 6 (J'PQ'X . V [Q'z 

= (PQ'X) ['Q'z . (PQ'X'z) P ( V'z) 
[*-4] h . Hp . D . a^S' . <S6\ . SP^V 

*-42. |-:.P, QeO.X-eOul.xC C'P'^' . p'(Tp^A Y\ : 

D:p = C'P'i nR[R\ a'PQ'X = PQ'X] .D.pC p'P'"'X . 

pcpQ"p c s'P''\ [*r3.r-H] 

r43. \-:¥AQ^'X.D.s'X'Q^'X^el.s'fp'X'Q^'Xel [*r0r02] 
r431. \-:Tp(lA.{Tp\^(lA 

Deni. 
|-.r43. Dh: fiTpX.D.ij,CX.^^X (1) 

|-.(1).*201-18. Dh.Prop 

*-432. h : Hp *203 . D . "I^i (P«)'X C /j'CTp*^ )'X 

[*-21 7] h :. Hp . P e /ci . yLt e (^V*.! )'X . /x {xl (T'p, \)| v . 

P ~ 6 z/ : D : g! z^ . D . a*S' . >SP^'P :. 
DI-:.Hp.Peyit.yL6e (rp*J.)'\ . P mill (P'O X : 

D : /x [^ (Tp, X)]v.'^\v.D.R€ V (1 ) 

[r431] t-:Hp.Pe/i.i;{^(Tp,X.)j/i.D.Pei; (2) 

h . (1) . (2) . D h :. Hp . D . P e i"^ P^'X : D : /^ e {Tp^Ayx . 

^M.Pe/^ (3) 
h . (3) . D h . Prop 

*-433. h : Hp *-42 . p = C'P^^ nR(R[ a'PQ'X = PQ'X) . 

D.'^(P<0'/3 = tl(P'^)*X 
Dem. 

h :: Hpr42 . D :. SeC'P'^' : D : S[a'PQ'X = PQ'X . 

y.'^z.ze a'PQ'X . ^ ['Q'z = PQ'X [ Q'z . 
(S'z)P{PQ'X'z) . V . (PQ'X'z)P(S'z) :. 



230 Miss Wrinch, On the exponentiation of well-ordered series 

[-r4] D I- :: Hp . D :. SeCP"^ .D . S e p :v: T e p .Dj ■ 

SP'iTiv.'g^U.UeX.UP^S: 

D h :: Hp . D :. T^eX . D,- . SP'^V : S eC'P'^ : 

DiSep.v.Tep.Dr-^'^P'^T: 



D I- :: Hp . D :. (T = ?7(FeX . D,- . UP'W) - max (P'iyp . 
SecTzD-.El max (P'^Yp . D . SP^ (max (P'^Yp) . 

max (P«) V epiv: i^'P« ^ = A . D . gT . .ST'-T . T e p : : 

D\-::lIl).D:.(T = U{Ve\. D,- . ?7P«F) - ^{P^p . 

D.aCP'^"p (1) 

h.(l).*205-193. DI-:.Hp.D:(7 = 0'(FeX.D,-- t^P'^F) 

-l'^(P«)V . D . ^(P«)'p u cr = ^(P'?)'p (2) 

h . *206-02 . D I- :. Hp . D : prec (P«)^X 

= ^' &( F 6 /,; . D . FP'^^ U) (3) 

h . (3) . :) h :. Hp . D : (7 = ^( Fe X . Dr ■ ^i"^'^^) 

- ^ (P«)'p . D . ^ (P«)'X = 1^ (P'-') Cp yj a) : 

|-.(2). Dh:.Hp.D:c7-f/(F6X.Dp.. f/^P^F) 

- rmJ (P«)> . D . prec (P<^)'X = nmJ (P''*)'p (4) 

I- . *-432 . D I- : Hp . D . i^ (P«)'X = A (5) 

h . (4) . (5) . *207-02 . D h : Hp . D . ^ (P^Yp = tl (P^)'^ 
r44. h : Hp r42 . p = C'P^ nR{R\- d'PQ'X = PQ'X) .pel. 

D . T'p = tl (P«)'\ [*205-lS . *-433] 

*-45. h : . Hp r42 . p - C"P« n P (P p d'PQ'X = PQ'X) : 

D : a'PQ'X =G'Q. = .p = I'PQ'X 
Dem. 

h :: Hp . D :. a'PQ'X = C'Q . D . PQ'X e G'P^ : 

Rep.D.a'R = a'PQ'X:: 

D I- :. Hp . D : a'PQ'X = C'Q . D . PQ'X = ^'p (1) 
h :. Hp . D : /^ = t'PQ'X . D . PQ'X e (7'P« :. 



Miss Wrinch, On the exponentiation of well-ordered series 231 

[*176-19] D f- :. Hp . D : /? = i'PQ'X . D . G'PQ'X ■-= C'Q (2) 
l-.(l).(2).DI-.Prop 

*-451. t-:.P,QeO.\CC*P«.X-'eOul .p\Tp^Ay\ = A. 

D . a'PQ'X =C'Q:D.E ! tl (P'?)'\ [**45-44] 

r46. f- :, P, Q e a . \ C 0'P« . \ ~ e u 1 . p'(Tp*Ay\ = A . 

D^ . E! limin (P^Yfi [*rll-21 8-451] 

r5. l-.Hpr42.D.maJ<2^a'PQ'X = A 

Pe?u. 
[*-04] h. Hp.^ 6 a*PQ*\. 

D.'^fX,.fl€(Tp^AyX.2Q{Qm'f^) (1) 

[r04] h.Hp.yLie(rp*^yx. 

D.ai..^ = TpV.Q'QmV.ca'PQ'x (2) 

1- . r21G . (1) . (2) . D I- . Hp . ^ e a'PQ'X . 

D.'^z'.zQz'.z'ea'PQ'X (3) 

I- . (3) . D h . Prop 

r51. l-:.Hpr42:D:a*PQ'\=C'Q. 

V . a^ . ^ e c"Q - a'Q, . a'PQ'X = Q'^ [*-5] 

-r52. \-:.P,Qen.C'QC Q'Q, . D : ya ~ e . /^ C G'P^ . 

D^ . E ! limin (P«)V [*r46-51] 

r53. h : P, Q e n . C'Q C Q'Q, . E ! P'Cnv'P« . D . P<^' e Ded 

[**-52-l] 

r531. \-:P,Qea.C'QC a'Q, .~B'Cnv'P'i = A . 

D . P« e semi-Ded [**-52-l] 

r5401. h :. P, Qe n : D : p = B'Cnv'P^ .^.Rep. 

■D,.I)'R = i'B'P [*17ryl9] 

*'541. f- :. P, Q e O . D : E ! 5'Cnv'P« . = . E ! 5'P [r5401] 

*-55. h : . P, Q e n . (7'Q C a^Qi : D : P« e semi-Ded : 

P« e Ded . = . E ! B'P [**-541-53-531] 
*-56. h : . P, Q e n . Nr'Q ^ « . D : P« e semi-Ded : 

P«eDed.= .E!7i'P [r55] 



232 Miss Wrinch, On the exponentiation of tuell-ordered series 

*-6. h : . Hp *-42 . p = C'P'^ r^ R[E\ a'PQ'X = PQ'X] . p ~ e 1 : 

D : E ! B'P . = . E ! max (P'^'Yp 
Dem. 
[*176-19] I- :. Hp . D :R\a'PQ'\ = PQ'X . Q'E = C'Q . 

B'{B\- a'PQ'X) = i'B'P . =^ . E - max (P^O^ (^) 

t- . (1) . D h :. Hp . D : E ! B'P . = . E ! max (P'O'p 
r61. h:.P,Qe-Q.ao. 

\ = P (as . sQ^a . P = //1pT"Q'^ e; (/2p t^Q*'^) • 

a € C'Q, - a'Qi : D : E ! limin (P'-')'^ . = . E ! B'P 
Dem. 

h:.Hp:D:-(aP,AS.P,,S'6X.P4=6'.P[^"4.'a 

h :. Hp . D : ya 6 {Tp^AYX . D . Q^iV Q « (1) 

h.(l).r04. Dh:.Hp.D.a'PQ^\c'$a . (2) 

I- . (2) . D h : Hp . D . p - e 1 (3) 

h . **-6-433 . D h :. Hp . D : E ! limin (P«)'\ . s . E ! P*P 
r62. V:.P,QeVL.r^\G'Q- Q'Qi . D - ^ C C"P« . X - e : 

D : E ! limin {P'^yx . = . E ! P'P [^Ol] 

-r63. h :. P, Q 6 O . Nr'Q > « : D : P« e Ded . = . E ! B'P . 

B'P = A. = . P«~esemi-Ded [*r61-l] 

r7. h :. P, Q 6 O . D :. P« 6 Ded . = . E ! P'P [**-56-63] 

r8. h :: P, Q 6 O . D :. P«e semi-Ded . = . E! P'P : = : Nr'Q > co 

Dem. 
h.*r63-7. Dh::P,Qefl.D:.Nr'Q>ft): 

D : P« e semi-Ded . = . E ! B'P (1) 
|-.r56. Dh::P,Q6f2.D:.Nr'Q^ft). 

D:P«esemi-Ded:E!P^P.v.p'^P = A (2) 
|-.(1).(2). DI-::P,Qeft:.Nr'Q>a): 

= : P« e semi-Ded . = . E ! B'P 



Miss Wrinch, On the exponentiation of luell-ordered series 233 

The definitions and method used in the earlier part of this paper 
(**"01 — •341) are suggested in Principia Mathematica *27G. There 
it is stated tentatively that 

g! p'iT^Ayx . D . ^YiTp^Ayx = min (P«)'X 
<- g! p\Tp^Ayx . D . PQ'X = prec (P^YX 

The first of these propositions is established in ***1 — •218 : the 
second seems to be untrue. If in the field of Q there is a term a 
with no immediate predecessor (as for example the term co if Q 
were the series of ordinals less than <w + 4), there is a X, a subclass 
of the field of P***, for which PQ'X is a relation covering with P-terms 
only the Q-terms which precede a (cp. *"61). In such a case PQ'X 
is not a P*^ term and so is not prec (P^yx. If P has a last term z, 
the relation agreeing with PQ'X as far as a and covering a and 
all subsequent Q places with z will be prec {P'^yx, and therefore 
the lower limit of X with respect to P^. 

Thus, while agreeing with the proposition if P and Q are well- 
ordered series and P has a last term, P^ is Dedekindian, and ex- 
tending it to the proposition if P and Q are well-orde7-ed series, 
P'^ is Dedekindian tuhen and only when P has a last term, we dis- 
agree with the conclusion that if P and Q are well-ordered series, 
P'^ with the addition of a term at the end is Dedekindian even if P 
has no last term. Instead we would substitute the propositions 
when P and Q are luell-ordered series, and Nr'Q ^ w, P^ with the 
addition of a term at the end is DedekiJidian whetJier or not P has 
a last term, and if Nr'Q •> o), P^ with the addition of a term at 
tJie end is Dedekindian luhen and only when P has a last term. 



234 Mr Neville, The Gauss-Bonnet Theorem 



^ 



The Gfinss-Bonnet Tlieorem for Multiply -Connected Jler/ions of 
a Surface. By Eric H. Neville, M.A., Trinity College. 

[Received 1 Dec. 1918: read 8 Feb. 1919.] 

Among the most delightful passages of differential geometry is 
the use of Green's theorem to prove the relation discovered by 
Bonnet between the integral curvature of a bounded region on 
any bifacial surface and the integrated geodesic curvature of the 
boundary. The fundamental equation is 

,ds+l\Kd^S= I'^^ds, 

.V 'as 

where the line integrals are taken round the whole boundary and 
the surface integral over the region contained, Kg is the geodesic 
curvature of the boundary, K the Gaussian curvature of the 
surface, and f an angle to the direction of the boundary from the 
direction of one of the curves of reference. Though there is no 
allusion to curves of reference on the left of this equation, not 
only do these curves appear explicitly on the right, but the use 
of Green's theorem implies that there does exist some system of 
curvilinear coordinates valid throughout the region and upon the 
boundary, an assumption of which it is difficult to gauge the exact 
force. The primary object of this note is to express Bonnet's theorem 
in a form purely intrinsic. 

In the case of a simply-connected region not extending to 
infinity, whose boundary has continuous curvature at every point, 
the value of J(d^/ds)ds is 27r*. If the region is simply-connected 
and does not extend to infinity, but the boundary is a curvilinear 
polygon, formed of a finite number of arcs of continuous curvature, 
the sum of the external angles must be added to the integral to 
make the total of 27r ; in other words, j{d^/ds) ds is then the 
amount by which the sum of the external angles falls short of 27r. 
In the particular case of a curvilinear triangle, the amount by 
which the sum of the three external angles fails short of 'Itt is the 
amount by which the sum of the three internal angles exceeds tt, 
and is called the angidar excess of the triangle. The name is 
adopted to serve a wider purpose : whether a connected region of 
a surface is bounded by a single closed curve or by a number of 

* See a paper by G. N. Watson, "A Problem of Analysis Situs"', Froc. Loud. 
Math. Soc, ser. 2, vol. 15, p. 227 (1916). 



for Multiply-Connected Regions of a Surface 235 

curves, the amount by which the sum of all the external angles 
of the boundary falls short of 27r is called the angular excess of the 
boundary. 

Whatever the number of curves forming the boundary of a 
region, the addition to the boundary of a simple cut, joining a 
point of the boundary either to a point of the cut or to a point of 
the boundary and described once in each direction, increases the 
sum of the external angles by 27r. If the cut divides the region 
into two parts, the angular excess of each part is the amount by 
which the sum of the external angles of that part tails short of 27r, 
and therefore the sum of the two angular excesses is the amount 
by which the sum of the external angles of the composite boundary 
falls short of 47r; this, being as we have just seen the amount by 
which the sum of the external angles of the original boundary falls 
short of 27r, is the angular excess of the original boundary. If on 
the other hand the cut leaves the region undivided, there is an 
actual decrease of 27r in the excess. It follows that if by a 
succession of n simple cuts the region is divided into m distinct 
parts, the sum of the angular excesses of the boundaries of the 
parts is less than the angular excess of the original boundary by 
2 (?i — in + 1) IT. Suppose now that each of these parts is simpl}^- 
connected and that there are no singular points of the surface ni 
the original region or upon its boundary. Then since Bonnet's 
theorem in its simplest form is applicable to each of the parts, 
addition of the sum of the integral curvatures of the parts to the 
sum of the integral geodesic curvatures of the boundaries of these 
parts gives the sum of the angular excesses of the individual 
boundaries. But the sum of the integral curvatures of the parts 
is the integral curvature of the original region, and the sum of the 
integral geodesic curvatures of the boundaries of the parts is the 
integral geodesic curvature of the original boundary, since an arc 
described once in each direction adds nothing to JKgds. Hence 
the sum of the integral geodesic curvature of the original boundary 
and the integral curvature of the bounded region is less than the 
angular excess of the original boundary by 2 {n — ni + 1) tt. This 
result affords a proof that if only the dissection has reached a stage 
at which every part is simply-connected, the difference n — m is 
independent alike of the form of the cuts and of their number. 
Since a simply-connected region is divided by one cut into two 
pieces, the integer used to measure connectivity is not n — m but 
n — in -h 2, and Bonnet's theorem in its most general form asserts 
that 

If a bounded bifacial region of any surface has finite con- 
nectivity k and neither extends to infinity nor includes tvithin it or 
upon its boundary any singularities of the surface, the sum of the 
integral geodesic curvature of the boundary and the integral curva- 



236 Mr Neville, The Gauss-Bonnet Theorem 



I 



ture of the region bounded is less than tlie angular excess of the 
boundary by 2(^' — 1) ir. 

In other words, the sum of the two integrals and the external 
angles of the boundary is 2 (2 — k) -k. 

Gauss' famous theorem on the integral curvature of a geodesic 
triangle, which may be regarded either as the simplest case or as 
the ultimate basis of Bonnet's theorem, is in no less need of modi- 
fication if the region contemplated is multiply-connected. 

If a geodesic triangle on any surface has internal angles A, B,C 
and connectivity k, and if the surface is regidar throughout the 
triangle and on its perimeter, the integral curvature of the triangle 
is A + B-\-G-{'2.k-l)7r. 

The application to the whole of a surface which, like a sphere 
and an anchor- ring, does not extend to infinity, but has no 
boundary, is interesting. A simple closed curve can always be 
drawn to divide such a surface into two distinct parts, and since 
its direction as the boundary of one part is opposite to its direction 
as the boundary of the other part, the sum of the external angles of 
the two boundaries is zero, and so also is the sum of their integral 
geodesic curvatures. It follows from Bonnet's theorem that, if 
there are no singular points on the surface and the connectivities 
of the two parts are i, j, the integral curvature of the complete 
surface is 2 (4 — i —j) it. Hence i +j is constant ; in order that a 
surface which, like a sphere, is cut by any simple closed curve into 
two simply-connected parts may be described as of unit con- 
nectivity, the connectivity is measured by the integer i+j—1, 
and 

If the connectivity of a bifacial surface which has no boundary 
and no singular points and does not extend to infinity is k, the 
integral curvature of the surface is 2 (3 — k) ir. 

A striking deduction made by Darboux from Bonnet's theorem 
may be mentioned here. If on a complete surface there is any 
family of curves such that the surface can be divided into a finite 
number of parts throughout each of which this family provides 
one set of curves of reference, the angle ^ of our first paragraph 
can be measured from the curve belonging to this family, and 
J{d^/ds)ds taken once in each direction over every part of an 
imposed boundary is necessarily zero. Hence 

For there to exist on an unbounded bifacial surface, which does 
not extend to infinity and is everywhere regidar, afainily of curves 
which covers the surface and is wholly withoid singularities, the 
surface must have integral curvature zero and must therefore be 
triply-connected. 

In conclusion the subject may be presented in another form. 
Let the angular excess of the boundary of a region of connectivity 
k reduced by 2(A;- l)7r be called the effective angular excess. If 



for Multiply-Connected Regions of a Surface ^ 237 

a simple cut which is added to the boundary does nut divide; the 
region, the anguh^r excess is reduced by 27r, and, since the con- 
nectivity is reduced by unity, the etfective angular excess is 
unaltered. If, on the other hand, the cut divides the region into 
parts of connectivities i, j, not only is the sum of the actual angular 
excesses of the boundaries of the parts the actual angular excess 
of the original boundary, but, since /; is i +j — 1, the sum of i — 1 
and J — 1 is ^' — 1 : the effective angular excess of the boundary of 
the whole is the sum of the effective angular excesses of the 
boundaries of the parts. Effective angular excess is therefore 
additive in precisely the same way as the surface integral of a 
single-valued function. If then Bonnet's theorem for a simply- 
connected region is expressed in the form that the sum of the 
integral curvature and the integral geodesic curvature is the 
effective angular excess, the restriction on the connectivity is seen 
at once to be superfluous. But to take this course implies a 
previous acquaintance with the theory of connectivity, whereas it 
is arguable that if Bonnet's theorem is used to establish the theory 
of connectivity the extent to which there is an appeal to intuition 
is materially reduced. 



238 Mr 8hah and Mr Wilson, On an empirical formula, connected 

On an empirical formula connected with GoldhacJis Theorem. 
By N. M. Shah, Trinity College, and B. M. Wilson, Trinity Col- 
lege. (Communicated by Mr G. H. Hardy.) 

[Received 20 January 1919 : read 3 February 1919.] 

§ 1. The following calculations originated in a request recently 
made to us by Messrs G. H. Hardy and J. E. Littlewood, that we 
should check a suggested asymptotic formula for the number of 
ways V (n) of expressing a given even number n as the sum of two 
primes. The formula in question is 

vin)^\{n) = 2Aj^^/^l^^^ (1), 

^ ^ ^ ^ (log ny p-2q — 2 ^ ^ 

where ??, = 2'^p«^^ ... (a^l) 

and A denotes the constant 

CO J- I 

p assuming, in this product, the odd prime values 3, 5. 7, 11, 13, .... 
The formula (1) was deduced from another conjectured asymp- 
totic formula, namely 

X A{m)A(m')^2An^^^ (2), 

where A (m) is the arithmetical function equal to log p when m is 
a prime p, or a power oi p, and to zero otherwise, and the summation 
on the left is extended to all pairs of positive integers m, m' such 
that 

m + m = n. 

Formula (1) arises from (2) by replacing in the latter A{m) and 
A {m) each by log n. It is natural, however, to expect a more 
accurate result if we replace A (m) and A {ni) not by log n but by 
\og^n, or, better still, if we replace the left-hand member of (2) by 

— ^ log X log {n — x)dx (3). 

1^ . 

The exact value of the expression (3) is found to be 

V (n) {(log ny - 2 log n + 2- i-tt^} (4). 

The various formulae thus obtained from (2) are, of course, all 
asymptotically equivalent ; but the modified formulae are likely to 
give more accurate results than (1) for comparatively small values 
of n. We used the formula 

v(n)^p(n)=2A-. -^. ^^^~i (5), 

^ ^ • ^ ^ (log ny-2\ogn p — 2 q-2 ^ ^ 

obtained by ignoring the constant 2 — ^tt'^ in (4). 



luith Goldhach's Theorem 239 

§ 2. For the numerical data used we are indebted to two 
different sources. The most complete numerical results are con- 
tained in the tables compiled and published* by R. Haussner, which 
give the values of v(n) for all values of n not exceeding 5000. 
Tables extending up to 1000 and 2000 had been calculated earlier 
by G. Cantor and V. Aubry. Further data, less systematic, indeed, 
than those of Haussner, but extending to considerably larger values 
of n, were given by L. Ripertf in a number of short papers in 
V Intermediaire des mathe'inaticiens. 

The values given for v()i) in the accompanying table differ, in 
several respects, from those given by Haussner or Ripert. In the 
first place, 7n + m and m + m are here counted as different decom- 
positions, whereas the above two writers regard them as identical ; 
secondly we do not (as do Haussner and Ripert) regard 1 as a 
prime ; and thirdly we increase the values of v (n) obtained from 
their tables by addition of the number of ways in which n may be 
expressed as the sum of two powers of primes, i.e. the number of 
ways in which 

n=p'^ + (f', 
where j) and q are primes, and either a or b is greater than unity. 
The last two modifications make, of course, no difference to the 
asymptotic formula, but it seems natural to make them when the 
genesis of the formula (1) or (5) is considered. 

As regards the choice and arrangement of the numbers n in the 
table, the smaller numbers — i.e. the numbers not exceeding 5000 
— are intended to be " typical " ; that is, they are specially selected 
numbers, taken in groups so as best to test or illustrate the accuracy 
of formula (1). Thus, for example, if the formula in question is true, 
a multiple of 6 may be expected, in general, to allow of an unusually 
large number of decompositions :|:. On the other hand a power of 2 
may be expected to allow of an unusually small number. The 
numbers below 5000 have therefore been selected in groups of four 
or five, all the numbers of each group being as nearly equal as 
possible ; and each group of numbers contains, in general, one 
highly composite number (i.e. 2.3.5.7,11....), one power of 2, 
and one number which is the product of 2 and a prime. 

For values of n exceeding 5000, such choice of" typical " numbers 
was, unfortunately, impossible without a large amount of fresh 
calculation. Ripert, indeed, selected his numbers according to a 
system, and they, too, occur, in general, in gToujJs of approximately 
equal magnitude; but he selected them with different objects, so 
that his numbers are, from our point of view, neither " typical " nor 
arbitrary. 

* Nova Acta tier Akad. der Natur/orscher (Halle), vol. 72 (1897), pp. 5-214. 
t See, for example, vol. 10 (1903), pp. 76-77, 16(3-167. 

X It was first pointed out by Cantor, on the evidence of his numerical results 
previously mentioned, that this is actually so. 

VOL. XIX. PART V. 17 



240 Mr Shah and Mr Wilson, On an empirical formula connected 

The accompanying table gives the number of decompositions — 
actual and theoretical — for thirty-five numbers ; the value found 
for the constant A was 0"66016. In the second column the first 
number is the number of decompositions, using prime numbers only, 
and the second the number of decompositions involving powers of 
primes higher than the first. 



§ 3. Table of decompositions. 




n 


V («) 


P («) 


v{n):p{n) 


30 = 2.3.5 

32 = 2" 
34 = 2.17 
36 = 22.32 


6+ 4= 10 

4+ 7= 11 
7+ 6= 13 
8+ 8= 16 


22 
8 
9 

17 


•45... 
1-38... 
1-44... 

•94 


210 = 2.3.5.7 
214 = 2.107 
216 = 23.33 
256 = 28 


42+ 0= 42 
17+ 0= 17 
28+ 0= 28 
16+ 3= 19 


49 
16 
32 
17 


•85 
1-07 

•88 
1-10 


2,048 = 211 
2,250 = 2.32.53 
2,304 = 28.32 
2,306 = 2.1153 
2,310 = 2.3.5.7.11 


50 + 17= 67 
174 + 26= 200 
134+ 8= 142 

67 + 20= 87 
228 + 16= 244 


63 
179 
136 

69 

244 


1-06 
1^11 
1^04 
r26 
1-00 


3,888 = 2*. 35 
3,898 = 2.1949 
3,990 = 2.3.5.7.19 
4,096 = 212 


186 + 24= 210 

99+ 6= 105 

328 + 20= 348 

104+ 5= 109 


197 

99 

342 

102 


1-06 

ro6 

1^02 
1-06 


4,996 = 22.1249 
4,998 = 2.3.72.17 
5,000 = 23.5* 


124 + 16= 140 
288 + 20= 308 
150 + 26= 176 


119 
305 
157 


1-18 

roi 

1-12 


8,190 = 2.32.5.7.13 

8,192 = 213 

8,194 = 2.17.241 


578 + 26= 604 
150 + 32= 182 
192 + 10= 202 


597 
171 
219 


1^01 

roe 

•92 


10,008 = 23.32.139 
10,010=2.5.7.11.13 
10,014 = 2.3.1669 


388 + 30= 418 
384 + 36= 420 
408+ 8= 416 


396 

384 
396 


1^06 
1^09 
1^05 


30,030 = 2.3.6.7.11.13 
36,960 = 25.3.5.7.11 
39,270 = 2.3.5.7.11.17 

41,580=22.33.5.7.11 


1,800 + 54 = 1,854 
1,956 + 38 = 1,994 
2,152 + 36 = 2,188 

2,140 + 44 = 2,184 


1,795 
1,937 

2,213 
2,125 


1-03 

1^03 

•99 

1-03 


50,026 = 2.25013 
50,144 = 25.1567 


702+ 8= 710 
674 + 32= 706 


692 
694 


1^03 

ro2 


170,166 = 2.3.79.359 
170,170 = 2.5.7.11.13.17 
170,172 = 22.32.29.163 


3,734 + 46 = 3,780 
3,784+ 8 = 3,792 
3,732 + 48 = 3,780 


3,762 

3,841 
3,866 


1^00 
•99 

•98 



ivith Goldbach's Theorem 241 

§4. Goldbach asserted that every even number is the sum of 
two pnmes, and this unproved proposition is usually called 'Gold- 
bach s Theorem'. It is evident that the truth of Hardy and 
Littlewood's formula would imply that of Goldbach's theorem, at 
any rate for all numbers from a certain point onwards. 

Previous writers, from Cantor onwards, had noted that the 
HTegularity m the variation of j/(«) depends on the structure of n 
as a product of pi-imes. In a short abstract in the Proceedings of 
the London Mathematical Society, Sylvester* suggested the formula 

, . 2n p-2 
'^'^-logn^p^ (6), 

where, in the product on the right p assumes all prime values from 
3 to Vw, except those which are factors of n. Sylvester gives but 
little indication as to how he arrived at the formula and indeed 
there is much m his paper which is not very clear. It is at once 
obvious that if 71, n' are two large, but approximately equal even 
numbers, the values furnished for the ratio v{n):v {n') by formulae 
(1) and (6) will be the same. For if 

n =2"-p'^ q^ ... 

and n = 2'^ p'^' q'^ 

both formulae will give, as an approximate expression for this ratio 
the quotient ' 

p-2 q-2'" I p -2 q -2"" 

The actual values of v {n) would however be different. For from 
formula (6) we should deduce 

\ognp-2q-2-j,'^lji-l- 

Now n ^-2= n Pil:z^ 






'A U (1-- 



p<\'n \ PJ 

where A is the same constant as in formula (1). Also it is known f 



that 

IN 2e-y 



n 1- 

P<sJn\ p) log/i 

^■,1 r ^1°"- f °"^«" ^^««''- ^oc-. vol. 4 (1871), pp. 4-6 (il/atft. Papers, vol. 2, pp 709- 
711). See also Math. Paj^ers, vol. 4, pp. 734-737. > I'r « c; 

t Landau, Handbuch der Lehre vuii der Verteilung der Primzahlen, p. 140. 

17—2 



242 Mr Shah and Mr Wilson, On an empirical formula connected 



so that (6) is equivalent to 

^ ^ (logw)^ j[9 — 2 g — 2 

Hence the asymptotic values furnished for v{n) by (6) and by (1) 
are in the ratio 2e~'>' : 1, i.e. in the ratio 1123 : 1. 

A quite different formida was suggested by Stackel*, viz. 

(log?i)2 0(n) 
where ^ (n) denotes, as usual, the number of numbers less than n 
and prime to n. This is equivalent to 



V (n) ' 



P 



.(9). 



(log ny p — lq — 1 

Since p/(p—l) is nearer to unity than (p — l)/(p — 2), the 
oscillations of v (n) would, if Stackel's formula were correct, be 
decidedly less pronounced than they would be if (1) were correct. 
As between the two formulae, the numerical evidence seems to be 
decisive. Thus the ratio z^(8190) : z/(8192) is 3-32, whereas ac- 
cording to (1) it should be 3"48, and according to Stackel's formula 
it should be 2*37. Stackel's i-esult is obtained by considerations of 
probability which ignore entirely the irregularity of the distribution 
of the primes in a given interval ii^N, and it is not surprising, 
therefore, that it should be seriously in error. 

On the other hand it should be observed that Sylvester's for- 
mula (7) gives, within the range of the table on p. 240, very good 
results, not much worse than those given by (5), and decidedly 
better than those given by (1). This is shown by the table which 
follows, in which decompositions into powers of primes higher than 
the first are neglected. 



n 


Formula (7) 
v{n):2e-y\{n) 


Formula (1) 
v{n) -.X {n) 


2,048 = 2" 
2,250 = 2.32.53 
2,304=28.32 
2,306 = 2.1153 
2,310 = 2.3.5. 7. 11 


•95 
1-17 
1-18 
1-17 
1-12 


1-06 
1-31 
1-33 
1-31 
1-26 


10,008 = 23.32.139 
10,010 = 2.5.7.11.13 
10,014 = 2.3.1669 


1-11 
1-12 
1-17 


1-25 
1-27 
1-32 


170,166=2.3.79.359 
170,170 = 2.5.7.11.13.17 
170,172 = 22.32.29.163 


1-06 
1-05 
1-04 


1-19 
1-18 
1-16 



Gottinger Nachrichten (1896), pp. 292-299. 



with Goldbach's Theorem 243 

§ 5. It has been shown by Landau* that 

Sv(/0~„-7r— ^ (10); 

1 ^ ^ 2(logn)2 ^ ^' 

and that Stackel's formula (8) is inconsistent with (10), and ac- 
cordingly incorrect. 

The same test can be applied to the formula (1) and Sylvester's 
formula (7). In fact Messrs Hardy and Little wood have shown f 
that (10) is a consequence of (1) : from which it follows, of course, 
that the asymptotic formula of the type of (10), furnished by 
Sylvester's formula, would be in error to the extent of a factor 
2e~'>'= ri23 ; that Sylvester's formula is therefore also incorrect ; 
and that if any formula of this type is correct, it must be (1). 

It may seem at first surprising that, in these circumstances, 

Sylvester's formula should give, for fairly large values of n, results 

actually better (as is shown by the results in the table on p. 242) 

than those given by (1). The explanation is to be found in the 

nature of the error term in (1). The modified formula (5), which 

we have already shown to be likely to give better results than (1), 

for moderately large values of n, differs from (1) by a factor of the 

type 9 

1+r^ +.... 
logn 

This factor does not affect the asymptotic value of v (n), but it 
makes a great deal of difference within the limits throughout 
which verification is possible : thus when n= 170,170 it is equal to 
1"166. When n= 10^", it is equal to 1"087, and its difference from 
unity is negligible only when n is quite outside the range of 
computation. It is only such values of n that would reveal the 
superiority of the unmodified formula (1) over Sylvester's formula. 

1 6. Shortly after the writing of the preceding sections had been 
completed, Mr Hardy informed us of the existence of a third pro- 
posed asymptotic formula for v (n), given more recently by V. 
Brun;|:. The formula to which Brun's argument leads is 

v{n)^2Bn^'~l^^ (11), 

p—z q—2 

where 5= { 1 - " ] ( 1 - " ) ( 1 -^ ) ... ( 1 -r 



=(-!) 


('-?) 


(^ 


2' 

7, 


h<\'u f 

= n (1 


-I)- 






7i = 3 V 







" GiHtinger Nachrichten (1900), pp. 177-186. 
t See their note which follows this paper. 

:!: Archiv for Mathematik (Christiauia), vol. 34, 1917, no. 8. See also § 4 of 
Hardy and Littlewood's note. 



244 Mr Shah and il/?- Wilson, On GoldbacJiS Theorem 

By an argument similar to that used in §4, in the reduction of 
Sylvester's formula, it may be shown that this is equivalent to the 
formula 

i.(n)~8^e-^y^ ,,^^...=4e-=yX(70 (12). 
(log nf p-2 q- 2 

Thus this asymptotic value for v (n), and the Hardy-LittleAvood 
value, are in the ratio 4e~-T : 1 = 1'263... : 1. Sylvester's is their 
geometric mean. 

The formulae (11) and (12) would furnish a quite close ap- 
proximation for V (n) for those values of oi on which it could be, in 
practice, tested. Thus, for n = 170,170, we find that 

v{n)/4^e-'yX{n) = -9S.... 

But the ultimate incorrectness of the formula may be proved in 
the same way as that of Sylvester's formula, namely by use of 
Landau's asymptotic formula (10). 

Brun knew of the memoirs of Stackel and Landau, but appears 
to have been unacquainted with Sylvester's work. 



Mr Hardy and Mr Littlewood, Note on Messrs Shah, etc. 245 



Note on Messrs Shah and Wilson's paper entitled: 'On an 
empirical formula connected with Ooldbach's Theorem '. By G. H. 
Hardy, M.A., Trinity College, and J. E. Littlewood, M.A., 
Trinity College. 

[Received 22 January 1919: read 3 February 1919.] 

1. The formulae discussed by Messrs Shah and Wilson were 
obtained in the course of a series of researches which have occupied 
us at various times during the last two years. A full account of 
our method will appear in due course elsewhere*: but it seems 
worth while to give here some indication of the genesis of these 
particular formulae, and others of the same character. We have 
added a few words about various questions which are suggested by 
Shah and Wilson's discussion. 

The genesis of the formulae. 

2. Let 

f{x) = %A (n) X'' = SA (n) e-"y = F{ij) 

and /, (x) = F, {y) = Ix^ (n) A (n) e"".", 

where A (n) is equal to logp when n is a prime p, or a power of ^j, 
and to zero otherwise, and x< (^0 i^ one of Dirichlet's ' characters to 
modulus (/'+. Also let 

X = xe'^'^''?, 

where p is positive, less than q, and prime to q ; and suppose that 
X tends to unity by positive values. 
It is known that 

n 

^X'^{v)A{v) = o{n\ 
1 

unless Xk is the ' principal ' character Xi> ^^ which case 

n n 

1 1 

It follows that 

(2-1) A(K)'^-- ^ 



l-x 

and 

(2-2) f^^''^ = ^{l^ ^'^^l^- 

* An outline of one of its most important applications is contained in a paper 
entitled ' A new solution of Waring's Problem ', which will be ijublished shortly in 
the Quarterly Journal of Mathematics. 

t See Landau, Haudbuch, pp. 391 et seq. 



■I 



246 Mr Hardy and Mr Littleivood, Note on 

Now 
(2-3) /(^) = 2A(?2)x"e2n/.'^^'9= S e=-^^^'^'"? 2 A(n)x>\ 

i = l n=,i 

If J is prime to q, we have* 
(2-4) S A (n) X- = -^ 'S X. ( j) /. (X), 

where %« is the character conjugate to %«, and ^(5') is the number 
of numbers less than and prime to q. It follows from (2'1) and 
(2-2) that 

(2-5) S A (n) x» ^ 144 r^- = X?-M ^ • 

n=j </)(g) 1-X <l){q)l-K 

If on the other hand j is not prime to q, the formula (2'4) is 
untrue, as its right-hand side is zero. But in this case A (??) = 
unless n is a power of q, so that 

(2-6) S Ain)K^^=^o(~). 

From (2-3), (2-5), and (2-6) it follows that 

(2-7) •^^^•)~l~x' 

where 

^ ^ ' <P{q)7 ^{q)7' ' 

the summation extending over all values of j less than and prime 
to q. The sum which appears in (2*71) has been evaluated by 
Jensen and Ramanujanf, and its value is /Li{q), the well-known 
arithmetical function of q which is equal to zero unless 5 is a product 
JJ1P2 ■•• Pp of different primes, and then equal to (— 1)p. Thus 



(^•«> f^^>-mTh.t- 



3. The sum 
(3-1) co{n)= t A(m)A(??0, 

■m + m'=n 
* Landau, I.e., p. 421. 

m 

t J.L.W.V. Jensen, 'EtnytUdtryk for den talteoretiske Funktion 2M(n) = M{my, 

1 
Saertryk af Beretning om den 3 Skandinaviske Matematiker-Kongres, Kristiania, 
1915 ; S. Eamanu jan, ' On certain trigonometrical sums and their applications in the 
theory of numbers', Trans. Camb. Phil. Soc, vol. 22, 1918, pp. 259-276. 

J If ^ (q) is zero, this formula is to be interpreted as meaning 



/w=o(r^) 



Messrs Shah and Wilsons paper 247 

which appears on the left-hand side of Shah and Wilson's equation 
(2), is the coefficient of a;'* in the expansion of [/(«))". And 

when a; ^ e2i?7ri7(? along a radius vector. Our general method ac- 
cordingly suggests to us to take 

n(n) = nt\^i(i^'e-'-^^P-il<i, 

where the summation extends over 5'= 1, 2, 3, ... and all values 
of jt) less than and prime to q, as an approximation to co (n). Using 
Ramanujan's notation, this sum may be written 

(3-2) n{n)=nl\^^l^%,(n). 

The series (3'2) can be summed in finite terms. We have 

(3'3) c,(7i) = SS/.(| 

the summation extending over all common divisors S of q and n*; 
and it is easily verified, either by means of this formula or by means 
of the definition of Cq{n) as a trigonometrical sum, that 

Cqq'{n) = Cq{n)Cg'(n) 

whenever q and q' are prime to one another. We may therefore 
write 

n(n)=n:ZAq = nUx-r!r, 
where the product extends over all primes -or, and 

since Aq contains the factor /j,(q) and A^^^., A ^-3, ... are accordingly 
zero. 

If n is not divisible by zj, we have c^ (n) = /u, (ot) = — 1 and 

A =_^ 1 = L__. 

while if n is divisible by zr we have 

Cw('0 = At(t3-) + t3-/A(l)= OT - 1, 

. _ 1 

CT — 1 

Hence 

* Ramanujan, I.e., p. 260. 



248 Mr Hardy and Mr Littlewood, Note on 

where 11' applies to primes which divide n and II" to primes 
which do not. 

It is evident that O (n) is zero if n is odd. On the other hand, 
if n is even, we have 



"W = 2»n|i-(-'-3y,}n 



-^)/|-.^-:. 



= 2^nn-^^ln^P-^ 



where tn- now runs through all odd primes and p through odd 
prime divisors of n. 

The formula w (n) ^- il (n) 

is formula (2) of Shah and Wilson's paper*. 

The incorrectness of Sylvester's formula. 

4. It is easy to prove that if any form tda of the type 

(4-1) fw (7^) ~ CO (n) 

be true, then G must be unity. In other words, our formula is the 
only formula of this type which can possibly be correct. This 
may be shown as follows. 
Let 

(4-2) f(^) = t'l^l 

where n runs through all even values; and let s — 1 = ^. The series 
is absolutely convergent if s > 2, ^ > 1. Replacing 12 (n) by its 
expression in terms of the prime divisors of n, and splitting up 
f{s) into factors in the ordinary manner, we obtain 

say, where A is the same constant as in Shah and Wilson's paper, 
and OT runs through all odd primes. 
Let 

+ (*) = n (i + r^--.) = n [^}^_) = (1 - 2-0 f (0. 

and suppose that ^^1. Then 



x(0.nifi+^-4,*"4Vfi+i-^)} 



'y\r{t) [\ OT-2 1- 



•37 



V(^-2)| ^'1(37-1)^-11 A' 
When fi(n) = 0, the formula is to be interpreted as meaning w(H) = o(/t). 



Messrs Shah and Wilsons paper 249 

and so 

(4-3) f{s) ~ 2^^ (0 -2(1- 2-0 ?(0 - ^1 = ,--2 • 

This is a consequence of our hypothesis : the corresponding 
consequence of the hj^pothesis (4"1) would be 

(4-31) /(*^>~^- 

On the other hand, it is easy to prove* that 
(4-4) &)(l) + ft)(2)+ ... +&)(7i)~i«'; 

and from this to deduce that 



<^(.) = 2 



on 



(n) 1 



w* s — 2 

when s—>2. This equation is inconsistent with (4"1) and (4'31), 
unless (7 = 1. 

It follows that Sylvester's suggested formula is definitely 
erroneous. 

It is more difficult to make a definite statement about the 
formula given by Brun. The formula to which his argument 
naturally leads is Shah and Wilson's formula (12); and this 
formula, like Sylvester's, is erroneous. But in fact Brun never 
enunciates this formula explicitly. What he does is rather to 
advance reasons for supposing that some formula of the type (4"1) 
is true, and to determine G on the ground of empirical evidence^. 
The result to which be is led is equivalent to that obtained by 
taking C= 1-5985/1-3203 = 1-2107 %. The reason for so substantial 
a discrepancy is in effect that explained in the last section of 
Shah and Wilson's paper. 

Further results. 

5. The method of § 2 leads to a whole series of results con- 
cerning the number of decompositions of n into 3, 4, or any number 
of primes. The results suggested by it are as follows. Suppose 

* Since SA (71) .r™-:; 

as a;-*-l, we have 2w(H)a;"= (SA («) a;"}-~ ,- - -', 

and the desired result follows from Theorem 8 of a paper published by us in 1912 
(' Tauberian theorems concerning power series and Dirichiet's series whose coefficients 
are positive', Proc. London Math. Soc, ser. 2, vol. 13, pp. 174-192). This, though 
the shortest, is by no means the simplest proof. 

The formula (4-4) is substantially equivalent to Landau's formula (10) in Shah 
and Wilson's paper. 

t Evidence connected not with Goldbach's theorem itself but with a closely 
related problem concerning pairs of primes differing by 2. See g 7. 

1 1-5985 is Brun's constant, while 1-3203 is 2A. 



250 Mr Hardy and Mr Littlewood, Note on 

that Vr (n) is the number of expressions of n as the sum of r primes 
Then if r is odd we have 

(5-11) v,(n) = o()i>-') 

if V is even, and 

if n is odd, p being an odd prime divisor of ??, and 

(513) B=n[i^^^^^^^. 

where tn- runs through all odd primes. On the other hand, if r is 
even, we have 

(5-21) Vr(n) = o{n''-^) 

if n is odd, and 

where 

(5-23) c=n|i- 



(^-ir 

if 71 is even. The last formula reduces to (1) of Shah and Wilson's 
paper when r = 2. 

We have not been able to find a rigorous proof, independent 
of all unproved hypotheses, of any of these formulae. But we are 
able to connect them in a most interesting manner with the famous 
' Riemann hypothesis ' concerning the zeros of Riemann's function 
f (5). The Riemann hypothesis may be stated as follows : ^(s) has 
no zeros whose real part is greater than ^. If this be so, it follows 
easily that all the zeros of ^(s), other than the trivial zeros s = — 2, 
s = — 4, ..., lie on the line <t = 'R{s) = ^. It is natural to extend 
this hypothesis as follows: no one of the functions defined, luhen a- > 1, 
hy the series 

n" 

possesses zeros luhose real part is greater than ^. We may call this 
the extended Riemann hypothesis. This being so, what we can prove 
is this, that if the extended Riemann hypothesis is true, then the 
formidae (5"11) — (5'23) a?'e true for all values of r greater than 4. 
The reasons for supposing the extended hypothesis true are 
of the same nature as those for supposing the hypothesis itself 
true. It should be observed, however, that it is necessar}", before 
we generalise the hypothesis, to modify the form in which it is 
usually stated; for it is not proved (as it is for ^{s) itself) that 
L{s) can have no real zero between ^ and 1. 



1 



Messrs Shall and Wilson's paper 251 

6. A modification of our method enables us to attack a closely 
related problem, that of the existence of pairs of primes differing 
by a constant even number k. 

We have 

2 A (n) A (n + k) r^''+'' = J- f '" \f{re^^) \ ' e'^'^ cW, 

where f(x) is the same function as in § 1, and r is positive and less 
than unity. We divide the range of integration into a number of 
small arcs, correlated in an appropriate manner with a certain 
number of the points e"P''^"J, and approximate to {/(j'e'")!^ on each 
arc by means of the formula (2-8). The result thus suggested is 
that 

^A{n)A (n + k) r- ^ ^^~ U (^ £ ^) , 

where A has the same meaning as in § 2 and p is an odd prime 
divisor of k. From this it would follow that 

(6-1) S A (v) A (v + k) - 2AnU P^) ; 

and that, if A^^. (?2) is the number of prime pairs less than /?, whose 
difference is k, then 

T.r / N 2ylri „ /p — 1\ 

(6-2) ^V'«~(i^^,n(P-2). 

This formula is of exactly the same form as (1), except that p is 
now a factor of k and not of n. In particular we should have 

,_ . , 2An 
(6-3) ^^(»)~(lo-g»r 

and 

(6-4) ^^"<»>~(Ttg-LV 

We should therefore conclude that there are about two pairs of 
primes differing by 6 to every pair differing by 2. This conclusion 
is easily verified. In fact the numbers of pairs differing by 2, below 
the limits* 

100, 500, 1000, 2000, 3000, 4000, 5000, 
are 

9, 24, 35, 61, 81, 103, 125; 

while the numbers of pairs differing by 6 are 

16, 47, 73, 125, 168, 201, 241. 

* To be precise, the numbers of pairs ( j), p') such that p' =p + 2 and p' does not 
exceed the limit in question. 



252 My^ Hardy and Mr Littlewood, Note on 

The numbers of pairs differing by 4, which should be roughly the 
same as those of pairs differing by 2, are 

9, 26, 41, 63, 86, 107, 121. 



7, 



Brun, ni his note ah'eady referred to, recognises the corre- 
spondence between the problem of §§ 2—4 and that of the prime- 
pairs differing by 2, and realises the identity of the constants in- 
volved m the formulae ; but does not allude to the more o-eneral 
problem of prime-pairs differing by k. He does not determme the 
fundamental constant A, attempting only to approximate to it 
empirically by means of a count of prime-pairs differing by 2 and 
less than 100000, made by Glaisher in 1878*. The value of the 
constant thus obtained is, as was pointed out in § 4, seriously in 
error. The truth is that when we pass from (6-1), which, when 
k = 2, takes the form 

2 A{v)A{v + 2)r^^An, 

to (6-3), the formula which presents itself most naturally is not 
(6-3) but "^ 

(7-1) i\r3(n)o.2yir--^. 

J (log^)- 

This formula is of course, in the long run, equivalent to (6-3) 
But 

(log xf (log ny \ ^ log n "^ (log nf "^ " ' 7 ' 

and the second factor on the right-hand side is, for n = 100000 far 
from negligible. Thus (6-3) may be expected, for such values of 
n, to give results considerably too small. 

}^C^^ *^^® *^® ^^^^®^^ ^"^^^* ^^ integration in (7-1) to be 2 we 
find that the value of the right-hand side for n = 100000 is to' the 
nearest integer, 1249, whereas the actual value of i\^., (92) is, accord- 
ing to Glaisher, 1224^ The ratio is 1-02, and the agreement seems 
to be as good as can reasonably be expected. 

Tir "Sf ^^l^^^ation of prime-pairs has been carried further by 
Mrsfetreatteild, whose results are exhibited m the following table: 

*'ri '^^■- • ^^e number of pairs below 100000 is 1225 

t iiie series is naturally divergent, and must be closed, after a finite number of 
terms with an error term of lower order than the last term retained 

^ Glaisher reckons 1 as a prime and (1, 3) as a prime-pair, making 1225 in all. 



Messrs Shah and Wilsons paper 



253 



11 


N,(n) 


2 A f" ''"■ 


Ratio 1 

1 


100,000 


1224 


1249 


! 

1-020 


200,000 


2159 


2180 


1-010 


300,000 


2992 


3035 


1-014 


400,000 


3801 


3846 


1-012 


500,000 


1562 


4625 


1-014 


600,000 


5328 


5381 


1-010 



8. In a later paper* Brun gives a more general formula relating 
to prime-pairs {p, p) such that p = ap + 2. This formula also 
involves an undetermined constant k. It is worth pointing out 
that our method is equally applicable to this and to still more 
general problems. Suppose, in the first place, that v{n) is the 
number of expressions of n in the form 

n = ap + hp, 

where pi and p' are primesf. We may suppose without loss of 
generality that a and h have no common factor. 

The results suggested by our method are as follows. If n has 
any factor in common with a and h, then 



"<">='' {(log. o-^}' 



and this is true even when n is prime to both a and h, unless one 
of n, a, b is even|. But if n, a and b are coprime, and one of them 
even, then 



2J. 



n 



P-i 



ab (log iif \p— 2 

where A is the constant of § 2, and the product is now extended 
over all odd primes which divide n or a or b. 



* ' Sur les nombres premiers de la forme ap + h\ Archiv for Mathematik, vol. 
24, 1917, no. 14. 

t We might naturally include powers of primes. 

+ These results are trivial. If n and a have a common factor, it divides hp', 
and is therefore necessarily p' , which can thus assume but a finite number of values. 
If n, a, h are all odd, either ^^ ox p' must necessarily be 2. 



254 Mr Hardy and Mr Littleiuood, Note etc. 

Similarly, suppose N(n) to be the number of pairs of solutions 
of the equation 

a})' — hp = h 

such that p' < n. It is supposed that a and h have no common 
factor. Then 

N(n) = o 



l(logw)- 

unless k is prime to both a and b, and one of the three is even. 
If these conditions are satisfied 

where p is now an odd prime factor of k, a, or h. 



Mr Harrison, The distribution of Electric Force, etc. 255 



The distribution of Electric Force bettueen two Electrodes, one of 
which is covered with Radioactive Matter. By W. J. Harrison, M.A., 
Fellow of Clare College. 

[Read 17 February 1919.] 

It has been shown by Rutherford* that it is probable that the 
ionisation due to an a particle per unit length of its path is in- 
versely proportional to its velocity, provided the velocity exceeds 
a certain minimum necessary to effect ionisation. It follows that 
the ionisation per unit time is constant at all points of the path. 

Suppose radioactive matter distributed uniformly over the sur- 
face of a large plane electrode assumed to be infinite in order to obtain 
simplicity in calculation. Consider the a particles projected from a 
point P of the electrode. These particles are projected equally in all 
directions, hence the rate of ionisation per unit volume at a point 
Q will be proportional to l/PQ^ provided PQ< R, where R is the 
range of the particles. The total rate of ionisation at a point Q 
distance x (j: < R) from the electrode will be proportional to 

■\/Ji---'e"27rrdr 



x- + r^' 

where r is the distance of a point P on the electrode from the foot 
of the perpendicular from Q. Now 

•sjitr—x^ 27' dr 



Jo 



x^ + r'' 

= log 



log (x- + r^) 
R 



sJlP-X' 





Hence rate of ionisation 

1 ^ 

^ = ^0 log -; . 



X 



The equations determining the distribution of electric force are 
given by Thomson, Conduction of Electricity through Gases, 1906, 
chap. III. The notation of this book is adopted as being sufficiently 
well known. The differential equation for the electric force X is of 
the form 

d'X^^ a (dX-'\' b , R „ 

_— =0, x> R. 

dx~ 

* Radioactive Substances and their Radiations, 1913, p. 158. 
VOL. XIX. PART v. 18 



256 Mr Harrison, The distribution of Electric Force hetiueen ttuo 

The numerical solution may be obtained for any particular 
values of the constants a, b, c, q^, R by approximate methods. In 
the absence of any definite experimental results with which to 
compare the calculations, the labour involved in integration is not 
worth undertaking. 

The case, however, of the saturation current is the most impor- 
tant, and the integration is simple. It is assumed that recombi- 
nation of ions does not take place in this case, and therefore the 
equations reduce to 

= 0, x>R. 

Write Si7eq,{^ + y\=K. 

Then, for x< R, 



for X > R, 



9 log 7. -\-ix- + Bx + G 



(vide Rutherford, Radioactive Substances, etc., p. 67), A, B, G are 
constants of integration. 
Now the conditions are 

(1) at ^ = 0, ni = 0, if ^ = be the positive plate, 

(2) at a; = J?, Wg = 0, 

(3) at a; = i^, n^ is continuous, 

(4) Sit x = R, X \B continuous. 

{vide Conduction of Electricity through Gases, chap. iii.). 

dX^ _ _ Sttj 
' ' dx kz ' 

(2) and (3) lead to the same condition, which is the same as 
(1), if 

i = eRqQ. 

Now since there is no recombination 



. [^ R 

1=1 eqolog - dx = eRqo. 
Jo ^ 



Electrodes, one of which is covered luith Radioactive Matter 257 



Hence conditions (1), (2), (3) are identical and determine B. 
Condition (4) supplies a relation between G and A, 

A^K{G-IR^). 
Hence 



X' = K 



^x~ log h f 



h 



Rx + BR' 



X' = K 



CO R. 



X ft-l ~f~ rCo 

0<x<R, where BR- = C, 

The constant B can be determined when the potential differ- 
ence between the electrodes is given*. 

The general character of these results can be shown by numerical 
calculation for the cases k, = L, l-25k,=k^, A;i = 1-25 A-., (corresponding 
to the case in which the positive ion moves more slowly, as usual, 
than the negative ion, and the radioactive matter is spread on the 
negative plate), and for distances R, 2R, SR between the electrodes, 
and for B = O'l, 0-5, I'O. In order that the current may be the satu- 
ration current it is necessary in practice that B should exceed 
a certain limit. This limit is dependent on the particular conditions 
of any given experiment. 

The distribution of the electric force X is shown on the graph 
below. The curves marked (1), (2), (3) are for the cases 
kj = 1-25 k.2, ki = ^'2, A.-2 = 1-25 k^, respectively. 

The potential difference V between the electrodes is given in 
the following table, d being the distance between the plates. 

V 
RKK^' 







ki = l-25k.2 


^1 = ^2 


k.2^1-25ki 


D=0-l 


d=R 

d=2R 

d=3R 


-343 

1-056 
2-034 


-379 
1-147 
2-193 


•412 
1-232 
2-343 


D = Ob 


d=R 

d=2R 

d=3R 


-725 
1-676 
2-841 


-743 

1-740 
2-963 


-762 
1-800 
3-078 


D = l-0 


d=R 

d=2R 

d=3R 


1-014 
2-203 
3-566 


1-027 
2-250 
3-663 


1-041 
2-298 
3-759 



These forms of X are not strictly valid in the immediate neighbourhood of the 
electrodes, as the natural agitation of the ions has been neglected in this theory 
Vide Pidduck, Treatise on Electricity, 1916, p. 505. 

18—2 



258 Mr Harrison, The distribution of Electric Force, etc. 




I '0 R Z'O R 

DISTANCE BETWEEN ELECTRODE^. 



3-0 R 



Mr Purvis, The conversion of saiv-dust into sugar 259 



The conversion of saw-dust into sugar. By J. E. PuRViS, M.A. 
[Read 17 February 1919.] 

The production of sugar from wood is well known. In the 
Classen process, saw-dust is digested in closed retorts with a weak 
solution of sulphurous acid under a pressure of between six and 
seven atmospheres. The products contain about 25 °/^ of dextrose, 
and other substances are pentose, acetic acid, furfurol and formal- 
dehyde. Cellulose material can also be converted into sugar by 
other acids. 

The following results were obtained by digesting saw-dust 
from ordinary deal with different acids of varying concentrations ; 
estimating the amount of sugar in the liquid in the usual way 
from the amount of cuprous oxide precipitated from Fehling's 
solution, and converting this oxide of copper to cupric oxide. The 
numbers were then calculated in terms of dextrose. 

(1) 25 grams of saw-dust were digested with 300 c.c. distilled 
water and 50 c.c. strong H2SO4 (1 c.c. H2S04 = 1*78 grms. H2SO4) 
for 5^ hours in a sand bath at a temperature just below the 
boiling point and the mixture was constantly stirred. This was 
then filtered ; the residue well washed and the filtrate made up to 
a litre ; 10 c.c. of the filtrate were neutralised with sodium 
carbonate and the cuprous oxide from Fehling's solution was 
precipitated, filtered, dried and ignited to cupric oxide. This gave 
0"215 grm. CuO which is equivalent to 39 °/^ of dextrose on the 
original amount of saw-dust. 

(2) 25 grams of saw-dust to which were added 500 c.c. of 
distilled water and 25 c.c. of strong H2SO4 of the same strength as 
in experiment (1) and digested for 5 hours under the same 
conditions. This gave 13 °/^ of dextrose. 

(3) 50 grams of saw-dust were digested with 500 c.c. of 
distilled water and 50 c.c. of the strong H2SO4 for 5f hours. The 
yield was 11 "5 % dextrose. 

(4) 25 grams of saw-dust were digested with 250 c.c. of tap 
water and 10 c.c. of strong H2SO4 for 2 hours. This yielded 10*5 7o 
dextrose. 

(5) 25 grams of saw-dust were digested with 720 c.c. of tap 
water and 10 c.c. strong H2SO4 for 2 hours. This produced 3*35 "/^ 
dextrose. 

(6) 50 grams of saw-dust were digested with 500 c.c. water 
and 50 c.c. N/1 HCl (= 1-825 grms. HCl) for 3 hours. This gave 
3-35 % dextrose. 



260 Mr Purvis, The conversion of saw-dust into sugar 

(7) 50 grams of saw-dust were digested with 500 c.c. water 
and 100 c.c. N/1 H2SO4 (= 2-45 grms. H.SO4) for 2 hours. This 
produced 1'82 °/^ dextrose. 

(8) 25 grams of saw-dust were digested with 700 c.c. water 
and 5 grams P0O5 for 12 hours at the temperature of the room 
(about 15° C), and then for 3 hours just below the boiling point. 
This gave 12'66 °/^ dextrose. 

The results show that the amount of sugar which can be 
obtained depends on the nature of the acid and its strength relative 
to the amount of saw-dust, and on the time of digestion. The 
greatest amount was obtained when the strongest sulphuric acid 
acted for a considerable time. In the other experiments not so 
much was obtained as by the Classen process. For the commercial 
production of sugar from such a cheap material as saw-dust the 
question to be decided would be the relative cost of the Classen 
process compared with the cost under the conditions of these 
experiments. That would include a comparison of the cost of 
the various acids and the recovery of these acids for further use. 
The conversion of sugar into alcohol and acetone presents no 
difficulty ; and it would be important to consider whether such 
useful chemical substances could not be produced from a waste 
product like saw-dust at a cheaper rate than by the present costly 
methods. 



Mr Purvis, Bracken as a source of potash 261 



Bracken as a source of potash. By J. E. Purvis, M.A. 
[Read 17 February 1919.] 

The Master of Christ's College, Cambridge, in the autumn of 
1917, had some correspondence with Mr J. A. A. Williams of 
Aberglaslyn Hall, Beddgelert, in regard to the use of bracken as a 
fertiliser. Mr Williams had burnt the bracken growing on a peaty 
soil on his estate at Beddgelert, ploughed in the ashes and obtained 
highly satisfactory crops of potatoes. It seemed to be of some 
importance to find out what amount of potash could be obtained 
from the ash; and in October 1917 a sample of bracken from the 
Botanic Gardens, Cambridge, was analysed. This grows on a poor 
sandy soil. 

It is known that bracken contains larger quantities of potash 
in the summer months than in the autumn and more complete 
investigations were deferred till the summer of 1918. Meanwhile 
in the April (1918) number of the Journal of Agriculture (vol. 25, 
no. 1, p. 1) Messrs Berry, Robinson and Russell published an 
article on " Bracken as a source of potash " which contained the 
results of the analyses of material collected from various districts 
in England, Scotland and Wales from May to October 1916, and 
from June to October 1917. The numbers show that the amount 
of potash is much higher in the summer months than in the autumn. 
For example, bracken gathered June 1st, 1917, from Harpenden 
Common, Rotharnsted, which is mainly gravel and clay, produced 
4"1 ° I ^ of potash (KoO) on the dried material and only 1'8 7o when 
gathered September 1st, 1917. The authors also considered that 
their evidence indicates a more rapid falling off of the potash from 
bracken growing on sandy and peaty soils than on heavier soils 
rich in potash : and that, therefore, its chances of success as a 
fertiliser would be greater in these heavier soils. 

In view of these results the investigations were continued with 
the bracken growing in the Botanic Gardens, Cambridge, and also 
with that on Mr Williams's Welsh estate. The following tables 
summarise the results. 

Generally, the numbers are of the same order as those obtained 
by Messrs Berry, Robinson and Russell, and confirm the opinion 
that in the summer months there is more potash than in the later 
months. Also there is a clear indication that, on an average, the 
Welsh peaty soil yields more potash than the Cambridge poor 
sandy soil. 



262 Mr Purvis, Bracken as a source of potash 

Cambridge Bracken. 



Date when sample 
was gathered 


Percentage of 
dry matter in 
fresh bracken 


Percentage of 

ash in 

dry matter 


Percentage of potash (K2O) in 


fresh bracken 


dry bracken 


16 October, 1917 


27-60 


7-51 


0-29 


0-82 


1 June, 1918 


15-34 


6-81 


0-46 


3-00 


2 July, 1918 


21-58 


5-02 


0-52 


2-45 


1 August, 1918 


30-26 


5-96 


0-50 


1-70 


31 August, 1918 


26-50 


7-86 


0-30 


1-07 


1 October, 1918 


29-06 


7-93 


0-33 


1-13 




Welsh Bracken 






3 June, 1918 


24-4 


6-55 


0-77 


3-19 


4 July, 1918 


25-8 


5-78 


0-83 


3-22 


31 July, 1918 


40-7 


3-84 


0-42 


1-45 


1 September, 1918 


30-97 


7-02 


0-53 


1-71 


3 October, 1918 


34-54 


4-82 


0-45 


1-32 



To estimate the cost of collection is difficult as the conditions 
of transit and labour are variable and estimates for one locality 
would be useless for another. It is evident, however, that bracken 
is a valuable source of potash : but its economic application as a 
fertiliser will be controlled by the requirements and conditions of 
the neighbourhood where it grows. 

I have to thank Mr Williams for supplying the Welsh bracken, 
and Mr Lynch, of the Cambridge Botanic Gardens, for samples 
from the gardens. 



Dr Shearer, The action of electrolytes on the electrical, etc. 263 



The action of electrolytes on the electrical conductivity of the 
bacterial cell and their effect on the rate of migration of these cells 
in an electric field. By C. Shearer, Sc.D., F.R.S., Clare College. 
(From the Pathological Laboratory, Cambridge.) 

{Read 17 February 1919.] 

If a thick creamy emulsion of the meningococcus or B. coli is 
made up in neutral Ringer's solution (that is, one in which the 
sodium bicarbonate is left out), and the conductivity measured by 
means of a Kohlrausch bridge and cell; it is found that its resistance 
is more than treble that of the same solution without the bacteria : 
that is the greater part of the resistance is due to the presence of 
the bacteria. 

This determination was made as follows: a 24 hour culture of 
the meningococcus or B. coli on trypagar (2-t plates) was washed 
off in a considerable quantity of Ringer's solution, centrifuged down 
and re washed several times in a similar manner to remove all traces 
of serum or any salts derived from the culture medium. The centri- 
fuged deposit was then made up to standard strength in neutral 
Ringer's solution, so that it was not too thick to be sucked up in a 
medium sized pipette and transferred to a Hamburger cell and its 
conductivity determined. It was found that the conductivity of 
such standard emulsions when measured under similar conditions 
of temperature was fairly uniform*. When sufficient care was 
taken to get the emulsions of the right thickness, resistances of 
110 ohms could be pretty constantly obtained. The same quantity 
of Ringer's solution alone had about 26"7 ohms resistance under 
the same conditions. 

If, however, in place of the Ringer's solution we make up the 
bacterial emulsions in pure sodium chloride of the same conducti- 
vity as that of the Ringer's solution, i.e. one in which the resistance 
is 26'7 ohms (which corresponds to a NaCl solution of about 0'85 °/^), 
we obtain as in the case of the emulsion in Ringer's solution an 
initial resistance of 110 ohms. Within a few minutes, however, this 
gradually drops and at the end of 30 or 40 minutes the emulsion 
now has the same conductivity as that of the bare sodium chloride 
solution without the bacteria, i.e. 26'7 ohms resistance. Thus pure 
sodium chloride of about the concentration as that present in the 
blood gradually destroys the resistance of the bacterial cell. If the 
bacteria are allowed to lie in this solution for several hours it will 
be found that at the end of this time, on subculture, they are 

••" All measurements were made at constant temperature 25° C. Resistance con- 
stant of conductivity cell = 29 8 x 10~^. 



264 Dr Shearer, The action of electrolytes on the 

dead. If they are only allowed to remain in the NaCl for a short 
time and then transferred to neutral Ringer again they immediately 
return to their normal resistance and grow freely on subculture. 

If when the resistance of the bacterial emulsion has fallen in 
NaCl solution a little CaCL is added it again regains its normal 
conductivity and is uninjured. Thus we get the usual antagonistic 
action of CaCla to NaCl. It was found that KCl, LiCl, MgCl^ 
acted like NaCl in reducing the resistance offered by the bacteria, 
while BaCls, SrClg have no action on the resistance but act like 
CaClg. Thus it is clear that in the bacteria as with so many other 
plant and animal cells the entrance of the ions of NaCl, KCl, 
LiCl, MgCla is prevented by the presence of very small quantities 
of CaCL, BaCla or SrClg. Bacterial emulsions made up in BaClo, 
SrCla and CaClg , having the same conductivity as Ringer's solution, 
showed no change in resistance on being kept in these solutions 
for some time, invariably remaining normal. 

The interest of these experiments consists in that they agree 
completely with the results obtained by Loeb, Osterhout and a 
large number of other workers on animal and plant cells. 

In Laminaria, Osterhout finds with CaCL and presumably also 
with BaCla and SrCl.. there is invariably a brief temporary rise in 
resistance when placed in these solutions of the same conductivity 
as sea-water which is followed by a gradual fall. With the bacterial 
cell no such preliminary rise can be distinguished, w^hile the fall 
due to the toxic action of the solution is much delayed and slower. 

In view of the remarkable action of tri-valent ions on artificial 
membranes as shown by the work of Perrin, Girard and Mines, and 
the action on the permeability of cell wall as shown by the work 
of Mines, Osterhout and Gray, it is of great interest to consider 
their action on the bacterial cell. 

While the tri-valent positive ion of lanthanium nitrate brings 
about a rapid rise of resistance in Laminaria according to Osterhout 
and in the Echinoderm egg according to Gray, when this salt is 
used in such dilution as not to affect the conductivity of the solu- 
tion itself, no such action can be distinguished in the case of 
bacteria by means of the Kohlrausch method. The resistance 
remains unchanged until it begins to fall on account of the in- 
creasing strength of the salt added. In the same way the positive 
tri-valent ions of CeCL, neo-ytterbium chloride and the tri-valent 
negative ions of sodium citrate appear to have no action in in- 
creasing or decreasing the resistance of the bacterial cell as deter- 
mined by the conductivity method. It should be pointed out that 
these salts can only be used in very dilute solutions. In the case 
of lanthanium nitrate this salt readily flocculates living bacteria 
Avhen used in stronger solutions than ^ y\j^ M. 

It would seem remarkable in view of the sharp action of La on 



electrical conductivity of the bacterial cell, etc. 265 

the Echinoderm egg when used in a strength of g^*^ M. that some 
similar action should not be found with bacteria, but repeated 
experiments with centrifuged solid bacterial deposits of both the 
meningococcus and B. coli using the same type of electrodes used 
by Gray for the Echinoderm egg and obtaining resistances as high 
as 150 ohms failed to show any initial rise of resistance. It was 
possible that in the case of bacteria, their enormous surface would 
render the preliminary rise of resistance so temporary that, before 
the electrodes could be placed in position and the bridge readings 
adjusted, it would be over and passed. To test this point a small 
quantity of La was added while the bridge telephone was kept to 
the ear, but in every instance no change could be detected. It 
would seem that the bacterial cell is normally in a state of 
maximum impermeability and that this can not be further increased 
by the presence of CaCL and the tri-valent salts. 

In distinction to the absence of effect of the tri-valent salts on 
bacteria as demonstrated by the conductivity method, is the marked 
action of these salts and especially lanthanium nitrate in changing 
the rate of migration of these cells in an electric field. This can 
be determined by the ultramicroscopic or still better the U tube 
method. 

If 10 c.c. of a thick growth of B. coli in spleen broth be run 
into a U tube under neutral Ringer's solution of the same conducti- 
vity as the broth, then on passing an electric current through the 
tube, the temperature being constant, an even rapid migration of 
the bacteria takes place towards the anode. 

That practically all bacteria carry a negative charge and migrate 
to the anode has been repeatedly confirmed by numerous workers, 
but what is of interest here is that this charge can be materially 
modified by various tri-valent salts, especially La. If to the 10 c.c. 
of B. coli emulsion in spleen broth run into the U tube in the 
above experiment 1 c.c. of a -^^ M. lanthanium nitrate solution 
is added, it will be found that the rate of migration of the 
bacilli under the same conditions of electric field and temperature 
is now halved. If 2 c.c. of the solution is added, little or no migra- 
tion takes place and the emulsion soon flocculates and is preci- 
pitated to the bottom of the tube. 

In terms of the Helmholtz-Lamb theory of the double electric 
layer the addition of the La has considerably altered the nature of 
the charge on the bacterial cell wall. The conductivity method 
however fails to show any change under this condition. This result 
is possibly of some interest in view of Mines' theory of the polarising 
action of certain ions on the cell membrane. It is of course possible 
that the resistances obtained in the conductivity experiments were 
too low to bring out the real changes taking place. 



266 Miss Haviland, The bionomics of Aphis 



The bionomics of Aphis grossulariae Kalt., and Aphis viburui 
Schr. By Maud D. Haviland, Bathurst Student of Newnham 
College. (Communicated by Mr H. H. Brindley.) 

[Read 17 February 1919.] 

Aphis grossulariae Kalt. is a serious pest 'of currant and goose- 
berry bushes in this country. It attacks the young shoots in May, 
and when present in numbers, it distorts them to such an extent 
that growth ceases and a dense cluster of leaves is formed, under 
which the aphides swarm. 

The bionomics of this aphis are incompletely known. It appears 
on red currants in May, and remains there until the middle or end 
of July. The sexuales have never been found. In 1912 Theobald 
(Journ. Econ. Biol., vol. Vli. p. 100) first pointed out its resemblance 
to Aphis viburni Schr., a common species, which is found on the 
guelder rose ( Viburnum opulus) in spring and summer, while the 
sexual forms have been recorded from the same plant in the autumn. 
Aphis viburni has a very characteristic appearance, owing to the 
row of lateral tubercles on the abdomen. Such tubercles are not 
very common among the Aphidinae, but they are prominent like- 
wise in Aphis grossulariae. In fact there seems to be no structural 
difference between the two species; though in spirit specimens, the 
guelder rose aphis frequently stains the alcohol dark brown, while 
the currant form has no such property. 

In May 1918, I had under observation some red and black 
currant bushes, and two guelder rose shrubs, which all grew close 
together. Early in the month all were free from aphid attack, but 
on May 31st three colonies, each consisting of a single winged 
female with a few new-born young, appeared on the guelder roses, 
and the same evening four sprigs of currant were likewise each 
infected. During the following week, numerous other winged forms 
appeared both on the guelder roses and on the currants. The 
method of attack was the same in both cases. The migrant crept 
into the axil of a leaf, and from thence her progeny gradually spread 
up the stem and along the midrib. About the same time, I found 
a Viburnum tree swarming with winged females of Aphis viburni 
in a shrubbery a hundred yards away; and as these were in- 
distinguishable from the migrants on the Viburnum and currants, 
I have little doubt that this was the source of infection. 

Assuming that A. viburni and A. grossulariae are identical, I 
began experiments to test how far the host plants were interchange- 
able. Unfortunately, owing to heavy rains, the experiments with 
the original winged migrants were all inconclusive, and during 



grossulariae Kalt., and Aphis viburni Schr. 267 

June and July I worked with alate and apterous individuals of later 
generations. The results are set out in the accompanying tables 
from which it will be seen that out of thirteen attempts to transfer 
A. viburni to Ribes rubrum, in only two cases did the resulting 
colonies survive more than ten days, while reproduction was very 
feeble and never occurred beyond the third generation. In one 
case (Table A, Number IX) an attempt was made to re-transfer the 
third generation back from the currant to the guelder rose, but 
the result was that the aphides all died within twenty-four hours. 

Similar attempts were made to transfer A. grossulariae from 
currant to guelder rose, but the colonies never survived more than 
six days, and reproduction was very feeble. Meanwhile the natural 
colonies on guelder rose and currant flourished from the end of 
May to the middle of August and end of July respectively. 

Aphis grossulariae has not been recorded from other food plants, 
but during June I observed three instances where winged migrants 
had established themselves on the flower heads of the Canterbury 
Bell {Campanula) and the resulting colonies persisted for two or 
three weeks. 

The conclusions suggested by the foregoing observations are 
that, as Theobald points out, A. grossulariae is probably identical 
with A. viburni. The first migrant from the birth plant ( Viburntmi) 
can form colonies either on Viburmim, which is the natural host, 
or else on Ribes. The descendants of the migrants to Viburnum 
may with some difficulty be established on currant although the 
resulting colonies are not so strong as those derived from an early 
migrant. On the other hand the descendants of the migrants to 
currant cannot be re-established on Viburnum. It seems as if in 
two or three generations some change takes place in the currant 
form which prevents it from flourishing on the guelder rose. One 
explanation is that there is some change in the constitution of the 
guelder rose plant — an increase of tannins for instance — and that 
the strain on guelder rose can gradually adapt itself to altered 
conditions which the newly transferred currant reared stock cannot 
tolerate. But this explanation is not wholly satisfactory because 
the dates show that unsuccessful transferences took place in the 
second and third generations while the plants were still young, 
while the most successful attempt was made in July when the 
shoots were mature. It is also worth noticing that while the more 
successful attempts were made with winged parents, yet in several 
of the Viburnum-io-cnvYajxt experiments, wingless females were 
found to feed and reproduce on the new host. 

Theobald {op. cit. p. 100) suggests that A. grossulariae maybe 
the alternating form of A. viburni, but says that he has twice 
failed to transfer the former to Vibui^num — a result confirming my 
own experiments in Table B. On the other hand, it is possible that 



Table A. 

Results of transference of Aphis viburni from Viburnum 

opulus to Ribes rubrum. 



Number 


Date of 
transference 


Forms transferred 


Death of 
last survivor 


Number of 

Generations born 

On new host 




I 


12 . VI . 18 


alate and apterous 


21 . VI . 18 


?2 


II 


13 . VI . 18 


alate 


17. VI. 18 


1 




III 


17 . VI . 18 


alate and apterous 


22 . VI . 18 


?2 




IV 


24. VI. 18 


apterous 


29 . VI . 18 


1 




V 


29 . VI . 18 


apterous 


2 . VII . 18 







VI 


13. VI. 18 


alate and apterous 


26 . VI . 18 


?3 




VII 


5 . VII . 18 


apterous 


6. VII. 18 


1 




VIII 


30 . VI . 18 


— 


9 . vii . 18 


2 




IX 


9 . VII . 18 


alate and apterous 


25. VII. 18 


3 




X 


6 . VII . 18 


apterous 


12. VII. 18 


2 




XI 


5 . VII . 18 


apterous 


6 . VII . 18 







XII 


6 . VII . 18 


— 


7 . VII . 18 







XIII 


9. VII. 18 




13. VII. 18 


?1 





Table B. 

Table of transference of Aphis viburni, self-established on 

Ribes rubrum, to Viburnum. 



Number 


Date of 
transference 


Forms transferred 


Death of 

last survivor 


Number of 

Generations born 

on new host 


I 


5 . VI . 18 


apterous 


12. VI. 18 


1 


II 


2 . VI . 18 


alate and apterous 


8 . VI . 18 


1 


III 


8 . VI . 18 


— 


10 . VI . 18 


— 


IV 


10 . VI . 18 


— 


14. VI. 18 


1 


V 


22 . VI . 18 


apterous 


24 . VI . 18 


— 


VI 


30 . VI . 18 


alate and apterous 


1 . VII . 18 


— 


VII 


1 . VII . 18 


apterous 


2. VII. 18 


— 


VIII 


24 . VII . 18 


alate and apterous 


25. VII. 18 


— 



Miss Haviland, The bionomics of Aphis grossulariae, etc. 269 

A. grossulariae is not the natural summer form of A. viburni, but 
is merely a casual parasite of the currant. In those of the Aphidinae 
which have a regular migration between two plants, the change is 
usually from a woody stemmed primary, to a herbaceous secondary, 
host; and if in the case of ^. viburni, the currant should be found 
to be the normal second host, it would be a remarkable exception to 
this rule. Perhaps we have here a form that has not yet adapted 
itself to the conditions of modern fruit growing. In a natural state, 
the aphides are probably able to follow the whole life cycle on 
Viburnum, but the spread of the cultivated currant has presented 
them with an increasing supply of alternative food which induces 
a change that makes a return to Viburnum impossible. Whether 
sex-producing forms can arise from the currant stock, and thence 
return to the guelder rose, is not known. If not, and the early date 
of the disappearance from the currant is against this view, we must 
consider that the infestation of the currant is an unfortunate 
accident in the history of the species, which entails a waste of 
migrating individuals upon a cultivated plant that might otherwise 
have perpetuated themselves on the natural host. However this 
does not mitigate the danger of the pest from a fruit grower's point 
of view, and infected Viburnum ought not to be allowed in the 
neighbourhood of currant bushes. 



Note on an experiment dealing with mutation in bacteria. By 
L. DoNCASTER, Sc.D., King's College. 

[Read 17 February 1919.] 

(Abstract.) 

It was noticed that the recorded ratio of occurrence in cases of 
meningitis of the four agglutination-types of Meningococcus corre- 
sponded very closely with the ratio of occurrence of the four iso- 
agglutinin groups of blood in a normal human population. It 
seemed possible, therefore, that by growing Meningococcus of one 
type in media containing human blood of ditferent groups, mutation 
to other types might be induced. Experiment showed that con- 
siderable differences in type of agglutination resulted, but it was 
concluded that this was caused by the sorting out of races of 
different agglutinability from a mass culture, rather than by true 
mutation. 



CONTENTS. 

PAGE 

On Certain Trigonometrical Series lohich have a Necessary and Sufficient 
Condition for Uniform Convergence. By A. E. Jollifpe. (Com- 
municated by Mr G. H. Hardy) 191 

Some Geometrical Interpretations of the Concomitants of Ttvo Qtiadrics. 

By H. ^Y. TuRNBULL, M.A. (Communicated by Mr G. H. Hardy) 196 

Some properties ofp{^n), the number of partitions ofn. By S. Ramanujan, 

B.A., Trinity College 207 

Proof of certain identities in combinatory analysis : (1) by Professor 
L. J. Rogers; (2) by S.- Ramanujan, B.A., Trinity College. (Com- 
municated, with a prefatory note, by Mr G. H. Hardy) . . .211 

On Mr Ramanujan's congruence properties of p (n). By H. B. C. Darling. 

(Communicated by Mr G. H. Hardy) 217 

On the exponentiation of well-order^ series. By Miss Dorothy Wrinch. 

(Communicated by Mr G. H. Hardy) . . . . . .219 

The Gauss-Bonnet Theorem for Midtiply -Connected Regions of a Siorfaee. 

By Eric H. Neville, M.A., Trinity College 234 

On an empirical formida connected with GoldhacK's Theorern. By N. M. 
Shah, Trinity College, and B. M. Wilson, Trinity College. (Com- 
municated by Mr G. H. Hardy) 238 

Note on Messrs Shah and Wilson^s pamper entitled: '■ On cm empirical 
formida connected xoith GoldhacK s Theory \ By G. H. Hardy, M. A., 
Trinity College, and J. E. Littlewood, M.A., Trinity College . 245 

The distribution of Electric Force betioeen tivo Electrodes, one of whixih is 
covered with Radioactive Matter. By "W. J. Harrison, M.A., Fellow 
of Clare College 255 

The conversion of soAV-diist into sugar. By J. E. Purvis, M.A. . . 259 

BracTcen as a sotorce of potash. By J. E. Purvis, M.A 261 

The action of electrolytes on the electrical conductivity of the bacterial cell 
and their effect on the rate of migration of these cells in an electric 
field. By C. Shearer, Sc.D., F.R.S., Clare College . . .263 

The bionomics of Aphis grossiilariae Kcdt., and Aphis viburni Schr. By 
Maud D. Haviland, Bathm'st Student of I^wnham College. (Com- 
municated by H. H. Brindley) 266 

Note on an expe7'iment dealing tvith mutcUion in bacteria. By L. Don- 
caster, Sc.D., King's College. (Abstract) . . . . , 269 



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PROCEEDINGS 

OF THE 



Colourimeter Design. By H. Hartridge, M.D., Fellow of 
King's College, Cambridge. 

[Received 7 October 1919; read 10 November 1919.] 

In a previous paper (1) I have described certain factors which 
affect the efficiency of the spectrophotometer. The colourimeter 
has been found to be similarly affected, so that various modifica- 
tions in the usual designs are indicated. 

The comparison field is in most instruments divided at a 
diameter, so that one half receives light which has passed through 
one limb, and the other half light that has passed through the other 
limb of the instrument. In a few designs the bull's-eye and the 
central strip fields have been employed. All these fields have 
the disadvantage that local stimulation of the retina may occur that 
sets up after image phenomena greater in degree in one part than 
in another, thus preventing accurate determinations. And, further, 
they do not make the best use of the effects of simultaneous contrast. 
A better type of field is the one which I have previously described 
in connection with the spectrophotometer, namely, one which is 
subdivided into a number of strips, of which alternate numbers 
receive light from the two limbs of the instrument. With this field 
the eye does not select any one part for examination, but tends rather 
to judge of the field as a whole. When the adjustment of intensity 
has been correctly made the whole field should become uniform. 
The effects of retinal fatigue therefore tend to become uniformly 
distributed. The contour of this type of field is of considerable 
length compared with its total area ; the conditions are therefore 
beneficial for the development of contrast. The absence of visible 
lines of junction still further increases this effect. 

The prisms A and B by which the beams of light through the 
two limbs of the instrument are combined at the compound field 

VOL. XIX. PART VI. 19 



272 



Br HarLridge, Colourimeier Design 



Eyepiece Cup - 



Ramsden eyepiece • 



Comparison field 



Comparison field 



Plunger ■ 



Staa;e - 




Tail piece to carry lamp 



f*ush-on terminals 
Horse-shoe foot 



Opal glass disc 
Watt lamp 



Dr Hartridge, Colourimeter Design 273 

described above, are similar in shape to those used in the spectro- 
photometer. They are shown in the diagram of the apparatus. It 
will be observed that the interface of the prisms is silvered, the 
metallic film being removed by means of a simple ruling machine, so 
that narrow strips of the silver alternate with strips from which the 
whole of the silver has been removed. Examination of the diagram 
will show that by this arrangement the field seen on looking down the 
eyepiece is formed of alternating narrow beams which have either 
been transmitted from one limb of the instrument through the 
spaces between the silver strips, or reflected from the other limb 
by the silver strips themselves. The lengths of the prisms A and 
JB should be such that the two entering beams have passed through 
equal lengths of glass. 

The troughs are adjustable on both limbs of the instrument, 
in colourimeters of usual design. This arrangement has the dis- 
advantage that if there should be any backlash in the micrometer 
mechanism which is used for adjusting the position of the movable 
troughs, or error in the setting of the scale, these will affect both 
the thickness of the pigment solution to be estimated, and also 
that of the standard. Such errors can be eliminated so far as the 
standard is concerned by the use of a special cell, the distances 
between the sides of which are determined by accurately ground 
distance pieces, which may be made of either glass or metal. 
Rustless steel would appear to be a suitable metal because it resists 
the corrosive action of ordinary solvents. 

I have shown that in the case of the spectrophotometer there 
are important reasons for the use of troughs with double compart- 
ments on both limbs of the instrument. In both ti'oughs the com- 
partment near the light source should contain the solvent only, 
the other being filled with the solution of the pigment. Double 
troughs should be used with the colourimeter for similar reasons, 
namely, (a) in order that absorption by the solvent may be com- 
pensated, since the thickness is the same on both sides of the 
instrument ; (b) that pigments accompanying the one under esti- 
mation may be compensated for; (c) that specific surface reflection 
at the sides of the troughs which contain pigment may be similar 
on both limbs of the instrument. With regard to the type of 
trough that should be employed I have previously considered the 
advantages of the double wedge trough in conjunction with the 
spectrophotometer. In the case of the colourimeter the plunger 
type usually employed has the advantage of not requiring calibra- 
tion with a micrometer microscope as wedge troughs do. The 
method of employing double compartment plunger troughs and 
standard troughs is shown in the diagram. In some colourimeters 
the troughs are bell mouthed, and are manufactured from black 
glass. These points are to be recommended. It should be noted, 

19—2 



274 Dr Hartridge, Colourimeter Design 

however, that reflection can still take place at the sides of the 
troughs, so that it is necessary carefully to restrict the light illu- 
minating the troughs to narrow vertical pencils of just sufficient 
diameter fully to illuminate the comparison fields. Since scattered 
or reflected light may increase the apparent brightness of one of 
the fields it is essential that this be reduced to a minimum. Special 
care should therefore be taken in designing the instrument to pre- 
vent the entrance of stray light, and to employ an illuminating 
system that will limit the entering beams to the narrow pencils 
above referred to. 

The illumination in the majority of colourimeters is obtained 
from the sky by means of a plane mirror. In some instruments this 
may be replaced at will by a finely matted white surface. The 
illumination therefore in either case consists of a large number of 
divergent pencils, which enter the lower ends of the troughs in all 
possible directions. Scattered light is therefore at a maximum. In 
the case of the microscope a similar practice used to be in vogue, but 
it has given way to the use of illuminating lens systems in which 
the corrections and alignment are well nigh as perfect as those 
used in the objective and eyepiece. Now, in the case of the spec- 
trophotometer I have shown that the beams illuminating the two 
limbs of the instrument should proceed from identical parts of the 
light source. This condition should be realised in the case of the 
colourimeter also. The arrangement of the illuminating apparatus 
is shown in the diagram. 

The light source is similar to that which I have applied to the 
microscope (2), consisting of a slab of white opal glass finely 
ground on both sides. This is lit from behind by means of a small 
half watt electric lamp, which obtains its current from a small 
accumulator or dry cell, or from the town supply through a suit- 
able resistance. The lamp is enclosed in a brass box, which is 
silver plated inside, and is finished dead-black outside so as to 
radiate heat. The life of the lamp is increased by connecting it 
with a press switch so that it is in circuit during observation only. 
The lamp box is attached to the tail-piece of the instrument so 
that it forms an integral part of the apparatus. The whole may 
thus be tilted or moved from place to place without requiring re- 
adjustment. Immediately above the opal glass is a metal dia- 
phragm, the aperture in which limits the surface exposed to a 
disc 4 mm. in diameter. Attached beneath the stage of the in- 
strument and 60 mm. above the diaphragm of the light source is 
a plano-convex achromatic lens of 26 mm. diameter and 60 mm. 
focal length. The divergent rays from each point of the source 
are rendered parallel by this lens, and at once pass through two 
achromatic plano-convex lenses of 18 cms. focal length and 14 mm. 
diameter. These lenses have a clear aperture of 12 mm. and form 



Dr Harfridge, Colourimeter Design 275 

a focussed image of the diaphragm of the light source, which is 
magnified in the ratio of the focal lengths of the lenses; since the 
ratio is 3 to 1 this image has a diameter of 12 mm. 

The beams that emerge through the lenses Tl and T'2 do not 
therefore anywhere exceed 12 mm. and the light does not spread, 
for this reason, to the sides of the troughs during its passage 
and therefore stray light is reduced to a minimum. The beam 
from the lens T2 passes vertically upwards through a hole in 
the stage to the standard trough which rests upon it. Having 
passed through both the layer of solvent and also that of the 
solution of pigment, the beam enters prism B', and is totally inter- 
nally reflected at its inclined surface on to the silvered strips of the 
comparison field. The beam that has passed through Tl is deflected 
by internal reflection at the right angled prism C which is cemented 
to it, and falls on the silvered sui'face between the two halves of the 
prism D, so that the beam is directed vertically through a second 
hole in the stage on to the lower fixed cup of the adjustable trough, 
which is filled with solvent. It then passes through the movable 
cup which contains the pigment, and enters the prism A to fall on 
the silvered strips of the comparison field. The passage of this 
beam through the intervals between the strips, and the reflection 
of the beam from the other limb of the instrument at the strips 
themselves, has already been described. It will be noted that the 
reflection of the one beam by internal reflection within the prism 
C, and by ordinary reflection within the prism D, causes this beam 
to compensate for the internal reflection and reflection at a silvered 
surface which occurs within prism B in the case of the other beam. 
As it has been found that silvered surfaces vary in the intensity 
of rays of different wave-length which they reflect, it is advisable 
that both mirror D and prism B be silvered with the same solution 
at the same time. 

The lengths of the paths of the beams through the instrument 
are found to be in the case of the left-hand beam an actual dis- 
tance of 19'5 cms., that is an effective distance of 18 cms. since 
2"2 cms. of glass is passed through ; in the case of the right-hand 
beam the total and the equivalent lengths are the same as those 
on the left. 

The comparison field therefore is illuminated by two super- 
posed images of the diaphragm of the light source, one of which 
has passed through the standard trough and the other through the 
adjustable trough. When the instrument is in correct adjustment 
these two images exactly coincide, so that if there should be any 
slight inequality between the intensity of illumination of different 
parts of the light source both images will be similarly affected, 
and therefore the match between their different parts will remain 
unchanged. Such a condition is not secured in the usual forms of 



276 Br Hartridge, Colourimeter Design 

colourimeter, since it is due to the particular method of illumina- 
tion described above. 

The eyepiece used in the du Bosq type of colourimeter consists 
of a Eamsden lens system, at the upper focal plane of which has 
been placed a diaphragm pierced with a small aperture. This has 
the effect of limiting the rays reaching the eye to those which 
have passed as approximately parallel bundles up the limbs of the 
instrument. To be effective the aperture has to be small, and this 
has the disadvantage of making the intensity of illumination of 
the fields somewhat low. When this type of eyepiece is in use it 
is found that the eye has to be inconveniently close to the aperture 
in order that the whole field shall be seen at one and the same 
time. This is due to the fact that the diaphragm is a considerable 
distance below the effective pupil of the eye, even when the eye 
has been placed as close as possible, and as a result some of the 
rays which spread out from the diaphragm may not enter the pupil. 
The difficulty is in fact similar to that met with in high power 
microscopic eyepieces of the Huygenian type. To avoid this diffi- 
culty a more elaborate type of eyepiece has been devised, in which 
an erecting lens system has been placed above the Ramsden ocular 
and its diaphragm (8). This causes a sharp image to be seen on 
looking down the eyepiece, and at the same time the image of the 
small aperture is formed at a considerable distance above the top 
lens, so that the eye does not have to be placed inconveniently 
close to the eyepiece in order to obtain a full view of the field. 
These improvements are obtained, however, at a certain sacrifice 
of definition, which is unimportant in the usual types of colouri- 
meter in which the fields are of simple design, but is of relatively 
greater importance if the more detailed type of field be used which 
has been described above. It will have been observed that in the 
colourimeter which I have described above the illuminating beams 
are formed by the special method of illumination employed. Under 
which circumstances it is found that the Ramsden disc of the ocular 
contains the overlapping focussed images of the restricting aper- 
tures of the lenses Tl and T2, which when the instrument is in 
correct adjustment exactly overlay one another. It is therefore 
unnecessary that the eyepiece should contain any diaphragm to 
restrict the beams, and therefore the difficulties introduced by 
such a diaphragm are not met with. The eyepiece itself should 
be achromatic and should slide in a tight-fitting jacket so that the 
observer may set it at the best focus. It should magnify about 3 
diameters. 

The angle at which the comparison field lies will be seen to be 
45 degrees. But since it is enclosed between two pieces of glass, 
the apparent angle to the eye is reduced in the ratio of the refi'ac- 
tive indices of glass and air. The apparent angle would therefore 



Dr Hartridge, Colourimeter Design 277 

be about 29 degrees. Now, the dimensions of the field seen by the 
eye are 8 mm. by 6 mm., the latter being in the direction of the 
slope. The apparent diiference of focus is therefore less than 4 mm., 
which would be equivalent to 12 mm. at a distance of 25 cms. 
Such a small change of focus would be at once met by a trifling 
change in the degree of accommodation of the eye, which would 
be effected subconsciously and involuntarily. No difficulty is to be 
met with therefore from this cause. 

The Mechanical System. 

The metal work of the colourimeter follows closely that of the 
microscope. The horse-shoe foot, stage and coarse adjustment all 
resemble those used in that instrument. The adjustment has a 
range of 40 mm. only, because, as will be shown later, the use of 
standard solutions of 20 mm. thickness makes a bigger movement 
than this unnecessary. An accuracy of one-quarter per cent, should 
be sufficient, and this is readily provided by a scale gi^aduated in 
half mm. and reading by a vernier to one-twentieths. The adjust- 
ment should have long, well-made V slides so as to eliminate lost 
motion. The scale should be attached to the moving member, the 
vernier being attached to the fixed. A simple lens and 45 degree 
mirror should make a magnified image of this visible to the ob- 
server. To the moving member is first screwed and afterwards 
sweated with soft solder a strong brass ring. To this is attached 
by means of a three-prong bayonet catch the ring fixed to the 
upper lip of the movable trough. The trough is cemented into a 
groove turned in this ring by means of plaster of Paris or Caemen- 
tium. Where plaster has been used the joint should be covered 
by a thin coat of Robiallac. The prisms and eyepiece are attached 
to a strong projection at the top of the pillar which forms the 
handle of the instrument. 

The removal of the troughs for filling and cleaning and their 
replacement is a simple process which should not take more than 
a few seconds. To remove the adjustable troughs, first swing the 
substage to one side ; this allows the lower trough to drop verti- 
cally through the hole in the stage until it can be removed. The 
upper trough is now gripped between the finger and thumb, and 
the trough rotated so as to free the bayonet catches ; this trough 
is then lowered through the hole in the stage and removed. The 
plunger and the troughs can now be cleaned, refilled and returned. 
The standard double trough simply rests on its side of the stage, 
so that its removal takes but a moment. 



278 Dr Hartridge, Colourimeter Design 

The Colourimeter in Practice. 

Experiment has shown that if two solutions of the same colour 
contain different pigments in solution, then the thicknesses re- 
quired for a match vary not only with the observer and with the 
quality of the light, but also with the same observer from time to 
time. It is for this reason that the technique has been introduced 
of using the same pigment for the standard as that required to be 
estimated. Thus creatinin is no longer estimated by comparing the 
colour which develops when picric acid and soda are added with the 
colour of a solution of potassium dichromate ; but a standard solution 
of creatinin is used, picric acid being added to it at the same time as 
it is added to the solution to be standardised. If, then, the thick- 
ness of the standard is 20 mm. and that of the unknown 17 mm., 
it is assumed that the strengths of the solutions are in the inverse 
ratio of those numbers. Such is not the case however, because 
the sodium picrate itself absorbs rays from the same part of the 
spectrum as does the sodium picramate, and therefore, although 
the light may encounter the same number of coloured radicals in 
both limbs of the instrument, yet the sodium picrate absorption is 
greater on one side than the other, because the fluids are not of 
the same thickness. It is principally for this reason that I have 
adopted an instrument in which double troughs are used, on both 
sides of the instrument ; the lower pair on both sides being filled 
with sodium picrate solution in the case taken above as example, the 
upper pairs containing the picric acid plus creatinin. In this way 
the number of picrate radicals is kept approximately constant, since 
the total thickness of sodium picrate solution is the same on both 
sides of the instrument. The balance is not perfect however, because 
a certain amount of picric acid is used up in forming the sodium 
picramate, and this amount cannot be ascertained without assum- 
ing that the estimation to be done has already been accurately 
performed. The problem is, in fact, represented by a simultaneous 
equation involving two unknowns. I find that the matter can be 
solved in the following manner. Having diluted both the standard 
and the unknown solutions with equal amounts of standard picric 
acid and soda solutions, and having allowed the colour to develop in 
the ordinary manner, an estimate is made of the relative strengths of 
the solutions in the colourimeter. Having found that, say, a 20 mm. 
thickness of the standard has the same tint as 13"4 mm. of the 
unknown solution, a fresh sample of the unknown is taken and 
13'4 c.c. of it diluted with water to bring the total to 20 c.c. The 
solution of the unknown has thus been brought to approximately 
the same concentration as the standard. (Where the approximate 
strength is known a preliminary dilution before making the initial 
estimation is beneficial.) The correctly diluted solution of the un- 



Dr Hartridge, Colourimeter Design 279 

known is now treated, ab initio, with fresh picric acid solution and 
soda, and is then estimated against the standard in the colourimeter. 
It is now found that a 20 mm. thickness of the standard has the 
same tint as one of, say, 19"85 of the unknown after dilution. The 
strength of the unknown is thus ascertained, with considerable 
accuracy, because the conditions of equilibrium under which the 
sodium picramate develops and exists, and the quantities of picric 
acid used up in the determination are approximately constant. 

It should be pointed out that the above technique presents no 
difficulties, and takes little longer than the ordinary method. The 
p)rinciple may with advantage be applied to all estimations made 
Avith the colourimeter. 

The Accuracy of the Colourimeter. 

Since colour is due to absorption the colourimeter depends for 
its utility on the fact that a change in the number of coloured 
radicals encountered by light causes a change in the retinal stimu- 
lus when that light falls on the eye. We may, therefore, arbitrarily 
state that the accuracy of the determinations depends, firstly, on 
the rate of change in the quality of the light which is passed 
through the pigment, and, secondly, on the acuteness of the per- 
ception of the eye for the change in quality of the light. The 
greater the rate of change and the greater the acuteness of percep- 
tion of that change, the greater will be the accuracy. Many bodies 
which absorb light do so selectively, that is, they have a gref^ter 
effect in one part of the spectrum than in another ; they therefore 
show colour, that is, they are pigments. Under ordinary circum- 
stances the greater the absorption the stronger the colour and the 
less the intensity of the transmitted light. As the concentration 
of a pigment is altered, and therefore the degree of absorption, the 
strength of colour and the brightness of the transmitted light both 
vary. The colourinietric determination, therefore, depends on the 
simultaneous occurrence of both these changes. The important 
questions that arise are : (1) on what do the magnitudes of these 
changes depend ? (2) which is the more important ? and (3) how 
can the changes be increased for a given alteration in concentra- 
tion ? A study of absorption band formation gives a definite answer 
to each of these questions as follows: (1) The changes for a given 
alteration of concentration are greater the flatter and broader the 
absorption band. If, therefore, there were two pigments of the same 
concentration and the same colour, i>ne of which had a sharp well- 
defined band, while that of the other was broad and flat, the latter 
pigment would be found to give the more accurate readings in the 
colourimeter. (2) Of the two changes, that of colour is usually the 
more important, particularly with pigments showing single absorp- 
tion bands. In pigments with multiple bands the intensity change 



280 Dr Hartridge, Colourimeter Design 

may be the more important : for example, a pigment absorbing to 
an equal extent in two complementary parts of the spectrum will 
cause the light to suffer no change in colour at all, while the in- 
tensity is altered. (3) The changes in the case of any one pigment 
can be increased by increasing the intensity of that part of the 
spectrum which is suffering change or by decreasing that of parts 
which do not show alteration. Of the two methods the latter is 
the easier to carry out and the more efficient. If colour filters are 
used they must be carefully adjusted according to the position in 
the spectrum of the absorption band of the pigment to be estimated. 
If a spectral illuminator is used the apparatus virtually becomes 
a spectrophotometer, and this elaboration is hardly necessary for 
ordinary work. The possibility should not be overlooked of the 
existence of alternative colour reactions to those at present in use 
in which pigments having less steep absorption bands are used and 
which therefore permit greater accuracy in their colourimetric 
estimation. 

The factors which influence the acuteness of perception of the 
eye remain for consideration. Firstly, it is clear since the accuracy 
of the determination depends on the correctness of the match ob- 
tained, that the eye should not be suffering from fatigue. The 
reading of small print and the exposure of the eyes to excessive 
light should, therefore, be avoided for a reasonable time before the 
determinations. The absence of refractional errors, eye strain, want 
of eye-muscle balance and the possession of good general health are 
all factors of importance. In my own case the period after tea is the 
best, provided that the morning's work has not been arduous. The 
presence of after images is most harmful for accurate estimations ; 
the best method of eliminating them is, I find, to look for a fcAV 
moments at a uniformly lit grey surface. All the above points may 
seem obvious ; it is however my experience to find that they are 
sometimes overlooked. The apparatus itself is best placed in a dark 
room, or at all events where the full light of a window cannot fall 
on the eye of the observer. In the latter case the eyepiece cup 
may be made deep with advantage, so as to protect the periphery 
of the retina from stimulation and thus bring about an increase in 
the diameter of the pupil. 

With regard to the use of colour filters, experiment shows that 
the theoretical conclusions arrived at above are amply justified, 
namely, that the accuracy of the determinations is increased if 
either the rays absorbed by the pigment are increased in intensity, 
or those not absorbed are decreased or removed altogether. The 
removal by means of colour filters is however usually attended by so 
great a diminution in the intensity of the light that a powerful 
source such as an arc lamp becomes necessary. It is a fortunate 
circumstance, therefore, that the retina should be even more sensi- 



Dr Hartridge, Colourimeter Design 281 

tive to change in shade than it is to change in intensity. I have 
found, further, that the point of greatest sensitiveness is obtained 
when the fields are nearly neutral in colour. Such a condition is 
obtained by the use of a suitable colour filter which absorbs in that 
part of the spectrum which is occupied by the complementary 
colour to that absorbed by the pigment. Suppose, for example, a 
yellow pigment is to be estimated, then a blue solution of a dye is 
placed in the path of the light from the source of such a thickness 
and concentration that the comparison field seen in the instrument 
is of a neutral grey colour. Permanent colour films between glass 
should be used if much work is likely to be done with any given 
pigment. Such a technique is very simple, and I find that in my 
hands it increases the accuracy of the determinations by about 
three times (when estimating sodium picrate), the method of mean 
squares being used to calculate the average error of the experi- 
mental determinations both with and without the complementary 
filter. The probable error of the determinations was found to be 
0"8 per cent., using home-made apparatus and the complementary 
screen. It should be possible to halve this amount if the precau- 
tions outlined above be taken and well-designed apparatus be used. 

Summary. 

(1) The comparison field seen on looking down the instrument 
should cause the greatest contrast and at the same time should not 
produce after images. 

(2) On both limbs of the instrument double troughs should be 
used, so that the thickness of pigment to be measured may be 
varied at will, while the absorption caused by other pigments 
remains constant. 

(3) An artificial light source should be used, and the lighting 
system be so designed that narrow beams are produced of just 
sufficient width as to completely illuminate the comparison field. 
The amount of reflected and scattered light may thus be reduced 
to a minimum. 

(4) If experiment shows that the change in colour produced 
by a given change in thickness or concentration of the pigment 
can be increased by modifying the relative intensity of different 
parts of the spectrum of the light source, then suitable colour filters 
should be prepared for use during the determinations. It was 
found in a test case that this modification alone increased the 
accuracy by three times. 

(5) The general design of the instrument should conform to 
microscopic practice, fixed troughs being supported by the stage 
and the movable trough actuated by the rack and pinion course 



282 Dr Hartridge, Colourimeter Design 

adjustment screw. The illuminating system should be fitted 
beneath the stage so that the instrument may be tilted or moved 
from place to place without disturbing the alignment. 

For certain purposes it may be found beneficial to employ 
smaller quantities of liquid than those required in the ordinary 
colourimeter. I find that a modification in the design of the 
troughs should make 1 to 2 c.c. of liquid sufficient ; and further, by 
modifying the optical system as well, as little as "001 c.c. could be 
worked with. It should be pointed out however that such quantities 
could only be employed with solutions of considerably greater con- 
centration than those usually estimated ; e.g. about ten times the 
usual concentration for 1 c.c, and one hundred times for '001 c.c. 

References. 

(1) Hartridge, Journ. Physiol, l, p. 101 (1915). 

(2) Hartridge, Joum. Qitekett Micro. Soc. Nov. 1919. 

(3) Kober, Journ. Biol. Ghem. xxix, p. 155 (1917). 



Mr Snell, The Natural History of the Island of Rodrigues 283 



The Natural History of the Island of Rodrigues. By H. J. Snell 
(Eastern Telegraph Company) and W. H. T, Tams^. (Communi- 
cated by Professor Stanley Gardiner.) 

[Read 10 November 1919.] 

Rodrigues lies some 350 miles east of Mauritius, and is a rugged 
mass of volcanic rock closely resembling Mauritius and Reunion. 
It is surrounded by a coral reef, the edge of which at the eastern 
end is within 100 yards of the beach, whilst on the north and south 
it extends outwards to a distance of three to four miles, and on the 
west to two miles. There is an irregular channel inside the reef 
close to the shore, extending round most of the island, sufficiently 
deep for boats at any state of the tide, and at the south-east end 
a small lagoon of three to ten fathoms, with a passage through 
the reef. The usual anchorage is Mathurin Bay, in the reef to the 
north. The reef is studied with islets, those nearer the shore being 
mostly of volcanic nature, and situated on the north and west, 
whilst the rest are of limestone, modern accumulations of debris, 
and situated on the south. 

The island itself is eleven miles long by five miles broad, and 
has an area of just over forty square miles. There is a central 
lofty ridge extending from east to west, with a break about one- 
third of its length from the west. The western bastion of the range 
is Mount Quatre- Vents, 1120 feet high, while at the eastern end 
is Grande Montaigne, 1140 feet. The highest point is Mount 
Limon (1300 feet), which lies with two other peaks a little out of 
the general line of mountains. The sides of these peaks are cut 
into numerous ravines, these being deeper and more frequent on 
the south side than on the north. At their upper ends these ravines 
are often bordered by perpendicular columnar basaltic cliffs, 
sometimes exceeding 200 feet in height, extensively cut into many 
coulees by small streams which often descend in a series of cascades. 

The volcanic ridge descends on the south-west gradually, and 
passes into a broad coralHne limestone plain, with occasional hills 
up to 500 feet high, indicating a comparatively recent elevation 
of at least a like amount. This tract of limestone is honeycombed 
with caves, in which stalactites and stalagmites are abundant. 
There are many holes and fissures, and often deep hollows occur, 
at the bottom of which lie large fragments of limestone in irregular 
heaps; these are apparently old caves, the roofs of which have 
fallen in. The floors of these hollows are covered with soil, often 

1 The second author is solely responsible for the names of the insects herein 
recorded. 



284 Mr Snell, The Natural History of the Island of Rodrigues 

witli lumps of volcanic rock on the surface. The limestone is not 
found along the northern or southern shores, except at their eastern 
extremity, where patches occur at the mouths of the valleys, 
occasionally at some distance from the shore. Some of the patches 
of limestone found in the volcanic region indicate an elevation of 
perhaps 500 feet, and the raised beaches on the south shore, some 
20 feet in height, may point to a further subsequent change of 
level. The position of old volcanic craters has not been accurately 
determined, but the main ones appear to have been situated 
about the Grande Montaigne and Mount Malartic, 

The island is comparatively dry, and during the warm season 
many of the streams are dried up, though they assume in the 
rainy season torrential proportions. The climate is like that of 
Mauritius. The rainfall is very irregular; during the north-west 
monsoon from November to April the weather is wet and warm, 
and early in this season there are frequently severe hurricanes. 
From May to October the south-east monsoon prevails, and the 
weather is then cool and dry. Fogs are rare, and climatic conditions 
render the island healthy to live in. 

Rodrigues was discovered in 1510, by a Portuguese commander, 
whose name it bears. In 1691 the Dutch landed several fugitive 
French Huguenots there, among whom was M. Fran9ois Leguat, 
who wrote an account of the island in 1708. The island was later 
cultivated by the French East Indian Company, and maize and 
corn were grown ; these, with dried fish, turtles and land tortoises, 
were exported to Mauritius. It was occupied by the British in 
1809, and made the base of operations against Mauritius. It is 
still cultivated as a garden for Mauritius, its main exports being 
beans, acacia seed, maize, salt fish, cattle, goats and pigs. The 
population is about 5000, mostly settled around Port Mathurin, 
the only town in the island. The people are mainly French Creoles, 
with a few Chinese and Indians, and are subject to the Government 
of Mauritius, which suppHes a Resident Magistrate. The island is 
a station of the Eastern Telegraph Company, connecting to Cocos- 
Keeling. 

Each family usually cultivates an acre or several acres of land, 
whereon they grow maize, sweet potatoes, haricot beans, pumpkins, 
various herbs, onions, etc. They depend, in fact, largely on their 
own plantations for food. At one time a species of mountain-rice, 
which does not require an abundance of moisture, was grown in 
large quantities, but its cultivation was abandoned owing to the 
depredations of small birds. Tobacco grows well. Haricot beans 
are still exported. There have lately been, however, only five ships 
per year, and these small sailing ships of 500 tons down to 100 tons 
register; this makes it very difiicult to market the produce of the 
island. The maize grown is barely enough for local consumption. 



Mr Snell, The Natural History of the Island of Rodrigues 285 

One of the most profitable products of this island is acacia 
seed, which is exported to Mauritius for cattle feeding. The acacia 
{Lucaena glauca), which was introduced about seventy years ago, 
now grows wild and flourishes everywhere, covering the ground 
for acres, and forming a dense almost impenetrable scrub, beneath 
which nothing will grow. The cattle and goats are exceedingly fond 
of the leaves and pods, and this is probably the reason for its 
spreading so extensively, the original plantation having been in a 
valley near Port Mathurin. Amongst other things which have been 
successfully grown may be mentioned coffee, vanilla, sugar-cane, 
oranges and lemons. Bananas and plantains, custard apples, 
strawberries and raspberries are found wild. Many other com- 
modities such as ginger, safran (turmeric) and arrowroot have also 
been grown. 

There is very little real pasturage in Rodrigues, the largest 
area being in Malgache Valley. Besides this there are barren tracts 
round the coast covered with coarse grass, which provides in- 
sufficient subsistence for the stock. Most of the inhabitants own 
goats and pigs, on which they rely for their milk and meat supply, 
and which are also exported. They were allowed to run wild, but 
measures have now been introduced by the Government to control 
them. Poultry, ducks and geese also thrive in the island. 

Rodrigues was originally covered with dense forests of lofty 
trees, with corresponding undergrowth. Indeed, according to 
early descriptions its vegetation partook of the nature of a regular 
tropical moist woodland. Here were to be found flightless birds, 
the Solitaires, and giant land tortoises. When Leguat saw this 
island first, the scenery was such as to call forth from him such 
designations as "a lovely isle," "an earthly paradise." To-day its 
grandeur and beauty have vanished. There remains a bare parched 
pile, on which it is difficult if not impossible to discover any corner 
in its original condition. Many agencies are responsible for this 
destruction and denudation. It has been swept by fire many 
times, accidentally and intentionally. The goats devour the young 
shoots and leaves of any vegetation within their reach. Pigs have 
done their share, especially with regard to the Latanier Palm 
(Pandanus), of the nuts of which they are very fond. Then there 
are the introduced plants, which have in many cases crowded out 
the native vegetation. A notable example is seen in the acacia, 
previously mentioned, which has spread into almost every valley 
in the island. A certain amount of destruction has been done by 
the inhabitants, who have cut timber over large tracts without 
discrimination. Though a check has been placed on this by the 
government, there still remains a source of destruction, in that the 
inhabitants are in the habit of acquiring year by year fresh tracts 
of woodland, the undergrowth of which they cut down and burn, 



286 Mr Snell, The Natural History of the Island ofRodrigues 

and here they plant their haricot beans. They utiHse a tract of 
land for one season, and abandon it the next. Thus the work of 
destruction continues. Many of the older inhabitants, at present 
living on the island, say that they remember large tracts, which 
are now almost bare except for a few Vacoas (Screw-pines), being 
originally covered with almost impenetrable forest, but nobody 
remembers the large expanse of coralline limestone at the south- 
western end of the island in any other than its present state, 
though there are unmistakeable traces, in roots and stumps em- 
bedded in the ground and charred by fire, shoAving that this region 
was also at one time completely afforested. The large rifts are often 
thirty feet or more deep, and fifteen to twenty yards wide, and 
contain many fine old indigenous trees which have escaped destruc- 
tion. The Valley of St Frangois, at the north-east end of the island, 
is perhaps the only other tract which has escaped destruction. 

The commonest trees in the island are the Vacoas or Screw-pines 
(Pandanus), of which there are two species, both endemic. Three 
other species have been recorded by various authorities, one being 
a native of Asia, and the other two Madagascar species. None of 
them occurs in Mauritius or Reunion, and the evidence of their 
occurrence in Rodrigues is faulty. There are three species of 
endemic palms, belonging to three genera, which are all Mascarene. 
Probably half the plants have been destroyed, but from what is 
left — 297 species of Phanerogams, and 175 species of Cryptogams 
(excluding Marine Algae) — it is clear that the endemic flora was 
large and of Mascarene aSinities. There are only about twenty 
species of ferns, the scarcity of this group being accounted for by 
the present dryness of the island, in confirmation of which it may 
be remarked that the tree-ferns of the other Mascarene islands 
are not represented. 

The present day fauna is not large. The extinct fauna has proved 
to be of very great interest, particularly in the case of the Solitaire 
(Pezophaps solitaria, Gmel.), the extinct Didine bird related to the 
i)odo of Mauritius. Considerable collections of the remains of this 
bird have been made from the limestone caves, where also the 
remains of other extinct birds and of the giant Land Tortoise have 
been found. Our main knowledge of the recent fauna is due to the 
labours of the naturalists attached to the Transit of Venus Ex- 
peditions carried out in 1874-5. 

The marine fauna is in general of the Indo-Pacific type. 

The only indigenous mammal found in the island is a fruit-bat, 
Pteropus rodericensis, Dobson, which is peculiar to Rodrigues. The 
introduced mammals, other than those already mentioned, are 
deer, rabbits, rats, mice and cats, the latter being left by the 
Dutch to destroy the rats. 

Sir Edward Newton, K.C.M.G., published a list of Rodrigues 



Mr Snell, The Natural History of the Island of Rodrigues 287 

birds in his "List of the Birds of the Mascarene Islands" {Trans. 
Norfolk and Nonvich Naturalists' Society, vol. iv, President's 
Address). 

The Fresh Water Fishes, as far as known, belong to species 
which inhabit the fresh waters of the Mascarene Islands generally, 
with the exception of two Grey Mullets, which were collected by 
the Transit of Venus Expedition, and were described as new. 

Further collections in certain groups have recently been made 
by Mr H. P. Thomasset and Mr H. J. Snell, who visited the island 
during the period August to November, 1918, with a view to im- 
proving our knowledge of the insect fauna. 

Mr Snell visited practically every part of the island, with the 
exception of the valley of St Francois, and a small district round 
the Riviere Coco. The best collecting ground he found to be un- 
doubtedly the Grande Riviere Valley, which he worked right up to 
Mount Limon. The islands on the reef were also visited, but con- 
tained very little of interest, as they have been burnt over in recent 
years, and are now covered with rough coarse grass and short 
scrub {Tournefortia, Pemphis, etc.). These islands, particularly 
Gombranil and Flat, were formerly nesting places for sea-birds, 
which seem to have disappeared, only a few white terns and 
boobies being found on Sandy and Coco Islands, which were some 
years ago planted with firs. 

In the deepest ravines were commonly seen the fruit-bats or 
flying-foxes, feeding on the flower of a kind of aloe, of which they 
seem very fond, and also on wild figs, mangoes, etc. Geckos were 
abundant in warm and sheltered spots, particularly in all habita- 
tions. Their eggs were frequently found in nests (usually composed 
of dry Sow-thistle bloom) under rocks and in crevices. Two species 
only have been recorded: Gehyra mutilata, Gray, and Phelsuma 
cepedianum, Gray; the latter is common in Madagascar, Mauritius 
and Reunion, but is rare in Rodrigues. Freshwater fishes were 
found in many of the streams, in which also eels were quite 
common. 

There are in the island a Land Planarian, Geojplana whartoni, 
Gull., and a Land Nemertean, Tetrastemma rodericanum, Gull. 
Both are peculiar to Rodrigues, but the former has not been ade- 
quately described. (Mr Thomasset subsequently obtained a Land 
Planarian from Mauritius, a new locality for these.) They were 
found under decaying logs, sometimes on the bark, under the 
bark, or in the wood; the Nemertean appeared to exist in far 
greater quantities than the Land Planarians, but they often live 
together in the same situation. Earthworms were not abundant. 
Amongst the Crustacea collected, large numbers of an Amphipod 
were found under stones, dead leaves, etc., wherever the ground 
was moist. In all the streams were to be found freshwater shrimps 

VOL. XIX. PART VI. 20 



288 Mr Snell, The Natural History of the Island of Rodrigues 

and a crayfish. Woodlice were abundant in deca}dng vegetable 
matter, the largest specimens being obtained from rotting banana 
stems. 

Myriapoda were common throughout the island. Large centi- 
pedes live on the corals on the west side of the island, attaining 
sometimes a length of twelve inches. Hardly a lump of debris can 
be turned over without disclosing one or more of these creatures. 
The Transit of Venus Expedition obtained twelve species of 
Myriapods, of which eleven were new. There is a single species of 
scorpion, Tityus marmoreus, Koch, and in addition the Transit of 
Venus Expedition obtained twenty-seven species of Arachnida, 
eleven being new ; unfortunately Mr Snell could not obtain a supply 
of alcohol adequate to preserve these. 

In the Insect collections among the Orthoptera, the Forficulidae 
are represented by eleven specimens, probably Anisolabis varicornis, 
Smith. Of the Blattidae, Periplaneta americana, Linn, and Leu- 
cophaea surinamensis, Fab. are among the five species previously 
recorded, whilst there are two other species in Mr Snell's collection 
at present undetermined. One species of Mantidae occurs in the 
island, viz. Polyspilota aeruginosa, Goeze, of wide distribution. Of 
the Gryllidae there are three species in the present collection: 
Acheta bimaculata, de Geer, found also in Africa and S. Europe; 
Curtilla africana, Beauv., found also in Africa, Asia, Australia, and 
New Zealand (introd.?); and a species of Ornebius near syrticus, 
Bolivar, but larger and more brightly coloured than the Seychelles 
specimens of this species. Besides the first of these, the Transit of 
Venus Expedition obtained three other species. Among the 
Phasgonuridae we have Conocephaloides differens, Serv. and 
Anisoptera iris, Serv., both previously recorded by the Transit of 
Venus Expedition. In addition the present collection contains a 
specimen of apparently another species of Anisoptera, resembhng 
A. conocephala, Linn., which occurs in Spain, Africa, and the 
Seychelles. There are two species of Locustidae: Locusta danica, 
Linn., a cosmopolitan species, and Chortoicetes rodericensis, Butl., 
described from Rodrigues, and not found elsewhere. 

The Neuroptera comprise a few specimens of a Termite, and 
specimens of one species of Hemerobiidae and of one species of 
Chrysopidae. It may here be mentioned that Dr H. Scott found a 
species of Termite working in the wood at the bottom of a fighter 
in Victoria harbour, Mahe, Seychelles. This indicates a possible 
explanation of the existence of Termites in such a locafity as 
Rodrigues, where any indigenous Termites would probably be 
exterminated by the fires which have repeatedly devastated the 
island. Until the Termites in Mr Snell's collection have been 
identified, no statement of course can be ventured regarding the 
distribution of this species. Mr Gulfiver, on the Transit of Venus 



Mr Snell, The Natural History of the Island of Rodrigues 289 

Expedition, secured one specimen of Myrmeleon obscurus, Rambur. 
This species was described from Mauritius, and is widely distributed 
in Africa. 

The Odonata consist of six species, as follows : 

Pantala flavescens. Fab., occurs in all the warmer parts of the 
world, but not in Europe. 

Tramea limhata, Desj., a very variable species of wide dis- 
tribution, described from Mauritius. 

Orthetrum hrachiale, P. de Beauv. Found elsewhere in Zanzibar, 
Congo, etc. 

Anax imperator mauricianus, Rambur. Agrees with a specimen 
in the Museum of Zoology, Cambridge, named by Campion. The 
species was also taken by Gulliver, on the Transit of Venus Ex- 
pedition. 

Ischnura senegalensis, Rambur. Widely distributed in tropical 
Asia and Africa. 

Agrion ferrugineum, Rambur. One specimen was taken by 
GulHver. The present collection contains several specimens. 

The collection of Hymenoptera, exclusive of Ants, contains two 
species of Tubulifera, eleven species of Aculeata, and approxi- 
mately 170 specimens (of about twenty species) of Parasitica. The 
two species of TubuUfera, for the identification of which I am 
indebted to Mr F. D. Morice of the British Museum of Natural 
History, are Chrysis {Pentachrysis) lusca, Fab., found also in India, 
Ceylon and Mauritius, and Philoctetes coriaceus, Dahlb., known 
also from East and South Africa. Of the Aculeata, the Formicidae 
are not yet determined, and a species of Halictus is at present 
unidentified. The remainder of the Aculeates are as follows : 

Megachile disjuncta, Fab. Common in India; recorded also 
from Mauritius. (M. lanata, Fab., is recorded by Smith as having 
been taken by Gulhver on the Transit of Venus Expedition.) 

Megachile rufiventris, Guer. Found elsewhere in East and South 
Africa, Mauritius and Seychelles; previously taken in Rodrigues 
by GulUver. 

Apis unicolor, Latr. Previously taken in Rodrigues by Gulhver. 
Found in the Seychelles, Amirantes, Chagos (Diego Garcia, Peros 
Banhos). Commoner in Madagascar. 

Odynerus trilobns, Fab. This species has not been previously 
recorded from Rodrigues. It is common and widely distributed, 
being known from Madagascar, Mauritius, Reunion and South 
Africa. 

Polistes macaensis, Fab. Previously taken by Gulhver and 
listed as P. hebraeus, Linn, There seems to have been considerable 
confusion over these names, as Cameron {Trans. Linn. Soc. (2), 
vol. XII, p. 71) hsts this species as P. hebraeus, Fab., stating that 
it is known from Rodrigues. Dr R, C. L. Perkins has, however, 

20—2 



290 Mr Snell, The Natural History of the Island of Rodrigues 

demonstrated the differences between the male P. macaensis and 
male P. hebraeus. (See Ent. Mo. Mag. (2), vol. xii, 1901, p. 264.) 
P. macaensis is known also from Seychelles, Amirantes, Chagos 
(Salomon Islands, Diego Garcia), and Mauritius. 

Scolia (Dielis) grandidieri, Sauss. I am indebted to Mr Rowland 
E. Turner of the British Museum of Natural History for the 
identification of this species. He states that the specimens under 
review are of "a form of D. grandidieri, Sauss. from Madagascar, 
with a few more punctures on the abdomen than in that 
species." 

Ampulex compressa, Fab., not previously recorded from 
Rodrigues. Common from Eastern Europe to China, and also in 
Africa. 

Passaloecus (Polemistus) macilentus, Sauss. Mr R. E. Turner has 
kindly identified this species for me. He states (in litt.) that "Mr 
Morice considers that Philoctetes coriaceus, Dahlb. is probably 
parasitic on this, as species of Passaloecus are often attacked by 
small Chrysids." The species was described from Madagascar. 

Sceliphron hengalense, Dahlb. ( = Peolpaeus convexus, Sm.). 
Mr Turner has confirmed my identification of this species. He 
adds: "This is probably an imported species, as species of the 
genus build mud nests on ships and are carried in that way from 
place to place." 

Trypoxylon errans, Sauss. Not previously recorded from 
Rodrigues. Found also in Mauritius and the Seychelles. 

There are approximately 750 specimens of Coleoptera, of pos- 
sibly 100 species; 640 specimens of Diptera, of at least seventy 
species; and 360 specimens of Hemiptera, of some forty-five 
species. These have not yet been critically examined. 

In the Lepidoptera, seven species of Butterflies were collected 
by Mr Grulliver on the Transit of Venus Expedition. Of these one 
species is not represented in Mr Snell's collection, viz. Hesperia 
forestan, Cr. The list of Butterflies is as follows : 

'^Melanitis leda, Linn. '\*Zizera lysimon, Hiibn, 

*Danais chrysippus, Linn. "f^Polyommatus boeticus, Linn. 

Precis rhadama, Boisd. *Tarucus telicanus, Lang. 

*Hypolimnas misippus, Linn. Parnara borbonica, Boisd. 
"f^Atella phalantha, Drury 

Among the Moths (Heterocera), exclusive of the Pyralidae, 
Tortricidae, and Tineidae, though Gulliver's collection contained 
only twelve species, five of these were species not represented in 
Mr Snell's collection, Mr Snell obtained three species of Sphingidae, 

* Of wide distribution. 

■]• Not previously recorded from Rodrigues. 



Mr Snell, The Natural History of the Island of Rodrigues 291 

one species of Arctiidae, twenty-five species of Noctuidae, and 
two species of Geometridae, as follows : 

^Acherontia atropos, Linn. "f^Erias insulana, Boisd. 
'f*Herse convolvuli, Linn. *An,ua tirhaca, Cr. 

"I" Hippotion aurora, Roth. & Jord. Achaea trapezoides, Guen. 
i*Utetheisa pulchelloides, Hamps. Achaea finita, Guen. 
'f*Chloridea obsoleta, Fab. *Parallelia algira, Linn. 

"f^Agrotis ypsilon, Linn. *Chalciope hyppasia, Cr. 

■f*Cirphis loreyi, Dup. "f^Mocis undata', Fab, 
■j" Cirphis leucosticha, Hamps. *Phytometra chalcytes, Esp. 

( = insulicola, Saalm.) *Cosmophila erosa, Hiibn. 

"f^Perigea capensis, Guen. "f^Dragana pansalis, Walk. 

^^Eriopus maillardi, Guen. ^*Magulaha imparata, Walk. 

*Prodenia litura, Fab. ^^Hydrillodes lentalis, Guen. 

*Spodoptera abyssinia, Guen. "f^Hypena masurialis, Guen. 

Athetis expolita, Butl. ^*Hyblaea puera, Cr. 

"fEublemma apicimacula, Mab. f*Craspedia minorata, Boisd. 

*Amyna octo, Guen. ^*Thalassodes quadraria, Guen. 

The five species collected by Mr Gulliver and not represented 
in the present collection are as follows: 

*Argina cribraria, Clerck. (Hypsidae). 

*Nodaria externalis. Walk, (redescribed as Diomea bryophiloides, 
Butl.) (Noctuidae). 

Pericyma turbida, Butl. (Noctuidae). Peculiar to Rodrigues. 
*Achaea catella, Guen. (Noctuidae). 
*Mocis repanda, Fabr. (Noctuidae). 

Butler listed a species as Laphygma cycloides, Guen., apparently 
in error, as Sir George Hampson has in his Catalogue placed the 
record under Spodoptera abyssinia, Guen. 

There are about 180 specimens, of some thirty species, of 
Micro-lepidoptera. These have not yet been worked out. 

The collections made by Mr Snell are of importance as showing 
more definitely the relations of Rodrigues with the other islands 
in the vicinity. Undoubtedly the fauna has, with the flora, suffered 
considerably from the devastating effects of the fires which have 
so frequently swept the island, but investigation of the collections 
of the groups not yet worked out, will undoubtedly show that con- 
siderable traces of the indigenous fauna still exist, and will serve 
to indicate with greater accuracy the affinities of Rodrigues with 
the neighbouring islands. 

*. Of wide distribution. 

f Not previously recorded from Rodrigues. 



292 Mr Snell, The Natural History of the Island of Rodrigues 

Bibliography. 

Legtjat, Franqois. Voyages et aventures en deux lies Desertes des Indes 

Orientales (1650-1698). 
Grant, C. History of Mauritius and the neighbouring Islands (1801). 
(Pridham, C?) An Account of the Island of Mauritius and its Dependencies 

(1842). 
Strickland and Melville. The Dodo and its Kindred (1848). 
HiGGiN, E. "Remarks on the Country, Products and Appearance of the 

Island of Rodrigues, with opinions as to its future Colonization." Joum. 

Boy. Geogr. Soc. xix, Pt I, 1849, p. 17. 
Newton, E. "Notes of a Visit to the Island of Rodrigues." Ibis, vol. i, (new 

series), 1865. 
Balfour, I. B. and others. "An Account of the Petrological, Botanical and 

Zoological Collections made in Rodrigues during the Transit of Venus 

Expeditions in 1874-5." Phil. Trans. Boy. Soc. vol. CLXvm (extra volume), 

1878. 
Oliver, S. P. (Edit.) The Voyage of Frangois Leguat of Bresse (2 vols. 1891). 

(Transcribed from the Original English Edition for the Hakluyt Society.) 
Gardiner, J. S. "Islands in the Indian Ocean, Mauritius, Seychelles, and 

Dependencies, 1914." In the volume on Africa, Oxford Survey of the 

British Empire. 
Newton, Sir E., K.C.M.G. "The Birds of the Mascarene Islands." Presi- 
dential Address. Trans. Norfolk and Norwich Naturalists' Society, vol. iv. 
Encyclopaedia Britannica. 



Miss Haviland, Note on the Life History of Lygocerus 293 



Preliminary Note on the Life History of Lygocerus {Procto- 
trypidae), hyperparasite of Aphidius. By Maud D. Haviland, 
Fellow of Newnham College. (Communicated by Mr H. H. 
Brindley.) 

[Read 10 November 1919.] 

Plant lice are frequently parasitized by certain Braconidae of 
the family Aphidiidae. The parasite oviposits in the haemocoele 
of the aphis, and the larva, during development, consumes the 
viscera of the host. At metamorphosis nothing remains but the 
dry skin, within which the Aphidius spins a cocoon for pupation. 

At this stage, the Aphidius itself is liable to be parasitized in 
turn by certain Cynipidae, Chalcidae, and Proctotrypidae. The 
two former are known to be hyperparasites, but the Proctotry- 
pidae have hitherto been considered doubtful, although some 
writers have suspected that they are hyperparasites of the Aphidius, 
and not parasites of the aphis. Gatenby in his paper: " Notes on the 
Bionomics, Embryology, and Anatomy of certain Hymenoptera 
Parasitica" (Journ. Linn. Soc. 1919, vol. xxx, pp. 387-416) says: 
". . .1 am inclined to support the view that the Proctotrypid is a 
parasite, and not a hyperparasite." 

The following is a summary of some observations made in the 
summer of 1919, on two Proctotrypids of the genus Lygocerus. 
I am much indebted to Professor Kieffer, who has kindly identified 
them for me as L. testaceimanus, Kieff., hyperparasite of Aphidius 
salicis, Hal., parasite of Aphis saliceti, Kalt., from the willow; and 
L. cameroni, Kieff., hyperparasite of Aphidius ervi, Hal., parasite 
of Macrosiphum. urticae from the nettle. The following notes 
probably apply to both species, but the observations were made 
more especially upon the latter. It was found also that in cap- 
tivity L. testaceimanus would oviposit on Aphidius ervi. The 
Proctotrypids do not confine their attacks to the Aphidiidae, but 
their larvae may also be found feeding on the larvae of other 
Chalcid or Cynipid hyperparasites of that family ; and indeed once 
or twice were observed upon dead pupae of their own species. One 
remarkable instance of hyperparasitism came under notice. An 
aphis {Macrosiphum urticae) was parasitized by an Aphidius (A. 
ervi). The latter had been hyperparasitized by a Chalcid, of species 
unknown, which immediately after pupation had been attacked 
by another hyperparasite, either Chalcid or Cynipid, whose identity 
is not yet determined. This second hyperparasite in turn had been 
attacked by Lygocerus cameroni, and the larva was in the second 
instar when the cocoon was opened. We may ask, where are the 
limits to this hyperparasitism? 



294 Miss Haviland, Note on the Life History of Lygocerus 

Lygocerus cameroni was fairly common round Cambridge in 
1919, from mid-July to the end of August. The female selects an 
aphis-cocoon containing a full-grown larva or newly transformed 
pupa of Aphidius, and runs round it with much excitement, 
tapping it with her antennae. Oviposition takes from 30-60 
seconds, the insect meanwhile standing either on the top of the 
cocoon facing the anterior end, or on the leaf behind, with her back 
to it. Either way, the ovipositor is brought into the angle of the 
host's body, as it lies curled inside. Sometimes two or three eggs, 
the result of successive ovipositions by different females, are 
found on the same host. 

The egg, which is laid on the upper surface of the abdomen of 
the Aphidius, measures -25 x -10 mm. It is translucent, white, 
and elliptical, with marked longitudinal striae of the chorion, and 
a minute stalk at one end. Treatment of the egg with lacto-phenol 
and cotton-blue showed the presence of bodies resembling the 
symbiotes from the pseudovitellus of Aphides. The egg hatches in 
about twenty hours. 

The larva of the first instar is a maggot shaped form, with 
thirteen body segments and a head furnished with two minute 
papillae. The mouth, which is circular and very small, contains 
two simple chitinous mandibles set well behind the hood-hke 
labrum and the labium. The mid-gut, which at this stage does not 
communicate with the rectum, is large and globose, and its con- 
tents tinge the transparent body pale yellow. Later on, when the 
host dies, they become brown. The tracheal system consists of two 
lateral longitudinal trunks, united by an anterior and posterior 
commissure. When newly hatched, there are two open spiracles 
between the first and second and on the fourth segments, but 
soon afterwards the spiracles of the third and fifth segments 
become functional. The larva is active and crawls over the host's 
body. This instar lasts from twenty to twenty-four hours, and the 
dimensions are about -45 x -22 mm. 

The larva of the second instar differs from that of the first 
chiefly in the size, which is -70 x -35 mm., and in the tracheal 
system. The ramifications of the latter are more numerous, the 
dorso- ventral branches of the second segment become visible, and 
the spiracular trunks of segments six, seven, and eight appear, 
though their spiracles are not open. The duration of this instar is 
about thirty-six hours, and at this time the host usually dies, and 
its body becomes blackened and shrunken. 

In the third instar, the papillae on the head disappear, the body 
becomes more globose, and the greater proportionate development 
of the three first segments causes the head to be bent round to the 
ventral side. The dimensions are about 1-00 x -75 mm. The spiracles 
of the sixth, seventh and eighth segments open, and the spiracular 



{Prodotrypidae), hyperparasite of Aphidius 295 

trunk of the second segment becomes visible. In addition, two 
short spiracular trunks can be made out on the ninth and tenth 
segments; but these never become functional, and they disappear 
in the later stages of development. This instar lasts from about 
thirty-five to forty hours. 

In the fourth instar, which lasts about two days, the Procto- 
trypid grows rapidly, and when mature measures 1-67 x -83 mm. 
The remainder of the host is quickly consumed, and, just before 
metamorphosis, the mid-gut opens into the rectum, and its con- 
tents are voided into the cocoon. The larva is active and wriggles 
about freely inside the aphis skin, aided possibly by a curious 
caudal appendage; and by these movements the faeces, together 
with the host's skin, are kneaded into a moist compact pellet on 
the ventral side of the body. 

The full grown larva is yellowish white, and each segment has 
a double row of short chitinous spines. The thorax is large and 
broad, while the abdominal segments taper away somewhat to the 
eleventh, which bears a short stout appendage furnished with 
spines. The head is turned completely under the thorax, and the 
tracheal system does not differ essentially from that of the pre- 
ceding instar. No larval antennae nor maxillary nor labial palpi 
seem to exist at this stage. 

Lygocerus does not produce silk, but pupates in the cocoon made 
previously by the Aphidius inside the skin of the aphis. The period 
of pupation is fourteen to sixteen days. When ready to emerge, 
the imago gnaws a hole somewhere on the upper side of the cocoon, 
and creeps out. So far, no parthenogenetic ovipositions have been 
observed, and two broods, certainly, and possibly more, may occur 
in the season. The life of the imagoes is generally five or six days, 
but they may live as many as ten. Examples in captivity were 
observed to feed on the sap oozing from cut leaves, and on honey- 
dew dropped by the aphides, but they seemed to live as long and to 
remain as vigorous when no food was supplied. 



296 Mr Warburton, Note on the solitary wasj), 



Note on the solitary wasp, Crabro cephalotes. By Cecil 
Warbueton, M.A., Christ's College. 

[Read 10 November 1919.] 

Last summer a small colony of C. cephalotes took possession in 
my garden of a log of elmwood which was kept as an example of 
a woodpecker's nest. The entrance hole of the woodpecker was 
there, and just below it the log had been sawn through so that the 
internal cavity could be examined. 

The first advent of the wasps was not noticed, but in the first 
week of August a wasp was observed entering the hole, and this 
led to an investigation of the log, which presented signs of boring 
in the half-decayed heart-wood. One of the wasps had attacked 
the log from the top and its operations could be noted with more 
or less exactness, but the others passed in and out by the wood- 
pecker's hole, and it was impossible to recognise individuals or to 
follow their work without constantly disturbing it by opening up 
the log, with the risk of inaccurately replacing the two halves. 
The log was nevertheless opened several times during the first half 
of August, but it was then thought better to let the wasps finish 
their work without further disturbance. 

That the wasps are not easily diverted from their labours the 
following facts sufficiently demonstrate. The log was moved several 
yards, to a spot more convenient for observation. The wasp 
working on the top (hereafter referred to as wasp No. 1) was 
captured in a glass tube and examined for identification, but on 
being liberated continued working as before. Close observation, 
with a hand lens, did not deter this wasp from entering its burrow 
without hesitation in the course of its operations, nor were the 
other wasps disconcerted by the removal of the lid on several 
occasions at an early stage of their work. As a rule no attention 
was paid to anyone sitting silently near the log, but it must be 
recorded that on one occasion a wasp returning with a fly appar- 
ently objected to the dress — light with dark spots — of a lady sitting 
near at hand, and after a close investigation from many points of 
view, retired instead of entering the log. To ascertain if wasp No. 1 
were at home or not I was in the habit of placing a stout straw in 
its burrow — protruding an inch or more. One would have thought 
that on returning home and finding such an object impeding its 
entrance the insect would manifest some perturbation and either 
refuse to enter or take some measures to remove the obstacle. It 
did nothing of the kind, but absolutely disregarded the straw, 
pushing past it even when laden with a fly. It was several times 



Crabro cephalotes 297 

ejected together with the frass from new tunnelling operations, 
but never otherwise. 

Continuous observation of work that went on for many hours 
a day for about three weeks was, of course, impossible, but on 
several days, especially during the week Aug. 18 — 25, operations 
were watched for spells of an hour or two at a time, and the exact 
times of ingress and egress carefully noted. The notes which 
immediately follow especially concern wasp No. 1. 

The hole was sometimes clear, sometimes choked with "saw- 
dust." After watching for a time the "sawdust" would be seen to 
heave up and form a mound over the hole. Then the wasp would 
emerge and proceed to remove the frass, butting it away from the 
neighbourhood of the hole with its head. Sometimes in the course 
of its excavations the wasp would emerge, fly away for a time, and 
return empty handed to resume its digging. 

On Aug. 19 it was seen to be carrying home flies, and the per- 
formance was watched for an hour, and the following times were 
noted: 

Returned with fly, 9.37, 9.48, 10.18, 10.31. 
Emerged, 9.40, 9.55, 10.25, 10.39. 

Thus four flies were caught in the hour, and the times spent in 
capturing three of them were 8', 23' and 6' respectively, while 
3', 7', 7' and 8' were occupied in packing the four flies into the 
burrows. To find, capture, paralyse and bring home the right kind 
of fly in six minutes strikes one as a remarkable feat. From further 
observations it appeared that the operation usually occupied about 
a quarter of an hour. None but "hover flies" (Syrphidae) were 
taken by any of the wasps, and the prey was generally Syrphus 
halteatus, a species almost as large as the wasp itself. It was, 
nevertheless, carried with perfect ease, arranged longitudinally, 
head foremost beneath its captor, and, I believe, venter to venter. 
No preliminary examination of the hole was ever made before 
carrying the fly in, such as Fabre has recorded in the case of some 
wasps. About noon on Aug. 21 this wasp apparently ceased 
workino;. There were no signs of activitv that afternoon nor the 
following morning. 

On Aug. 22 about 3 p.m. a wasp (wasp No. 2) was seen to come 
out of the woodpecker's hole and alight on the top of the log, 
which it proceeded to explore. It found No. I's burrow and 
entered it for a short distance, after which it flew away. Nothing 
further was noted till the evening of Aug. 23, when on returning 
home at 5.30 I noticed a heap of frass on the top of the hole. At 
6.20 a wasp arrived and after pointing at the main entrance, 
seemed to change its mind and alighting on the top, entered No. 
I's hole. Its behaviour convinced me that it was not No. 1, but it 



298 Mr Warburton, Note on the solitary wasp, 

might very well be wasp No. 2. Anyhow it entered the burrow, 
and by 7.50 it had turned out more "sawdust" containing several 
of the flies so carefully stored up by wasp No. 1 ! The explanation 
that first occurred to one was that the wasp wanted to dig, and 
naturally found it easier to work w^here someone had been before. 
Such a defective instinct would, however, militate against the 
preservation of the race. Moreover there were no further develop- 
ments, and No. 2 remained satisfied with undoing some of No. I's 
work. A wild suggestion did occur to me, which I will give for 
what it is worth. Is it possible that one of those working from the 
interior became aware of operations from the outside which might 
imperil the results of its own labours, and proceeded to put a 
stop to them? 

With regard to the remaining wasps, which entered by the 
woodpecker's hole and worked from the inside, the following notes 
may be given. 

The earlier hasty inspections of the interior showed that the 
cavity of the woodpecker's nest was being gradually filled with the 
"sawdust" of their workings, and conspicuous on the "sawdust" 
were a number of Syrphid flies, apparently dead. At the final 
investigation at the beginning of October about a hundred and 
twenty of these derelict flies were found in the central cavity, and 
as there were certainly not more than six wasps at work at any 
time, and as two were early captured and retained for identification, 
it is probably safe to estimate the average numbers of the wasps 
responsible for discarding them at five. This allows twenty-four 
discarded flies to each wasp — about six hours strenuous labour by 
each insect entirely wasted! As wasp No. 1 was never seen to 
discard a captured fly this phenomenon was apparently attributable 
to the conditions prevailing inside. There all the burrows com- 
menced with a horizontal boring at the junction of the two sections 
of the log, at some little distance from the main opening. After 
alighting at the main entrance they had, therefore, either to fly 
across or to crawl round the central cavity, and it seems as though 
a number of flies had been accidentally dropped. It would be 
quite in keeping with what has been observed in the case of allied 
insects that a wasp which had accidentally dropped a fly should 
make no attempt to retrieve it, but should simply go away and 
catch another. These discarded flies were in any case very useful 
as evidence of the particular prey selected by Crabro cephalotes. 

At the beginning of October some of these flies had been 
reduced to fragments by other predaceous creatures, but of 113 
recognisable specimens 60 were S. halteatus. 

My friend Mr N. D, F. Pearce very kindly undertook to identify 
the remainder for me and he finds among them five species of 
Syrphus, three of Platychirus, two of Melanostoma, and one of 



Crabro cephalotes 299 

Rhingia, Catabomba and Helophilus respectively. No family of 
flies except the Syrphidae was represented. The complete list is as 
follows : 

Syrphus balteatus 60 

S. luniger 5 

S. vitripennis 4 

S. corollae 4 

S. auricoUis 3 

S. albistr ictus 1 

Platychirus albwianus 9 9 

P. scutatus ? 2 

P. peltatus 1 

Melanostoma mellinum 7 

M. scalar e ? 2 

Rhingia campestris 13 

Catabomba pyrastri 1 

Helophilus pendulus 1 

113 

Early in October the log was thoroughly explored, and an 
attempt was made to follow out the windings of the galleries, 
but the extreme friability of the decaying heart-wood made this 
very difficult. 

The first thing that struck one was the absence of any attempt 
to seal or mask the tunnels which were entirely open to any 
chance intruder. Indeed a family of wood-lice was found three 
inches down the tunnel of wasp No. 1. There was nothing to prevent 
any enemy from entering. While at work the wasps had never 
manifested any interest in other insects in the neighbourhood of 
their burrows, nor did they finally make any provision for keeping 
them out. While watching the operations of wasp No. 1 a few 
insects had been seen to enter the tunnel, including Phoridae, one 
of which was secured, and a Muscid fly ( ? Tachina) and an Ichneu- 
monid which unfortunately evaded capture. 

The main tunnels were clear, and penetrated the wood for 
several inches, with abrupt turnings on no definite plan. From 
these proceeded side galleries in which were found " sawdust," the 
debris of flies, and the brown cocoons containing the fully-fed wasp 
larvae. Sections of the log showed that these were dotted here 
and there throughout the soft heart- wood precisely hke the raisins 
in a Christmas pudding. 



k 



300 Mr Aston, Neon Lamps for Stroboscopic Work 



Neon Lamps for Stroboscopic Work. By F. W, Aston, M.A., 
Trinity College (D.Sc, Birmingham), Clerk-Maxwell Student of 
the University of Cambridge. 

[Read 19 May 1919.] 

For the accurate graduation and testing of revolution indicators 
and similar technical purposes the stroboscopic method is probably 
the most reliable. This depends on the fact that if a rotating disc is 
illuminated N times per second by very short flashes, a regular 
figure drawn symmetrically on the disc will appear at rest when 
the number of revolutions of the disc per second is some exact 
multiple or submultiple of N depending on the number of sides of 
the regular figure. 

The value of N — in practice 50 — can be s^t and easily kept 
extremely constant by the use of an electrically driven tuning-fork 
so that the success of the method rests principally upon the 
illuminating flashes ; its accuracy will depend upon their shortness 
of duration and brightness; its convenience as a practical method 
upon their brightness and quality as affecting the eye of the 
observer. 

The first experiments were tried with naked Ley den jar sparks 
obtained from the secondary of an ordinary ignition coil, the 
tuning-fork being introduced into the primary circuit as an 
interrupter. These showed the principle of the method to be 
excellent but spark illumination left much to be desired; it was 
noisy, feeble in intensity, and being mostly of short wave-length, 
caused rapid and excessive eye-strain even when used in a dark 
room. 

The remarkable properties of Neon seemed to offer an almost 
ideal solution of the illumination problem. A form of lamp to 
replace the spark was therefore devised which appeared likely to 
give good results and several of these were filled from the author's 
stock of Neon at the Cavendish Laboratory. The success of these 
lamps was immediate, eye-strain disappearing completely. The 
present paper is a description of the lamps and their behaviour 
during continuous use. 

The Form of Lamp. 

The original form of the lamp, which it has not been found 
necessary to alter materially, is shown in the sketch. As, in the 
discharge in Neon, nearly all the light is in the "Positive Column" 
and its brightness increases with the current density, the lamp 
was designed to give a positive column as long and narrow as 



Mr Aston, Neon Lamps for Stroboscopic Work 



301 



possible consistent with the potential available in the spark, and 
consists essentially of two relatively large spaces containing the 
electrodes connected by a very long capillary tube which is the 
counterpart of the filament in an ordinary glow lamp. In the lamps 



flNOoe: 



-h 



C<^THOOE 

Neon vacuum lamp for Stroboscopic work. 
Two-thirds actual size. 

in use the filament is about 60 cm. long by 1 mm. diameter and is 
coiled up inside the space containing the anode. This was done for 
convenience and strength, but it has another and important 
advantage, for this type of construction is strongly unsymmetrical 
to the discharge, allowing it to pass much more easily in the direc- 



I 



302 Mr Aston, Neon Lamps for Strohoscojnc Work 

tion indicated in the figure than in the opposite, hence it effectually 
stops the "reverse" current from the secondary of the coil. 

Other important results depending on the length of the fila- 
ment will be discussed later, it should be roughly one hundred 
times the length of the spark the coil is capable of giving in air 
when running on the tuning-fork break. 

It is hardly necessary to state that the shape into which the 
filament is wound is not in the least essential and could be varied 
to any extent in lamps for special purposes. 

The electrodes are of aluminium and may be of any form so 
long as they are not too small. 

Method of Filling Lamps. 

As Neon, like the other gases of the Helium group, has the 
remarkable property of liberating gas from aluminium electrodes 
which have been completely run in for other gases, the operation 
of filling necessitates the contamination of a comparatively large 
volume of Neon, so that this can only be done economically and 
conveniently where liquid air is available for re-purifying. 

So far all the lamps have been filled on the author's Neon 
fractionation apparatus at the Cavendish Laboratory^, The gas 
for filling is contained in charcoal cooled in liquid air. A quantity 
is admitted to the exhausted lamp which is then sparked at a 
pressure of 1 to 3 mm. with a small coil for a time. The dirty gas 
is then pumped off with a Toepler mercury pump, a fresh supply 
of pure gas admitted and the tube run again. These operations 
are repeated until spectroscopic and other observations show the 
desired conditions of purity have been reached and are not altered 
seriously by prolonged running. The full charge of 5 to 10 mm. of 
gas is now let in and the lamp sealed off. The whole operation takes 
about 3 hours, three lamps being filled at once. The pressure, 
purity and time of running in are all matters of some nicety as 
will be seen from consideration of the life of the lamp. 

Life of the Lamps. 

Apart from accident the lamps are serviceable until the pressure 
of gas within them becomes too low for the spark to light them 
adequately. Their life appears to consist of two distinct periods, 
the first during which chemically active impurities derived from 
the electrodes and walls of the tube are being slowly and completely 
eliminated (at least as far as a spectroscopic observation goes) and 
the second during which sputtering of the cathode takes place and 
the inactive Neon itself slowly disappears until the pressure gets 
too low for use. During the first period the luminosity steadily 

1 V. Lindemann and Aston, Phil. Mag. sxxvii, May 1919, p. 527. 



Mr Aston, Neon Lamps for Stroboscopic Work 303 

improves, remaining almost constant afterwards till near the end 
of the second period when it rapidly decreases. 

The first set of lamps were filled with very carefully purified 
Neon at 1-2 mm. pressure and run till sputtering had commenced 
before being used; they may therefore be considered to have had 
no first period at all. These lamps had a life of 500-1000 hours. 

Experiments soon showed that the less preliminary running 
and the higher the pressure of filling the longer the life would be, 
but on the other hand, if the preliminary running is not sufficient 
the impurities derived from the electrodes turn the light' of the 
lamp a dull grey and render it absolutely useless and pressures 
above 10 mm. are not advisable as these increase the spark 
potential of the lamp too much. 

One lamp was actually so nicely balanced in these respects 
that though it became grey and useless after about 1 hour's use it 
completely recovered its original brightness after a day's rest. This 
is clearly a case of carbon compounds being given off by the elec- 
trodes while running, which are reabsorbed on standing and there 
is little doubt that were it worth while very prolonged running 
would render this lamp quite satisfactory. Very slow production 
of gases from the electrodes is advantageous, as prolonging the 
first period of the life, so that these should be of a fairly solid 
pattern. 

So far, the best results have been obtained from a batch of 
lamps filled at about 10 mm. pressure, some with pure Neon, 
some with a mixture of Neon and about 10 per cent. Helium. 

One of the latter had a working life of well over 3000 working 
hours. Helium disappearing from its spectrum after the first few 
hundred. 

As there is every reason to assume that for any given lamp the 
life is determined by the total number of coulombs passed through 
it, the light obtained per coulomb should be arranged to be a 
maximum. This will be the case when the filament is made as long 
as possible, consistent with the potential available from the coil. 

Cause of Disappearance of Gas from the Lamps. 
The exhaustion of gas by continuous running has long been 
observed in the case of spectrum discharge tubes. It is doubtless 
allied to the phenomenon of "Hardening" in X-ray bulbs, but 
difi'ers from the latter in that under the relatively high pressures 
in spectrum tubes, and the Neon lamps under consideration, the 
mean free-path of a charged molecule is so small that it can only 
fall freely through a potential of a few hundred volts and so never 
attain the very high velocities reached in the X-ray bulbs which 
are supposed to cause the gas molecules to become permanently 
embedded in the glass walls. 

VOL. XIX. PART VI. 21 



304 Mr Aston, Neon Lamfs for Strohoscopic Work 

The disappearance of gases of the HeHum group in spectrum 
tubes is invariably associated with sputtering of the electrodes 
which, at high pressures, only takes place when the gas is spectro- 
scopically free from chemically active gases. It is generally sup- 
posed that the gas so disappearing remains embedded or adsorbed 
in the layer of sputtered aluminium on the sides of the tube near 
the cathode, the idea of true chemical combination not being 
acceptable without very rigorous proof. 

In order to obtain information on this point, a completely run 
out specimen of the first batch of lamps, which was of course very 
heavily sputtered, was taken for test. First the sputtered cathode 
end was gradually heated to near the softening point of the glass 
(when it cracked) without any substantial or apparent increase in 
the internal pressure of Neon. The end was then cut ofi, broken into 
small pieces and heated in a quartz tube in a high vacuum apparatus 
provided with a spectrum tube. At a temperature about the 
softening point of the glass a good deal of gas was released which 
showed the hydrocarbon spectrum (but may nevertheless have 
contained some Neon as this is easily masked) ; this gas was pumped 
off and on heating further to a red heat, as the glass started to 
melt, Neon was given off, the spectrum showing quite clearly. 

Apparatus for measurement and analysis of the gas so released 
was not available, but it is hoped to repeat this interesting experi- 
ment, which shows definitely that the Neon is contained either in 
the sputtered aluminium or very near the surface of the glass so 
that it is released by heat. 

Use of other Gases instead of Neon. 

Ordinary chemically active gases give very feeble illumination, 
CO being about the best. Helium gives a bright discharge but not 
nearly so valuable in quality for visual work as Neon ; its presence 
as an impurity in the latter gas renders the discharge more rosy 
red but up to 10 per cent, does not affect its brightness seriously. 
Mercury vapour as used by C. T. R. Wilson in his photography of 
ionisation tracks would probably give very bright flashes, but the 
fact that the lamp has to be kept very hot is a serious objection. 

Reason for Superiority of Neon. 

The brilliant orange-red glow of the discharge in Neon is com- 
posed almost entirely of lines in the region 5700-6700 a.u. and is in 
such striking contrast to sunlight that strohoscopic observations 
can even be done in broad daylight if necessary, the ordinary 
appearance of the rotating disc having merely a grey background 
added, looking bluish by contrast. 

The actual amount of light radiated per unit of energy, i.e. 



Mr Aston, Neon Lamias for Stroboscopic Work 305 

the real efficiency of the discharge in Neon, is not markedly greater 
than that in e.g. mercury vapour, but the apparent efficiency is 
enormously enhanced by the fact that it consists so largely of red 
light. Victor Henri and J. L.desBancels have shown (" Photochemie 
de la Retine," Jl. Phys. Path, xiii, 1911) that the Fovea Centralis of 
the eye is immensely more sensitive to red light than the outlying 
portions of the retina^, thus a Neon lamp as a source of general 
illumination is very disappointing, but when viewed directly 
appears surprisingly bright. As the spinning disc of the stroboscope 
subtends a comparatively small angle the Fovea is the only part 
of the observer's eye used in testing, which is probably the reason 
for the eye strain with the spark. 

Nature and Duration of the " Working Flash.'" 

If one analyses the flash of a short spectrum type Neon tube in 
a rotating mirror it is seen to consist of two separate parts, an 
extremely short flash followed by a flame or "arc." The first is 
probably due to the simultaneous ionisation of the gas throughout 
the whole length of the tube, the second to the further carriage of 
current by the ions formed during the first. The structure of the 
latter, which appears to consist of bright striations travelling from 
anode to cathode at velocities of the order of that of sound in the 
gas, is of great theoretical interest and is at present under investi- 
gation. Discussion of its nature is needless in the present paper 
for its duration being of the order of thousandths of a second it is 
useless for stroboscopic work and, by the employment of a suffici- 
ently long filament tube, it can be eliminated altogether. In a 
lamp properly proportioned to the power of the coil in use the 
v.'hole energy of the discharge is absorbed in the first flash. In 
order to get some idea of the duration of this "working flash" the 
following experiment was performed. 

A plain mirror, silvered outside to avoid double images, was 
mounted vertically on the axis of a large centrifuge and the image 
in this of the Neon lamp at a distance of 3 metres was observed 
by means of a telescope with a micrometer eye-piece. Each 
division in the micrometer subtended 4-2 x 10^^ radians and when 
the centrifuge was running at 3500 revolutions per minute corre- 
sponded to 5-75 X 10~' seconds. 

The lamps were run with the tuning-fork attachment used in 
actual testing and were viewed directly and also through ground 
glass with a V-shaped slit to be certain of getting the effect of the 

' The difference of retinal effect between red and green light can be easily ob- 
served by looking at an ordinary luminous wrist watch in the faint red light of 
a photographic dark room. On shaking the watch so sluggish is the green light in 
recording its position on the retina compared with the red that the figures seem 
to be shaken completely off the dial, giving a most curious and striking effect. 

21 — 2 



306 Mr Aston, Neon Lamj)s for Sirohoscopic Work 

total duration of the flash. In neither case was the fuzziness of the 
image of a measurable order. After careful observation under 
good conditions the conclusion of three observers was, that it was 
probably less than one-tenth of a division and certainly less than 
one-fifth. This gives the maximum duration of the working flash 
as one-ten-millionth of a second, so that it can be taken as perfectly 
instantaneous for the purpose employed. 

Other Technical Afflications . 

Of the many uses besides measuring velocity of rotation to 
which Neon lamps may be put with advantage in engineering and 
other problems it is sufficient to mention two in which they have 
been very successful. Any rapidly rotating mechanism such as an 
airscrew, if illuminated by a lamp the break of which is operated 
mechanically at each revolution, will appear at rest, flicker being 
small at speeds well over 1000 r.p.m., so that strains or movement 
of parts can be examined with great accuracy under actual working 
conditions. 

A still more striking effect can be obtained by illuminating a 
high speed internal combustion engine by a lamp whose break is 
operated mechanically at e.g. 99 breaks per 100 revolutions of the 
engine shaft by the use of a creeping gear. The engine then appears 
to be rotating quite smoothly at one-hundredth its normal speed 
so that such instructive details as the movements of the valves 
and springs, the bouncing of the former on their seats, etc., can be 
studied with ease. 

It is of course necessary for the speed of rotation to be fairly 
rapid to give appearance of continuity to the eye and in conse- 
quence one cannot apply this method to the analysis of such a 
thing as the movement of a chronometer escapement. 

As the technical importance of Neon lamps is rapidly on the 
increase it is very desirable that liquid air engineers in this country 
should consider the erection of a fractionating plant for recovering 
the gas from the air (which contains -00123 per cent, by volume) 
such as has been used with such success by Mons. Georges Claude 
of Paris, to whom the author is indebted for the Neon with which 
these experiments were performed. 



Mr Harrison, The pressure in a viscous liquid etc. 307 



The ^pressure in a viscous liquid moving through a channel unth 
diverging boundaries. By W. J. Harrison, M.A., Fellow of Clare 
College, Cambridge. 

[Read 24 November 1919.] 

If non- viscous liquid is flowing along a tube having a cross- 
section which is increasing in area in the direction of flow, the 
pressure will also increase, in general, in the same direction. On 
the basis of this remark an explanation has been given of the 
secretory action of the kidneys. The author's attention was drawn 
to this explanation by Dr Ffrangcon Roberts. The physiological 
aspect of the question and a more detailed numerical consideration 
will be dealt with by Dr Roberts and the author in a separate 
paper. 

In the present paper two problems are considered, viz. the flow 
of liquid in two and three dimensions when the stream lines are 
straight lines diverging from a point. 

Two-dhnensional 'problem. 

Let the boundaries of the channel be ^ = ± a, where (r, d) are 
two-dimensional polar coordinates. The motion in which the stream 
lines are straight lines passing through the origin has been ob- 
tained by G. B. Jeffery^. With a slight change of notation the 
results of his solution are as follows. 

Let the velocity at any point be ujr, where m is a function of d 
only. Then 

2 A ^^** , 

U^ = — 4:VU — V -77i5- + a, 

dO^ 

where v is the kinematic coefiicient of viscosity, and a is a constant 
of integration. Whence 

u = — 2i^ (l — m^ — m^k^) — Qvkhn^ sn^ {md, k), 

where k and m are constants, which may be determined from the 
conditions that u must vanish at ^ == ± a, and that the total rate 
of flux may have a given value. Instead of the latter condition it 
is simpler to assume that the velocity is given for ^ = 0, i.e. u = Uq 
for 6 = 0. 

Thus the conditions are 

— 2v {1 — m^ — m^k^) = Uq, 

(1 — ni^ — m^k^) + 3kh)i^ sn^ {ma, k) = 0. 

1 Phil. Hag. (6), vol. xxix, p. 459. 



308 Mr Harrison, The pressure in a viscous liquid 

These may be written 

m2 = (1 + uJ2v)l{l + B), 
'I + uJ2v\^ J 1 + F 



sn' 



(^-^1?)-^ 



3F (1 + 2i//wo) ' 

If the values of Uq and a be given, the last equation serves for 
the determination of h. Writing h-y = l/k, the equation has the 
same form in k-^ as in Jc. Hence, if k is a solution, l/k is also a 
solution. Therefore, of real values of k, it is only necessary to 
consider such that satisfy ^ ^ ^ 1 . 

Treat a as small, and assume that ( — = — %-^ — ) a is also small. 

V l + k^ J 

(1 + 7(^2)2 

We have a^ = ^-.^ .^ \, , — tk— . • 

3P (2 + 2v/uq + uJ2v) 

The least value of a for a given value of Uq/2i', if k is real, is given 
hj k = 1. In this case, if Uq/2p = 1, a^ = ^, a = -58. This value of 
a is not small enough for the approximation to hold good. Put 
^ = 1 and 2v/uq = 1 in the original equation, and we find a = "65, 
approximately. For smaller values of a, k will be a complex 
imaginary quantity. As uJ2v is either increased or decreased, a 
real value for k can be obtained for smaller values of a. 

It will be found sufiicient for the purposes of the present paper 
to restrict the consideration of the solution to the ranges of values 
of a and Uq/2v for which k has a real value. We proceed to discuss 
the pressure variation in the case for which k is real ; the variation 
in the case for which k is complex can be inferred by considerations 
of continuity. 

Let J) be the mean pressure at the point (r, 9) in the liquid, and 
p its density. We obtain from, the two-dimensional polar equations 
of motion 

u^ _ Idp V d'^u 
~^^ ~pdr^r^W' 
_ \ dy 2vdu 

^'^^^ ^-~2^-2^aP + ^(^) 

substituting for ^^ from the differential equation satisfied by u. 



Also ^=J^+/(y) 



p "'^ 



moving through a channel with diverging boundaries 309 

Hence - = — - — --^ + C, where C is a constant. Now the lateral 
p r^ zr^ 

stress in the liquid is pee , where 

p p r 

Hence pgg is independent of d, and is the normal stress (of the 
nature of a tension) exerted by the liquid on the boundary. If 
a is negative the normal pressure on the boundary decreases as 
the channel widens, and if a is positive the normal pressure 
increases. 

Now by substitution of the solution for u given above in the 
differential equation satisfied by ii, we find 
a = 4i;2 [_ 1 + m^ (1 - F + J^)] 
= iv^ [-! + (! + uJ2vf (1 + k^)/{l + k^fl 

(!) Writing a = 0, we can immediately discriminate between 
those cases for which the pressure on the boundary decreases and 
those for which it increases. 

If a = 0, we have 

1 + uJ2p - (1 + k^r/{i + k^)^, 

and 1 + 2i//mo = (1 + k^f/{{l + k'^f - (1 + k^)^ 

Hence s„^ \( A + ^Y., 4 = (1 + P)* - (1 +^e)t 
iV(l + k^y/ ) sk^ (1 + F)* 

The following diagram shows how the value of uJ2v for which 
Pffg is independent of r varies with a, for those cases in which k is 




310 



Mr Harrison, The pressure in. a viscous liquid 



real. It clearly indicates that when a is small the critical value of 
Uq/2v may be somewhat large. 

If a > 7r/4, the lateral pressure increases for all values of Uq. 

(2) It is a simple matter to discuss the variation of the pressure 
when Uq/2i> is large. We have, approximately 

uJ2v 



sn' 



1 + k^ 
3F = 



m" = 



a = 



1 + P' 



(1 + F)3 ' 

k will be real provided F > |, and, corresponding to real 
values of k, a will be small. 

In the absence of viscosity, so that u = Uq for all values of d, 



1^69 = 



2r2 



C. 



1 + k^ 
Thus the lateral pressure increases at a rate which is ■ ,^3 

of the rate for a non- viscous liquid. 

The following table will indicate the character of the results 
when k is real. 



uJ2. 


a 


{l+k^)/{l+k^r 


100 


10° 30' 
9° 30' 


•30 

•27 


1000 


1° 3' 
0°57' 


•30 

•27 


10,000 


0° 6' 


•28 



m 



For larger values of a than those given above, and for the corre- 
sponding values of uJ2v, k is unreal. 

When a is small there is apparently an approximation which 
JefEery gives, viz. 

u= — 2v {1 — m^ — m^k^) — Qvk^m'^d^, 



1 



moving through a channel with diverging boundaries 311 



where 


= - (1 - m2 - m^F) - 


- Skhn^a^, 


leading to 


Qvm^ (1 — m2),„„ 
= «„ (1 - ff'ja?). 


-«2) 


This gives 


a = — 2vuja'^, 




and 


Pee ^Uq , ^, 

_ — ..9...9. + ^ • 





/3 a^r" 

Thus the lateral pressure apparently decreases for all values of 
Uq. But if 

Uq= — 2v {I — m^ — m^k^), 

and = - (1 - w2 - yn^) - Skhn^a^, 

we have Uq/2u = 3khn^a^, 

and therefore ma is not necessarily small. Hence the approxima- 
tion is only valid for values of Uq/2v below some limiting value. 
If this condition be satisfied the expression for pgg given above is 
an approximation to its value for small values of a. 

Three-dimensional problem. 

Let the boundary of the channel be ^ = a, where {r, 9, (f)) are 
polar coordinates. This problem has been considered by Prof. 
A. H. Gibson^. In his solution Cartesian and Polar Coordinates 
are confused, and he assumes that the stream lines are straight lines 
diverging from the origin, a state of motion which is impossible if 
the inertia terms are retained in the equations of motion, as he 
retains them. One result of these errors is that in his solution p is 
a function of 6 although the preliminary assumption is virtually 
made that p is independent of 6. His expression for the pressure 
appears to be quite wrong. 

Assume, in the first place, that the stream lines are straight 
lines diverging from the origin, so that u = f{d)jr^, v = 0, w = 0. 
The polar equations of motion reduce to 



du 1 dp 

or p or 



dht 2 du cot 6 du 1 c^u 2u 
dr^ r dr r^ dd r^ dd^ r^ 



^ _ 1 dp 2v du 
pr^^^dd' 

= - -^ ^ 

p d(f) 

1 Phil. Mag. (6), vol. xviii, p. 36, 1909. 



312 Mr Harrison, The pressure in a viscous liquid etc. 

We have 

i| = |-V^ [cot «./'+/"], 
1 dp _2v ., 

Hence eliminating f, 

^ + ,-4 [/'" +/" cot d -/' cosec2 e + 6/'] = 0. 

Therefore ff = 0, and 

/'" +/" cot 9-f' cosec2 d+6f' = 0. 
Hence/' (6) = 0, and the boundary conditions cannot be satisfied^ 
since u becomes independent of 6. 

For slow motion, or any motion in which the inertia terms can 
be neglected, we have 

/'" +/" cot e -f cosec2 d + 6/' = (1). 

A first integral is 

/"+/'cot0+6/+C = O (2). 

The solution of (2) suitable for the present purpose is 

/(6') = Z)(2-3sin2^)-iC. 
Let f{e)-^u„ 6=0, 

f{e) = 0, d^a. 
We have D = uJ3 sin^ a, 

(7 = 2 (2 — 3 sin^ a) ^o/sin^ a. 
Hence u = Uq (sin^ a — sin^ 6)/r^ sin^ a. 

Integrating the equations of motion, we have 

f=-|^(/'cot^+r) + i^x(^) 

^"d P^=p+F,ir). 

Hence ,^ ^ '^V(^) + ^^^ + 5, 

and Vm^_19^_b. 

p 3 r* 

The lateral pressure will continually increase as the channel 

widens if C be negative, that is, if sin a > (f)^, or a > 54° 45'. If 
a < 54° 45', for sufficiently small values of Uq the pressure will 
continually diminish. 



Mr Gray, The Effect of Ions on Ciliary Motion 313 



The Effect of Ions on Ciliary Motion. By J. Gray, M.A., 
Fellow of King's College, Cambridge. 

[Read 10 November 1919.] 

The ciliary mechanism of the gills of Mytilus edidis has been 
described by Orton^. There are at least four distinct sets of cilia. 
. whose movements form a complex but highly coordinated system 
by which food particles are filtered from the sea-water and passed 
up to the mouth. This coordinated system is entirely free from 
any nervous control and continues for many days in detached 
portions of the gill. These gill fragments therefore form an 
admirable material for the physiological study of ciliary motion. 

The effect of the hydrogen ion on ciliary action is very easily 
studied. Normal sea- water has a Ph of about 7-8; when the con- 
centration of hydrogen ions is increased to about 6-5 rapid cessation 
of movement occurs. In sea-water of Ph 6-7 the rate of ciliary 
movement is checked at first, but within f-l| hours complete 
recovery takes place. If gill fragments whose cilia have been 
stopped by the more acid solution are returned to normal sea- 
water, complete recovery takes place in less than 20 minutes 
although the cilia may have been motionless for several hours. 
A large number of experiments have been performed from which 
it is clear that if the concentration of hydrogen ions is only slightly 
greater than normal, the cells can react to the environment and 
recovery take place in the acid solution. In stronger acid, however, 
recovery only takes place on removing the gills to a more alkaline 
solution. In still stronger acid the cells become opaque and are 
killed. 

Gills which are exposed to an abnormally high concentration 
of hydroxyl ions behave in a remarkable manner. In such solu- 
tions ciliary action is either not affected at all or proceeds at 
an abnormally rapid rate, but the individual cells of the ciliated 
epithelia break away from each other and move about in the 
solution owing to the movement of their cilia. Since such cells 
are no longer in their normal environment, it is impossible to 
determine any upper limit of hydroxyl ions which will permit 
normal ciliary action to go on. 

Since the hydrogen ion has a most marked effect on ciliary 
activity,. it is necessary to adjust the hydrogen ion concentration 
of all artificial solutions during a study of the effects of various 
salts on ciliary action. In the case of the salts of the alkali metals 
this is satisfactorily performed by the addition of an appropriate 

1 Journ. Marine Biol. Assoc, vol. ix, p. 444 (1912). 



314 Mr Gray, The Effect of Ions on Ciliary Motion 

buffer such as sodium bicarbonate. In the case of the salts of the 
alkaline earths it is impossible to obtain pure isotonic solution of 
the same hydrogen ion concentration as sea-water, and it is there- 
fore necessary to compare the effects of the pure solutions with 
that of sea-water whose hydrogen ion concentration is abnormally 
high. 

A number of experiments have been performed which prove 
that sodium, potassium, calcium and magnesium are all necessary 
to maintain gill fragments in a normal state of ciliary activity 
for a protracted period, viz. four days. If one or more metals are 
omitted, the individual cells of the ciliated epithelia show the same 
disruptive phenomenon as in sea-water of abnormally high con- 
centration of hydroxyl ions. Solutions containing only one metal 
show this phenomenon to a very marked degree although they 
may be more acid than normal sea-water; the effect of solutions 
containing two metals is less marked than that of solutions contain- 
ing only one metal, but more marked than that of solutions con- 
taining three metals. No evidence was obtained of specific ion 
action or of antagonistic action between monovalent and divalent 
ions. 

These experiments afford another example of the intense action 
of the hydrogen ion upon physiological activity and of its reversible 
nature if the acid treatment is not too severe. The same action of 
acids is found in the activity of the heart and in the movement of 
spermatozoa. 



Mr Saunders, Photosynthesis and Hydrogen Ion Concentration 315 



A Note on Photosynthesis and Hydrogen Ion Concentration. By 
J. T. Saunders, M.A., Christ's College. 

[Read 10 November 1919.] 

Last April (1919) I was testing the hydrogen ion concentration 
of the water of Upton Broad, a small broad in Norfolk. I had 
determined the hydrogen ion concentration of the water of the 
broad itself to be 8-3 and I found this varied very Httle whether 
the water was taken from the surface or the bottom, from near the 
edge or the centre of the broad. The determination of the hydrogen 
ion concentration was made by the use of standard solutions and 
indicators as recommended by Clark and Lubs. 

When however the water in the shallow lodes and ditches 
surrounding the broad was tested, great variations in the hydrogen 
ion concentration occurred. The water became more acid as soon 
as the broad was left and the ditches entered. At one end of the 
broad where the water was shallow, not more than 18 inches deep, 
and when there was no wind to mix it with the open waters of the 
broad which was 6 feet deep, the hydrogen ion concentration 
would fall to 8' 15. In the lode itself the hydrogen ion concentration 
was 7-65. After boiling and rapidly cooling, water from the middle 
of the broad and from the shallows both showed a hydrogen ion 
concentration of 8'4, while that from the lode after the same treat- 
ment was 8"15. 

At one point in the lode, however, I found surprising varia- 
tions. Dippings of water from the same place gave readings of the 
hydrogen ion concentration varying from 7-7 to 8-6. At this point 
there was a certain amount of Spirogyra growing and I found that 
if I took water from the centre of a mass of Spirogyra I could get 
a reading as high as 9-0. 

I took some of the Spirogyra back with me and placed it in 
test-tubes in tap-water which I coloured with indicator solutions. 
The hydrogen ion concentration was 7-2 at the commencement of 
the experiment. After standing the test-tube in a window in sun- 
light the hydrogen ion concentration rose after an hour to 8-6 and 
in two hours the phenolphthalein indicator had turned bright 
pink, indicating a hydrogen ion concentration of more than 9-0. 
I had no standard solutions with me which I could use to test 
higher values than 9-0 so that I was unable to determine accurately 
the ultimate result. I left the test-tubes until the next morning, 
when I found the hydrogen ion concentration had fallen to 7-6. 
After again placing the test-tubes in sunlight the hydrogen ion 
concentration rose above 9-0. 

On my return to Cambridge I repeated these rough experi- 
ments. It is easy to prove that the rise in alkalinity is not due to 
alkali dissolved out of the glass, nor is it due alone to the abstrac- 



316 Mr Saunders, Photosynthesis and Hydrogen Ion Concentration 



tion of the dissolved carbon dioxide out of the water. The hydrogen 
ion concentration of the Cambridge tap-water which I used for 
these experiments was 7-15 when the water was tested immediately 
after being drawn from the tap. On standing at a temperature of 
13° C. the hydrogen ion concentration rises to l-i. After boihng 
and rapidly cooling the hydrogen ion concentration was 7-9 and 
bubbling through air free from carbon dioxide produced the same 
result. By incubating tap-water for 36 hours at a temperature of 
40° C. and then cooling the hydrogen ion concentration could be 
made to rise to 8-15, but in no case did the value of the control 
tap-water approach near that of the tap-water containing Spiro- 
gyra filaments. 

The following is a record of a typical experiment. The Spirogyra 
was placed in 25 c.c. of tap- water in a boiling tube and exposed to 
light at a window. Control boiling tubes containing tap-water 
only were used. All these tubes were half immersed in a glass bowl 
of running water so that the temperature was maintained fairly 
constant. 









Hydrogen Ion 










concentration 




Date 


Time 
(G.M.T.) 


Temp. 




Remarks 


1 








Control 


Spirogyra 




1. V. 19 


11-10 a.m. 


14-0° C. 


7-15 


7-15 


DuU day. 




12-10 p.m. 


13-0° C. 


7-4 


8-3 






1.10 p.m. 


12-5° C. 


7-4 


8-6 






2.10 p.m. 


12-5° C. 


7-4 


8-6 






3.10 p.m. 


12-5° C. 


7-4 


8-8 






5.30 p.m. 


13-0° C. 


7-4 


8-5 





I have tried using Elodea instead of Spirogyra and it gives 
much the same result. 

Both in darkness and in daylight the contents of the living cell 
of Spirogyra show an acid reaction when stained with neutral 
red. When Spirogyra is killed by heating to 40° C. and then placed 
in tap-water the hydrogen ion concentration falls considerably 
since the cell membranes are broken or dead and the contents of 
the cell are now free to pass out into the water. 

In a large pond the mass of the plants in proportion to the 
water is not sufficiently great to affect the hydrogen ion concen- 
tration very much. I have however found slight variations. On 
one occasion I noticed a fall in the hydrogen ion concentration of 
0-1 after several dull days and a subsequent rise of 0-2 after sunny 
days. This variation may possibly be due in some degree to the 
photosynthetic activity of the plants present. 



Mr Aston, Distribution of intensity 317 



The distribution of intensity along the positive ray parabolas of 
atoms and molecules of hydrogen and its possible explanation. By 
F. W. Aston, M.A., Trinity College (D.Sc, Birmingham). Clerk- 
Maxwell Student of the University of Cambridge. 

[Read 19 May 1919.] 

No one working with positive rays analysed by Sir J. J. 
Thomson's method can fail to notice the very remarkable intensity 
variation along the molecular and atomic parabolas described by 
him under the term ' beading.' It will be sufficient for the reader 
to refer to Plate III of his monograph on the subject {Rays of 
positive electric, p. 52) to realise how striking these can be. 
Beadings at points corresponding to energy greater than the normal 
have been quite satisfactorily accounted for by multiple charges 
{I.e., p. 46), but the ones with which this paper is concerned have 
a smaller energy than the normal, actually half, and fractional 
charges are presumably impossible. Nevertheless they seem 
capable of a simple explanation and an opportunity of putting 
this to the test occurred recently while making some experiments 
to determine the best form and position of the cathode pre- 
liminary to the design of an apparatus to carry the analysis to 
higher degrees of precision. 

The observations were made with an apparatus essentially of 
the form now well known {I.e., p. 20) the discharge tube being 
arranged to be removable with the minimum trouble to change 
or move the cathode. As no camera suitable for photographic 
recording was immediately available or necessary a willemite 
screen and visual observation was employed. This form has many 
obvious disadvantages and in addition, owing to the enormous 
difEerence in sensitivity between the parabolas of hydrogen and 
those due to heavier elements the latter can only be seen with 
difficulty. It has however one notable advantage, namely that 
sudden and even momentary changes in intensity can be observed 
and correlated in time with changes in the discharge or in the 
intensity of other lines. As no accurate measurements were 
intended a large canal ray tube was employed so that the H^ and 
H^ parabolas could be easily seen even with the less effective types 
of cathode. 

It was soon realised that the appearance on the screen was in 
general the sum of two superposed effects which could be only 
unravelled like the writings on a palimpsest by eliminating one of 
them. This by good fortune it was found possible to do under 
certain conditions. For the sake of clearness it is proposed to 



I 



318 



Mr Aston, Distribution of intensity 



consider these two extreme types and their explanation before 
going on to describe the conditions under which they may be 
attained or approached. In the diagrams the fields of electric and 
magnetic forces are horizontal and such that positive ions will be 
deflected to the right and up, negative ones to the left and down. 
Brightness is roughly indicated by the width of the parabolic patch 
drawn. 




O 




Fig. 1. Atomic Type. 



Atomic type of discharge. 

Fig. 1 illustrates the first or 'Atomic' type in which apparently 
the whole of the discharge is carried up to the face of the cathode 
by ions of atomic mass. Those which pass through the fields 
without collision produce the true primary streak on parabola 
m=l, the head of which corresponds in energy to that obtained 
by the charge e falling through the full potential of the discharge. 
Now the pressure in the canal ray tube is never negligible being on 
the average at least half that in the discharge tube, and the 
ionisation along its length very intense so that in passing through 
it a large number will collide with electrons, atoms or molecules. 
The collision and capture of a single negative electron will result 



along positive ray parabolas of atoms and molecules of hydrogen 319 

in a neutral atom striking the screen at the central undeflected 
spot while the capture of two will cause the faint negative 
parabolic streak a^ as has already been described (I.e., p. 39). 

But besides these forms of collision by which the velocity of 
the atom is practically unaffected there is distinct evidence that 
it may collide with and capture another hydrogen atom. If the 
atom struck is negatively charged the resulting molecule will 
strike the central spot but if it is neutral and the collision is 
inelastic the resulting positive ray will have the same momentum 
(the^ atom struck being relatively at rest) but double the mass so 
that it will strike the molecular parabola at a point the same height 
above the JT-axis as would the atom which generated it. Molecular 
rays formed in this manner will therefore form the streak b^ 
which, allowing for the geometrical difference in the curves will 
show a similar distribution of intensity to a^. Collision with a 
positively charged atom wiU obviously be unlikely to result in 
capture and those with heavier atoms will be referred to later. 
It is to be noted in connection with the brightness of these 
secondary streaks a^ and 63' which may conveniently be called 
'satellites' to distinguish them from the 'secondary lines' already 
fully described {I.e., p. 32), that a^ is always very much fainter than 
its primary but b^ can be equally bright. 

This atomic type of discharge with its pendant bright arc on 
the molecular parabola corresponding to similar momentum and 
half normal energy is most beautifully illustrated in Fig. 29 of 
Plate III already referred to. It was this photograph which 
suggested the above theory of its explanation. 

Molecular type of discharge. 

The extreme form in which the whole discharge is carried up 
to the cathode by ions of molecular mass is unattainable so far 
in practice and is probably impossible but its share in the illumina- 
tion of the screen can be deduced by eliminating the superimposed 
atomic type and is indicated in Fig. 2, 

The principal feature is a short and very bright spot of light b^ 
on the molecular parabola at the point corresponding in energy 
to a fall through the full potential of the discharge. It will be 
shown that all the ions causing this are probably generated in the 
negative glow. Besides this there are two symmetrical and equally 
bright positive and negative satellite patches ag ^^^ (^2 on the 
atomic parabola but of half the normal energy. The proposed 
explanation of these is somewhat similar to that considered by 
Sir J. J. Thomson {I.e., p. 94) and is as follows. The collision with 
and capture of a single negative electron by a positively charged 
molecule will not necessarily merely neutralise it and cause it to 

VOL. XIX. PART VI. 22 



320 



Mr Aston, Distribution of intensity 



hit the central spot but may result in it splitting into two atoms 
one with a positive one with a negative charge. The energy of 
impact may be itself capable of causing this, if not some other 
cause, e.g. radiation, may effect the dissociation. In any case it 
would give exactly the observed result, i.e. two bright patches 
lying symmetrically on the extension of the line joining the 
primary spot to the origin at twice its distance from the latter, 
corresponding to half the mass but the same velocity. 













•^ 




__i 






^ 




/c. 







Fig. 2. Molecular Tyi^e. 

The general appearance on the screen when both types of 
discharge are present is indicated in Fig. 3. 



Effect of different forms of cathode. 

Experiments were performed with plane, concave and convex 
cathodes. Convex cathodes are the least efficient in producing 
bright effects but give the molecular type with the least atomic 
blurring. Concave ones are most efficient and throw the maximum 
energy into the atomic type which can be obtained practically 
pure with them under a moderate range of conditions. The original 



along positive ray parabolas of atoms and molecules of hydrogen 321 



shape of cathode {I.e., p. 20) may be said in a sense to combine 
both forms and was designed to give long and bright parabolas 
at the same time allowing the discharge to pass easily at very low 
pressures. The present results however lead one to recommend a 
concave cathode similar to those used in X-ray focus tubes but 
pushed further forward into the neck of the bulb, for though this 
form requires a rather higher pressure this objection is more than 
counterbalanced by the great increase in efficiency. Plane cathodes, 
as was expected, give effects midway between the other forms. 



O 




/ 



Fig. 3. General Type. 

Under very exact conditions of pressure, etc. it is possible to 
obtain the pure atomic type with plane cathodes but no conditions 
have yet been found under which convex ones will give it. 

These results seem to indicate that atomic ions are formed by 
the passage of the stream of cathode rays through the Crookes 
dark space molecular ones tending rather to be formed in the 
negative glow. The axial intensity of the cathode stream is 
enormously increased by the concavity of the cathode while that 
of the negative glow does not appear to be affected to anything 
like the same extent. 

22 2 



322 Mr Aston, Distribution of intensity 

Behaviour during change of pressure. 

The pressure in a freshly set up bulb always increases with 
running owing to the liberation of gas by heat etc. so that the 
changes due to gradual alteration of pressure can be observed 
most conveniently by exhausting highly, starting the coil and 
watching the events on the screen. Thus using a concave cathode 
of about 8 cms. radius of curvature set just in the neck of the 
discharge bulb the following sequence of events was observed. 
At very low pressures with a potential of about 50,000 volts the 
parabolas are very faint but correspond to the general type, the 
primary streak a^ and spot h-^ being much brighter than their 
satellites (doubtless due to few collisions). As the pressure rises 
the discharge becomes curiously unsteady the spots on the screen 
become much fainter and change with flickering into the pure 
atomic type (Fig, 1), 6i having practically disappeared. This form 
of discharge which is evidently abnormal lasts for a certain time 
depending on the rate of increase of pressure. Then with absolute 
suddenness h-^ flashes out intensely bright and with it appear at 
the same instant its satellites a^ and a^. At the same time the 
current through the bulb increases, the discharge settles down and 
the negative glow makes its appearance. As far as it was possible 
to judge the satellites a^, and a^ are of equal brightness and generally 
much brighter than the negative atomic satellite «!. 

The appearance of the discharge bulb while the pure atomic 
type is shown on the screen is difficult to describe but quite 
characteristic and different from the general. Near its critical 
upper limit of pressure it was found possible to effect the change 
to the general type by bringing a magnet near the cathode and 
so disturbing the discharge. On removing the magnet the discharge 
at once reverted to the atomic type. This form of controlled 
change from the one to the other gave an excellent opportunity 
of testing the invariable association between the primary spots 
and their appropriate satellites. 

Possible cause of disappearance of primary molecular rays. 

It is unlikely that change of pressure is itself the determining 
factor in the disappearance of the molecular type. This seems to 
be due to some disturbance in the discharge by the cathode stream 
(not caused by the diffuse one given by a convex cathode) which 
makes the formation of the negative glow impossible. 

The facts so far may be brought into line fairly well by the 
somewhat speculative assumption that molecular rays can only 
originate freely in parts of the discharge where the electric force 
is very small, e.g. the negative glow, ionisation by more violent 



alo'ng positive ray parabolas of atoms and tnolecules of hydrogen 323 

means in strong fields tending to cause simultaneous disruption 
of the molecule into its atomic constituents. This agrees with the 
observed fact that in general molecular arcs, or at least true 
primary molecular arcs, are shorter than atomic ones. It would 
also mean that a very short arc infers as origin a molecule capable 
of disruption. If this is so it offers interesting confirmatory 
evidence, if such were needed, that the substance X^ is molecular 
as this body often makes its appearance on the photographic plate 
as a short arc. 

Effects with heavier elements. 

The inelastic collision of a hydrogen atomic positive ray with 
the atom of a heavy element would clearly result in the formation 
of a molecular ray of such low velocity that it might not be 
detected by a screen or plate and would in any case be deflected 
completely off the ordinary photograph. 

The visual evidence on the screen although faint leaves little 
doubt that the formation of satellite arcs also takes place by 
atoms of heavier elements colliding to form molecules. There is 
also some evidence of this in many of the photographs, thus in 
Fig. 26 (I.e., p. 46) taken with oxygen all four maxima are suggested. 
In Fig. 17 (p. 26) the satellite on the molecular parabola caused 
by the capture of oxygen atoms by carbon atomic rays (or vice 
versa, but this is less likely) is unmistakable, in fact attention is 
called in the text to this remarkable increase in brightness. 

Should the above theory of collision with capture prove 
correct the formation of compound molecules by this means opens 
an extremely interesting field of chemical research. Another 
important question raised is in what form the energy of the 
collision is radiated off by the rapidly rotating doublet formed. 

In conclusion the author wishes to express his indebtedness to 
the Government Grant Committee for defraying the cost of some 
of the apparatus used in these experiments. 



324 Sir Joseph Larmor 

Gravitation and Light. By Sir Joseph Larmor, St Johirs 
College, Lucasian Professor. 

[Read 26 January 1920.] 

1. Newton's provisional thoughts on the deep questions of 
physical science were printed at the end of the second edition 
of the Opticks in 1717. As he explains in the Preface " . . .at the 
end of the Third Book I have added some questions. And to shew 
that I do not take Gravity for an Essential Property of Bodies, 
I have added one Question concerning its Cause, chusing rather to 
preface it by way of a Question, because I am not yet satisfied 
about it for want of Experiments." In the first and next following 
Queries he gives formal expression to the idea that "Bodies Act 
upon Light at a distance and by their action bend its Rays. ..." 

What was thus propounded in general terms as an explanation 
of the diffraction of light in passing close to the edge of an obstacle, 
assumed a more definite but different form in the hands of the 
physically-minded John Michell*; in Phil. Trans. 1767 he insisted 
that the Newtonian corpuscles of light must be subject to gravita- 
tion like other bodies, therefore that the velocities of the corpuscles 
shot out from one of the more massive stars vrould be sensibly 
diminished by the backward pull of its gravitation, and thus that 
they would be deviated more than usual by a glass prism, a supposi- 
tion which he proposed to test by experiment. He also speculated 
that the scintillation of the stars might be due to the small number 
of corpuscles which reach the eye from a star, amounting perhaps 
to only a few per second. 

The forces, of molecular range, that would have to be con- 
cerned, on the lines of Newton's Query, in the diffraction of light 
would be of course enormously more intense than gravitation : but 
the other Newton-Michell theory of the gravitation of light rays 
is paralleled in both its aspects with curious closeness in certain 
modern physical speculations. 

It will be observed that this notion of light being subject to 
gravitation makes its velocity exceed the limiting velocity c, which 
on electrodynamic theory could not be attained by any material 
body. But there need not be a discrepancy there: for the limit 
arises because a material body is supposed to acquire more and 
more inertia, belonging to energy of its motion, without limit as 
its velocity increases, whereas the quantum of energy in the hypo- 
thetical light-bundle presumably would remain sensibly the same — - 
at any rate we would be free to make hypotheses in absence of 
any knowledge. 

* See Memoir of John Michell (of Queens' College), by Sir A. Geikie, Cambridge 
Press, 1918. 



Gravitation and Light 325 

Forty years a.oo there was a phase of strong remonstrance in 
this country against the famihar uncritical use of the phrase 
centrifugal force. The implication was that the term force should 
be restricted to intrinsic unchanging forces of nature, which are 
determined physically by the mutual configuration of the system 
of bodies between which they act: these forces are then held 
responsible for the accelerative effects specified by the Newtonian 
second law of motion. In this sense, centrifugal force so-called 
would not be a force of nature, but would be the reaction postulated 
in the scheme of the Newtonian third law to balance an imposed 
centripetal acceleration. 

This formative principle, the Newtonian third law, of balance 
everywhere between appHed forces and reactions against palpable 
changes of motion, as amplified in the Scholium an]iexed to it — 
which so widely reached forward towards modern theory as 
Thomson and Tait especially have remarked — would then assert 
that the forces of nature that act on the framework of a material 
body and the forces of reaction that are thereby induced in it, 
form together a system of forces that preserve statical equilibrium 
in relation to the constraints of that framework, as tested by the 
principle, also Newtonian in its origin, of virtual work. This 
became in time the Principle of d'Alembert (1742), who did not 
invent it, but exhibited its power and developed its method by 
applying it to a great dynamical problem of unrestricted form, 
that of the precession of the equinoxes. As a preliminary to its 
solution he had to develop in general terms the equations of static 
equilibrium of a system of forces considered as applied to a single 
rigid body such as the Earth, that is, to create a formal science of 
Statics: and it may be said to be the mode of development rather 
than the principle itself that constitutes his essential contribution 
to general dynamical theory. Cf. the historical introductions in 
Lagrange's Mecanique Analytique. 

2. The principle of the relativity of force has recently become 
prominent again, and pushes along further on the same lines; it 
now even puts the question — Are there intrinsic forces of nature 
at all? May not all force, including universal gravitation, be ex- 
pressible as reaction against acceleration of motion, just after the 
manner of the obviously unreal centrifugal? On such a view, 
wherever there is a force of gravitation in evidence, its presence 
must be replaced by an acceleration common to all of the material 
bodies at each place and relative to our frame of measurement, 
of amount equal and opposite to the intensity of the force. That 
would be the end of the matter, if any frame of reference could 
be found to satisfy this condition. There being then no forces left, 
the Principle of Least Action would make orbits simply the shortest 
paths in the frame. Newtonian uniform space and time certainly 



326 Sir Joseph Larmor 

could not permit this transformation: nor could the fourfold 
uniform continuum of interlaced space and time of the earlier 
relativity theory be adapted to it. Will such a fourfold, deformed 
into a non-uniform and therefore non-flat heterogeneous space, 
permit it? This is the problem raised by Einstein's idea of the 
relativity of gravitational force. Perhaps it goes even further, and 
asks whether if this will not do, there can be some other corpus 
of abstract differential relations invented, that will transcend 
the notion of spacial continuity altogether but will in compen- 
sation for that formidable complexity succeed in effecting this 
object. 

In any case we may recognise that this merging of all the forces 
of nature into spacial relations satisfies one requirement which is 
not quite the claim that is explicitly made for it. The question 
is immediately insistent; why should intrinsic forces be measurable 
with Newton in terms of second gradients of type (Ps/dt^ and not 
by a more complex formula involving others as well? The answer 
supplied by the theory would be that the idea of the curvature of 
a deranged space is expressed by a measure which does not involve 
higher gradients. 

It is interesting to reflect nowadays that in referring to the 
doctrines of action at a distance in the preface to the Electricity 
and Magnetism, in 1873 Maxwell classifies them as "the method 
which I have called the German one," and that notwithstanding 
Helmholtz's very powerful critical work on Maxwell's theory, be- 
ginning in 1870, that description remained substantially true until 
after Maxwell's death in 1879. Though he lived for nine years 
longer he seems to have taken no part in these discussions with 
exception of a reference to Helmholtz in connexion with Weber's 
theory {Treatise, § 254), but worked chiefly at the development of 
the theory of stresses in gases regarded as molecular media, and 
so in some respects parallel to his theory of an electric medium. 
He seems to have been content to leave his electric scheme to 
germinate and expand in the fulness of time. In connexion with 
the recent efliorts to transcend both action at a distance and an 
aethereal medium, his explanations, in an Appendix to the Memoir 
on the determination of the ratio of the electric units, Phil. Trans. 
1868 and the critical chapter on ' Theories of Action at a Distance' 
in the Treatise, §§ 846 — 866, are far from being obsolete. 

This hypothesis as to gravitation, which asserts that it is 
essentially of the same nature as the apparent increase of weight 
which is experienced by an observer going up in a lift with ac- 
celerated motion, naturally involves many consequences, and 
raises questions regarding the relation of gravitation to physical 
agencies such as light, the answer to which may be ambiguous until 
yet further postulates intervene. 



Gravitation and Light 327 

Thus in the preliminary stage it occurred to Einstein that the 
period of a train of light waves would be no longer uniform 
throughout its course. Let us consider a mass of hydrogen gas 
at P, say in the Sun, sending light- waves to an observer Q, both 
being situated in a region in which there is a field of gravitation 
of intensity represented by </, directed from Q to P. In terms of 
the postulate of the relativity of that force this statement would 
mean that the spacial frame to which the underlying events are 
referred is rushing as a whole from P toward Q with acceleration g. 
Let V be the velocity of the frame at the instant when a specified 
light- wave passes any intermediate point Q' : by the time this 
wave has reached Q the velocity of the frame as a whole has risen 
to V -\- g.Q'Qjc approximately, where g is mean intensity along 
the range from Q' to Q. Thus to the accelerated observers the 
waves emitted become longer with distance traversed, in the ratio 
^ + 9 -Q Q/c^, owing to this velocity of recession from the source : 
that is, the apparent wave-length undergoes change so that 
during the progress from Q' to Q it is altered in the ratio 1 — SF/c^, 
where 87 is the rise of potential (or fall of gravitational potential 
energy) along that path. 

The period of the light will thus appear to be increased to 
different observers on the line PQ, all of them travelHng along 
with the same acceleration g, in different degrees according to their 
positions. This is what will happen if the observers and their space 
and optical instruments form a world of their own rushing past, 
or through, an underlying actual world, with this acceleration g, 
instead of the actual world rushing past them with the opposite 
acceleration produced by a force of gravitation. For these alter- 
natives are not now the same: the finite velocity of propagation c 
is constant with respect to the actual underlying world, not the 
observers' moving space. If the radiating hydrogen belongs to the 
actual underlying world, and the spectroscopes of the observers 
belong to their own spacial scheme that is imposed on that world, 
this description is complete: the period of each wave as apparent 
to observers along its path will increase as the wave travels away 
to places of lower gravitational potential. The spectral lines of 
solar hydrogen as observed on the earth ought to be displaced 
towards the red, by the amount corresponding to the total fall 
of potential between Sun and Earth. But the postulate of two 
worlds seems to be here necessarily involved. Which of them would 
a mass of radiating hydrogen situated half-way to the Sun belong 
to?* The larger Doppler-Fizeau effect due to the motion of the 
source itself relative to the observers' frame has not here been 

* All the bodies in the space, being subject to the same gravitation, would 
move along with it: the waves of light alone would seem to be regarded as inde- 
pendent: yet they have energy and so inertia. 



328 Sir Joseph Larmor 

mentioned: that is included satisfactorily in tlie earlier uniform 
relativity formulation. 

This relation of light to gravitation is thus one of the questions 
raised by the postulate of the relativity of that universal force. 
Einstein answered in 1911* in one way, that the spectrum of solar 
hydrogen, when compared with terrestrial hvdrogen which is con- 
nected with the observer, should be displaced slightly towards the 
red: but it is a question whether the consistent development of 
that train of ideas would not rather require that it be not displaced 
at all. 

In connexion with his later formal theory of gravitation the 
same effect is described as due to varying local scales of time, 
which seem to be carried without change, by the pulsations of the 
rays, from the place of their origin to all the other parts of the 
universe: whereas in the above the apparent period f changes as 
the ray advances. The observers along the ray are supposed to 
be in communication with one another. In so far as their space 
moves forward as a whole it is not stretched or shrunk: in that 
case it can be only their scales of apparent duration of time that 
are lengthened localh'^ by a factor, the inverse of 1 — V jc^. This 
involves that the scale of apparent velocity in the unchanged space 
will be altered in the direct ratio: and rays of light in a field of 
varying potential, if they were paths of stationary time, might be 
thought to be deflected. But fundamentally the path of the ray 
is determined by the number of wave-lengths in its course being 
made stationary, as compared with neighbouring courses: and this 
is, in the present case, not the same as minimum time of transit, 
for apparent time has lost its uniform scale while space has not. 

Thus the path of a ray would be determined by the condition 
that SSs/A summed along it shall be stationary: but if there is 
correspondence between the two systems of reference which 
changes all lengths around each point in the same ratio then hsjX 
will be everywhere the same in both systems. The circumstances 
of the path would thus not be altered by this change of view 
regarding gravitation, and there ought to be no special deviation 
of the rays involved in it. 

But if g is not uniform along the path r of the ray, is a 
shrinkage of the accelerated apparent space involved? The answer 

* His exposition which has here been paraphrased is in Ann. der Physik, 35, 
1911, §3, p. 904. 

The argument of this and the next two paragraphs is based on the implication 
that in a theory of transmission by contact, radiation like other things, the so- 
called clocks included, must conform to local measure: the alternative, described 
at the end of the paper,- that racUation is extraneous in so far as it imposes an 
absolute scale of space-time of its own on the whole cosmos, was here taken to be 
excluded in advance from this type of theory. 

•j- Measured on a fundamental scale. 



Gravitation and Light 329 

is given that, passing to the general problem, the demands of the 
universal gravitational correspondence (to be evolved immediately, 
infra) require that the apparent space of the observers must be 
constructed so that S/^ — c"^ht^ where c is a function of r shall be 
invariant. This requires slight warping of the fourfold space, so 
that the section in the plane r, t is curved away from its tangent 
plane. But is the warped element of extension ^r' .c'ht thereby 
altered only to the second order from its corresponding previous 
normal value Sr.cS^? If that be so, the scale of t must be altered 
in the inverse ratio to the scale of velocity c' or (what is the same 
in another aspect) of time t : and in fact it is partly this secondary 
change of scale of r that modifies the astronomical gravitation, as 
will presently appear. 

The answer to this question might at first be imagined to be as 
follows : any change in the element of surface may be made in two 
stages, a stretching on the original plane and a displacement along 
the direction normal to that tangent plane: it is only the former 
that can produce a first-order effect: but this is only an apparent 
change, a mere alteration of coordinates, because in it the curvature 
of the plane is conserved, so it cannot affect the concatenation of 
relations or events which alone counts : the latter does affect them, 
e.g. disturb the law of gravitation, but only to the second order. 

But as will appear presently this relation of conservation of 
extent is between coordinate systems that most closely correspond, 
so is a real imposed condition which cannot be adjusted by 
change to another set in the fiat. It is the expression of, or at any 
rate is involved in, a restriction that in the containing fivefold 
the distance between corresponding points on the two systems is 
everywhere small, so that approximate methods can apply con- 
sistently throughout, of which otherv/ise, in making continuations 
in an uncharted extension, there would be no guarantee. 

3. Now let us survey this problem of transcending gravitation 
from the other side, on which it originated. With Minkowski the 
very incomplete relativity of electrodynamics, referring only to 
uniform translatory convection, crystallised into the complete pro- 
position that events occur in a uniform fourfold of mixed space 
and time, determined by the consstitutive spacial equation 

Here c has nothing to do with the velocity of radiation : it is simply 
the dimensional factor, prescribing a scale of measurement, that 
is needed to make time homogeneous with length and may be 
taken as unity. Gravitation remains outside this electrodynamic 
scheme, being formulated in the different Newtonian reckonings of 
space and time. Can it be forced in, either exactly or approxi- 
mately? 



dt = 



330 Sir Joseph Larmor 

The complete circumstances of the orbits in a field of force of 
potential energy —V per unit mass (in a gravitational field V is 
TiSm/r) are condensed into the single variational Least Action 
equation of Lagrange-Hamilton, 

with integration between limits of time fixed and unvaried. This 
suggests comparison with the equation for the shortest or most 
direct path in a modified fourfold involving Euclidean space com- 
bined with a measure of time varying from place to place: for 
that equation is 

Sjda=0 where Sa^ ^ Sx'- + Sy^ + Sz^ - c'^Bt^ 

in which c' is a function of x, y, z. Let us write 

C'2 = C2 (1 -f K), 

where K _is very small on account of the greatness of c. The 
equation is now 

or approximately up to the fourth order 



■&--i---im-m-m. 



dt^ 0. 



The time-limits being unvaried the first term — c^ can be omitted : 
thus this variational equation of most direct path coincides with 
the previous orbital equation if 

- |Zc2 = F. 

Thus the forces are absorbed into a varying scale of time; and the 
motion being now free under no force, the orbit is, as was antici- 
pated, a geodesic or straightest path. The orbits have become 
however straightest paths, not in their original Newtonian separ- 
ated space and time, but in the uniform space-time fourfold of 
relativity as slightly deranged by the not quite constant scale of 
time. 

Thus the orbits in any field of attraction have actually been 
fitted into the mixed space- time frame of electrodynamic relativity, 
at the expense of doing slight violence to that frame, by making 
the measure of time vary from place to place while the positional 
specification remains uniform. 

But this transformation does more than is needed. It ought 
somehow to be restricted to the one universal force of nature, that 
of gravitation with its inverse-square law. It is here that the 
special feature of the Einstein theory seems to come in. For 



Gravitation and Light 331 

velocities beyond actual astronomical experience, not small com- 
pared with that of light, mass comes to depend on speed; thus it 
is not any longer available as a definite dynamical constant. On 
the earlier uniform relativity it emerged however definitely in 
another way as a feature of every permanent collocation of energy 
and proportional to its amount E, equal in fact to Ejc^. This 
follows immediately if Least Action is fundamental. Thus it is 
grouped energy that possesses located momentum: and it is this 
energy that has to gravitate, mass confined to matter alone having 
proved inadequate to a Least Action formulation in the mixed 
space-time of universal limited relativity. Dynamical principles 
had therefore to take the form of a theory of conservation of energy 
and of abstract momentum as they travel through a medium, at 
the same time receiving additions by the operation of an internal 
stress to which the medium is to be subject. In other words, 
general dynamics cannot be more detailed than a mere description 
of the migration of energy and of momentum in a medium under 
the influence of some internal system of stress adjusted to fit the 
equations as simply as possible. This stress is what has to stand 
for or represent the agencies of nature. The theory is borrowed 
and generalised from the Maxwellian theory of stress in the aether, 
which was an isolated, apparently rather accidental, feature that 
did not fit well into the substance of Maxwell's scheme, because in 
fact it could not be connected with a strain expressive of its 
origin. Now however, inertia of bodies having failed as the standard 
measure of force, energy and momentum, and a postulated ad- 
justing stress entirely at our choice, are promoted to occupy the 
vacant place. Only it is not called a stress: the idea of a physical 
medium is avoided, so it is named an algebraic tensor. There is 
no law of elasticity involved, or relation of stress to strain, such 
as makes elastic problems determinate. Thus the scheme may 
have accidental features, is perhaps far from being unique. Another 
parallel to it is Maxwell's theory of stresses in a gas due to varying 
temperature: but that continuous theory could never have been 
constructed in definite form without the foundation of the be- 
haviour of the individual molecules. 

When however the fourfold frame is very nearly flat, the rela- 
tions of energy-momentum-stress appear to fall in with the law of 
gravitation, with energy as the source of its potential instead of 
matter. 

When the deranged spacial frame nowhere differs much from 
the flat, it may be expected that the extent of its fourfold element 
will be altered from the value for coordinates of the corresponding 
type on the flat only to the second order, for the same kind of 
reason as applies in comparing a slightly deranged plane sheet with 
the original plane. In fact, if the displacement is everywhere small, 



332 Sir Joseph Larmor 

this extent taken over a small region would have a stationary 
value for the flat, changing in the same direction on both sides 
of it. Cf. supra, p. 329. Thus for a spherically symmetrical field 
the constitution of the fourfold must be determined in polar 
coordinates by the equation 

Sct2 - {c/c'f 8/2 + {rSdf + (r sin O^f - c'^Si, 

showing that the positional part of the extension is very slightly 
non-uniform and so not quite Euclidean. It appears to be this 
secondary feature, not the energy-momentum-stress tensor con- 
ditions, that modifies gravitation from the Newtonian law. 

The expositions of relativity do not mention an extended 
fourfold, which would be foreign to the cardinal idea that space 
is constructed from physical origins, only in so far as it is needed — 
even though it has to be implied that it is reproduced unerringly 
each time. But the instrument of such construction or continua- 
tion of a metric space is an infinitesimal linear measuring rod 
supposed to have complete free mobility without change of in- 
trinsic length : and it would seem to be a tenable view that such 
a mobile apparatus must determine an underlying flat space of 
higher dimensions* in which the physical system may be supposed 
imbedded. 

It is to be noted here that a surface defined intrinsically in the 
Gaussian manner by the distance relation on it 

Ss^ =fSp^ + 2gSpSq + hSq^, 

remains the same surface when the coordinate quantities p, q are 
changed to others p , q' which are any assigned functions of them 
both, so that 

Ss^=.f'8p'-^ + 2g'8p'8q' + h'8q'^, 

provided 8s is measured by the same infinitesimal unchanging 
measuring rod extraneous to the surface in both cases. These two 
equations represent the same surface, only the generalised co- 
ordinates of the same point on it are changed from {p, q) to {p, q'). 
The intrinsic curvatures are the same from whichever form they 
be calculated: if one form represents a flat, so does the other. On 
this definition by an intrinsic differential relation surfaces are 
indistinguishable, if one can be bent to fit the other without 
stretching. So in the Riemann theory of spaces of more than two 
dimensions it is the functional forms of the coefficients in the 
quadratic function of differentials and the mobile absolute mea- 
suring rod that determine the nature of the space; any transforma- 
tion of coordinates changes the coefficients (or potentials in the 
gravitational formulation) but so that the space remains un- 

* For a radial field it need be of onlj' one more dimension. 



Gravitation and Light 333 

changed, being only referred as regards the same points to the 
other generalised coordinates. But the apparent extent ^pdq does 
alter when the coordinates are changed, and it would be a limita- 
tion to keep it constant. See Appendix infra. 

The feature that remains unfathomed as yet is the fact that 
the velocity of transfer of energy of radiation in undisturbed regions 
of space is equal to the merely dimensional constant that renders 
time comparable with space on the fourfold frame of reference : it 
at any rate suggests a dynamical origin for that mixture of the 
effective relations of time with those of space *. 

The locus in the fourfold in which a never changes and so ha 
vanishes has some claim to be called the 'absolute,' in a sense 
parallel to the ' absolute ' of Cayleyan geometry which for Euclidean 
space is represented by the equation x^ + y^ + z^ = 0. Everywhere 
on this locus S.s = c'ht ; thus velocity of displacement is everywhere 
c', and the rays in it are the paths of shortest time with this 
velocity. It separates the disparate regions in which ha measures 
real distance when time is unvaried and in which iSct measures real 
time when position is unvaried. 

4. It would appear (as infra, p. 335) that if we are prepared to 
replace a field of potential energy of gravitation or any other type of 
universal force by a field of varying time-scale without change of the 
uniform scale of space, on the lines sketched above, this formal 
change ought not sensibly to affect radiation either as regards its 
path or its period. To each element of extent there would be a cor- 
responding element, and all events and measures in one pass over to 
the other according to rule. 

But we now pass from kinematic discussion of frames of refer- 
ence to physical considerations. If we are to assert, in agreement 
with the doctrine of relativity plus Least Action, that inertia is a 
property of organised energy and proportional to it, therefore not 
solely of matter, and if we are to admit with Einstein, in the same 
and other connexions, that light is made up of small discrete 
bundles or quanta of energy, it would appear to follow that each 
bundle is subject to gravitation. Therefore if a bundle comes on 
from infinite distance with velocity c, when it has reached a 
place of potential V near the Sun its velocity c must be given by 

ic'2 - F - ic2, 

in other words, is increased in the ratio 1 + V jc^. It will swing 
round the Sun in a concave hyperbolic orbit, and as the result, 
the direction of its motion will suffer deflection away from the Sun 
by half the amount that has been astronomically observed. 

This reasoning would not be estopped by the principle that c is 
the upper limit of possible material velocities: for that is because 

* See final paragraphs. 



334 Sir Joseph Larmor 

a moving body acquires energy and therefore inertia without hmit 
as its speed approaches c, whereas the energy of a Hght quantum 
is not supposed so to increase. 

This is all on the older notions: the velocity c is far too great 
for the new approximate gravitation analysis to be applicable. 
But the idea of wavefronts and phases must also be introduced 
somehow. If we imagine a row of these corpuscles of energy coming 
on abreast, the more distant ones would fall behind in swinging 
round the Sun and their common front would become oblique to 
their direction of motion, the exactly transverse directions being 
now the loci of equal Action not of equal time. If we superposed 
the Huygenian principle of propagation normal to the front, the 
orbital deflection would thereby be just cancelled by the swinging 
back of the front which would retain its direction : and there would 
be no deflection of direction of propagation. But such ideas are 
plainly incoherent. 

The earlier development of Einstein sketched above* was 
driven on other grounds to conclude that light must gain energy 
in a field of gravitation, but the gain was named potential energy. 
In the finally developed theory there seems to be no longer energy 
of motion or other types : energy becomes a single analytic scalar 
in what is left of the field of interplay of momentum, energy and 
stress. 

These earlier considerations have doubtless crystallized into 
the formal theory of which also the result has been illustrated 
above, in a way which transforms the variational equation of free 
orbits in ordinary space and time into the variational equation of 
straightest lines in a non-uniform space-time fourfold given differen- 
tially. The coordinates are carried over unchanged in values, into 
this fourfold, but their differentials no longer express in it direct 
measurements of length and time; these are now imported in the 
Riemann manner as regards any element of arc or interval of 
time by the value of the absolute element 8a. As compared with 
the underlying absolute time determined by Scr, the element of 
apparent time St of a gravitational world, which is taken over into 
its expression is variable, proportional to c'"^, with locality. 

The quantities x, y, z, t which are the measures of space and 
time as apparent in the world of gravitation are now mere co- 
ordinate quantities in the new differentially given world in which 
there are elements of absolute length and time both measured by 
Sct. The final expression for Scr^ with radial symmetry 

,f Sr2+ ... + ... - (^y^^S^^ 
shows that the element of apparent time in the gravitational world 

* Ann. der Physik, 35, 1911, §2, p. 902. 



Gravitation and Light 335 

is the unchanging element of absolute time divided by c' Ic, or that 
the scale of apparent time is variable with locality in the ratio cjc' : 
also that the scale of apparent radial length is variable in the 
ratio c jc: and therefore the scale of radial velocity is variable as 
their quotient c^jc"^. How then with respect to the velocity of 
rays of light whose absolute value is the same as the dimensional 
constant c? Referred to these variable scales its apparent value 
along any element of arc ought to be changed at the same rate 
as any other velocity along that element of arc would be changed, 
if rays are not to remain outside the correspondence between 
hx, 8y, Sz, St representing time-space in the apparent gravitational 
world and the same quantities, now elements of mere coordinates in 
a difEerentially given world in a curved space-time which has 
absorbed gravitation. This maintenance of correspondence is 
secured if we determine the ray- velocity along any element of arc 
by making Scr = : and the modified theory of radiation for the 
apparent space of gravitation must be such as can accept this 
value of the velocity of propagation *. The correspondence takes 
over the same values of the coordinate differential elements. In 
the apparent gravitational world they represent its space and time, 
in the new world differentially specified, they belong to mere 
coordinates: absolute elements of space and of time are there ex- 
pressed by 8o-, but a relation of scales can be established from the 
formula which expresses Sct^. 

The transformation which changes orbits into geodesies in the 
difEerentially given space-time does not turn rays into rays: their 
velocity is too great and moreover their minimum property is 
relative to their locus 8a = 0. But if the ray is supposed to have 
a constant underlying absolute period of pulsation and a constant 
absolute wave-length (and therefore to be a straight line in an 
auxiliary uniform fivefold) its apparent period in the gra^dtational 
world must vary with locality as (c'/c)"^, also its apparent element 
of length inversely as the scale of length pertaining to its direction 
on that locality, and its apparent velocity as before specified. Its 
apparent path in the gravitational world will correspond to the 
true absolute path Sfda/X^ == 0, therefore will be given by 

Sjds/X = 0, 

complications being avoided as fortunately t is not involved ex- 
plicitly in these equations. But at the same place the scales of 
apparent 8s and apparent A would alter on the same ratio owing 
to the presence of gravitation : therefore its influence is eliminated 
in the quotient, and the path is not affected by the gravitation, 
is the same whatever be its intensity. A ray passing near the Sun 
ought not to be deflected on this view: an observed deflection, 

* On this and the following paragraphs, cf. however the end of the paper. 
VOL. XIX. PART VI. 23 



336 Sir Joseph Lannor 

whicli a priori was well worth looking for, would seem to await 
explanation on other lines. 

Again would there be an observable change of periods of 
spectral lines according as the vibrating source was at the Sun or 
at the Earth? The underlying absolute periods of radiating 
hydrogen molecules would be always and everywhere the same: 
thus the apparent period in the gravitational world would vary 
inversely as the local scale of time, and be longer at the Sun. 
But this is a local apparent period. The waves sent out from the 
solar molecule are observed at the earth: we have seen that their 
length changes as they progress, being inversely as the local scale 
of length, and their speed changes also, so that their period changes 
inversely as the local scale of time. Thus when they have reached 
the Earth their period conforms to the local scale and would agree 
with that of the radiation of a similar terrestrial molecule. In fact 
if complete correspondence is established*, element for element, 
as above, all periods or intervals of time measured at any element 
are changed in the same ratio depending on the locality alone. 
Any other conclusion would make the pulsating rays into signals 
establishing absolute time throughout the apparent universe, 
which could hardly be a result of a theory of relativity. 

The condition 8ct = prescribes a definite ray- velocity for each 
element of arc, the same forwards as backwards, only when Scr^ 
involves St'^ but no products of St with other differentials: in 
other cases it gives two velocities, not equal and opposite, and 
this spacial scheme of rays seems to fail. If rays are to be pro- 
perties of the space a very severe restriction is thus imposed on 
the form of 8cr^, but one which seems to be satisfied for the slight 
modifications that would be involved in the actual gravitation of 
experience. 

In the modifications of the expression for 8cr^ which absorb 
gravitation the coefficients do not involve the time explicitly: 
therefore the ray-paths are fixed in the space, and it almost looks 
as if they were guides imposed by the nature of the space alone, 
as thus modified, for the alternating energies of radiation to run 
along. 

Any inference that because a ray is fixed in space, as many 
waves must run in at one end as run out at another, would be at 
variance with the very notion of relativity, by providing a scale 
of absolute time throughout the universe. Such an argument 
seems to amount in more general form essentially to this: when 
the expression for 8a^ does not contain t explicitly it will make no 

* As has been estabKshed for the more general case in a beautiful analysis by 
Prof. Th. de Bonder, of Brussels, Comjptes Rendus, July 6, 1914, Archives du 
Musee Teyler, Haarlem, vol. iii, 1917, pp. 80-180. [It is merely continuity with 
non-gravitational fields, and not correspondence, that is established.] 



Gravitation and Light 337 

difference to the cosmos if t is everywhere increased by the same 
constant: therefore the scale of time must be everywhere the same 
— which excludes any possibility of local scales of time, A change 
of origin of measurement for time is not the same as progress of 
events in time, unless the scale of time is everywhere the same. 

The matter may be put from a different angle as follows. To 
obtain the time of transit of a ray from P to Q it is not possible 
to add elements of heterogeneous local times such as 8^*. What 
can be done is to find the true underlying time of transit. If this 
homogeneous true time is delayed at the start, at one end of the 
path at P, it is delayed by an equal amount at arrival at the other 
end, as the equations of transit do not involve this time explicitly: 
hence apparent times at the two ends are delayed not by equal 
amounts, but by amounts inversely as their local scales, so that 
a ray cannot (as has been impKed) transmit apparent time along 
its path. 

The alternative development is, as above, that 8ct^ being the 
underlying unchanging standard there are local scales of time, and 
local scales of length which may involve direction, and therefore 
also of velocity (including that of the rays) which is their quotient. 
The path of a ray from point to point is determined by making 
the number of wave-lengths from the one to the other minimum, 
that is by Sjds/X = : but Ss and A are both altered to the same 
scale; thus there is no alteration due to gravitation in the varia- 
tional equation determining the ray-path, so that it would suffer 
no deflection. The essential feature in the argument is that, 
whether rays may be regarded as the limiting case of free orbits 
or not, their specification has been postulated so that the ray- 
velocities correspond in the same way as all other velocities in 
the two frames. 

Appendix. — On Space and Time. 

Let us try for a closer realization of these abstract positions. 
The Gauss-E,iemann theory for an ordinary curved surface will be 
wide enough to serve as an illustration. The theory involves 
coordinates p, q: they must represent something. The very least 
we can do for them is to regard the surface as twofold extension 
dotted over with points, so that the coordinates express their 
order of arrangement according to some plan of counting them 
with respect to this extension in which they lie. There is no metric 
idea at all in this numeration, and nothing to distinguish one 
surface from another. Now bring in an infinitesimal unchanging 

* Yet it is just such elements of quasi-time d.v^ that are added together, ■infra 
p. 343. It is the so-called shifting clock-time and absolute time running parallel 
that are the source of all this confusion. 

23—2 



338 Sir Joseph Larmor 

measuring rod, which can make play in each element of extension 
represented by SpSq and also be transferred from place to place: 
and we can thereby impart or rather superpose metric quality on 
the twofold which hitherto was purely positional or rather tactical. 
The simplest plan is to follow Euclid, on the basis of the Pytha- 
gorean theorem, and expressing absolute length according to 
measuring rod by a symbol Ss, to impose a scale-relation of form 

8s^ = Sp^ + Sq^. 

But this metric cannot be applied consistently over a curved 
surface, unless it is of the very special type that can be rolled out 
flat: for other surfaces it is necessary to have the more general 
type of relation 

S52 =fSp^ + ^g^pBq + hSq^ 

in which/, g, h are functions of the coordinates p, q. 

This specification of an imported metric thus determines the 
surface: starting from a given small region of it, the form of the 
surface in an outer threefold space can be gradually evolved by 
prolongation so as to fit in with consistent application of this 
metric. It is this idea of prolongation of a non-uniform manifold, 
equivalent to its geometrical continuation within a flat one of 
higher dimensions, that was Riemann's contribution to the ideas 
of geometry. But the manifold itself is supposed to be given only 
tactically or descriptively; and it is the metric that is imposed on 
it that, by its demand for consistency in measurements, deter- 
mines for it a form, as located in a higher flat manifold. This form 
is expressed in detail analytically by the ' curvature ' at each place, 
as specified by a set of functions (one in the case of a surface) of 

the successive gradients of the set /, g,h, If we keep the system 

self-contained by avoiding the immersion of it in a uniform 
auxiliary manifold of higher dimensions, our resource is to deter- 
mine the curvature as the simplest set of functions that are invariant 
for local changes of coordinates. But, in order of evolution at any 
rate, this invariance may be held to be only a derived idea. 

In any case the nature of the non-uniform manifold, as thus 
determined by a metric imposed on formless space, has nothing 
to do essentially with the coordinates p, q, ... to which it may 
happen to be referred: it is settled by the algebraic form of the 
functions/, g, h, ... expressed in terms of jj, q, ..., or in geometric 
terms by the 'curvature' as so expressed. 

As a consequence, if we transform a surface from internal or 
intrinsic coordinates p, q, to others p', q', which are assigned 
functions of the former, so that we obtain 

§§2 = f'8p'^ + 2g'8p'8q' + h'8q'^ 
and construct the surface implied in this new equation by the 



4 



Gravitation and Light 339 

process of continuation, it will prove to be just the same surface 
as before. Whether it is expressed in terms of p' , q' or of jp, q is 
intrinsically of no consequence : the coordinates are of no account, 
it is only the functional forms of/, g, h that are essential. 

This last statement, developed in terms of the criterion of 
invariance in order to avoid a representation by immersion in a 
uniform geometrical manifold of dimensions higher than the given 
four of space and time, appears to cover the general relativity 
of Einstein. The/, g, h, ... can be named the potentials which deter- 
mine the space. In the special relativity, before gravitation was 
absorbed into the metric of extension, all spaces were flat, so 
/, g, h, ... were constants ; which is all that is left, for that particular 
case, of these relations of invariance. 

In this flat fourfold, relativity implied merely that a physical 
system is determined by its own internal relations, so that the 
position that may be assigned to it in the fourfold is of no account, 
any more than is the position of a surface or a system of bodies 
in space. In the later general relativity the manifold must be 
supposed given descriptively by coordinates, which represent 
numerical counts arranged to suit the number of dimensions that 
are involved : it only gains internal form when a metric is imposed 
upon it. If the Euchdean metric 

§s^ = Sp^ + Sq^+ ... 

is imposed it becomes a Euclidean space everywhere uniform and 
also flat, in which bodies are mobile without change of form. If 
a metric varying with position is imposed, the expressions in this 
manifold of the metric relations of nature will become complicated, 
and the relations so changed be described as a modified set of laws. 

The original non-metric continuum might be marked for 
instance by gradations of colour: the colour-scheme of Newton as 
developed by Young, Helmholtz, and Maxwell, is the standard 
example of a non-metric threefold extension. 

May we not here have refined down to the unresolvable essence 
of space, as the mere possibihty of descriptive continuity of three- 
fold type which is an essential feature in our mental world ? Within 
this a priori datum of threefold uncharted pure continuity we may 
construct types of charted spaces almost without limit, by imposing 
metrics of various types. Any particular space is not however 
determined by the system of coordinates of reference p, q, ... but by 
the variable coeSicients f, g, h, ... of the imposed metric expressed 
as functions of them. But yet it is only under special conditions 
when it is uniform and flat that finite difl'erences of these co- 
ordinates can be involved, this being part of the expression of the 
mobility of solid bodies in the space. It is in this narrower sense, 
that "the system of coordinates is accidental, that relativity has 



340 . Sir Joseph Larmor 

now expelled general metric ideas of position. Would it be entirely 
wrong to assert that local or sectional relativity has been retained 
for nature, so far as this order of ideas extends, by transferring the 
laws of nature into a space-time frame which itself no longer 
possesses that quality? 

The distinction has thus been made between an ultimate idea 
of space as mere threefold continuity, marked but uncharted, and 
the metric that may be imposed on it by which it becomes a frame 
fit for the purposes of description of nature. There is only one 
space: but its practical aspect, whether Euclidean or elliptic or 
merely heterogeneous, depends on the metric that we choose to 
assign to it. The metric would thus appear to pertain more closely 
to the order of nature for which it is to form the most convenient 
frame for description, than to space itself. For space is primarily 
bare threefold continuity; though a set of descriptive coordinates 
jp, q, ... is unavoidable as a foundation of thought, any set is as 
valid as any other. For ultimately, the count or census of the points 
or marks that pervade the continuity and render it descriptively 
given to us, is the same count however it be made. May we say 
that the insistent, originally uncritical, notion of relativity reduces 
itself ultimately into this postulate, that as nature is presented to 
us, it is such that in mental operations we need attend only to 
one portion of the spacial continuity at a time? This makes the 
onefold time, or rather mere temporal succession as representable 
by the 8a of Minkowski, the fundamental feature*, which however 
diverges spacially into a manifold: according to Hamilton long 
ago, algebra was the science of pure time. 

In the above, space is given by a manifold array of points, of 
which the coordinates p, q, ... express one of the varieties of 
numerical census. Is then space-time absolute, or is it continually 
being constructed by physical science as it ranges over the void, 
for its own purposes, just to the extent that it may be required? 
May we say that the formless manifold is the fundamental feature, 
that the array of points and their census do not need to be 
definite in any respect a priori, and that the metric which is 
imposed on it and makes it into a definite working type of space 
is related to the physical world and so is to be regarded as evolved 
in connexion with our organic description or mapping of nature, 
and to be just as permanent? 

What remains of the original notion of relativity after this 
sifting of ideas would then coincide with the principle of Newton, 
Faraday and Maxwell, originated by Descartes, that the operations 
of nature are elaborated in fourfold extension according to a scheme 
purely differential, that is by transmission from element to element 

* The spacial sign here attached to 8(r^ is an accident of the order of exposition. 



J 



Gravitation and Light 341 

of the cosmos, in no case leaping across intermediate elements as 
action at a distance would imply. The early stage of formulation 
of the confused notion of relativity is the postulate that position 
and change of position are purely relative: the final solution is to 
abolish the idea of immediate ^m/e change of position altogether. 
But that does not imply that a portion of the cosmos can evolve 
itself without constant interference from all the rest. 

To a question as to what is gained by absorbing gravitation in 
space an answer would be that it need make no difference as regards 
gravitation ; but if other relations of an assumed space- time fourfold 
(e.g. stress-tensor theory) have to go in also in a simple way, it 
may be convenient or even necessary to assist them by choosing 
a space which requires some alterations of the recognised laws of 
gravitation and, if these suggested discrepancies are verified, that 
may presumably have a claim to be the real type of space. The 
aim is not primarily to reduce gravitation to a quality of space, — 
perhaps is not even relativity, which has evaporated, — but is to get 
it out of Newtonian space and time into the mixed space-time 
fourfold which was strongly suggested by the form of the Max- 
wellian electrodynamic relations of free space, and would make 
that scheme valid for great velocities of convection beyond ex- 
perience, even up to the speed of light. 

An expansion of the Einstein ideas on general relativity has 
been worked out by H. Weyl {Ann. der Physik, 59, 1919) in which 
a further metric scale of vector character appears to be imposed 
on a non-uniform space-time, which has here been itself ascribed 
to the imposition of a Gauss- Riemann metric on the formless 
spacial threefold that is inherent in the mind. There would seem 
to be no formal obstacles to such piling up of metric upon metric, 
in an unlimited play of thought. 

The physical analysis perhaps not very remote to this new 
elaboration of metric is, as I think Prof. Schouten remarks, a 
theory of an elastic aether in which at each point p, q, ... a vector 
displacement ^, 17, ... of the element of the medium is supposed, 
involving a strain and an elastic stress determined in terms of 
the strain by assigned laws. Only it is to be remembered that 
time is now in a fourth dimension, in which the historical world- 
process is all spread out once for all; so that the feature of elastic 
wave propagation becomes a static relation. The idea that the 
single fundamental electric vector is represented by a superposed 
metric is thus correlative with the usual dynamical hypothesis 
that electric force is a stress in an aether. It thus affords another 
illustration of this kind of speculation: the interlacing of space 
and time for purposes of electrodynamics having upset the his- 
torical development of dynamical principles on a Newtonian basis 
of separate space and time, order has to be re-constituted by 



342 Sir Joseph Larmor 

piecing together a cognate analytical scheme on a symmetrical 
fourfold basis which tries to make no difference between them. 

It is not improbable that these remarks merely turn over 
ground that has already been explored by cultivators of hyper- 
geometry. But it may be claimed that the interest of this range 
of ideas extends far beyond the analytical technique, and that their 
naive expression in a form of language outside its conventions may 
prove to be helpful in other regions of speculation. 



The argument above has been based on the supposition that the 
mathematical analysis must establish a complete correspondence, 
element for element, between the activities in the new space-time 
and in the Newtonian space and time. That however is not the case. 
There is a gravitational correspondence into which radiation and 
its rays do not enter. As regards the latter no conclusions could be 
drawn at all, except in the special circumstances in which the 
coordinate X/^ that stands nearest to time * does not enter explicitly 
into the quadratic expression determining the space. If that is 
postulated the equations of propagation of radiation have their 
solutions periodic as regards x^^^ treated as a quasi-tim.Q, therefore 
every beam of radiation carries with it a scale of X/^^ throughout 
its course |. Moreover, if the spacial quadratic contained hx^ in a 
product term, the velocities of the waves of radiation in forward 
and backward directions would not be the same : their half difference 
would thus be the local velocity of the frame of reference in that 
direction. Where hx/^^ does not occur in the first power, the frame of 
reference is thus fixed locally with respect to the waves of light 
and their assumed underlying uniform fourfold extension with 
regard to which they are propagated. 

Thus, under these postulated circumstances of x^ not occurring 
explicitly in So-^, the mere fact that isotropic vibratory radiation 
exists with its absolute velocity c is sufficient, not merely to de- 
termine absolute measurements both in space and time, at every 
locality in the extension, but also to determine the rate of motional 
change of the coordinates as referred to the uniform space-time of 
the radiation. It is gravitational correspondence, subject to this 
general control of the whole range of space-time by observations 
of light, with its isotropic and uniform qualities, that has led to 
verifiable conclusions. Cf. letter in Nature, Jan. 22, 1920: also 
Monthly Notices R. Astron. Soc. 

* That is the one coordinate the square of whose differential is affected in dcr- 
with a negative sign, which marks it off from the others. 

t It is the alleged measurement of this abstract coordinate x^ by a travelling 
clock, which connotes a physical system, that is a main source of confusion. 



Gravitation and Light 343 

We have absorbed gravitation into space and time by distorting 
the latter from its essential Newtonian uniformity: but there can 
be no illusion about the matter either way, for the theoretical 
measuring bar of the differential spacial theory is not our only 
instrument; in the practical world rays of light provide the essential 
isotropic measures, and the spectroscope is always available to 
reveal to us what spacial adjustments have been made, in relation 
to the underlying frame with regard to which the propagation of 
light is isotropic and has its standard absolute velocity. Light, 
instead of conforming to local relativity, imposes its own absolute 
space-time*. 

The argument may be directed tow^ards yet another type of 
conclusion, as follows. When change is made from Newtonian 
space and pure time to the uniform space-time fourfold, the 
equation of a straight path is altered from h^ds = to SJ(Zct = 0. 
The free orbits in any field of force of potential energy function 
— F can readily be altered so as to preserve continuity with this 
change, as above, that is, so that where F becomes negligible they 
tend to straight lines: they are then given by 

h\{d<j^ + 2Vdt^Y = ^. 

The interpretation is at hand, to regard them as the analogues of 
straightest paths in a modified space-time, referred to a set of 
coordinates represented now by colourless symbols x^, x^, Xz, x^ 
and given in terms of them by 

8a2 = Sa;i2 + Sx^^ + Sx^^ - c^ (1 - 2c-2F) Sx^^ 

As Sct^ does not here involve x^^ explicitly, the differential equations 
of propagation of free radiation, as expressed in this space-time 
in terms of these coordinates, have solutions involving the quasi- 
time, x^ only in the form e'^^*: therefore the radiation from any 
source, however far it has travelled, retains the same period in 
regard to x^ as it had at the start. Around a radiating molecule the 
extension can be taken as practically uniform: therefore the 
interval of absolute time is equal to (1 — c'W) hx^. It follows thus 
from the periodicity as regards x^ that the periodic time of a ray 
alters as it travels so as to be proportional to 1 — c~^V . If the ray 
belongs to a definite molecular period at the Sun, it has changed 
when it reaches the Earth so as to agree no longer with that period 
as reproduced by a local vibrator. 

All this is true only to the first order, but it applies to any law of 
potential, and is irrespective of any special energy-tensor theory. 
The point to be brought out is that if influence of gravitation on 

* Prof. Eddington in a recent article, Quarterly Review, Jan. 1920, seems not to 
disagree with this conclusion: at any rate he contemplates the possibility of an 
aether. 



344 



Sir Joseph LanTwr, Gravitation and Light 



spectral periods were definitely disproved, then it would appear 
that any hope of bringing orbits into direct relation with the 
electrodynamic space-time fourfold must be abandoned altogether*, 
on the threshold. This drastic conclusion is perhaps an argument in 
fa^^our of the existence of the effect. 

The other two verifiable effects, the influence on the planetary 
perihelia and the deviation of light passing near the Sun, arise in 
part from first order and in part from second order causes. Unlike 
the previous one, their exact verification is thus a test of the special 
theory of Einstein, or the equivalent Least Action formulation. Its 
original recommendation was that it restricts the universal forces 
of nature to the one type of gravitation: possibly it would be 
difficult to imagine ways in which there could be room for any 
different result. 



* A formulation of the original Nordstiom type, starling from d^Vda = 0, is to 
some degree an exception. 



C. T. R. Wilson, On a Micro-voltameter 345 



On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney 
Sussex College. 

[Read 19 May 1919.] 

Experiments were described with a mercury voltameter, in 
which one elctrode consists of a sphere of mercury deposited on 
the end of a fine platinum wire and measured by means of a 
microscope. Quantities of electricity varying from a few hundred 
electrostatic units to about one coulomb may be measured by it. 
The almost instantaneous change of size of the drop when a 
capacity of one tenth of a microfarad, charged to 1 volt, is dis- 
charged through the instrument is easily observed. A magnet 
inserted in or removed from a coil connected to the terminals of 
the voltameter produces an easily measured effect. Experiments 
were also mentioned which suggest the possibility of its application 
in measurements of much smaller electrical quantities. 



346 R. Whiddington, The self-oscillations of a Thermionic Valve 



The self -oscillations of a Thermionic Valve. By R. Whid- 
dington, M.A., St John's College. 

[Read 19 May 1919.] 

(Abstract.) 

It lias been found possible to produce oscillations of almost any 
frequency from a three electrode vacuum valve, without employing 
the usual capacity-induction circuits. Thus a. valve with two 
suitable batteries, one in the anode circuit, another in the grid 
circuit, will produce quite powerful oscillations, whose frequency 
will be determined by the value of the grid potential. 

The phenomenon can be explained by supposing that the oscil- 
lations are due to surges of mercury ions closing in on the filament 
from the grid with a frequency given by the approximate formula 

2 2^ T/ 

n = . V 

md^ ' 

e • 
where — is the usual charge to mass ratio, d is the radial distance 

m o ^ 

filament to grid and V is the positive grid voltage. 

Experiments conducted so far indicate that the monatomic 
Hg ion with one live charge is mainly responsible. 



PROCEEDINGS AT THE MEETINGS HELD DURING 

THE SESSION 1918—1919. 

ANNUAL GENERAL MEETING. 

October 28, 1918. 
In the Comparative Anatomy Lectui'e Room. 

Prof. Marr, President, in the Chair. 

The following were elected Officers for the ensuing year : 

President : 

Mr C. T. R. Wilson. 

Vice-Presidents: 
Dr Doncaster. 
Mr W. H. Mills. 
Prof. Marr. 

Treasurer: 

Prof. Hobson. 

Secretaries : 

Mr Alex. Wood. 
Mr G. H. Hardy. 
Mr H. H. Brindley. 

Other Members of Council: 
Dr Shipley. 
Prof. Biffen. 
Mr L. A. Borradaile. 
Mr F. F. Blackman. 
Prof. Sir J. Larmor. 
Prof. Eddington. 
Dr Marshall. 
Prof. Baker. 
Prof. Newall. 
Dr Fenton. 

The following was elected an Associate of the Society : 
G. A. Newgass, Trinity College. 

The following Communications were made to the Society: 

1. Proof of certain identities in combinatory analysis. By Prof. L. J. 
Rogers and S. Ramanujan, B.A., Trinity College. 

2. Some properties of p (n), the number of partitions of n. By 
S. Ramanujan, B.A., Trinity College. 



348 Proceedings at the Meetings 

3. On the exponentiation of well-ordered series. By Miss D. Wrinch. 
(Communicated by Mr Gr. H. Hardy.) 

4. On certain trigonometrical series which have a necessary and 
sufficient condition for uniform convergence. By A. E. Jolliffe. 
(Communicated by Mr Gr. H. Hardy.) 

5. Some geometrical interpretations of the concomitants of two 
quadrics. By H. W. Turnbull, M.A. (Communicated by Mr G. H. 
Hardy.) 

6. On Mr Ramanujan's congruence properties of p (n). By H. B. C. 
Darling, B.A. (Communicated by Mr G. H. Hardy.) 

7. On the correct Generic Position of Dacrydium Bidwillii Hook f. 
By B. Sahni, M.A., Emmanuel College. (Communicated by Professor 
Seward.) 



February 3, 1919. 
In the Balfour Library. 

Mr C. T. R. Wilson, President, in the Chair. 

The following were elected Fellows of the Society : 

S. R. U. Savoor, B.A., Trinity College. 
S. C. Tripathi, B.A., Emmanuel College. 

The following was elected an Associate : 
P. W. Burbidge, Trinity College. 

The following Communications were made to the Society : 

1. The Gauss-Bonnet Theorem for multiply-connected regions of a 
surface. By E. H. Neville, M.A., Trinity College. 

2. On the representations of a number as a sum of an odd number of 
squares. By L. J. Mordell. (Communicated by Mr G. H. Hardy.) 

3. On certain empirical formulae connected with Goldbach's 
Theorem. By N. M. Shah and B. M. Wilson. (Communicated by 
Mr G. H. Hardy.) 

4. Note on Messrs Shah and Wilson's paper entitled : On certain 
empirical formulae connected with Goldbach's Theorem. By G. H. 
Hardy, M.A., Trinity College and J. E. Littlewood, M.A., Trinity 
College. 



Proceedings at the Meetings 349 

February 17, 1919. 
In the Comparative Anatomy Lecture Room. 

Mr C. T. E. Wilson, President, in the Chair. 
The following Communications were made to the Society : 

1. Note on an experiment dealing with mutation in bacteria. By 

Dr DONCASTER. 

2. Electrical conductivity of bacterial emulsions. By Dr Shearer. 

3. The bionomics of Aphis grossulariae, Kalt., and Aphis viburni. 
Shrank. By Miss M. D. Haviland. (Communicated by Mr H. H. 
Brindley.) 

4. (1) The conversion of saw-dust into sugar. 
(2) Bracken as a source of potash. 

By J. E. Purvis, M.A., Corpus Christi College. 

5. Terrestrial magnetic variations and their connection with solar 
emissions which are absorbed in the earth's outer atmosphere. By 
S. Chapman, M.A., Trinity College. 

6. The distribution of Electric Force between two electrodes, one 
of which is covered with radioactive matter. By W. J. Harrison, M.A., 
Clare College. 



May 19, 1919. 
In the Cavendish Laboratory. 

Mr C. T. R. Wilson, President, in the Chatr. 

The following were elected Fellows of the Society : 
E. V. Appleton, M.A., St John's College. 
W. G. Palmer, M.A., St John's College. 
S. P. Prasad, B.A., Trinity College. 

The following was elected an Associate : 

Mrs Agnes Arber. 
The following Communications were made to the Society: 

1. (1) Use of Neon Lamps in Technical stroboscopic work. 

(2) The distribution of intensity along the positive ray parabolas 
of atoms and molecules of Hydrogen and its possible 
explanation. 
By F. W. Aston, M.A., Trinity College. 

2. On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney 

Sussex College. 

3. The self-oscillations of a Thermionic Valve. By R. Whiddington, 
M.A., St John's College. 



INDEX TO THE PROCEEDINGS 

with references to the Transactions. 



M 



Abel's Theorem and its converses (Kienast), 129. 

Amos, A., Experimental work on clover sickness, 127. 

Aphidius, Life History of Lygocerus (Proctotrypidae), hyperparasite of 

(Haviland), 293. 
Aphis grossulariae Kalt., Bionomics of (Haviland), 266. 
Aphis viburni Schr., Bionomics of (Havtland), 266. 
Appleton, E. v.. Elected Fellow 1919, May 19, 349. 
Arbek, a., Elected Associate 1919, May 19, 349. 
Aston, F. W., Neon Lamps for Stroboscopic Work, 300. 
The distribution of intensity along the positive ray parabolas of atoms 

and molecules of hydrogen and its possible explanation, 317. 
Axiom in Symbohc Logic (Van Horn), 22. 

Bacteria, Mutation in (Doncaster), 269. 

Bailey, P. G., see Punnett, R. C. 

Bessel functidns of equal order and argument (Watson), 42. 

Bessel functions of large order (Watson), 96. 

Bionomics of Aphis grossulariae Kalt., and Aphis viburni Schr. (Havtland), 266. 

BORRADAILE, L. A., On the Functions of the Mouth-Parts of the Common 

Prawn, 56. 
Bracken as a source of potash (Purvis), 261. 
Brindley, H. H., Notes on certain parasites, food, and capture by birds of 

the Common Earwig (Forficula auricularia), 167. 
Buchanan, D., Asymptotic Satellites in the problem of three bodies. See 

Transactions, xxii. 
Burbidge, p. W., Elected Associate 1919, February 3, 348. 

Cambridgeshire Pleistocene Deposits (Marr), 64. 

Caporn, a. St Clair, The Inheritance of Tight and Loose Paleae in Avena niida 

crosses, 188. 
Cells, Action of electrolytes on the electrical conductivity of (Shearer), 263. 
Chapman, S., Terrestrial magnetic variations and their cormection with solar 

emissions which are absorbed in the earth's outer atmosphere. See 

Transactions, xxii. 
Colourimeter Design (Hartridge), 271. 
Convergence, Uniform (Jolliffe), 191. 
Convergence, Uniform, concept of (Hardy), 148. 



Index 351 

Convergence of certain multiple series (Hardy), 86. 

Corals, Reactions to Stimuli in (Matthai), 164. 

Crabro cephalotes. Solitary wasp (Warburton), 296. 

Cubic Binomial Congruences with Prime Moduli (Pocklington), 57. 

Darling, H. B. C, On Mr Ramanujan's congruence properties of ^ {n), 217. 

See MacMahon, P. A. 

Dirichlet, Theorem of (Todd and Norton), 111. 

DoNCASTER, L., Note on an experiment dealing with mutation in bacteria, 269. 
DuTT, C. P., On some anatomical characters of coniferous wood and their 
value in classification, 128. 

Electric Force between two Electrodes (Harrison), 255. 
Electrometer, A self-recording, for Atmospheric Electricity (Rudge), 1. 
Empii'ical formula comiected with Goldbach's Theorem (Shah and Wilson), 

238. 
Exponentiation of well-ordered series (Wkinch), 219. 

Eish-freezrng (Gardiner and Nuttall), 185. 

Forficula auriciiluria, Common Earwig, parasites, food, and captm-e by birds 
of the (Brindley), 167. 

Gardiner, J. Stanley, and Nuttall, G. H. F., Fish-freezing, 185. 

Gauss-Bonnet Theorem (Neville), 234. 

Gibson, C. Stanley, Elected FeUow 1918, May 20, 189. 

Goldbach's Theorem (Shah and Wilson), 238. 

Gravitation and Light (Larmor), 324. 

Gray, J., The Effect of Ions on Cihary Motion, 313. 

Green, F. W., Elected Fellow 1916, November 13, 126. 

Hardy, G. H., On a theorem of Mr G. Polya, 60. 

On the convergence of certain multiple series, 86. 

Sir George Stokes and the concept of uniform convergence, 148. 

See Rogers, L. J., and Ramanujan, S. 

Hardy, G. H., and Littlewood, J. E., Note on Messrs Shah and Wilson's 

paper entitled: On an empirical formula connected with Goldbach's 

Theorem, 245. 
Hargreaves, R., The Character of the Kinetic Potential in Electromagnetics. 

See Transactio7is, xxii. 
Harrison, W. J., The distribution of Electric Force between two Electrodes, 

one of which is covered with Radioactive Matter, 255. 

The pressure in a viscous liquid moving through a channel with diverging 

boundaries, 307. . 
Hartridge, H., Colom-imeter Design, 271. 
Haviland, M. D., The bionomics of Aphis grossulariae Kalt., and Aphis 

viburni Schr., 266. 

vol. XIX. part VI. 24 



352 Index 

Haviland, M. D., Preliminary Note on the Life History of Lygocerus (Procto- 

trypidae), hyperparasite of Aphidius, 293, 
Hill, M. J. M., On the Fifth Book of Euclid's Elements (Foui'th Paper). See 

Transactions, xxn. 
Horn, see Van Horn. 

Hydrodynamics of Relativity (Weatherburn), 72. 
Hydrogen Ion Concentration (Saunders), 315. 

Identities in combinatory analysis (Rogers and Ramanitjan), 211. 

Ince, E. Lindsay, Elected Fellow 1918, February 18, 189. 

Intensity along the positive ray parabolas of atoms and molecules of hydi'ogen 

(Aston), 317. 
Ion, Hydrogen, Concentration (Saunders), 315. 
Ions, Effect of, on Ciliary Motion (Gray), 313. 

JoLLiFFE, A. E., On certain Trigonometrical Series which have a Necessary 

and Sufficient Condition for Uniform Convergence, 191. 
Jones, W. Morris, Elected Associate 1916, October 30, 126. 1 

KiENAST, A., Extensions of Abel's Theorem and its converses, 129. 

Lake, P., Glacial Phenomena near Bangor, North Wales, 127. 

Shell-deposits formed by the flood of January, 1918, 157. 

Larmor, J., Gravitation and Light, 324. 

Liquid, Viscous, pressiire in a (Harrison), 307. 

LiTTLEWOOD, J. E., see Hardy, G. H. 

Logic, Primitive Propositions of (Nicod), 32. 

Logic, Symbolic, an axiom in (Van Horn), 22. 

Lygocerus (Proctotrypidae), hyperparasite of Aphidius, Life History of 

(Haviland), 293. 
Lynch, R. I., Elected Fellow 1916, November 13, 126. 

Exhibition of the Fruit of Chocho Sechium edule, 127. 

Mackenzie, K. J. J., see Marshall, F. H. A. 

MacMahon, p. a., On certain integral equations, 188. 

MacMahon, p. a., and Darling, H. B. C, Reciprocal Relations in the Theory 

of Integral Equations, 178. 
Madreporarian Skeleton (Matthai), 160. 
Marr, J. E., Submergence and glacial cUmates during the accumulation of the 

Cambridgeshire Pleistocene Deposits, 64. 
Marshall, F. H. A., and Mackenzie, K. J. J., On extra mammary glands and 

the reabsorption of milk sugar, 127. 
Matthai, G., Is the Madreporarian Skeleton an Extraprotoplasmic Secretion 

of the Polyps?, 160. 

On Reactions to Stimuli in Corals, 164. 

Micro-voltameter (Wilson), 345. 



Index 353 

Modular Functions (Mordell), 117. 

Moduli, Prime, Quadratic and Cubic Binomial Congruences with (Pockling- 

ton), 57. 
Mordell, L. J., On Mr Ramanujan's Empirical Expansions of Modular 

Functions, 117. 
Multiple series. Convergence of certain (Hardy), 86. 
Mutation in bacteria (Doncaster), 269. 

Neon Lamps for Stroboscopic Work (Aston), 300. 

Neville, Eric H., The Gauss-Bonnet Theorem for Multiply-connected Regions 

of a Surface, 234. 
Newgass, G. a.. Elected Associate 1918, October 28, 347. 
NicoD, J. G. P., A Reduction in the number of the Primitive Propositions of 

Logic, 32. 
Norton, H, T. J., see Todd, H. 
NuTTALL, G. H. F., see Gardiner, J. Stanley. 

Oldham, F. W. H., Elected Fellow 1917, February 5, 127. 

Palmer, W. G., Elected Fellow 1919, May 19, 349. 

Partitions of n, Some properties of p {n), the number of (Ramanujan), 207, 
Phase, Limits of applicability of the Principle of Stationary (Watson), 49. 
PoCKLiNGTON, H. C, The Du-ect Solution of the Quadratic and Cubic Binomial 

Congruences with Prime ModuU, 57. 
P6lya, G., see Hardy, G. H. 
Prasad, S. P., Elected Fellow 1919, May 19, 349. 
Prawn, Common, Functions of the Mouth-Parts (Borradaile), 56. 
Primitive Propositions of Logic (Nicod), 32. 

Proceedings at the Meetings held during the Session 1916-1917, 125. 

1917-1918, 187. 
1918-1919, 347. 
PuNNETT, R. C, and Bailey, P. G., Inheritance of henny plumage in cocks, 126. 

Some experiments on the Inheritance of weight in Rabbits, 188. 

Purvis, J. E., The conversion of saw-dust into sugar, 259. 

Bracken as a source of potash, 261. 

Quadratic and Cubic Binomial Congruences with Prime Moduli (Pockling- 

TON), 57. 

Quadncs, Geometrical Interpretations of the Concomitants of Two (Turn- 
bull), 196. 

Ramanujan, S., Elected Fellow 1918, February 18, 189. 

On the expression of a number in the form ax'^ + hy'^ + cz^ + du^, 11. 

Empu'ical Expansions of Modular Functions (Mordell), 117. 

On certain Trigonometrical sums and their applications in the theory of 

numbers. See Transactions, xxii. 



354 Index 

Ramanujan, S., On some definite integrals, 188. 

Some properties of ^ (n), the number of partitions of n, 207. 

8ee Rogers, L. J. 

Reciprocal Relations in the Theory of Integral Equations (MacMahon and 

Daeling), 178. 
Relativity, Hydrodynamics of (Weatherburn), 72. 
Rodrigues, Natural History of (Snell and Tams), 283. 
Rogers, L. J., and Ramanujan, S., Proof of certain identities in combinatory 

analysis, 211. 
RuDGE, W. A. D., A seK-recording electrometer for Atmospheric Electricity, 1. 



the I 



Sahni, B., On an Australian specimen of Clepsy drop sis, 128. 

Observations on the Evolution of Branching in the Ferns, 128. 

■ ■ On the branching of the Zygopteridean Leaf, and its relation to 

probable Pinna-nature of Oyropteris sinuosa, Goeppert, 186. 

• The Structure of Tmesipteris Vieillardi Dang, 186. 

— — • On Acmopyle, a Monotypic New Caledonian Podocarp, 186. 
Saunders, J. T., On the growth of Daphne, 126. 

A Note on Photosynthesis and Hydrogen Ion Concentration, 315. 

Savoor, S. R. U., Elected Fellow 1919, February 3, 348. 

Saw-dust, Conversion of, into sugar (Purvis), 259. 

Shah, N. M., and Wilson, B. M., On an empirical formula connected with 

Goldbach's Theorem, 238. 
Shearer, C, The action of electrolytes on the electrical conductivity of the 

bacterial cell and their effect on the rate of migration of these cells in 

an electric field, 263. 
Shell-deposits formed by the flood of January, 1918 (Lake), 157. 
Snell, H. J., and Tams, W. H. T., The Natiu-al History of the Island of 

Rodrigues, 283. 
Stokes, Sir George, and the concept of uniform convergence (Hardy), 

148. 
Symbolic Logic, An Axiom in (Van Horn), 22. 



Tams, W. H. T., see Snell, H. J. 

Theorem of Dirichlet (Todd and Norton),. 111. 

Theorem of Mr G. Polya (Hardy), 60. 

Thermionic Valve, Self-oscillations of a (Whiddington), 346. 

Todd, H., and Norton, H. T. J., A particular case of a theorem of Dirichlet, 

111. 
Trigonometrical Series which have a Necessary and Sufficient Condition for 

Uniform Convergence (Jolliffe), 191. 
Tripathi, S. C, Elected Fellow 1919, February 3, 348. 
TuRNBULL, H. W., Some Geometrical Interpretations of the Concomitants oi 

Two Quadrics, 196. 

Van Horn, C. E., An Axiom in Symbohc Logic, 22. 



i 

hlet, I 



I 



Index 355 

Waeburton, C, Note on the solitary wasp, Crabro cephaloies, 296. 

Wasp, Crabro cephalotes (Warburton), 296. 

Watson, G. N., Bessel functions of equal order and argument, 42. 

The hmits of applicability of the Principle of Stationary Phase, 49. 

Bessel functions of large order, 96. 

Asymptotic expansions of hypergeometric functions. See Transactions, 

xxn. 
Weatherburn, C. E., On the Hydrodynamics of Relativity, 72. 
WmDDiNGTON, R., The self-oscillations of a Thermionic Valve, 346. 
Wilson, B. M., see Shah, N. M. 
Wilson, C. T. R., Methods of investigation in atmospheric electricity, 126. 

On a Micro-voltameter, 345. 

Wood, T. B., The siu-face law of heat loss in animals, 126. 
Woods, H., The Cretaceous Faunas of New Zealand, 127. 
Wrinch, D., On the exponentiation of well-ordered series, 219. 

Yamaga, N, Elected Associate 1916, November 13, 126. 



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I 



CONTENTS. 

PAGE 

Colourimeter Design. By H. Hartridge, M.D., Fellow of King's 

College, Cambridge. (One Fig. in text) 271 

The Natural History of the Island of Rodrigues. By H. J. Snell 
(Eastern Telegraph Company) and W. H. T. Tams. (Communi- 
cated by Professor Stanley Gardiner) 283 

Preliminary Note on the Life History of Lygocerus {Proctotrypidae), 
hyperparasite of Aphidius. By Maud D. Haviland, Fellow of 
Newnbam College. (Communicated by Mr H. H. Brindley) . 293 

Note on the solitary wasp, Crabro cephalotes. By Cecil Warburton, 

M.A., Christ's College 296 

Neon Lamps for Stroboscopic Work. By F. W. Aston, M.A., Trinity 
College (D.Sc, Birmingham), Clerk-Maxwell Student of the Uni- 
versity of Cambridge. (One Fig. in text) 300 

The pressure in a viscous liquid moving through a channel loith diverging 
boundaries. By W. J. Harrison, M.A., Fellow of Clare College, 
Cambridge. (One Fig. in text) 307 

The Efect of Ions on Ciliary Motion. By J. Gray, M.A., Fellow of 

King's College, Cambridge 313 

A Note on Photosynthesis and Hydrogen Ion Concentration. By J. T, ■ 

Saunders, M.A., Christ's College 315 

The distribution of intensity along the positive ray parabolas of atoms 
and molecules of hydrogen and its possible explanation. By F. W. 
Aston, M.A., Trinity College (D.Sc, Birmingham), Clerk-Maxwell 
Student of the University of Cambridge. (Three Figs, in text) . 317 

Gravitation and Light. By Sir Joseph Larmor, St John's College, 

Lucasian Professor 324 

On a Micro-voltameter. By C. T. R. Wilson, M.A., Sidney Sussex 

College 345 

The self-oseillations of a Thermionic Valve. By R. Whiddington, M, A., 

St John's College 346 

Proceedings at the Meetings held duiing the Session 1918 — 1919 . 347 

Index to the Proceedings with references to the Transactions . . 350 



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I 



CONTENTS. 

VOL. XX. 



PAGE 



On the term by term integration of an infinite series over an infinite range 
and the inversion of the order of integration in repeated infinite 
integrals. By S. Pollard, M.A., Trinity College, Cambridge. 
(Communicated by Prof. 6. H. Hardy) 1 

Note on Mr Hardy^s extension of a theorem of Mr Polya. By Edmuxd 

Landau. (Communicated by Prof. G. H. Hardy) .... 14 

Studies on Cellulose Acetate. By H. J. H. Fenton and A. J. Berry . 16 

An examination of Searle's method for determining the viscosity of very 
viscous liquids. By Kurt Molin, Filosofie Licentiat, Physical 
Institute, Technical College, Trondhjem. (Communicated by Dr 
G F. C. Searle.) (Four figs in Text) 23 

Preliminary Note on Antennal Variation in an Aphis (Myzus ribis, 
Linn.). By Maud D. Haviland, Fellow of Newnham College. 
(Communicated by Mr H. H. Brindley) ..... 35 

The effect of a magnetic field on the Intensity of spectrum lines. By H. P. 
Waran, M.A., Government Scholar of the University of Madras. 
(Communicated by Professor Sir Ernest Rutherford.) (Plates I 
and II and one fig. in Text) ........ 45 

Further Notes on the Food Plants of tite Common Earwig (Forficula 

auricularia). By H. H. Brindley, M.A., St John's College . . 50 

Lagrangian Methods for High Speed Motion. By C. G. Darwin . . 56 

A hifilar method of measuring the rigidity of wires. By G. F. C. Searle, 
Se.D., F.R.S., University Lecturer in Experimental Physics. (Five 
figs, in Text) 61 

The Rotation of the Non-Spinning Gyrostat. By G. T. Bennett, M.A., 

F.R.S., Emmanuel College, Cambridge ...... 70 

Proof of the equivalence of different mean vahies. By Alfred Kienast. 

(Communicated by Professor G. H. Hardy) ..... 74 

Notes on the Theory of Vibrations. (1) Vibrations of Finite Amplitude. 
(2) A Theorem due to Routh. By W. J. Harrison, M.A., Fellow of 
Clare College ■ 83 

Experiments with a plane diffraction grating. By G. F. C. Searle, 
Sc.D., F.R.S., University Lecturer in Experimental Physics. (Ten 
figs, in Text) 88 

The Shadow Electroscope. By R. Whiddington, M.A., St John's College. 

(One fig. in Text) 109 

Mathematical Notes. By Professor H. F. Baker and C. V. Hanumanta : 
On the Hart circle of a spherical triangle . . . . . .116 

On a property of focal conies and of bi circular quartics . . . 122 
On the construction of the ninth point of intersection of two plane cubic 
curves of which eight points are given ...... 131 



vi Contents 



Mathematical Notes (continued); 

On a proof of the theorem of a double six of lines by projection from 
four dimensions. (Three figs, in Text) . . . . . .133 

145 

147 



On transformations urith an absolute quadric .... 

On a set of transformations of rectangular axes. (One fig. in Text) 

On the generation of sets of four tetrahedra of which any tivo are 

mutually inscribed .......... 155 

On the reduction of homography to movement in three dimensions. (One 

fig. in Text) 158 

On the transformation of the equations of electrodynamics in the 
Maxwell and in the Einstein for ms . . . . . . .166 

On the stability of periodic motions in general dynamics . . . 181 

On the stability of 7'otating liquid ellipsoids ...... 190 

On the general theory of the stability of rotating masses of liquid . . 198 

Sur le principe de Phragmen-Lindelijf Par Marcel Riesz. With Note 

by G. H. Hardy 205 

A note on the nature of the carriers of the Anode Rays. By G. P. 

Thomson, M.A., Fellow of Corpus Christi College .... 210 

Proceedings at the Meetings held during the Session 1919 — 1920 . . 212 

The Problem of Soaring Flight. By E. H. Hankin, M.A., Sc.D., late 
Fellow of St John's College, Cambridge, Chemical Examiner to 
Government, Agra, India. (Communicated by Mr H. H. Brindley.) 
With an Introduction by F. Handlby Page, C.B.E., F.R.Aer.S. . 219 

Preliminary Note on the Superior Vena Ca,va of the Cat. By W. F. 

Lanchester, M. a., King's College, and A. G. Thacker . . . 228 

A Note on Vital Staining. By F. A. Potts, M.A., Trinity Hall. (One 

fig. in Text) 231 

Preliminary Note on a Cynipid hyperparasite of Aphides. By Maud D. 
Haviland, Fellow of Newnham College. (Communicated by 
Mr H. H. Brindley) 235 

A method of testing Triode Vacuum Tubes. By E. V. Appleton, M.A., 

St John's College. (Two figs, in Text) 239 

The Rotation of the Non-Spinning Gyrostat. By Sir George Greenhill 

and Dr G. T. Bennett 243 

On the representation of the simple group of order 660 as a groiop of linear 
substitutions on 5 symbols. By Dr W. Burnside, Honorary Fellow 
of Pembroke College ......... 247 

On the representation of algebraic numbers as a sum of four squares. By 

L. J. Mordell. (Communicated by Professor H. F. Baker) . . 250 

On a Gaussian Series of Six Elements. By L. J. Rogers. (Communicated 

by Professor G. H. Hardy) 257 

Note on Ramanujan^s trigonometrical function Cg(n), and certain series of 

arithmetical functions. By Professor G. H. Hardy .... 263 

On the distribution of primes. By H. Cramer, Stockholm. (Communicated 

by Professor G. H. Hardy) 272 

Note on the parity of the number which emimerates the partitions of a 

number. By Major P. A. MacMahon ' . 281 

Note on constant volume explosion experiments. By S. Lees, M.A., St 

John's College. (Two figs, in Text) 285 



Contents vii 

PAGE 

On the Latent Heats of Vaporisation. By Eric Keightley Rideal, 

M.A., Trinity Hall .... 291 

Oil the fimction [x]. By ViGGO Brun (Drobak, Norway). (Communicated 

by Professor G. H. Hardy) 299 

A theorem concerning summahle series. By Professor G. H. Hardy . . 304 

Standing Waves parallel to a Plane Beach. By H. C. Pocklington, M.A., 

St John's College 308 

The Origin of the Disturbances in the Initial Motion of a Shell. By R. H. 

Fowler and C. N. H. Lock. (One fig. in Text) . . . .311 

Tides in the Bristol Channel. By G. I. Taylor, F.R.S. (Four figs, in Text) 320 

Expenments ivith Rotating Fluids. By G. I. Taylor, F.R.S. . . . 326 

Experiments on focal lines formed hy a zone plate. By G. F. C. Searle, 
So.D., F.R.S., University Lecturer in Experimental Physics. (Five 
figs, in Text) 330 

The Tensor Form of the Equations of Viscous Motion. By E. A. Milne, 

B.A., Trinity College 344 

Insect Oases. By C. G. Lamb, M.A 347 

A Note on the Hydrogen Ion Concentration of some Natural Waters. By 

J. T. Saunders, M.A., Christ's College 350 

The Mechanism of Ciliary Movement. By J. Gray, M.A., Balfour Student, 

and Fellow of King's College, Cambridge. (Three figs, in Text) . 352 

A Note on the Biology of the '■ Crown-GaW Fungus of Lucerne. By 

J. Line, M.A., Emmanuel College. (Seven figs, in Text) . . . 360 

On some Alcyonaria in the Cambridge Museum. By Sydney J. Hickson, 
M.A., F.R.S., Professor of Zoology in the University of Manchester. 
(One fig. in Text) 366 

The Influence of Function on the Conformation of Bones. By A. B. 

Appleton, M.A., Downing College. (Five figs, in Text) . . . 374 

Animal Oecology in Deserts. By P. A. Buxton, M.A., Fellow of Trinity 

College, Cambridge 388 

Venational Abnormalities in the Diptera. By C. G. Lamb, M.A. (Four- 
teen figs, in Text) 393 

The Cooling of a Solid Sphere with a Concentric Core of a Different 

Material. By Professor H. S. Carslaw. (Three figs, in Text) . 399 

Symbolical Methods in the theory of Conduction of Heat. By Dr T. J. I'a. 

Bromwich, F.R.S. (Two figs." in Text) 411 

0?!. the effect of a magnetic field on the intensity of spectrum lines. By 
H. P. Waran, M.A., Government of India Scholar of the University 
of Madras. (Communicated by Professor Sir Ernest Rutherford, 
F.R.S.) (Three figs, in Text and Plate III) 428 

On a property of focal conies and of bicircular quartics. Qj C. V. Hand- 
MANTA Rao, LTniversity Professor, Lahore. (Communicated by 
Professor H. F. Baker) 434 

Convex Solids in Higher Space. By Dr W. Burnside, Honorary Fellow 

of Pembroke College ......... 437 

Note on the Velocity of X-ray Electrons. By R. Whiddington, M.A. 

(One fig. in Text) 442 

A Laboratory Valve method for determining the Specific Indxictive 

Capacities of Liquids. By R. Whiddington, M.A. (One fig. in Text) 445 



viii Contents 



The Theoretical Value of Stitherland' s Constant in the Ki-netic Theory of^ 
Gases. By C. G. F. James, Trinity College, Cambridge. (Communi- 
cated by Mr R. H. Fowler.) (One fig. in Text) . . . .447 

On the Stability of the Steady Motion of viscous liquid contained between 
two rotating coaxal circidar cylinders. By W. J. Harrison, M.A., 
Fellow of Clare College, Cambridge 455 

The soaring flight of dragon-flies. By E. H. Hankin, M.A., Sc.D., Agra, 

India. (Three figs, in Text) 460 

The Gluteal Region of Tarsius Spectrum. By A. B. Appleton. 11! 

(Plate IV) 4^^ 11 

An unusual type of mcde secondary characters in the Diptera. By C. G. 

Lamb, M.A. (Four figs, in Text) 475 

A Note on the Mouth-parts of certain Decapod Crustaceans. By L. A. 
BoRRADAiLE, M.A., Fellow and Tutor of Selwyn College, Cambridge, 
and Lecturer in Zoology in the University ..... 478 

An Apparatus for Projecting Spectra. By H. Hartridge . . . 480 

Note on true and apparent hermaphroditism in sea-urchins. B}' J. Gray, 

M.A., Balfour Student, Cambridge University 481 

On Certain Simply -Transitive Permutation-Groups. By Dr W. Burnside, 

Honorary Fellow of Pembroke College 482 

Proceedings at the Meetings held during the Session 1920 — 1921 . . 485 

Index to the Proceedings with references to the Transactions . . 492 



PLATES. 

Plates I— III. To illustrate Mr Waran's papers .... 48, 433 
Plate IV. To illustrate Mr Appleton's paper 474 



CORRECTION. 

MoRDELL, p. 250, line 5, after conjugate numbers insert in the reed 
conjugate fields. 



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PROCEEDINGS 

OF THE 

Cambriirgc ISljtlasapIjkal Bomi^. 



On the term by term integration of an infinite series over an 
infinite range and the inversion of the order of integration in 
repeated infinite integrals. By S. Pollard, M.A., Trinity College, 
Cambridge. (Communicated by Prof. G. H. Hardy.) 

[Received 1 January, 1920. Read 8 March, 1920.] 

The problem for infinite series. 

1. The problem to be solved is that of determining conditions, 
under which the equation 

00 /•=» /"» 00 

2 Un(oc)dx==l S Un(oi;)dx, (1) 

n=l J a J an=l 

is true. It is discussed in detail in Bromwich's Infinite Series, 
pp. 452-455, where various conditions are given. All these con- 
ditions will be found to involve uniform convergence, the fact 
being that the infinite integrals there considered are obtained as 
limits of Riemann integrals and, in the theory of the latter, con- 
siderations as to the validity of the equation 
rb m rb oo 

lim I 2 M„ (x) dx = j S Uji (w) dx, (2) 

almost always involve uniform convergence. Thus conditions for 
term by term integration over an infinite range, being built up 
from the conditions for term by term integration over a finite 
range, involve uniform convergence. 

Now the condition of uniform convergence is by no means a 
necessary one : it occurs because of the lack of power in the 
methods of the Riemann theory. Much wider conditions can be 
obtained by the use of the Lebesgue theory. It is the object of 
this paper to give these. 

VOL. XX. PART L 1 



2 Mr Pollard, On the term by term integration 

Conditions for passage to the limit under the sign of 
integration, the range of integration being finite. 

2. We give, for the sake of reference, the two principal 
elementary conditions. 

(C 1) If u^ (sc) is positive for a^x ^b; n = l, 2, 3 . . ., then if 
either side of (2) is finite the equation holds, and if either side is 
infinite both are. 

V 

(C 2) Ifl'^Un (oc) I < i|r (a;) for a-^a:^b,v=l,2,S,..., tuhere 
n=l 

'>^ is summable in (a, b), then both sides of (2) eccist and are finite 
and equal*. 

Resume of theorems of double limits. 

3. As the use of double limits is fundamental in the theory 
about to be developed, we give a short summary of the results 
required. 

(a) If the double limit lim S^, y exists, and lim 8x, y exists 

for all sufiiciently large y ; then lim (lim S^, y) exists and is equal 

to the double limit. Similarly for the limit lim (lim S^^y). 

(^) If Sx, y is increasing in x and y, and any one of 
lim Sx,y, lim (lim *S^a;, j/X lim (lim>Sa;, j,) 

exist; then all three exist and are equal. 

(y) If 8x, y can be expressed as the difference of two functions 
S'x, y, S"x, y ^^ch of which is increasing in x and y and 

lim {S'xy + S"xy) 
X-»-oo , y^"Xi 

exists and is finite ; then 

lim Sx,y, lim (lim Sx,y\ lim (lim Sa;,y) 

all exist and are finite and equal. 

The condition (7) is especially convenient when 

S.,y=n''fil,v)d^dv. 

J aJb 

* De la Valine Poussin, Cours d^analyse infinites imale, t. i. , 3rd Ed., p. 264, 
theorems iii and 11. 



of an infinite series over an infinite range 3 

For if lim f" ^ \f{l v)\d^dr) 

exists and is finite, then S^^y satisfies the condition of (7). We 
have in fact 

^x, y — ^ x,y ^ x,yj 
fx ry 
where S'x,y= I \f{^,v)\d^dr}, 

J aJb 

S\y=r ("[\f{^,V)\-f{lv)]d^dv, 
J aJ b 

and both S'cc,y, 8"x,y are increasing in x and y and have a finite 
double limit — the former by hypothesis and the latter because 

Note. The above results still hold when either or both of the 
variables x, y take only positive integral values. 

Definition of infinite integrals. 

4. Let /(a;) be any function which is summable in (a, X) for 
all X greater than a. 

If lim f{x) dx, 

X-*-ix J a 

where the integral is taken in the sense of Lebesgue, exists and 
is finite, we say that 



J 



f (x) dx 



converges and attribute to it the value of the limit. 

This definition is evidently consistent with and more general 
than that usually given, where /(a:;) is assumed to be integrable in 
Riemann's sense in (a, X). It has the special advantage of not 
being restricted to functions which are bounded in every (a, X). 
And we lose nothing by adopting it, as the two theorems on which 
the theory of infinite integrals rests, the first and second mean 
value theorems, are still true when we abandon the restriction 
that f{x) is to have a Riemann integral and make only the 
assumption that /(a;) is summable*. 

General theorems. 

rX m 



rX m 
5. I. If the double limit lim I 2 m„ (x) dx exists and 

m-^-oo, X-*-<x>J an=l 

(a) I Un (x) dx converges for all n, 
J a 



Ibid. t. II. 2nd Ed., p. 53. 



4 Mr Pollard, On the term by term integration 

00 

(b) 2 u^ (x) converges for X'^a, 

n=\ 

rX m rX CO 

(c) lim S Un (oc) dx, 2 ?/„ {x) dx, 

m-*-ao J a n=l J an=l 

exist and are equal for all X ; then both sides of {!) exist and are 
equal. 

rX m 

Proof. Write S u^ {cc) dx = Sm, x, 

J a n=l 

and let lim Sm,x = S. 

m-*-'x>,X^'OD 



Since I w„ («) dx converges for all n 

J a 

rX m 
lim % u^{x)dx, i.e. lim Sm,. 



X 

exists for all m. Hence 



lim (lim >S^,x) ='S^ (3)» 

Wl^-oo X-^cc 

by(«)- . 

In virtue of (b) and (c) 

rX m ^ 

lim 1 2 Un (^) dx 

m^-as J a n=l 
rX 00 
exists and is equal to 2 u^ (x) dx. 
Jan=l 

Thus lira Sm, x exists for X ^ a. 

Hence lim (lim Sm,x) exists and is equal to S. 

X-^-CC 77l-*-00 

rX 00 
Taking lim S^, x in the form 2 u^ {^) dx, we see that 

m-*-oo J an = l 

/-■» 00 

S Un (^) dx = S. (4) 

J an=l 

And (3) and (4) give us our theorem. 

00 

II. J/ 2 I w„ (x) I converges for x^a and the double limit 

rXm 



rXm 

lim 2 I w„ (a;) I dx 

■00, X->-<x) J an=\ 



exists and is finite; then without further condition both sides of (1) 
exist and are finite and equal. 



o/ an infinite series over an infinite range 5 

rx m 
Proof. By (7), if lim I S | w„ {x) \ dx exists, so does 

«-»■», X-*-ao J a n=X 
rX m 

lim S w„ {x) dx. 

m-^-oo , X-*-<x> J an=l 
CX m 

Also Sm, x=\ ^ \Un{x)\dx^ 8, 

J a n = l 

for all X and m. But Sm,x increases with X for each m. Hence 

rx m 

% \u„(x)\ 
J an=l 



rX m 

lim I 2 I w„ (ic) I dx 
X^oo J an=l 

exists for each m and is less than 8. 
And therefore 

rX f rx m rX m-1 "j 

lim 1 I u„ (x) I dx = lim j I S 1 1*„ (x) \dx — I % \Un{x)\dx> , 
X^-co J a X^i-co [J an=l J a n=l J 

exists for each m, i.e. 1 | ^^^ (/») | dx and therefore I ?/.„ (x) dx con- 

^ a J a 

verges for each m. This is (a) of (I). 

Again, 8m, x increases with m for each X. 

Hence lim S.m,,x exists and is finite for each X. So from 

(C 1) [X 00 

X \Un (x) I dx 
Jan=l 

00 

is finite. Thus 2 | m„ (x) I is summable in (a, X). But 

mm 00 

71=1 n = l 71 = 1 

and so by (C 2) 

rx m rx CO 

lim I S Un (x) dx, I 2 w„ (a;) c?^, 

m-*-oo .' a n. = l ./a7i.= l 

exist and are finite and equal. This is (c) of (I). Now (b) of (I) 
is satisfied by hypothesis. 

Thus all the conditions of (I) are satisfied and so both sides of 
(1) exist and are equal. 

Deductions from the general theorems. 
6. A. // u^{x) = (f>(x)f„{x), 

where 2 /„ (x) converges for x'^a, 

n = l 

V 

I ^ fn{x)\< G, for x'^a and all v, 

n = l 



6 Mr Pollard, On the term by term integration 



and I ^ {x) I dx converges; then both sides 0/ (1) exist and are 

J a 

finite and equal. 
B. If either of 

% i \un(cc)\dx, 2 I M„ (x) I dx, 

w = l J a J a n = l 



eadst and are finite; then both sides of (1) exist and are finite and 
equal. 



rX m rx <x> 

C If lim S Un (x) dx, 'S. Un (^) dx, 

m-»-oo .' a«,= l ■!an=l 

exist and are finite and equal, and 

2 Un (x) dx 

J a 
converges uniformly for a^^ x, and each 

u^ (x) dx 

J a 

converges; then both sides of(l) exist and are finite and equal. 

V 

D. 7/ I 2 M„ («) I < -^x (^) for a^ x^ X and all v, where -^x 
n=l 

is summable in {a, X), and 

rx 

S Un (x) dx 

J a 

converges uniformly for a^ X, X being arbitrary, and each 

Un {x) dx 
J a 

converges; then both sides of {1) exist and are finite and equal. 

D is a special case of C obtained by making use of (C 2). 
A, B, D may be regarded as generalisations of theorems A — C, 
pp. 452-455 of Bromwich's Infinite Series. 

Proofs. A. If m' > m, 

m' m' m 

we have 2 /„ {x) = X fn{x)-X fn {x), 

and therefore 

m' Til' m 

I 2 fn{x)\^\ Xfn{x)\ + \ ^ f,{x)\^2G. 



of an infinite series over an infinite range 7 

Hence 

rX' m' rX' m' 

I S Un {x)dx\^\ I S </> {x)f^ (x) I dx 

JX n=m+l J X n=m+l 

rX' m' 

< \<^{x)\t \fn {OC) I dx 

J X n = m 

^2g[ \d> {x) I dx. 

J X 

Now, given any positive number e, we can, since \<f) {x)\dx 

J a 
converges, find Xo such that 

\(f){x)\dx< € 

J X 

forX, X'>Xo. Hence 

rX' to' 
I 1 S ii„ (x) dx\< € 

Jx n = m+l 

for X, X' > Xf) and all rn, m . Thus the double limit 



exists. Further 



rx m 
lim / S i<„ ix) dx 
•■»,X-*"X J a n = \ 



n=l x=l 

and G^ 1 (ic) I is summable in C«, X) for all X greater than a. 
Thus by (C 2) 

CX m rX 00 

lim X Un (x) dx, i u„ (x) dx 

J a n=l J a n=l 

exist and are equal and finite. 

All the conditions of (I) are now satisfied, and our theorem 
follows. 

B. If we write r^ m 



rX m 
Sm,X= I 2 \Un{x)\dx, 
•I a n=l 



then lim 8m,x exists and is either finite or positive infinity. 

In the first case our theorem follows at once by (II). 
In the second case, both the repeated limits 

lim (lim Sm,x), lim (lim 8m, x), 

m-^cD X^'x> X^oD m-*oo 

are infinite. Suppose now that 

S I I M„ (x) I dx 

»=1 J a 



8 Mr Pollard, On the term by term integration 

exists and is finite. Then lim (lim ^«t,x) exists and is finite, 
and we get a contradiction. And if 

^ \Un{x)\dx 
J an=l 

exists and is finite, then so does 

rx m 
\ 2 I ti„ {x) I dx 

J a 7i=l 

for all X greater than a. Hence as in theorem II 
rX oa rX TO 

I "E \ Un {w) \ dx = lim X I Wn (^) I ^^> 

J a 71=1 Wi-»-oo J a 71 = 1 

and it follows that 

r"" 00 

E \un\x)\dx= lim (lim S^^x), 

Ja n = l X-*-oo 7j,-*-oo 

and we again get a contradiction. 

Thus the first case alone is possible, and this is the case in 
which our theorem is true. 

C. Write r u,{w)dx = g,{X), 

- « 

m 
tgn{X)^Sm,X. 

71=1 

00 

Since S g^ (X) converges uniformly for a ^ X, given e > we 

71=1 

can find No such that 

I i gAX)\<^, {X>a,N^N-,). 

n=N+l 

Thus \Sm,x- i 9n (X) I < e, {X^a, m ^ N^). 

71 = 1 

00 

Hence if lim S gn{^) exists and is finite, so does lim aS'^^^ x , 

X-*-^ n=l 7?l-»Q0 , JC-».Xi 

and the two are equal. Now 

I i: g^ (ZO -i g, (Z") I ^ i f g, (Z) - % g, (Z") j 

tt=l 71=1 71=1 n=l 

+ 1 i gn{X')\ + \ i 5'n(X")I 
n=N+l n=N+l 

^1 f 5r,(Z')-i5'«(X")l + 26. 

71=1 71=1 

But since lim g^ (Z) (= u^ (w) dx) exists and is finite for 

X-*oo J a 



o/ an infinite series over an infinite range 9 

N 

each n, so does lim S gn{^) ^-nd we can find X^ such that 
1 f 5r„ (Z') -I g, {X") 1 < 6. {X', X" ^ Zo). 

71=1 »=1 

Hence 1 I ^„ (X) - 2 ^„ (Z") | < 3e, (Z', Z" ^ Z„) 
and therefore, by the general principle of convergence 

00 

lim 2 gr, {X) 

X-»-t» 71=1 

exists. Thus lim *S^„i,x exists. The other conditions of (I) are 
satisfied by hypothesis and our theorem follows. 

The problem for infinite integrals. 
7. We have to determine conditions under which the equation 

dx\ f{x,y)dy==\ dy \ f{x,y)dx (5) 

J a J b J b J a 

is true. The methods adopted above apply almost without change 
and we get conditions almost identical with those already given. 
We quote them without "proof, as the proofs can be made up 
immediately on the lines of those already given. 

As regards the nature oif{x, y), we assume throughout that 
f {x, y) is summable in the region 

{a^x^X, b^y^Y), 
for all Z ^ a, Y ^b ; so that, by Fubini's theorem*, the repeated 

rX rY fY rX 

integrals dx \ f{x,y)dy, I dy \ f{x,y)dx exist and are 

J a ■ b J b J a 

equal to the double integral. 

General theorems. 

rX rY 



and 



8. I'. If the double limit lim I j f{x, y) dxdy exists 

(a) I f{x,y)dx, converges for y^b, * 

(b) I f{x,y)dy, converges for x '^ a, 

rX rY rX r^ 

(c) lim dx f{x, y) dy, dx f{x, y) dy, 

F-w-oo J a J b J a J b 

* De la Valine Poussin, Integrales de Lebesgue etc., p. 53. 



10 Mr Pollard, On the term by term integration 

exist and are finite and equal for X ^ a ; then both sides of (5) 
exist and are equal. 

IT. If the double limit lim \ f{x, y)\dxdy exists 

X-*-ao , Y-^x J a J b 

and is finite; then without further condition both sides of (5) exist and 
are finite and equal. 

Deductions from the general theorems. 
9. A'. // f{x,y) = <^{x)d{x,y), 

where j 6 {x, y) dy \ < G for x^a, y ^b, 

Jh 

f{x, y) dy converges for x^a,' 

'b 

and \(f>{x)\dx converges; then both sides of (5) exist and are 

J a 

finite and equal. 
B'. If either of 

1 dx \f{x, y) I dy, dy \f{x, y) \ dx, 

J a J b J b J a 

exist and are finite; then both sides of {5) exist and are finite and 
equal*. 

rX rY rx r=° ■ 

C. If lim dxl f{x,y)dy, dx \ f{x,y)dy, 

Y-*-oo J a J b J a J b 

exist and are finite and equal, and 

dy fix, y) dx 

Jb Ja 

converges uniformly for a^ X, and 

o 

f{x, y) dx 



/ 

J a 



converges for y ^ b ; then both sides of (o) exist and are finite and 
equal. 

* This is de la Valine Poussin's theorem. See Bromwieh, Infinite Series, p. 457. 
The hypothesis given by Bromwieh to the effect that both the integrals 



/"OO /"OO 



are convergent is unnecessary, the existence of one (the one necessary to the 
existence of the repeated integral) is sufficient. That of the other is implied by 
the existence of the double hmit, see Note 2. 



o/ an infinite series over an infinite range 1 1 

D'. If \r fix, y)dy\^^x{x) for a^ x^ X, 6 ^ F, 

J b 

ivhere yjrx is summahle in (a, X), and 

\ dy \ f {x, y) dx 
Jb J a 

converges uniforndy for a ^ X, X being arbitrary, and 

f{x, y) dx 



converges for y^b; then both sides of {b) exist and are finite and 
equal. 

10. Note 1. Results B are especially valuable, as they are 
easy to remember and convenient to apply. The power of the 
Lebesgue theory is shewn very clearly here in that by using it 
we are enabled to make the hypothesis which ensures the exist- 
ence of the double limit* ensure also the passage to the limit under 
the sign. 

Note 2. It is well to be precise as to the meaning of the word 
"exists" as used in connection with repeated Lebesgue integrals. 

Suppose /(^, y) is measurable in x, y in the rectangle 

^a^x^X\ 
Kb^y^Yj' 

We know that the function y (a:;, y) considered as a function of 
X, is measurable in (a, X) for each y in (6, Y) a set of zero measure 
being excepted. It may not, however, be summable in (a, X), i.e. 

f{x,y)dx 



may not exist, for all ?/ concerned. But, if/(^, y) is summable 
over the rectangle, i.e., if the double integral 

rx rY 

fix, y)dxdy 

■ b 

exists ; then it can be shewn that 

rx 
fix, y)dx 



f 

J a 



exists for all values of y in (b, Y), save possibly those of a set of 
measure zero. 

CO ,-co 

* The existence of 2 I j u^^ (x) \ dx implies the existence of the double limit 
n=lj a 

by (7) of §3 ; and in addition, by the use of (C 1) on | u„(x) | , it will be found to 
imply the validity of the passage to the limit under the sign. 



12 Mr Pollard, On the term by term integration 

Now in the Lebesgue theory the integral of any summable 
function over a set of zero measure is zero, and consequently we 
may neglect a set of measure zero without affecting the value of _ 
the integral. Hence when we are faced with the problem of find- l{ 
ing the value of a function which is indefinite or infinite at the 
points of a set of measure zero, we simply neglect these points 
and find the value of the integral over the residue. This is taken 
to be the value of the integral over the original set. 

With the above convention it is true that, if 

I /(^, y) dxdy 
Ja Jb 

exists, so does dy f{x, y) dxdy, 

. h J a 

although there may be points in (6, Y) at which the single 

integral 

rx 
f{x,y)dx 



f 

J a 



does not exist. 

It is always to be understood in dealing with repeated 
Lebesgue integrals (finite or infinite) that the inner integrals 
need only exist at all the points of the range of integration of the 
outer integral save those of a set of measure zero. 

Let us apply the foregoing remarks to theorem B'. Suppose 

rY rx 

dy \ \f{^,y)\dx 

■lb J a 

exists. Then we know that 

lim 11 fix, y) I dxdy 

X-*-x , F-9.20 J a J b 

exists. It follows that 



I dx\ \f{x,y)\dy, 

J a Jb 

on of Y, is bounded £ 
lim / dx \f{x,y)\dy 

T-^x J a J b 



considered as a function of Y, is bounded as F tends to infinity. 
Thus 

rx rY 



Y- 
exists and is finite. It follows that 



\f{x, y) \ dy 

b 

converges at all the points of (a, X) save possibly those of a set of 



of an infinite series over an infinite range 13 

measure zero, because if it did not the above limit would be 
infinite ; and so, for our purposes 



J a J b 



exists. 

Our convention has enabled us to infer the existence of the 
inner integrals from the existence of the double limit. 

Note 3. A thorough treatment on different lines of the subject 
of this paper will be found in two papers by Prof W. H. Young : 

(1) " On the change of order of integration in an improper 
repeated integral," Trans. Gamh. Phil. Soc, xxi. p. 361. 

(2) "The application of expansions to definite integrals," 
Proc. Lond. Math. Soc, ix. (1910), p. 463. 

In this paper we content ourselves with giving simple 
generalisations of well-known results with proofs depending on 
comparatively elementary theorems. There is no attempt to 
obtain comprehensive results. 



14 Mr Landau, Note on Mr Hardy's extension of a theorem, etc. 



Note on Mr Hardy's extension of a theorem of Mr Poly a. 
By Edmund Landau. (Communicated by Prof. G. H. Hardy.) 

[Received 10 December 1919. Read 26 January 1920.] 

In a recent note in these Proceedings* Mr Hardy has estab- 
lished an improved form of a theorem of Mr Pdlya, viz. : 

Suppose that g {x) is an integral function, and M (r) the maxi- 
mum of I ^ (^) I for \x\^r. Suppose further that g {x) is an integer 
for x = 0, 1, 2, 3, ..., and that 

M{r)=o{2'). 
Then g (x) is a polynomial. 

As Mr Hardy remarks at the beginning of his note, it is 
sufficient (after the analysis given already by Mr Pdlya), to prove 
the one formula 

^,2-r _ — ^ — =0(1). 



— TT 



n (2n - s cos 6) 

Mr Hardy's proof of this formula may be replaced by the following 
shorter proof 
Since 

nl2^n^ = nl2^n(^^-^,2^nn^l=OWn), 

n(2n-s) ^ ^' 

s=l 

it is enough to prove 

'[ir{e,n)de==o{^), 

where 

ylr(e,n)=u( ^ ^^ ~ ' . 
^ ^ ' s=i\2n — scos6 

Now 

l-cos6/_2-24 + ---^2 24^2 24 ~ 12 ' 
for — TT < ^ ^ TT, and 



1 + 2/ 

* Vol. XIX. (1919), pp. 60-63. 



Mr Landau, Note on Mr Hardy's extension of a theorem, etc. 15 
for ^ y ^ 1. Hence 

1-7? 1 1 



1-^cos^ l+_^(l-cos^) ^+va-cos0) 

1 -7] 

^ g-i^ (1-cosfl) < g-Tll*^ 

for ^ t; ^ |, — TT ^ 6 ^tt; and so 

1 — — - 02 n 

-^{d, n) = n — — ^ e~487tif = e-TyV(«+i)9^ ^ e-?Vne^^ 

*='l-^cos6' 
In 

for — TT ^ ^ ^ TT and ?? = 1, 2, 3, .... Therefore 

[" A/r (^, n)de^j e- ^^^''dd = ('-^) . 

GoTTlNGEJf, 4 December 1919. 



16 Dr Fenton and Mr Berry, Studies on Cellulose Acetate 



Studies on Cellulose Acetate. By H. J. H. Fenton aii( 
A. J. Berry. 

[Read 8 March 1920.] 

The enormous demand for cellulose acetate and the serious, 
shortage of acetone and certain other materials used in the manu- 
facture of aeroplane dopes during the war originated a systematic 
research on cellulose acetate, especially as regards the behaviour 
of this material towards solvents and its chemical properties 
generally. The research has been pursued in a number of directions, 
the most important of which have been (a) substitutes for acetone 
as solvents, (b) the preparation of cellulose acetate and a study of 
the influence of the mode of preparation on the properties of the 
resulting product, and (c) the analytical chemistry of cellulose ace- 
tate. Most of our experiments, especially those relating to aeroplane 
dopes were necessarily of a technical character, but as a few results- 
of general chemical interest have been obtained in the course of 
the work, we have thought it desirable to give a brief account of 
them in the present communication. 

Solvents. 

At the time of the difficulty caused by the serious shortage of 
acetone we were urged to discover efficient substitutes for this, 
solvent for use in aeroplane dopes. It should, in passing, be 
observed that the properties of acetone make it an ideal solvent: 
its conveniently low boiling point, rapid solvent action on cellulose 
acetate, non-poisonous character, and, in normal times, cheap 
and abundant supply. All other liquids which have so far been 
suggested show a deficiency in. some one or other of these 
particulars. 

In August, 1917, we suggested that in case of emergency the 
three following solvents might be employed, viz. aeetaldehyde^ 
acetonitrile, and nitrobenzene with certain additions. Quite early 
in the investigation (October, 1916) we suggested acetic acid and 
ethyl formate as solvents. We also suggested the use of cyclo- 
hexanone and of beechwood creosote as substitutes for tetrachloro- 
ethane or benzyl alcohol as high boiling solvents. We were never 
informed whether these solvents were actually employed. It is- 
remarkable that at considerably later dates, patents have been 
taken out for the use of both acetaldehyde and cyclohexanone as- 
dope constituents. (British Patent 131647, July 4th, 1918 (acet- 
aldehyde) and Ibid. 130402, February 15th, 1918 (Cyclohexanone).) 



Dr Fento7i and Mr Berry, Studies on Cellulose Acetate 17 

Our experiments have demonstrated that the destructive effect 
of acids upon fabrics is dependent on the strength of the acid in 
the physico-chemical sense. Hitherto it had been supposed that 
esters were objectionable as dope constituents on account of the 
possibilities of free acids resulting from hydrolysis. This, however, 
we found not to be the case. As far as weak acids only are concerned, 
tensile strength determinations gave excellent results; and fabrics 
doped with acetic acid as the principal solvent compared most 
favourably with others. 

In our experiments a large number of liquids have been 
examined, not only from the purely practical point of view, but 
also from a desire to obtain if possible some information with 
regard to possible relationships between the nature of the liquid 
and its solvent action. It is of course impossible to define strictly 
the solubility of cellulose acetate in any given solvent owing to the 
colloidal nature of the products. The term "positive" is used in 
the following lists to imply that the liquid named has the property 
of gelatinizing cellulose acetate and subsequently converting it 
into a clear homogeneous "sol" without the aid of heat. All the 
results were obtained with a sample of the material which yields 
54 per cent, of acetic acid on cold alkaline saponification. 

Positive. 

Liquid ammonia, liquid sulphur dioxide, liquid hydrogen 
cyanide, acetaldehyde, benzaldehyde, salicylaldehyde, acetone, 
methyl ethyl ketone, suberone, acetonitrile, propionitrile, formic 
acid, acetic acid, butyric acid, formamide, ethyl formate, ethyl 
oxalate, ethyl malonate, etbyl acetoacetate, aniline, phenyl- 
hydrazine, ortho-toluidine, piperidine, pyridine, tetrachloroethane, 
nitrobenzene*, nitromethane, cyclohexanone, guaiacol, chloro- 
form*. 

Although cellulose acetate is insoluble in water and in absolute 
ethyl alcohol, a mixture of these two liquids dissolves it freely 
on boiling. On cooling, however, precipitation takes place almost 
completely. 

Negative. 

Liquid air, liquid ethylene, liquid nitrous oxide, liquid hydrogen 
sulphide, benzene, toluene, turpentine, carbon disulphide, carbon 
tetrachloride, alcohol, ether, ethyl chloride, acetal, dimethyl 
acetal, nickel carbonyl, and many other liquids. 

No general conclusion can be drawn as regards the chemical 
nature of a liquid and its solvent action on cellulose acetate. It is, 

* Nitrobenzene requires certain additions. Chloroform had only a partial 
solvent action on this specimen of the material. 

VOL. XX. PART I. 2 



18 Dr Fenton and Mr Berry, Studies on Cellulose Acetate 

however, worthy of note that there appears to be some relation 
(with undoubted exceptions) between the dielectric constant and 
solvent action. 

Influence of methods of preparation upon the properties 
of cellulose acetate. 

The materials obtained by acetylating cellulose with acetic 
anhydride diluted with acetic acid in presence of various catalysts 
such as concentrated sulphuric acid, ferric sulphate, ortho tolui- 
dine bisulphate, may show considerable variations in properties 
depending upon the temperature, length of time of acetylation, 
and numerous other factors. When cellulose is acetylated and 
the product at once precipitated by water, it is nearly insoluble in 
acetone. Various methods have been adopted in order to convert 
the product so obtained into an acetone-soluble modification. The 
most widely used of these methods is that of Miles. This consists in 
heating the acetic acid solution of the cellulose acetate with water 
in rather greater quantity than that required to combine with the 
residual acetic anhydride. Sodium acetate may also be added to 
react with the catalyst if still present. The results are usually 
supposed to be due to chemical hydration. 

In our experiments, cellulose was acetylated under the influence 
of various catalysts, and the effect of treatment, by the Miles 
process was subjected to a critical examination. The most marked 
effects of this process are the changes in solubility in acetone and 
chloroform, most cellulose acetates being soluble in chloroform 
and insoluble in acetone before the treatment. This change in 
physico-chemical properties was found to be accompanied by a 
fall in the acetyl number. In one case the untreated cellulose 
acetate with an acetyl number of 60-9, yielded a product after the 
Miles process carried out at 100° for 48 hours with an acetyl 
number of 46-7. In another case when the treatment was carried 
out at the same temperature for 23 hours, the acetyl number fell 
from 60-5 to 50-4. The specific gravity of the cellulose acetate is 
also greatly reduced after the treatment. The influence on the 
heat test is not well marked but the decomposition point appears 
to be lowered somewhat. 

In our view these results are to be ascribed to partial hydrolysis 
of the cellulose esters, not to hydration as is commonly supposed*. 
Apart from the diminution of the acetyl number already mentioned, 
we have carried out a series of experiments which have demon- 
strated that cellulose acetate does not form a hydrate. These 

* Oux view that the effect of the Miles process is essentially hydrolytic and not 
due to chemical hydration has been expressed subsequently by Ost {Zeifsch. 
angeic. Chem. 1919, xxxii, 66, 76, and 82). 



Dr Fenton and Mr Berry, Studies on Cellulose Acetate 19 

experiments originated in connexion with our determinations of 
tlie water contained in commercial samples of cellulose acetate. 
As is well known, the water is readily expelled by exposure of the 
material over concentrated sulphuric acid in a desiccator or by 
heating to 100°. It has frequently been supposed that the approxi- 
mately constant proportion of 5 or 6 per cent, of water usually 
met with indicates a definite hydrate. In order to obtain positive 
information on this point, we determined the pressure-concentra- 
tion relationship in the manner originally adopted by van Bemmelen 
in his well known researches on silicic acid {Zeitsch. anorg. Chem. 
1896, XIII. 233). Weighed quantities of the material were exposed 
in a series of exhausted desiccators over sulphuric acid of various 
determined concentrations, and the corresponding vapour pressures 
were found by reference to Landolt and Bornstein's tables. The 
weights were found to be constant after 24-48 hours, and the 
pressure concentration relationship showed that no chemical 
hydration occurs. The phenomenon is to be regarded as one of 
adsorption, probably with subsequent difiusion, and is precisely 
similar to the absorption of water by cellulose itself. (Compare 
Masson and Richards (Proc. Roy. Soc. 1906, lxxviii. 421), Trouton 
and Pool {Ibid. 1906, lxxvii. 292) and Travers {Ibid. 1906, lxxviii. 
21, and 1907, lxxix. 204).) 

Characterization and Analysis of cellulose acetate. 

In the technical analysis of cellulose acetate, it is usual to 
examine the product by the heat test, solubility, acidity, and 
viscosity of the solutions, in addition to the determinations of 
acetyl (as acetic acid), copper reducing power, water, ash, and 
impurities. We have made an exhaustive investigation of various 
methods of carrying out these determinations, especially of the 
acetyl number, and have also carried out many ultimate analyses 
for carbon and hydrogen in some commercial specimens of the 
material. 

The methods of determining the acetyl group may be classified 
under the two heads of alkaline saponification and acid hydrolysis. 
In the former the substance is saponified by excess of standard 
alkali, either at the ordinary temperature or at some higher tem- 
perature, and the excess of alkali determined by titration. In the 
latter, the substance is hydrolysed by strong acid, usually sulphuric 
or phosphoric, and the resulting acetic acid separated by steam 
distillation (Ost), or alcohol is added and the resulting ethyl acetate 
distilled off and collected in excess of standard alkali (Green and 
Perkin). The following is a summary of the principal results 
obtained in our experiments. 

(1) Cold alkaline saponification (Ost and Katayama, Zeitsch. 
angew. Chem. 1912 (25), 1467). A known weight of the substance 

2 2 



20 Dr Fenton and Mr Berry, Studies on Cellulose Acetate 

is soaked with alcohol, then a measured volume of normal alkali 
is added and allowed to stand for 24 hours. The excess of alkali 
is then determined by standard acid. The mean result was 54 per 
cent, of acetic acid calculated for the dry substance. 

(2) Cold alkaline saponification (Boeseken, van der Berg and 
Kerstjens, Rec. Trav. Chim. 1916, xxxv. 320). The substance is 
treated with strong aqueous potash for one or two days. A measured 
excess of normal hydrochloric acid is then added, the liquid then 
boiled for a moment to expel carbon dioxide and the resulting 
solution titrated with baryta water. The mean result calculated as 
above was 53-5 per cent, of acetic acid. 

(3) Hot alkaline saponification (Barthelemy, Moniteur Scienti- 
fique, 1913 (3), ii. 549). In this method the saponification is effected 
by heating the substance with normal soda for about 16 hours at 
85°. The excess of alkali is then determined by titration with 
standard acid. Several experiments were made in which the condi- 
tions were subjected to considerable variations as regards length 
of heating and amount of excess of alkali. The extreme variations 
in the acetyl number calculated as above were 60-0 and 62-1 per 
cent. 

(4) Hot alkaline saponification (Green and Perkin, Trans. 
Chem. Soc. 1906, 812). The saponification is carried out at the 
boiling point with semi-normal alcoholic soda and the excess of 
alkali titrated by standard acid. Our experiments yielded results 
of 60 per cent, of acetic acid, the extreme variations being 58-2 
and 61-9 per cent. These numbers are in agreement with those of 
Green and Perkin (loc. cit.). 

It is evident that the methods of hot alkaline saponification 
invariably yield results which are considerably higher than those 
obtained by cold saponification. There can be little doubt that 
the higher results are due to the action of alkali on the regenerated 
cellulose. Support to this contention was obtained by digesting two 
equal weights of filter paper with 50 c.c. of normal soda for two 
days, one at the ordinary temperature, the other at 85°. In the 
former case no alkali was consumed, while the heated product 
showed a loss of nearly 2 c.c. of normal alkali on titration. 

(5) Acid hydrolysis (Ost, loc. cit.). The substance is first 
digested with 50 per cent, (by volume) sulphuric acid. After 24 
hours the liquid is diluted considerably and the acetic acid separated 
by steam distillation, and titrated with baryta water. In our 
experiments phosphoric acid was substituted for sulphuric acid in 
order to avoid error due to possible formation of sulphur dioxide. 
The results varied from 51-5 to 55-0 per cent, of acetic acid. 

(6) Acid hydrolysis (A. G. Perkin, Trans. Chem. Soc. 1905, 107). 
In this method the cellulose acetate is treated with ethyl alcohol 
and sulphuric acid, and the resulting ethyl acetate distilled into 



Dr Fenton and Mr Berry, Studies on Cellulose Acetate 21 

excess of standard alkali. The ester is then saponified and the excess 
of alkali determined by titration. In our experiments phosphoric 
acid was used instead of sulphuric acid for the reason already 
mentioned. The results varied from 52-2 to 54-4 per cent, of acetic 
acid. 

In our opinion, preference should be given to the method of 
cold alkaline saponification of Ost. Not only are the results more 
uniform, but they agree well with those obtained by acid hydro- 
lysis. The latter methods are exceedingly tedious to carry out. 
We have also carried out some experiments with the use of hot 
baryta water as a saponifying agent and subsequent gravimetric 
determination of the barium, the results averaging 57-58 per cent, 
of acetic acid. 

The materials met with in commerce known as cellulose acetate 
are most probably mixtures or solid solutions of various acetates, 
not definite chemical individuals. If, however, it were desired to 
represent cellulose acetate as a chemical individual, the results of 
our analyses of a number of specimens do not correspond with the 
formula of the triacetate C6H7O2 (0C0CH3)3 which is commonly 
supposed. They agree better with the formula of a pentacetyl 
derivative of C12H20O10 and still better with that of a heptacetyl 
compound of CjgHgQOig. 

Thus 

Carbon Hydrogen Acetic acid 

CcH-Oa (OCOCH3)3 requires 50-0 5-5 62-1 per cent. 

C12H15O5 (OCOCH3)5 „ 49-4 5-6 560 

C.sHaA (OCOCHs)^ „ 49-2 5-64 53-8 

Our most reliable results average carbon 49-2, hydrogen 5-5, and acetic 
acid 54 per cent. 

Certain authors have stated that sodium ethylate may be used 
for the determination of acetyl in cellulose acetates. In investi- 
gating this reaction, we were surprised to find that ethyl acetate 
was always produced along with a yellow sodium derivative of 
cellulose. Quantitative experiments were performed in which the 
ethyl acetate was distilled into an excess of standard sodium 
hydroxide, and after saponification determined with standard 
acid. The residue was washed with alcohol to remove the unaltered 
sodium ethylate and this solution was titrated with standard 
acid. The residue was then treated with water to decompose the 
sodium compound and titrated also. It was found that the quantity 
of acetic acid converted into ethyl acetate to that becoming sodium 
acetate appears to depend to some extent on the proportion of 
sodium ethylate employed. The results can be explained, if the 
average commercial cellulose acetates are represented by the 
formula C12H15O5 (OCOCH3)5, by the equation: 



22 Dr Fenton and Mr Berry, Studies on Cellulose Acetate 

C12H15O5 (0C0CH3)5 + CgHsONa + 4C2H5OH 

= CiaHigOgONa + 5CH3COOC2H5 

which may be taken to represent the main reaction. 

In support of this, the yellow sodium compound from a similar 
experiment, after thorough washing with alcohol, was digested 
for several hours in a reflux apparatus with excess of methyl iodide, 
and the methoxy group in the resulting product determined by 
Zeisel's method. The result obtained was 9-2 per cent, of methoxyl , 
in agreement with that calculated for the formula C12H19O9OCH3. 

The adsorption of basic dyestuffs by cellulose acetate. 

Certain dyestuffs, such as gentian violet are adsorbed in con- 
siderable quantities from aqueous solution by cellulose acetate, 
the solid being coloured blue. Cellulose, it is true, also adsorbs 
the dye, but to a much smaller extent, and the solid becomes 
violet. This property may be utilized to identify unaltered cellulose 
in commercial preparations of cellulose acetate. Methyl orange 
gave negative results, but methyl red was adsorbed in considerable 
quantity, the solid becoming red. Free dimethylaminoazo benzene 
gave negative results, but the hydrochloride of this base was 
strongly adsorbed, the solid cellulose acetate assuming a pinkish 
yellow colour and the colour of the aqueous solution being almost 
completely discharged. 

The authors desire to express their grateful thanks to Mr J. W. 
H. Oldham, M.A., of Trinity College, for much valuable assistance 
in connexion with this investigation. Mr Oldham has also carried 
out a large number of experiments on the influence of the mode of 
preparation upon the resulting properties of cellulose acetate, and 
it is hoped that his results when completed may form the subject 
of a future communication. 



Mr Molin, An examination of SearWs method, etc. 23 



An examination of Searle's method for determining the viscosity 
of very viscous liquids. By Kurt Molin, Filosofie Licentiat, 
Physical Institute, Technical College, Trondhjem. (Communicated 
by Dr G. F. C. Searle.) 

{Read 9 February 1920.] 

§ 1. The determination of the coefficient of internal friction in 
very viscous liquids has been the object of measurements by many 
different methods. A review of these will be found in Reiger*. 
A number of more recent methods are given by Kohlrauschf, and 
among them is a method of Searle'sJ. An examination of this 
method is the object of the present paper. 

In his paper, "A simple viscometer for very viscous liquids," 
Dr SearleJ gives an account of a viscometer he has constructed. 
The method consists in causing a vertical cylinder to rotate within 
a coaxal cylinder containing liquid, and in determining the angular 
velocity of the inner cylinder for a known value of the driving 
couple. The couple is produced by the weights of two loads acting 
on a drum by two threads. The time, T seconds, of one revolution 
of the cylinder is found, and the length, I cm., of the inner cylinder 
immersed in the liquid is observed. 

Newton's statement is that 

f--^Tn' <1) 

where/ is the force per unit area which acts against the direction 
of motion and at right angles to the normal, n, to the surface, 
dV/dn is the velocity gradient, and rj is the coefficient of viscosity. 
In this statement the motion of the liquid is supposed to take place 
parallel to a fixed plane. Treating the liquid as incompressible, 
and modifying (1), by substituting the rate of shearing for dV/dn, 
so as to suit the case of rotation, we obtain the following formula: 

gD (a2 _ 62) fMT\ ^ (MT\ 

Here D is the effective diameter of the drum, a and h are the radii 
of the cylinders, and M is the mass of each of the two loads, which 
are required to move the inner cylinder with the constant angular 
velocity Q, such that 2ttJQ. = T. 

* R. Reiger, Ann. d. Phys., 19, p. 985, 1906. 

t F. Kohlrausch, Lehrbuch d. praktischen Physik, xii. Aufl., p. 268. 

% G. F. C. Searle, Proc. Cambridge Phil. Soc, 16, p. 600, 1912. 



24 Mr Molin, An examination of Searle's method 

The angular velocity of the liquid about the axis of the cylinders, 
at a distance r from the axis, is given by 

_ 277 62 /^ ^ 

^ T ' a^-bAr^~ 

When r = b, the radius of the inner rotating cylinder, 

oj = 1^ = 27r/r, 

and when r = a, the internal radius of the outer fixed cylinder, 
CO =0. This problem was first treated, not quite accurately, by 
Newton. The above results were given substantially by Stokes *, 
and are also given by Lambf and by SearleJ. 

The rate of shearing, rdco/dr, varies somewhat as r increases 
from b to a, as is shown by the formula 

doi 27r 2a%^ 

r 



dr T ' (a2 _ §2) ^2 • 

We have only taken into account the friction between the 
coaxal cylindrical layers of the liquid and not the friction between 
the horizontal layers in proximity to the bottom surface of the 
movable cylinder, and have not considered the conditions that 
arise near that surface. In practice, only the lower end of the 
rotating cylinder is exposed to viscous action ; Dr Searle makes an 
allowance for this end by writing 

^^^•r+i' (2) 

where I is the length by which the height, /, of the liquid, in the 
simple theory, must be increased, in order that the increase of 
couple shall correspond to the viscous action in proximity to the 
end surface and the edge of the rotating cylinder. 

Dr Searle gives a graphical method of determining k. The 
values of MT are plotted against I, and he says, "It will be found 
that the points lie on a straight line, which cuts the axis of I at 
a distance k from the origin." Dr Searle adds "If the corresponding 
total load hung from each thread be M grammes, it will be found, 
on repeating the observation with various loads, that MT is 
constant for a given level of liquid. This result confirms the 
fundamental assumption that the viscous stress at each point is 
proportional to the rate of shearing of the liquid." 

* G. G. Stokes, Brit. Ass. Report, p. 539, 1898. 
t H. Lamb, Hydrodynamics, Third Ed., p. 546, 1906. 
loal ^' ^' ^' ®^*^^®' ^°^- ^^f-' P- 602. Compare C. Brodman, Wied. Ann., 45, p. 163, 



for determining the viscosity of very viscous liquids 25 

§ 2. In my experiments I used Dr Searle's viscometer, as 
supplied by Messrs W. G. Pye and Co., Cambridge*. I determined 
the viscosity of treacle, as Dr Searle refers to a determination of 77 
for that liquid. I found 26 = 3-74 cm., 2a = 5-01 cm., and 
D = 1-95 cm. Since g = 982 cm. sec. -2 at Trondhjem, the con- 
stant C has the value 

C = 3-070 ± 0-035. 

From the data given by Dr Searle, I find for the constant of the 
instrument used by him, Cg = 3-153. 

In my instrument the rate of shearing for radius r is given by 

^_ _27r 15-80 
'^ dr~ T ' r^ ' 

§ 3. To examine how MT depends upon M, when I is kept 
constant, six series of observations were taken with six values of I 
varying from 10-0 to 2-15 cm., and in each series M was made to 
vary from 5 to 205 grammes. 

Since the viscosity of highly viscous substances diminishes very 
rapidly as the temperature increases, as was shown by Reigerf and 
by Glaser J for values of rj of the magnitudes 4-8 x 10^ to 67-2 x 10®, 
and by Ladenburg§ for '>7 = 1-3 x 10^, great care must be taken 
to keep the temperature constant. The apparatus was, therefore, 
placed in a thermostat with electric temperature regulation, and 
a very constant temperature of 19-8° C. was /naintained. The 
apparatus was left in the thermostat for 24 hours before the 
measurements were begun, and, during the short time a rotation 
trial was in progress, only the outer wooden door of the thermostat 
was opened, since one could see into the thermostat through the 
inner glass door. The final measurements were all carried out in 
the course of a day; the observations were made at intervals of 
about 10 minutes, so that the unavoidable disturbances of tempera- 
ture, due to the manipulations, might have time to disappear. 

In other respects the measurements were carried out in ac- 
cordance with Dr Searle's II instructions. The revolutions were 
timed by aid of a stop-watch and the times were taken for different 
numbers of revolutions with odd numbers up to 9, as well as the 
average time for one revolution. As no decrease in the time of a 
single revolution could be noticed as the rotation continued, the 
divergences from the mean lying within the limits of the errors 

* Catalogue of Scientific Apparatus manufactured by W. G. Pye and Co., 
List No. 120, p. 39, 1914. 

t R. Reiger, loc. cit., p. 998. 

X H. Glaser, Ann. d. Phys., 22, p. 719, 1907. 

§ R. Ladenburg, Ann. d. Phys., 22, p. 309, 1907. 

II G. F. C. Searle, loc cit., p. 603. 



26 Mr Molin, An examination of Searle's method 



of observation, there was no observable acceleration. AVe may 
conclude that, even for the greatest values of M, the viscosity of 
the liquid remained sensibly constant, in spite of the fact that 
some potential energy was converted into heat. 

The values of T found in these experiments are given in 
Table 1. 

Table 1. 

Time, in seconds, of one revolution of cylinder. 



M 


Z = 10-0 


Z = 8-45 


Z=:7-65 


Z = 5-50 


Z = 3-30 


l = 2-\5 


grm. 


cm. 


cm. 


cm. 


cm. 


cm. 


cm. 


5 












129-0 


7 








114-6 




50-7 


10 


120-4 


108-7 


100-2 


71-3 


44-7 




12 


93-3 


85-0 


77-0 


54-3 






15 


71-5 


65-3 


59-0 


42-3 


26-6 


19-5 


20 


52-8 


47-0 


41-7 


29-7 






25 


4M 


35-9 


32-2 


23-6 


14-5 


11-1 


30 


33-3 


29-6 


26-4 


19-0 






35 


28-8 


24-8 


22-2 


16-3 


10-0 




40 


251 


21-7 


19-2 


14-0 






45 


22-0 


19-0 


171 


12-3 






55 


17-9 


15-2 


13-6 


101 


6-3 


4-6 


65 


14-9 


12-6 


11-4 


8-3 






75 


12-8 


10-9 


9-8 


7-2 


4-5 


3-4 


105 


9-1 


7-7 


7-0 


5-1 






155 


61 


5-2 


4-S 


3-5 






205 


4-6 


3-9 


3-6 









The results have been plotted in the form of six curves each 
for one value of I, as in Diagram 1. The curves are represented in 
the form T {MT, M )^.eonst. = 0. 

From the diagram it is clear that the function T {MT, T)i = 
does not represent a family of straight lines parallel to the ilf-axis, 
and that each of the six curves has a hyperbolic appearance. When 
M approaches a certain lower limit Mq, MT tends to infinity. 
The area covered by the group of curves can be divided by a 
parabolic boundary curve into two departments, in one of which 
MT is sensibly constant for a given value of I. 



§ 4. I have, further, examined how MT depends upon I, when 
M is kept constant, and have found that the function 

F {MT, Z)^,=eonst. = 



for determining the viscosity of very viscous liquids 27 



represents, not a single straight line*, but a family of approxi- 
mately straight lines. Each line can be represented by the equation 
MT =al + p. For this group of curves d {MT)ldl tends to a definite 
value as M increases, i.e. the curves approach a certain border line 



MT 








IhagTa; 


Til- 
















I 








Wn;i 


i)-o. 


* 












t 


^ 






















l^ 




'~~~~ 






\\M 




l=1QO 










1 














\ 


\ 


-^ 




/ / 


(1-5,3) 




T=a^5 










\ 




--- 




/ / 






1=765 












/ * 


















V 






J 




(1-H.3) 




















■^ 


r' 




il-3.'^] 




1=5,50 


- 


















«K>0 




V 


il 


[' 






(i-:i,35) 














\ 

\ 


•v.,^^^/' 












]=S.SO 










\ 


/ 


















\ 


■"--^ 


/ 






fl-hl 




1=Z.15 













/ 














/ 


/ 
















N 


y 












Calculat 


e^forl- 


3. 








^ 























o lO 10 30 it-o so €0 70 60 90 100 110 1Z0 r3o i4o ;5o 160 170 180 190 200 aw 

which is comparable with Searle's straight line. The coefficients 
a and ^ have been calculated for each line by the method of least 
squaresf , using the formulae 

1.1 . HMT - 6111 . IslMT ^ 1.1 . i:iMT - IIMT . ZV- 



(SZf - 6S/2 



(S^)2 - 6SZ2 



* G. F. C. Searle, loc. cit., p. 604. 

t F. Kohlrausch, Lehrbuch d. jn-aktischen Physik, p. 13, 1914. 



28 



Mr Molin, An examination of SearWs method 



the various observations being regarded as having equal weights. 
The values of a and ^ have been thus calculated for seven different 
lines, and the results are given in Table 2. 

Table 2. 
Values of a, § and k. 



M grm. , 


a 


^ 


k cm. 


12 


105-86 


78-40 


0-740 


15 


102-64 


65-61 


0-639 


20 


99-29 


53-54 


0-539 


35 


108-80 


47-79 


0-439 


65 


90-78 


43-83 


0-482 


75 


90-27 


43-63 


0-483 


105 


89-79 


45-47 


0-506 



1 



When I = 0, then MT = ^, and Table 2 shows how ^ varies with M. 
The curve thus extrapolated for ^ = is marked "Calculated for 
Z = 0" in Diagram 1. 

When MT = 0, we have ^ = | ^ | = | /3/a | , where k is the 
correction for the lower end of the rotating cylinder. 



0,7 
0,6 

qs 

Q3 



\ 










Diagram % 














^ 




























\ 






2(k, 


n]=q 






















'x 






























o • 






— o— 


— «— 











































10 ZO 30 ^O 50 60 70 80 90 100 110 ISO ISO 1^^o I^ gr. 



Diagram 2 shows how k depends upon M. 
The facts here recorded show that equation (2) should be 
replaced by 

M,T 



-n^c 



i + k,' 

where k^ is the value of k corresponding to the load M^. 



.(3) 



for determining the viscosity of very viscous liquids 29 

If the value of h^ corresponding to M^ is read off from the 
curve of Diagram 2, the viscosity 17 can be calculated by equa- 
tion (3). The values of k found from Diagram 2 have been used 
in forming Table 3. 

Table 3. 

Values of M^T/il + k^). 



M 


Z = lO-0 


i = 8-45 


1 = 1 -m 


? = 5-50 


Z = 3-30 


grm. 


cm. 


cm. 


cm. 


cm. 


cm. 


10 


111-7 


118-0 


118-5 


] 13-4 


109-5 


12 


104-2 


111-0 


110-0 


104-5 




15 


101-2 


108-0 


106-8 


103-2 


1010 


20 


99-3 


104-5 


102-0 


98-6 




25 


97-9 


100-1 


99-1 


98-5 


95-1 


30 


96-5 


99-0 


97-2 


95-4 




35 


95-6 


97-0 


95-6 


94-7 


92-5 


40 


950 


970 


94-5 


93-7 




45 


94-5 


96-0 


94-3 


92-5 




55 


93-3 


94-0 


91-5 


92-4 


91-6 


65 


92-4 


92-0 


90-7 


89-7 




75 


92-0 


91-7 


90-4 


89-7 




105 


91-1 


90-2 


90-4 


901 




155 


90-3 


89-2 


90-6 


89-6 




205 


90-0 


89-3 


89-3 







From Table 3 it appears that the area in Diagram 1 in which 
equation (3) holds good is restricted to that part of the diagram 
to which the parabolic boundary curve is convex. From the values 
of MT derived from Table 1 and plotted in Diagram 1, the equation 
of the parabola is found to be M'^ = 11-26 {MT). I have not been 
able to give the parabola any definite physical interpretation, and 
it ought to be regarded as representing a diffuse limit region. But 
it is only when we pay regard to this, that we obtain values of 7) 
differing from each other by amounts lying within the limits of 
experimental error*. To make a comparison with the values of 
M and I which Dr Searle has used, I have, in Diagram 1, plotted 
(the broken hne) his values of i/Tf (strictly speaking, MTjC, 
which are comparable in magnitude with my values of MT) 
against M. 

Dr Searle has pointed out to me that the effect shown in 
Diagram 1 might conceivably be due to pivot friction. I have 
carefully considered this possibility. Before the liquid was put 
into the apparatus, I adjusted the pivots so that the rotation due 

* Compare G. F. C. Searle, loc. cit., Table II, p. 606. 

I Calculated from Table 1, G. F. C. Searle, loc. cit., p. 605. 



30 



Mr Molin, An examination of Searle's method 



to the weights of the two empty pans (5 grm. each) was so rapid 
that I was hardly able to measure, for instance, 3T by using a 
stop watch. I have, therefore, not been able to take account of 
any pivot friction. This cause of error would, at any rate, produce 
effects much smaller than those actually found. 



§ 5. From the results for M = 205 grm. given in Table 1 we 
find the mean value 

-q = TIA:-1 dyne sec. cm.-^, 

for the temperature of + 19-8° C. To show how t] depends upon 



3^0 

( 3« 


























^j 


1 












Diagrams. ja 

3« 


r 

4 

— 1— 












1 


- 








1 
1 

1 








S>«nt 1. 


ViS 


i 










* 


Vtt 


\ 






320 


— r^ 


\ 


3. 


\ 




■Q 


315 
310 
305 


io 


• 


— i: nngular veiociL^ oi the "^-q 
Rotating C^inder. 


qi GX 






\ 








'\ 


« 


L Tf(i3- 


90'r 




° 


\ J 






290 
285 
280 






•o\ 


A 










• 


\ 


« 












\ 


^j 


' * 


































.> 


^^ 






^ 




e 






e 










270 
























■ 








, 


A 





0,1 Qi 03 0^ q5 Q« Cr7 <;» <^9 ^O -p V^ t^ y^ \5 1/5 ^7 1^ 

the angular velocity O = 27t/T, the values of 17 and Q, obtained 
from the first three series, have been plotted in Diagram 3. The 
curve drawn among the plotted points suggests that the relation 
between -q and Q. can be expressed in the form 

7] = 274-7 + <f) exp (- AO^). 

To find the constants cf), A and x, I considered the equation 

log, [rj - 274-7) - log,^ - A^^' (4) 



/or determining the viscosity of very viscous liquids 31 

When the values of loge (-7 — 274-7) were plotted against O, the 
curve was roughly a straight line. Hence x may be taken as unity, 
and thus the number of constants to be found is reduced to two. 
By the method of least squares, I obtained logf<^ = 4-375 and 
A = 5-694, and thus 



r^ig.g = 274-7 + 79-44 e-^-''^'*". 



.(5) 



Equation (5) expresses the results of the observations when 
Q. exceeds 0-1, but not for smaller values of Q. 

§ 6. Experiments carried out at different temperatures showed 
that the curves representing the function 



T [MT, M\ 







are of the same character as those given in Diagram 1. Table 4 
gives the values of 7] found for various temperatures. In these 
experiments I was 10-0 cm. ; and, at each temperature, six different 
loads were used, in order that I might be able to decide with 
certainty that the values of M, used in calculating the value of 17 
for each temperature, lay in the area to the right of the parabolic 
boundary line of Diagram 1. The same value of k, viz. the limiting 
value 0-48 cm. shown in Diagram 2, was used in calculating the 

Table 4. 
Values of 7] at various temperatures. 



Temp. 


V 


Temp. 


■n 


t°C. 


Dyne sec. cm."^ 


t°C. 


Dyne sec, cm. ^ 


19-8 


274-7 


8-75 


1950 


18-0 


415 


6-2 


2700 


13-0 


860 


60 


2750 


11-8 


1140 


2-8 


4970 


11-6 


1200 







various values of rj. These values are not claimed to be exact. 
In these experiments it was very difficult to keep the temperature 
constant during each series of observations, and thus a deter- 
mination of k for each temperature was out of the question. From 
the curve of the function -q =f{t), shown in Diagram 4, it follows 
that I drj/dt I rises rapidly as -q increases; this tallies with what 
was said above. 



"32 Mr Molin, An examination of Searle's metJiod 



4000 




9 10 Tl 1Z 13 IV IS IS 17 18 19 20 



jTemptX- 



§ 7. I thought it would be interesting to compare the results 
given by Searle's method with those obtained by Poiseuille's 
method. The utility of the latter method for very viscous liquids* 
is proved by the investigations of Kahlbaum and Eaberf for 
values of rj in the neighbourhood of 40, and by LadenburgJ for 
r] = 1-3 X 10^. Fausten§ has found that the length of the dis- 
charge tube must exceed 45 cm., if the simple Poiseuille formula 

is to represent actual facts. In the formula 

h = Height of liquid corresponding to difierence of pressure 
between ends of tube. 

R = Internal radius of tube. L = Length of tube. 

p = Density of liquid (= 1-4103 ± 0-0003 grm. cm.-^at 19-8° C). 

m = Mass of liquid discharged. t = time of discharge. 

For shorter tubes, Hagenbach's* correction must be employed; 
otherwise the value obtained for 77 will be too high. As the liquid 
flows out into the air in an even jet, it carries kinetic energy with 
it; in order to allow for this, the value of t] given by Poiseuille's. 

* H. Glaser, Eriangen Diss., 1906. 

t G. W. A. Kahlbaum and S. Raber, Acta Ac. Leap., 84, p. 204, 1905. 

X R. Ladenburg, Ann. d. Phys., 22, p. 298, 1907. 

§ A. Fausten, Bonn. Diss., 1906. 



for determining the viscosity of very viscous liquids 33 

formula must be multiplied, according to Hagenbach*, by a cor- 
recting factor slightly less than unity. As the thermostat could 
only accommodate tubes shorter than 45 cm., Hagenbach's correc- 
tion was calculated, but was found to be negligible. Ladenburgf 
points out that both Hagenbach's and Couette's corrections to 
Poiseuille's formula can be entirely ignored for liquids such that t] 
is of the magnitude 1-3 x 10^. 

The discharge vessel consisted of a wide glass cylinder; through 
the bottom of this was bored a hole through which the discharge 
tube was connected with the interior of the cylinder. The whole 
apparatus was placed in the thermostat and the same temperature, 
19-8° C, was" maintained as was used in the earlier experiments, 
AVhen a tube whose internal radius was about 0-26 cm. was used, 
the liquid did not issue in a continuous jet but in drops. The 
values obtained for -q are given in Table 5. The mean value is 
r] = 271-1. The value obtained by Searle's method, viz. 274-7, 
differs from that obtained by Poiseuille's method by 1-3 per cent.; 
the agreement may be regarded as good. 



Table 5. 
Values of 7] by Poiseuille^s method. 



Rem.. 


Zcm. 


h cm. 


m grm. 


t sec. 


■n 


0-3168 


46-48 


49-36 
49-93 


54-421 
53-568 


1790 
1757 


269-9 
272-3 



§ 8. The influence of the base of the rotating cylinder can be 
eliminated, without determining k, by using the relation J 

77 = C . ^ -, = Cy, 

n ~ h 

provided that the points corresponding to M^T^ and M^T^ He to 
the right of the parabolic boundary line in Diagram 1. If we put 
Zj = 10-0 cm., we obtain the results given in Table 6. 



* F. Kohlrausch, Lehrbuch d. praktischen Physik, pp. 264 — 269, 1914. 

t R. Ladenburg, loc. cit., p. 298. 

i Compare C. Brodman, loc. cit., p. 163. 



VOL. XX. PART L 



34 Mr Molin, An examination of Searle's method, etc. 

Table 6. 
Values of y. 



MT 


944 


804 


734 


537 


342 


251 
215 


I 


10-0 


8-45 


7-65 


5-50 


3-30 


J 




90-2 


89-3 


90-5 


90-0 


88-4 



Wlien the various values are given the same weight, the 
mean value of y is 89-7, and then 17 = 2754-. 

§ 9. Diagram 3 and formula (5) show that 77 cannot be re- 
garded as independent of Q. unless Q. exceed a certain value, in 
this case 0-9. Since Q. is related to the rate of shearing rdw/dr, 
according to the formula 



it follows that 7y is a function of the rate of shearing. Hence, the 
assumption on which formula (1) is based, viz. that rj is independent 
of the rate of shearing, seems to be unjustifiable for small values 
of the rate of shearing, at least in the case of the highly viscous 
liquid used in these experiments. 






Miss Haviland, Note on Antennal Variation in an Aphis 35 



Preliminary Note on Antennal Variation in an Aphis (Myzus 
ribis, Linn.). By Maud D. Haviland, Fellow of Newnham 
College. (Communicated by Mr H. H. Brindley.) 

[Read 8 March 1920.J 

In 1918, during an investigation of the life-history of the Red 
Currant Aphis, Myzus ribis, Linn., it was observed that consider- 
able variation occurred in the antennae of the winged partheno- 
genetic females; and the evidence pointed to the conclusion that 
this variation was induced by the food^. Antennal variation in 
certain Aphididae has been studied by Warren ^ Kelly^ Ewing- 
and Agar^. Warren's experiments on Hyalopterus trirhodus 
showed some diminution of the correlation co-efficient in passing 
back from parent to grandparent. Kelly, for Aphis rumicis, con- 
sidered that somatic variations of the parents were not inherited 
by the offspring. Ewing, who bred eighty-seven generations of 
Aphis avenae, concluded that the variations were not transmitted 
to the offspring. 

Agar found some evidence of a partial inheritance of individual 
variations in Macrosiphum antherini, but he showed that this 
might be due to causes other than true inheritance. 

Myzus ribis is a common pest of red currant bushes. The 
sucking of the aphides upon the leaves tends to cause red galls or 
blisters, within which the plant lice continue to feed and reproduce. 
The fifth and sixth antennal segments of the winged partheno- 
genetic females normally bear two sense organs of unknown 
function — one on the distal third of Seg. v., the other on the 
proximal third of Seg. vi. It was observed in 1918 that, in indivi- 
duals reared on red blistered leaves, these sensoria were placed 
comparatively close to the articulation of Segs. v. and vi. On the 
other hand, if the aphides were fed upon green unblistered leaves, 
the sensoria were placed further away from the articulation. 

For the sake of brevity, the first type of antenna will be referred 
to hereafter as the Red (or R) type, and the second as the Green 
(or G) type; but every degree of transition may exist between the 
two extreme types. 

The experiments of 1918 were incomplete, and were conducted 
with a polyclonal population. They were repeated in 1919 with a 
monoclonal population, but the results are still far from being- 
conclusive owing to the small numbers available in some genera- 
tions. Only the winged forms show the required character. The 
production of these forms is probably governed by environmental 
factors which at present are imperfectly understood, and, for some 

3—2 



36 Miss Haviland, Preliminary Note on Antennal 

reason, in the population used in 1919, it was unusually low. It is 
hoped to repeat and extend the range of the experiments in 1920. 
The character chosen is the distance between the sensoria of 
antennal segments v. and vi. and the articulation of these 
two segments, expressed as the percentage of the width of the 
head between the eyes. The ratios are shown separately for 
each segment, with a dividing line to represent the articulation. 

^, ,„ , , ,, , Seg. VI. = 19% of the head- width 

Thus J# denotes that -^ ■^, — j— j — ^ — . ,^, . 

^ Seg. V. = 8% of the head- width 

Each generation is designated by combinations of two letters: 
E, (= red leaves) and G (= green leaves) and numerals, which 
express its complete ancestry. Thus ^^^^ denotes the fourth 
generation from the fundatrix of the population, and the F^. 
generation after transference to Green leaves after two consecutive 
generations on Red blistered leaves. In the transferred generations, 
the aphides were removed to the new environment when less than 
twelve hours old. The individuals for transference were selected 
wholly at haphazard. Thus, if a brood mother Eg gave birth to 
four young in the day, two were transferred to red blistered leaves, 
and two to green leaves, and so on in equal numbers from day to day. 

The pure Red (RRR, etc.) lines, and pure Green (GGG) lines 
were used as controls. The latter unfortunately became extinct in 
the third (Gg) generation. Hence for later generations the next 
longest unbroken line on green leaves (R2G0, etc.) had perforce to 
be taken as the control, though as it had been fed for the first two 
generations upon red leaves, it cannot be regarded as wholly * 
satisfactory. In Table 1, the curves of error of the ratios of genera- 
tions R2, R4 and R2G4 are shown. Rg is the common ancestral 
generation. The mode of the curve of R4 tends to shift to the left, 
i.e. the ratios of the antennal segments to the head-width are 
smaller. For the sake of clearness, in the graph only the curve of 
R4 is shown, but those of Rg, R5 and Rg, though with a smaller 
number of individuals, are almost identical with it. The curves of 
the ratios of R2G1 and R2G2 are very similar to their red controls. 
The R2G3 generation produced very few winged individuals, but 
these indicate a somewhat greater range of variation in Seg. vi. 
The curve of R2G4, as shown in the graph, has a marked tendency 
to shift to the right, indicating that the ratio of the antennal 
joints to head-width has increased, and this tendency is maintained 
in the succeeding generations, R2G5 and R2Gg. The position in 
the generation series does not account for the change in the 
antennal structure, for the modes for the six Red generations are 
nearly identical. 

So far we have considered only the modes. The mean ratios 
of the different generations are dealt with in the succeeding tables. 



I 



Variation in an Aphis (Myzus ribis, Linn.) 37 

Table 2 shows the mean ratios of the successive generations in 
four lines of descent, including the red and green controls. The 
extinction of the green control line was unfortunate, and in future 
experiments it will be very desirable to obtain a pure green line. 
At present the explanation that suggests itself of the variation of 
the RgGrg. . . line is that the influence of red feeding persists for at 
least two, and probably three generations after removal to different 
food, and this is somewhat confirmed by the R4G1. . . etc. line. 

Tables 3, 3a, 4, 4a and 5, 5a, give the effect of transference 
upon the mean ratios of the first, second, and third generations 
respectively, and below each is an analysis of the ratio of each 
segment, indicating its increase or decrease over previous genera- 
tions and the controls. 

Examination of the figures seems to show that the ratios of the 
first generation after transference vary irrespectively of the 
parental ratio. 

In transference to Red, the ratio of Seg. v. increases over that 
of the parental ratio, but in Seg. vi. it decreases (Table 3). In 
transference to Green, the results for both segments are quite 
inconclusive as regards the parental ratio (Table 3a). In the 
second generation after transference to Red, the results are like- 
wise inconclusive for both segments (Table 4). After transference 
to Green, the ratio of Seg. v. shows a tendency to rise above, 
and Seg. vi. a tendency to fall below, the parental and grand- 
parental ratios (Table 4a). 

In the third generation after transference to Red, the ratio of 
Seg. v. rises above the ancestral ratios, and that of Seg. vi. falls 
(Table 5). After transference to Green, the ratio of Seg. v. rises 
above those of the ancestral generations, and that of Seg. vi. rises 
in one case and falls in the other (Table 5a). 

These results are inconclusive, but examination of the control 
ratios shows that, with occasional exceptions, the ratio of a genera- 
tion with a mixed ancestry tends to rise above that of the Red 
control, but remains below that of the Green. Many more experi- 
ments in transference are required, and a much larger number of 
individuals must be examined before any conclusion can be 
reached; but at present the evidence suggests that the antennae 
of Myzus ribis are modified according to the food supplied, and 
that the effect induced by feeding in one generation is discernible 
in the succeeding three or four generations. It is difficult otherwise 
to explain the difference between the ratios of Rg and R2G4, and 
between Rg and R4G3, which, translated into the terms of human 
relationship, would be third cousins, and first cousins once removed, 
respectively, for all were produced by parthenogenesis, and, 
except for the food, reared side by side under identical environ- 
mental conditions. 



38 



Miss Haviland, Preliminary Note on Antennal 



Table 1. Curves showing the ratio of the distance of the sensoria 
from the articulation of antennal Segments V and VI to the 
width of head. The lower curves refer to the fifth, and the upper 
to the sixth segment. 



— B2 generation 



= E^Gi 




Table 2. Mean ratios of the successive generations of the lines^ 



Cto 



i?3 ..., JR/jG^ ..., Hcfii ..., and R2GQR1 



II 


III 


IV 


V 


VI 


VII 


VIII 


(^2¥ 


GsU 
















R2 ^^0- 


R3¥- 


R4¥ 


R5 ¥- 


Re -¥- 


— 


— 


)! i) 


» 5> 


)) }■> 


R4G1 ""7" 


R4G2 V 


^iGsB 


— 


1) H 


RA¥- 


E2G2 1§ 


R2G3 n 


R2G4 t"5 


R2G5 M 


RgGe ft 


)> )) 


>5 ?) 


)) )) 


55 55 


R2G3R1 n 


R2G3R2 V 


— 



J 



Variation in an Aphis (Myzus ribis, Linn.) 



39 



Table 3. Mean ratios of the first generation transferred from Green 
leaves to Red blisters, ivith an analysis below. 
+ = increase over ancestral ratio 
- = decrease from „ ,, 

= identical with ,, ,, 



Generation 


Parental Generation 


Red Control 


Green Control 


Ratio 


Ratio 


Ratio 


Ratio 


GiRi -V- 


G no winged forms 


R24? 


G2¥ 


G2R1 f* 


G, ^^ 


R3¥ 


Gsfl 


R2GjRi \f 


R2G1 ^^- 


R4¥ 


R2G2 i§ 


GlR2¥ 


GjRi -gi- 


R3¥ 


G3ff 


R4GXR1 ¥ 


R4G1 ■\-- 


R6¥ 


R2G4 ^ 


R2G3R1 f§ 


R2G3 \i 


R6¥ 


R2G4 f f 


R2G4R1 i\ 


^i^i 15 


Rc¥ 


R2G5 f 1 



Segment V 



Generation 


Variation from 

Parental 

Ratio 


Variation from 

Red Control 

Ratio 


Variation from 

Green Control 

Ratio 


GiRi 


no winged forms 


— 





G2R1 


+ 


+ 


- 


R2GiRj 


+ 


+ 


- 


G1R2 








- 


R4GiRj 


+ 





- 


R2G3R1 





+ 


- 


R2G4Rj^ 


- 


+ 


— 





Segment VI 




Generation 


Variation from 

Parental 

Ratio 


Variation from 

Red Control 

Ratio 


Variation froln 

Green Control 

Ratio 


GiR, 


no winged forms 


+ 


- 


G2R1 


- 


+ 


- 


R2GxRi 


- 





- 


G1R2 


- 


+ 


- 


R4GjRi 


- 


+ 


- 


R2G3R1 





+ 


- 


R2G4R1 


— 


+ 


— 



40 



Miss Haviland, Preliminary Note on Antennal 



Table 3a. Mean ratios of the first generation transferred from Red 
blisters to Green leaves, with analysis as in Table 3. 



Generation 
Eatio 


Parental Generation 
Eatio 


Green Control 
Eatio 


Eed Control 
Eatio 


ExGi \«- 


R no winged forms 


G2¥ 


R,^^ 


R2G1 ¥ 


R2 ¥ 


G-3 ft 


R3¥ 


GiRiGi -V- 


GfiRi "V" 


Gaff 


R3¥ 


RsGi^ 


R3 -¥ 


- G3 f f 


E4 ¥ 


GiRA¥ 


G1R2 ¥ 


R2G2 1§ 


E4¥ 


R4G1 ¥ 


R4 ¥ 


K2G3 15 


1 



Segment V 



Generation 


Variation from 

Parental 

Eatio 


Variation from 

Green Control 

Eatio 


Variation from 

Eed Control 

Eatio 


RiGi 


no winged forms 





— 


R2G1 


- 


- 


- 


GiRiGi 


+ 


- 


+ 


R3G1 





- 





GjR2Gi 


+ 


- 


+ 


R4G1 


- 


- 


- 



Segment VI 



Generation 


Variation from 

Parental 

Eatio 


Variation from 

Green Control 

Eatio 


Variation from 

Eed Control 

Eatio 


RjGi 


no winged forms 


_ 


_ 


R2G1 


- 


- 


+ 


GjRiGx 


- 


- 


+ 


R3G1 





- 


+ 


G^RgGj 





+ 


+ 


R4(]ri 


+ 


+ 


+ 



Variation in an Aphis (Myzus ribis, Linn.) 



41 



Table 4. Mean ratios of the second generation after transference 
from Green leaves to Red blisters, with analysis as in Table 3. 



Generation 
Ratio 


Parental 

Generation 

Ratio 


Grand-parental 

Generation 

Ratio 


Red 

Control 

Ratio 


Green 

Control 

Ratio 


G,R2 V- 
G2R2 if 
R2G3R2 V- 


GiRi -^ 
G2Ri U 
R2G3R1 f g 


Gj no winged forois 

G2 ¥ 
R2G3 H 


R3¥ 

Rg¥ 


G3tt 
G3 If 
K2G5 f § 



Segment V 



Generation 


Variation from 

Parental 

Ratio 


Variation from 

Grand-parental 

Ratio 


Variation from 

Red Control 

Ratio 


Variation from 

Green Control 

Ratio 


G,R2 
G2R2 
R2G3R2 




+ 


no winged forms 

+ 




+ 


— 



Segment VI 



Generation 


Variation from 

Parental 

Ratio 


Variation from 

Grand-parental 

Ratio 


Variation from 

Red Control 

Ratio 


Variation from 

Green Control 

Ratio 


(hR2 
R2G3R2 


— 


no winged forms 



+ 
+ 


— 



42 



Miss Haviland, Preliminary Note on Antennal 



Table ia. Mean ratios of the second generation after transference 
from Red blisters to Green leaves, with analysis as in Table 3. 



Geueratiop 
Ratio 


Parental 

Generation 

Ratio 


Grand-parental 

Generation 

Ratio 


Green 

Control 

Ratio 


Red 

Control 

Ratio 


R2G2 i§ 
R4G2 \^ 


R,Gi -V- 
R4G1 '^-f- 


R2 4P 
R4 V- 


G3 ti 
R2G4 ft 









Segment V 






Generation 


Variation from 

Parental 

Ratio 


Variation from 

Grand-parental 

Ratio 


Variation from 

Green Control 

Ratio 


Variation from 

Red Control 

Ratio 


R2G2 
R4G2 


+ 
+ 


+ 
+ 


- 


+ 

+ 



Segment VI 



Generation 


Variation from 

Parental 

Ratio 


Variation from 

Grand-parental 

Ratio 


Variation from 

Green Control 

Ratio 


Variation from 

Red Control 

Ratio 


R2G2 
R4G2 


- 


+ 


- 


+ 
+ 



Variation in an Aphis (Myziis ribis, Linn.) 



43 



Table 5. Mean ratios of the third generation after transference from 
Green leaves to Red blisters, with analysis as in Table 3. 



Generation 
Ratio 


Parental 

Generation 

Ratio 


Grand-parental 

Generation 

Ratio 


Great-grand - 

parental 

Generation 

Ratio 


Red 

Control 

Ratio 


Green 

Control 

Ratio 


G1R3 ¥ 


G1R2 ^' 


G,R, i^i 


110 winged 
forms 


K4V- 


GsM 



Segment V 



Generation 


Variation 

from 
Parental 

Ratio 


Variation 

from 

Grand-parental 

Ratio 


Variation from 
Great-grand- 
parental 
Ratio 


Variation 

from Red 

Control 

Ratio 


Variation 

from Green 

Control 

Ratio 


G1R3 


+ 


+ 


no winged 
forms 


+ 


- 



Segment VI 



Generation 


Variation 
from 

Parental 
Ratio 


Variation 

from 

Grand-parental 

Ratio 


1 
Variation from ! Variation 
Great-grand- ' from Red 
parental Control 
Ratio Ratio 


Variation 

from Green 

Control 

Ratio 


G1R3 


- 


- 


1 
no winged + 
forms 

1 


- 



44 Miss Haviland, Note on Antennal Variation in an Aphis 

Table 5a. Mean ratios of the third generation after transference 
from Red blisters to Green leaves, with analysis as in Table 3. 



Generation 
Ratio 


Parental 

Generation 

Ratio 


Grand- 

parental 

Generation 

Ratio 


Great-grand- 
parental 
Generation 
Ratio 


Green 
Control 
Ratio 


Red 

Control 

Ratio 


E2G3 u 
K4G3 !l 


R2G2 \% 
R4G2 -V- 


R2G1 -v- 

R4G1 5^ 


R4 V 


Gsff 
RsGsfl 


R5 ¥ 
Rg¥ 



Segment V 



Generation 


Variation 

from 
Parental 

Ratio 


Variation 

from 

Grand-parental 

Ratio 


Variation from 
Great-grand- 
parental 
Ratio 


Variation 

from Green 

Control 

Ratio 


Variation 

from Red 

Control 

Ratio 


R2G3 
R4G3 




+ 


+ 
+ 


+ 
+ 


- 


+ 
+ 



Segment VI 



Generation 


Variation 

from 
Parental 

Ratio 


Variation 

from 

Grand-parental 

Ratio 


Variation from 
Great-grand- 
parental 
Ratio 


Variation 

from Green 

Control 

Ratio 


Variation 

from Red 

Control 

Ratio 


R2G3 


— 


— 


— 


— 


+ 


R4G3 


+ 


+ 


+ 


+ 


+ 



LITERATURE REFERRED TO IN THE TEXT. 

(1) Agar, W. E. (1914). "Experiments on Inheritance in Parthenogenesis," 
Phil. Trans. Roy. Soc, Series B, vol. ccv, pp. 421-487. 

(2) EwiNG, H. E. (1916). " Eighty-seven generations in a parthenogenetic 
pure line of plant lice," Biol. Bull., vol. xxxi, No. 2, pp. 53-112. 

(3) Haviland, Maud D. (1919). " On the Life History and Bionomics of 
Myzus rihis, Linn.," Proc. Roy. Soc. Edinburgh, vol. xxxix, pt. 1 (No. 8), 
pp. 78-112. 

(4) Kelly, J. P. (1913). "Heredity in a Parthenogenetic Insect," Ainer. 
Nat., vol. XLVii, pp. 227-234. 

(5) Warren, E. (1901). "Variation and Inheritance in the Parthenogenetic 
Generations of an Aphis, Hyaloptenis trirhodus, Walk.," Biometrika, 
vol. I. 



Mr Waran, Ejfect of magnetic field on Intensity of spectrum lines 45 



The effect of a magnetic field on the Intensity of spectrum lines. 
By H. P. Waran, M.A., Government Scholar of the University of 
Madras. (Communicated by Professor Sir Ernest Rutherford.) 

[Read 8 March 1920.] 

[Plates I and II.] 

Since the discovery of the Zeeman effect the main attention 
has been directed to the detailed study of the phenomenon of the 
small change of wave length suffered by a monochromatic radiation 
in a magnetic field. The question whether a magnetic field affects 
the spectrum as a whole has not received much attention. 

While wiorking on the Zeeman effect with a mercury discharge 
tube run by an induction coil as the source, a small portion of the 
capillary tube being subjected to a magnetic field of about 5000 
c.G.s. units as shown in Fig. 1, the light was observed to suffer a 
change in intensity and also in colour opposite the pole pieces 
when the field was thrown on. A spectroscopic examination revealed 
the existence of some selective changes in the spectrum in addition 
to the increased brilliancy of the general spectrum. It was also 
noticed that the changes taking place varied considerably with 
the pressure, at a low pressure the tube showing little change 
visually but greater changes in the general spectrum. Attention 
was concentrated on the latter. 

In the case of mercury which was the first spectrum investigated, 
the tube, containing a trace of residual air at very low pressure, 
gave the principal mercury lines, viz. : 

5790-66, 5769-6, 5460-7, 4916-0, 4358-34 
and the principal hydrogen lines 

6563, 4861-5 and 4340-7. 

On applying the magnetic field, however, marked changes were 
observed, including a new set of lines at 

5426, 5679, 5872 and 5889, 
and a very strong red line at 6152, brought out prominently by 
the field. Mercury lines have been recorded at these wave lengths 
and these lines brought out are probably due to mercury. The 
behaviour of the line 6152 was very remarkable. It was invisible 
under ordinary conditions but showed up brilliantly in the magnetic 
field, the effect being practically instantaneous. Exhausting the 
tube still further and increasing the current through the tube to 
about 5 m.a. Four faint lines appeared at wave lengths 
6234, 6152, 6123 and 6072, 



46 



Mr Waran, The effect of a magnetic field 



and corresponding to these wave lengths mercury lines are recorded 
by* Stiles, Eder, Valenta, Arons and Hermann. But Arons and 




Fig. 1. 

Hermann have not recorded the line 6152, while Stiles records it 
as of equal intensity with the line 6234. Eder and Valenta have 
not observed the latter lines at all, but record the line 6152 as 

* Kayser, Handbuch der Spectroscopie, Band v. p. 538. 



on the Intensity of spectrum lines 47 

one of veiy great intensity. Examining the efEect of the magnetic 
fields on these four lines, it is very interesting to note that the 
line 6152 alone increases about five times in brilliancy while the 
others if they suffer any change at all, decrease in intensity. It is 
also interesting to note that this line 6152 seems to be the same 
line that becomes so greatly enhanced when the tube contains a 
trace of helium as observed by* Collie. It seems very difl&cult to 
excite this line unless at least a trace of helium is present in the 
ajjparatus and at this stage it is not possible to suggest any 
explanation of its abnormal behaviour. 

In addition to these very prominent changes there are also 
many minor changes, among which is the disappearance of a faint 
trace of continuous spectrum, as well as of some of the nebulous 
bands and lines, the remaining lines being quite sharp on a dark 
background. 

The abnormal behaviour of the mercury spectrum in the visible 
region (the ultra violet spectrum has not yet been investigated) 
'suggested the study of other spectra and the spectrum of helium 
was next examined. 

The discharge tube contained hydrogen and a slight trace of 
mercury vapour as impurity and the hydrogen lines and the 
prominent mercury lines were also visible. The effect of the 
magnetic field in this case was to enhance the helium lines very 
considerably, leaving the hydrogen lines practically unaffected or 
even slightly reduced in intensity. In this spectrum there were 
also a few faint lines not yet identified definitely which remain 
quite unaffected by the magnetic field. In the further study of 
the helium spectrum, the gas was contained in a separate tube 
from which any small quantity of it could be introduced into the 
discharge tube. At a pressure of 1 mm. of mercury the addition 
of a small trace of helium produced no perceptible effect on the 
spectrum of residual air which showed the prominent hydrogen 
lines and the nitrogen bands, but no trace of any of the helium 
lines. But on switching on the magnetic field, the helium lines 
flashed out prominently and disappeared again as soon as the field 
was turned off. The effect is shown in the accompanying photo- 
graphs (Plates I and II). In a plate taken with a greater percentage 
of helium the lines are visible without the magnetic field, but a 
great enhancement of these lines with the field is evident, and a 
dense new fine at 49334 a.u. is also noticed which has not yet 
been definitely identified. 

The spectrum of neon was also studied, and in a tube kindly 

lent to me by Dr Aston, there was a trace of hydrogen also present, 

showing the three principal hydrogen lines. Here also the effect 

of the field was to enhance very considerably the neon lines, 

* Proc. Roy. Soc. 71, 25, 1902. 



48 Mr Waran, The effect of a magnetic field 

leaving the hydrogen lines comparatively unaffected, so that by a 
casual examination of the spectrum the hydrogen and the neon 
lines can be distinguished from one another. 

The oxygen spectrum is rather difficult to excite when mixed 
with other gases. Yet a mixture of hydrogen, oxygen and a trace 
of helium was tried with success and here again the monatomic 
helium lines were brouglt out by the magnetic field, leaving the 
diatomic oxygen and hydrogen lines comparatively unaffected as 
shown in the photographs. 

From these experiments the natural inference follows that in 
a mixture of the monatomic and diatomic gases, the monatomic 
gases alone seem to be selectively affected in a peculiar way 
resulting in their spectrum lines alone being very considerably 
enhanced or brought out prominently even when not visible at 
all previously. By this method minute traces of the monatomic 
gases when mixed with other diatomic gases can be detected. 
On this view we might also explain the abnormal mercury line 
6152 and others as due to the radiation from the monatomic 
atom while the other lines may be classified as belonging to the 
molecule. 

Examining the spectrum of the atmospheric air at low pressure 
in this way the effect of the magnetic field is to bring out new lines 
which are not present without the magnetic field, as shown in the 
photographs. As far as their wave lengths have been determined, 
though one or two of them fit in fairly well with lines catalogued 
as belonging to oxygen and nitrogen, yet there are others which 
are difficult to identify while the absence of other stronger lines of 
oxygen and nitrogen make even these two or three fits inconclusive. 

Another interesting point noted in these experiments is the 
varying degrees of enhancement under the influence of the field 
for lines belonging to the same element helium. Preston has 
shown that the Zeeman effect is of the same magnitude for lines 
belonging to the same series, but differs in different series. Simi- 
larly we might expect the degree of enhancement of the lines in 
the magnetic field to depend on the series to which the line belongs. 

The exact nature of this phenomena and the mechanism of the 
reaction that brings about these novel changes in the general 
spectrum is not yet definitely known and it is not desirable to 
attempt an explanation until the study of the spectrum has been 
extended to the ultra violet. 

The current in the tube was usually about 3 m.a. and the effect 
of the field was to decrease the current by about 20 to 30 per cent. 
The changes of intensity observed cannot be attributed to this 
since the reduction of the current by a spark gap in series only 
brings about a proportionate decrease in brilliancy of the general 
spectrum. 



Phil. Soc. Proc. Vol. xx. Pt. i 



(a) Hydrogen and 
Mercury 




(6) Hydrogen and 
Helium 



(c) Hydrogen and 
Helium, low 
percentage 



(d) Hydrogen and 
Helium in 
larger per- 
centage 




(e) Neon 




On 



Oft 



(/) Air 



Ullltl I t K II 



On 
Off 



(gr) Air 




On 



Off 



Fig. 2. Photographs showing the enhancing effect of the field. The small lateral 
shift is due to the camera slider, and in («) the mercury line 6152 is incUcated 
by the dot, while in the other cases the lines that newly turn up are indicated 
by the arrows. 



I 



Phil. Soc. Proc. Vol. xx. Pt. i. 



Plate II 



O H, 



(1) Air, Oxygen, Hydrogen 
and trace of Helium 



(2) Oxygen, Hydrogen and 
trace of Helium 




(3) Excess of Helium and 
trace of Air and Oxy- 
gen 




On 



OfE 



(4) Oxygen, Hydrogen and 
Helium and trace of 
Air 




On 



Off 



Fig. 3. Photographs showing the effect in mixtures of gases studied. 



on the Intensity of spectrum lines 49 

It may be of interest to note that in solar spectroscopy the 
spectrum of the sunspots is found to difTer in many respects from 
that of the photosphere, considerable numbers of enhanced lines 
occurring in the sunspot spectrum. The existence of a powerful 
magnetic field in sunspots has been demonstrated by the Zeeman 
effect and possibly the differences in the spectrum of the sunspot 
and the photosphere may be attributed to this new effect of the 
magnetic field on the spectrum. 

The further study of this effect and the examination of other 
spectra are in progress. 

Cavendish Laboratory, 
Cambridge. 



VOL. XX. PART I. 



50 Mr Brindley, Further Notes on the Food Plants 



Further Notes on the Food Plants of the Common Earwig (For- 
ficula aiiricularia). By H. H. Brindley, M.A., St John's College. 

[Read 8 March 1920.] 

In a paper pubHshed in the Proceedings of the Cambridge Philo- 
sophical Society, xix, Part 4, July 1918, p. 170, I recorded certain 
observations in August and September, 1917, on the food plants of 
the Common Earwig, with the view of obtaining more exact infor- 
mation than was then available as to the damage likely to be 
done by this species in a flower or kitchen garden. The paper also 
epitomised recent literature on the subject, a consideration of 
which had revealed a considerable amount of diversity and want 
of exact information as to the favourite food plants of earmgs in 
the British Isles. The observations made by myself were on earwigs 
kept in captivity in connection with a statistical enquiry as to the 
variation of the forcipes which is still in progress. The observations 
in 1917 were on earwigs from St Mary's, Isles of Scilly, and those 
recorded in the present paper were made in the second half of the 
year 1918 on a collection from the Bass Rock, which swarms with 
earwigs. The animals were all adults and were kept in large glass 
dishes bedded with sand slightly damped occasionally. Earwigs re- 
main healthy in a soaked substratum if the ventilation is good, but 
in captivity in a warm room without circulation of air they suffer 
heavy mortality from fungoid attack, as I have already recorded 
{Proc. Camb. Phil. Soc, xvii, Part 4, Feb. 1914, pp. 335-338). The 
fungus appears to be usually Entomofhthora forflculae (Picard, 
BuU. Soc. Etude Vidg. Zool. Agric. Bordeaux, Jan.-Apr., 1914, 
pp. 25, 37, 62). The importance of ventilation and of normal tem- 
perature is well illustrated by the far fewer fungoid attacks and 
the low mortality when the new Insect House belonging to the 
Cambridge Zoological Laboratory became available in 1919. It is 
at present too early to say how far an improvement is obtainable 
in the survival of eggs and young which it is hoped to rear in the 
spring in normal outside temperatures in the Insect House. Earwigs 
offer a great contrast to cockroaches as regards desire for water; 
the latter thrive in captivity for months in ^ warm room on food 
which is entirely dry, while earwigs certainly visit water to drink, 
as I have seen in both the captive and wild conditions. I have 
previously recorded {Proc. Zool. Soc. Lond., Nov. 1897, p. 913) 
how Stylofyga orientalis in captivity seems to pay no attention to 
a damp sponge when that is the only source of moisture. We have 
however to bear in mind that the Common Cockroach is probably an 



of the Common Earwig (Forficula auricularia) 51 

immigrant from warmer countries of the East. The earwigs under 
observation during the past three summers had no animal food save 
that afforded by those which died. In order to obtain information 
as to preference for one kind of plant above another they were 
usually given three different species, taken haphazard, at a time, 
for a period of two or more days. 

In the following summary the observations of 1917 and 1918, 
with a few made in 1919, are combined. The dates when the different 
foods were given are noted, as in the latter part of September, when 
the animals tend to become lethargic, and in the succeeding two 
months the desire for food is much lessened, even in the artificial 
temperature of a laboratory. The capital letters after the names of 
the plants indicate those which were given at the same time, and 
the numbers appended indicate the preference exhibited by the 
earwigs: e.g. in food group M, M^ was attacked more than M^, M^ 
more than M^; in group F, F^ after two plants indicates that they 
seemed to be attacked equally, and more readily than F^: while 
in group Q, Q*' indicates that the plant offered was not attacked at 
all. Similarly for the other groups. 

24-26 Aug. '17. Alkanet, Blue {AncJmsa sp.) C-: leaves not attacked; petals 

gnawed considerably. 
27-29 Aug. '17. Anemone, White Japanese {Anemone japonica) D^: leaves 

not attacked ; petals eaten moderately. 
1-23 Sept. '17. Apple {Pyrus Mains) F^: rather unripe fruit with skin whole 

was not attacked; but when cut across was gnawed moderately: 24-28 

Sept. '18, leaves holed. 
24-28 Sept. '18. Ai-tichoke, Jerusalem {Helianthus tuberosus) W: leaves holed 

and edges gnawed down to midrib; tuber, cross sUce attacked vigorously 

and its buds also devoured. 
20 Sept.-5 Oct., 3-17 Nov. '18. Asparagus {Asparagus officmalis) OS T": leaves 

gnawed a little ; fruit not attacked. 
26-31 Aug. '18. Aster, Mauve China {Callistephics chinensis) K-: leaves not 

attacked ; petals and flower buds much eaten. 
6-11 Sept. '18. Aster, Pink China {Callistephus chinensis): leaves shghtly 

nibbled; petals much eaten ; ^OM;ers used as a refuge. 
15-20 Sept. '18. Balm, Pale Mauve {Melissa officinalis) J^ : leaves not attacked : 

petals of buds devoured. 
22-23 Aug. '17. Bean, Dwarf {Phaseolus vulgaris) B^: leaves nibbled ver;^ 

shghtly. 
30-31 Aug. '17. Bean, Scarlet Runner {Phaseolus multiflorus) E^: leaves, 

floivers and pods apparently neglected: 16-18 Oct. '18, leaves holed a good 

deal and edges gnawed down to veins. 
20-28 Oct. '18. Beard Tongue, Scarlet {Pentstemon sp.) R": leaves and flowers 

not attacked. 
22-23 Aug. '17. Beet {Beta vulgaris) B^: leaves much attacked, especially the 

petioles, which were opened out and their pith devoured. 
20-24 Sept. '18. Bell Flower, White {Campanula sp.) K^: leaves not touched; 

petals completely devoured. 
31 Aug.-6 Sept. '18. Bindweed, Common {Convolvulus sp.): leaves much holed. 
11-13 Sept. '18. Blackberry {Rubus fruticosus): vipe fruit well gnawed. 

4—2 



52 Mr Brindley, Further Notes on the Food Plants 

30-31 Aug. '17. Cabbage, Garden {Brassica oleracea capitata) W : leaves gnawed 

down to midrib and veins and ends of veins eaten off. 
2-5 Oct. '18. Canterbury Bell, Blue {Campanula medium) N^: leaves and petals 

well devoured. 
6-7 Sept. '18. Carrot {Daucus Carota): root not attacked where covered by 

skin, but cut end was much gnawed. 
6-11 Sept. '18. Celery {Apium graveolens) H^: leaves holed and their edges 

gnawed. 
29 Sept.-3 Oct. '18. Cherry {Prunus [Cerasus] sp.) M^: leaves not attacked. 
20-23 Oct. '18. Chickweed {Stellaria media) R^: edges of leaves gnawed 

sUghtly. 
31 Aug.-6 Sept. '18. Chrysanthemum, Garden {Chrysanthemum indicum): 

flower buds used as refuge, tips of petals apparently somewhat nibbled: 

31 Aug.-6 Sept. '18, purple variety: edges of leaves much nibbled; flower 

buds used as refuge, tips of petals apparently somewhat nibbled: 31 Aug.- 

6 Sept. '18, white variety: leaves not attacked; petals much eaten. 
20-24 Sept. '18. Clematis, White {Clematis sp.) K^: leaves, a few eaten off at 

ends and edges gnawed here and there ; flowers entirely devoured. 
23-27 Oct. '18. Cluvia miniata (Natal): leaves not attacked; petals gnawed a 

little along edges. 
15-20 Sept. '18. Cornflower {Centaurea Cyanus) .P: leaves well eaten, only 

midrib ]eit; flowers entirely devoured. 
29 Sept.-3 Oct. '18. Cups and Saucers {Cobaea scandens) M^: petals nibbled 

a httle. 

27 Oct.-3 Nov. '18. Dandelion {Taraxacum oflicinale): petals of ray florets 

entirely devoured. 
26-31 Aug. '18. Elephant's Ear, Pink {Begonia sp.): leaves much gnawed along 

edges and also holed ; flowers thoroughly devoured. 
2-5. Oct. '18. Fern, Male {Lastraea fllis-mas) 0°: leaves not attacked. 
15-20 Sept. '18. Feverfew {Pyrethrum sp.) J^: leaves gnawed down to midrib; 

flowers apparently not attacked. 
21-28 Sept. '18. Fig {Ficus Carica): leaves not attacked; fruit neglected when 

whole, but cross section was well gnawed. 
7-15 Oct. '18. Fox-glove {Digitalis purpurea) P^: leaves holed. 
6-11 Sept. '18. Fuchsia, Crimson Garden {Fuchsia sp.) H^: neither leaves or 

flowers were attacked. 

28 Sept.-2 Oct. '18. Geranium, Scarlet {Oeranium sp.) L^: petals eaten a httle. 
20-24 Sept. '18. Gesnera, Orange and Pink {Gesnerasp. )K}: leaves not attacked; 

petals entirely devoured. 
24-26 Aug. '17. Golden Rod {Solidago sp.) C^: leaves gnawed at edges here 

and there ; flowers apparently not attacked. 
2-5 Oct. '18. Gooseberry {Ribes grossularia) O^: leaves not attacked. 
11-15 Sept. '18. Hawthorn {Crataegus oxyeantha) P: neither leaves ov flowers 

were attacked. 
24-31 Aug. '18. Hollyhock, Dark Crimson {Althaea rosea): leaves not attacked; 

flower buds used as refuge, petals apparently eaten to some extent. 
10-20 Aug. '18. Honeysuckle {Lonicera sp.) G^: leaves not attacked; fruit 

gnawed considerably. 
7-20 Oct. '18. Hydrangea, Pink {Hydrangea sp.) Q": neither leaves ov flowers 

were attacked. 
7-15 Oct. '18. Larkspur, Garden variety {Delphinium sp.) Q^: leaves gnawed 

thoroughly down to midrib. 
3-6 Nov. '18. Leek {Allium porrum) T^: leaves gnawed deeply towards base. 
6-15 Sept. '18. Lettuce, Cabbage {Lactuca sativa): stem aljundantly gnawed 

and bored; leaves of "heart" entirely devoured. 
7-27 Oct. '18. Lupin {Lupinus polyphyllus) S^: leaves gnawed to some extent. 



of the Common Earwig (Forficula auric ularia) 53 

3-17 Nov. '18. Mallow [Malvus ? sylvestris): leaves holed and edges gnawed 

down to veins. 
23 Oct.-17 Nov. '18. Marguerite, White-rayed [Chrysanthemum leucanthemum) 

S^, U^: petals of ray florets well gnawed. 
20-21 Aug. '17. Marrow, Vegetable (Cucurbita ovifera) A^: leaves thoroughly 

devoured. 
20-21 Aug. '17. Michaelmas Daisy (Aster sp.) A^, 'i>i^: leaves hardly touched, 

if at Sill; floivers also neglected. 
11-15 Sept. '18. Mignonette (Reseda odorata): leaves gnawed down to midrib; 

flowers attacked but slightly or not at all. 
16-18 Sept. '18. Mint (Mentha sp.): leaves, edges and ends nibbled; flowers 

entirely devoured. 
20-23 Oct. '18. Xavew (Brassica campestris) R^ : leaves holed and edges gnawed 

a little; petals moderately attacked. 
3-17 Nov. '18. Nettle (Urtica dioica) U^: leaves well gnawed down to veins. 
31 Aug. -6 Sept. '18. Onion (Allium Cepa) L": inflorescence used as refuge, but 

apparently not eaten. 
7-15 Oct. '18. Pansy (Viola tricolor) P^: leaves nibbled slightly. 
10-20 Aug. '18. Parsley, Garden (Carum Petroselinum) G^: inflorescence 

nibbled moderately. 
29 Sept. -3 Oct. '18. Peach (Prunus [Amygdahis] sp.) N^: leaves gnawed 

moderately. 
28 Sept.-2 Oct. '18. Periwinkle, Blue (Vinca sp.) L^: leaves and petals gnawed 

moderatelv. 
22-23 Aug. '17. Phlox, White (Phlox Drummondi) B^: leaves apparently not 

attacked; petals much gnawed and pollen found in gut of earwigs. 
1-3 Sept. '17. Plum (Prunus communis) F^: fruit well eaten. 
23-31 Aug. '18. Poppy, Garden (Palaver sp.): dried fruits very popular as 

refuges; some were holed to obtain entrance. 
1-18 Sept. '17. Potato (Solanum tuberosum) F^: tuber in skin was neglected, 

but slices were thoroughly gnawed. 
28-29 Aug. '17, 20-23 Oct. '18. Primrose, Evening, yellow variety (Oenothera 

sp.) D^; leaves not attacked; petals eaten thoroughly; pods neglected. 
7-15 Oct. '18. Privet (Ligustrum vulgare) Q'^: leaves holed and edges gnawed; 

fruits not attacked. 
20-21 Aug. '17. Radish, Horse (Raphanus sativus) A^: leaves nibbled 

slightly. 
27-29 Aug. '17. Raspberry (Rubus idaeus) D": leaves not attacked, but earwigs 

assembled in crowds on their hairy undersides. 
22-28 Sept. '18. Red hot poker (Kniphofla sp.) : cut end of stem gnawed; 

lea ves and petals not attacked. 
11-15 Sept. '18. Rest-harrow (Ononis sp.) P: apparently neither leaves or 

floivers were attacked. 
30-31 Aug. '17. Rhubarb (Rheum officinale) W: leaves well gnawed. 
24-26 Aug. '17. Rose, White garden variety (Rosa sp. ) C^ : haves not attacked ;' 

petals devoured. 
7-10 Oct. '18. St John's Wort (Hypericum sp.) P^: leaves holed and their edges 

gnawed ; floioer buds not attacked. 
31 Aug. -6 Sept. '18. Scabious, Crimson Garden (Scabiosa atro-purpurea): 

leaves much holed ; floivers apparently not attacked. 
23-27 Oct. '18. Scotch Kale (Brassica oleracea acephala) S^: leaves holed a very 

little ; curled margins a favourite refuge. 
10-24 Aug. '18. Sea Kale (Brassica oleracea acephala) G^: leaves holed and 

gnawed away from edges to between veins. 
6-11 Sept. '18. Snapdragon, Scarlet (Antirrhinum sp.): leaves gnawed moder- 
ately; petals apparently holed to some extent, also used as refuge. 



54 Mr BrincUey, Further Notes on the Food Plants 

23-30 Oct. '18. Sow thistle [Sonchus oleaceus): leaves holed slightly; flower 

huds not attacked. 
3-17 Nov. '18. Strawberry (Fragaria vesca) W: leaves holed a little. 
31 Aug.-6 Sept. '18. Tomato (Lycopersicum esculentum): leaves and vipe fruit 

gnawed thoroughly. 
14-15 Sept. '18. Valerian, Red Garden ( Valeriana sp. ) : edges of leaves gnawed 

moderately ; petals entirely devoured. 
21-24 Aug. '18. Vervain, Blue (Verbena sp.): leaves nibbled slightly, haiiy 

undersides used for assembhng; petals entirely devoured. 
24-31 Aug. '18. Vetch, Mauve and White garden varieties (Vicia sp.): leaves 

attacked very slightly, if at all; petals entirely devoured. 
23 Oct.-3 Nov. '18. Violet, Single and Double garden varieties {Viola sp.): 

leaves holed and edges gnawed moderately. 
3-17 Nov. '19. Wartweed (Euphorbia helioscopia) T^: edges of leaves gnawed 

very shghtly. 
15-18 Sept. '18. Wormwood (Artemisia sp ): leaves not attacked. 



These observations are of course subject to tbe drawback that 
in captivity animals which normally feed daily may take unusual 
food with apparent eagerness because no other is available; but 
the above record probably indicates normal preferences over a 
certain range of common plants, and also that some are disliked 
by earwigs; thus Wartweed was left entirely untouched for many 
days in the absence of any other food, the animals attacking potato 
tuber ravenously as soon as this was substituted. It seems natural 
that such stiff and dry foliage leaves as those of Raspberry, Haw- 
thorn, and Cherry, should escape attack, and there is no doubt that 
the more succulent leaves are preferred. The list of plants affords 
some information which may facilitate the destruction of earwigs 
when they become a pest by the indications obtained as to plants 
which are popular as refuges, and also by the mode in which the 
attack on leaves is made; thus, some leaves seem to be attacked 
by holing as well as by gnawing along the edges, and others only 
by the latter method. There is no doubt that earwigs have pre- 
ferences among the common plants of a flower or vegetable garden, 
and that if numerous they are likely to become a pest. In certain 
cases, as for instance, chrysanthemums, the actual damage done 
seems to be exaggerated by common report. 

Since the epitome of recent literature on the subject in my 
previous paper {Proc. Camb. Phil. Sac, xix, Part 4, 1918, p. 170) 
was written. The Review of Applied Entomology has recorded 
attacks on beets and sugar-beets in Denmark sufficiently serious 
to obtain mention by Lind and others in their Report on Agri- 
cultural Pests in 1915 {Beretning fra Statens Forsogsvirksomhed i 
Plantekultur, Copenhagen, 1916, pp. 397-423). 

As regards the carnivorous habit of F. auricularia, lean roast 
mutton without other food was given for several days to the ear- 
wigs under observation in 1918 and was gnawed sparingly, while 



of the Common Earwig (Forficula auricularia) 55 

mutton suet substituted for it was eaten readily and extensively. 
In the Journal of the Bombay Natural History Society, xxvi, No. 2, 
May, 1919, p. 688, F. P. Connor records an unnamed earwig at 
Amara catching moths in its forcipes and in one case nibbUng its 
prey. F. Maxwell Lefroy {Indian Insect Life, p. 52) remarks: 
"The function of the forcipes is a mystery that will be cleared up 
only when their food habits and general hfe are better under- 
stood." They are very possibly "frightening" as well as defensive 
organs. Pemberton {Hawaiian Planters' Record, Honolulu, xxi, 
No. 4, Oct. 1919, pp. 194-221) mentions the benefit to cane fields 
arising from the destruction of the leaf-hopper parasite Perkin- 
siella optabilis by the black earwig Chelisoches morio. 

The importance of nocturnal observations on the feeding habits 
of Forficula auricularia to a satisfactory understanding of the 
economic effects of this insect in gardens, urged in my previous 
paper, may be referred to again. 



56 Mr Darwin, Lagrangian Methods for High Speed Motion 

Lagrangian Methods for High Speed Motion. By C. G. Darwin. 
[Read 8 March 1920.] 

1. In the later developments of Bohr's* spectrum theory, it 
is necessary to calculate the orbits of electrons moving \vath such 
high velocities that there is a sensible increase of mass. The selection 
of the orbits permitted by the quantum theory almost necessitates 
the treatment of such problems by Hamiltonian methods. Working 
on these lines Sommerfeldf and others have calculated with a very 
high degree of success those spectra which involve the motion of 
a single electron. But the application of the Hamiltonian function 
involves a knowledge of the momentum corresponding to any 
generalized coordinate, and in the formulation of most problems 
the momenta are not known a priori but must be calculated from 
the corresponding velocities. In other words the formation of the 
Hamiltonian function must in general be preceded by that of the 
Lagrangian. An exception occurs in precisely the problems referred 
to above; for, the electromagnetic theory furnishes directly values 
for the momentum and kinetic energy of a moving electron in 
terms of its velocity, and the velocity can be eliminated between 
them so as to obtain the Hamiltonian function. But in even slightly 
more complicated cases this simple relation is destroyed — thus the 
problem of a single electron in a constant magnetic field can only 
be solved by introducing the artificial conception of rotating axes 
• — and in general it will be necessary to follow the direct course of 
finding the Lagrangian function in terms of the generalized velocities, 
and then deducing from it the momenta and the Hamiltonian 
function in the usual way. 

If more than one particle is in motion another difficulty enters. 
For the interaction of two moving particles depends on a set of 
retarded potentials and the effect of the retardation is readily seen 
to be of the same order as the increase of mass with velocity. The 
calculation of the retardation can only be carried out by expansion 
and so the results are only approximate. This is not surprising since 
the methods of conservative dynamics cannot apply to such effects 
as the dissipation of energy by radiation, effects inevitably required 
• by the electromagnetic theory, though they do not occur in actuality. 
We can also see from the fact that these radiation terms are of 
the order of the inverse cube of the velocity of light, that it will 
be useless to expand beyond the inverse square. 

* N. Bolir, Kgl. Dan. Wet. SelsL, 1918. 

t A. Sommerfeld, Ann. Phys., vol. 51, p. 1, 1916. 



Mr Darwin, Lagrangian Methods for High Speed Motion 57 

2. We first consider the motion of a single electron in an 
arbitrary electric and magnetic field varying in any manner with 
the time and position. If m is the mass for low velocities, the 
momentum is known to be mv/^, where ^ = V 1 — v^jc^. Starting 
from this we have quasi-Newtonian equations of motion of the 
type 

lir*}-^^ '^■^'- 

The force F^. is given from the field E, H as the vector eE + ^ [v, H], 

where v is the velocity vector of the particle's motion. E and H 

can be expressed in terms of the scalar and vector potentials in 

1 3A 
the form E = — grad (t> ~ ^.^^ and H = curl A. 

C ct 

Then if r^ is the vector x, y, z we have as the vector equation of 
motion 

It \t '4 ^~'' ^''^ '^ ~ c W + c ^'1' '""'^ ^^ •••(^■^^' 

where ^^ = V 1 - V/C^. 

Let q be any one of three generalized coordinates representing 
the position of the particle. Take the scalar product of (2-2) by 

i=r^. Then since ^ = -^, we have 
oq cq cq 



dii d (nil . )\ d (m 



dq'dtX^^'^^U dt\^. 









dt dq dq 

'9r _ . ,\_ d4> 



where "Wq = j;^ — ^ ^^^ Lagrangian operator. 



Again - gj [J-, grad .^ j = - e^ ^ = e^Bc/,. 

The remainder can be reduced to 

CV''~dq)~c[dq'lt) ^-'^^' 

dA dA dA . dA . 8A . 

where -j7 = ^7 + ^^+^-y+'^^ 

dt dt ox Cy '^ oz 



58 Mr Darwin, Lagrangian Methods for High Speed Motion 

and so is the total change of A at the moving particle. (2-3) can 

be reduced to — ^ IBq (fi, A), 

Thus the whole equation of motion can be derived from a 
Lagrangian function 

L=- m,C^^, - e[<j> + g, (ii, A) (24). 

This is valid for any fields of force including explicit dependence 
of ^ and A on the time. The first term in L, which reduces to the 
kinetic energy for low velocities, differs from it in general. It is 
very closely connected with the "world line" of the particle. 

3. To treat of the case where several moving particles interact 
we shall start by supposing that there is a second particle present 
undergoing a constrained motion so that its coordinates are imagined 
to be known functions of the time. The same will then be true of 
the potentials it generates. The motion of e^ will then be governed 
by (2-4) if ^ and A are expressed in terms of the motion of e^. These 
potentials are given by 

/ _ ^2 a _ ^ ^2 /o.-i \ 

In these expressions r^ = (ig — ij)^ and the values are to be retarded 
values. If the time of retardation be calculated and the result 
substituted in (3-1) we obtain 

/_e2, 62 \ i^^+{i^,r^-Ti) (f^, r^ - i,)^ \ ._e.,i^ 

where now ij, la refer to the same instant of time, cf) is an approxi- 
mation valid to C"^, but the value of A has only been found to 
the degree C~^ on account of the further factor C~^ in (2-4) which 
is to multiply it. Then substituting in (2-4) we obtain 

r _ ,^ P2/P ^1^2 6162 ( r2^+ (r2,r2-ri) - 2 (fi, f^) 
L-- m,C Id, --y-^, I 



The equations of motion are unaffected by adding to L the expres- 
sion - mgC^^a + ^ ^2 ^'^"^'~^'^ - The first is a pure function of 

the time and so contributes no terms to the equations of motion. 
The second contributes nothing because for any function / we have 



Mr Danvin, Lagrangian Methods for High Speed Motion 59 
The new form of L then reduces to 

L = - m,C-^^, - m,C^^, - f + |gi j^^^ 

I (ri,ra -ri)(fa,r2-ri) | ^^.^^^ 



From the complete symmetry of this form the roles of e^ and eg ^^^-y 
be interchanged. Further from the covariance of the operator IB 
for point transformations, both may be included in the dynamical 
system, so that if q is any generalized coordinate involving both 
Tj and ig, the equations of motion will be of the form "313 gL = 0. 

For the sake of consistency, as the last term in (3-4) is only an 
approximation valid to C~^, the first two should be expanded only 
to this power. The first term will give 

- m^C^ + lm,i^^ + g^ mj^\ 

Generalizing our result to the case of any number of particles 
in any external field we have 

L = ^lm,i,^ + 2 g^, m,i,^ - He.cf. + 2 ^, (r,A) - SS '^^ 

+ si; ^^ I^AiA^ + (ri,r2-ri) (^2>r2-ri) ) ^ /3.5)_ 

The double summations are taken counting each pair once only. 
4. The transition to the Hamiltonian now follows the ordinary 

r) T 

rules. We find momenta f = -^ and solve for the g-'s in terms of 

the 2^'s. This can be done in spite of the cubic form of the equations 
in the g's by use of the approximation in powers of C. The Hamil- 
tonian function will then he H = Hpq — L and the equations of 

motion will be the canonical equations q = ^—, p = — -o— • K Pi 

be the momentum corresponding to Tj , the Hamiltonian in these 
coordinates will be 

«- ^ 2lJ - ^^ sit' + ^^'^ - ^ CS^ <""*» + ^^ t 



_ ss ^1^2 [ (Pi>P2) ^ (Pi,r2-ri) (P2,r.,-ri) 

All the applications of general dynamics, such as the Hamilton 
Jacobi partial differential equation, follow from this. As in ordinary 
dynamics, many problems can be conveniently solved in the La- 



60 Mr Darwin, Lagrangian Methods for High Speed Motion 

grangian form. The solution will usually depend on finding integrals 
corresponding to coordinates which do not occur explicitly in L 
and if cj) and A do not involve the time explicitly there is also the 
energy integral. This has the form 



^^^r/+2.,<^+SS^?^^ 



Simfi^ + Sf '-^l r/ + 2ei</. + SS -^-^ 



This completes the development of the method. Its direct applica- 
tions are naturally somewhat limited, since, even with the large 
order terms only, there are comparatively few problems that are 
soluble. A problem of some interest that can be solved completely 
is the motion of two attracting particles, where their masses have a 
finite ratio*. 

* A discussion of this problem by the present writer will be found in Phil. Mag. , 
Vol. 39, p. 537 (1920), together with a somewhat fuller account of the general theory. 



Dr Searle, A bifilar method of measuring the rigidity of wires 61 



A bifilar method of measuring the rigidity of wires. By G. F. C. 
Searle, Sc.D., F.R.S., University Lecturer in Experimental 
Physics. 

[Read 3 May 1920.] 

§ 1 . Introduction. In this method the couple due to the torsion 
of two similar wires is balanced against the couple due to the load 
carried by the wires and arising from bifilar action. 

The method is hardly suitable for accurate measurements of 
rigidity, but, as an exercise in the use of a bifilar suspension, it has 
proved useful at the Cavendish Laboratory. 

§ 2. Bifilar cowple. We first consider two light flexible strings. 
Let the strings AB, CD, each I cm. in length, hang from two fixed 
points A, C, which are at a distance 2a^ cm. apart in a horizontal 
plane. The lower ends B, D of the strings are attached to a rigid 
body of mass M grm., the points B, D being 2^2 cm. apart. The 
centre of gravity of the body is symmetrical with regard to B and 
D and thus the tensions of the strings are equal. The line BD will 
then be horizontal. If, now, a couple, whose axis is vertical, is 
applied to the body, the body will be in equilibrium when the 
couple due to the obliquity of the strings balances the applied 
couple*. 

In Fig. 1, A', B', C, D' are the projections of A, B, C, D on 
a horizontal plane. In our symmetrical case, A'C, B'D' bisect each 
other in 0. When the body has turned through 
6 radians from the zero position, in which the 
strings are in the vertical plane through A'C, qt 
then B'D' will make an angle 6 with A'C nT 
Let ON be the perpendicular from on A'B'. 
Let the tension in each string be T dynes. 

If the vertical distance of BD below AC 
is h cm., the vertical component of the tension is Th/l, and the 
horizontal component is T .A'B' jl. Since the weight of the body 
equals the sum of the vertical components, 

Mg = 2Thll. 

The horizontal component of the tension at B acts along a line 
whose projection is A'B', and hence its moment about the vertical 

* For the general theory of the bifilar suspension, see Maxwell, El. and Mag.^ 
Vol. n, § 459; A. Gray, Absolute Measurements in El. and Mag., Vol. i, p. 242;. 
Kohlrausch, Physical Measurements (1894), p. 226. 





62 Dr 8earU, A bifilar method of measuring the rigidity of ivires 



axis through is T .A'B' .ON/l. Since the moment due to the two 
tensions equals that of the appHed couple, G dyne-cm., 

G^2T. A'B'. ON/l. 
But A'B' .ON is twice the area OA'B' and thus is fz/Zg sin 6. We 
thus obtain 

_^ _ A'B'. ON _ a^a2 sin d 

Mg- 



h 



h 




Fig. 2. 



Since h = {P - A'B'^}^, we see that, when A'B' is small compared 
with I, we may put h = I, and so obtain 



G=''-^^^.Mg. 



•(i; 



In the examples of § 7, ^ never differed from I by as much as 
1 in 4000. 



Dr Searle, A bifilar method of measuring the rigidity of wires 63 



II 



§ 3. Ap'paratus. This is shown diagrammatically in Fig. 2. 
The wires are soldered into torsion heads S, T, which pass through 
a board XY held in a firm support. 

The lower ends of the wires are soldered into screws which pass 
through "clearing" holes in the bar EF, and are secured with nuts. 
The heads of the screws are made with "flats" to fit a spanner. 
Before the screws are secured to EF, the torsion heads are set to 
zero; the screws are then secured to EF so that, when the bar is 
only subject to the action of the wires and of 
gravity, the flats on both screws have the same 
directions as when the wires hung freely. 

The distance BD is, as near as may be, equal 
to AC. 

The load is carried by a knife-edge forming 
part of the hnk iV, Figs. 2, 3. The knife-edge rests 
in a V-groove in a plate, P, fixed to EF by screws 
passing through slots. By adjusting P, the tensions 
can be equalised; the notes emitted by the wires 
when plucked have the same pitch when the ten- 
sions are equal. 

A weight W (a few kilogrammes) is suspended by the rod Q 
from the link N. A slot in the lower cross-piece of N allows Q to 
be put into place; the nut drops into a recess. The weight should 
be so attached to Q that it cannot turn about a vertical axis 
relative to Q with any freedom; otherwise it will be difficult to 
reduce the system to rest. 

The bar may be fitted with two pointers K, L, and the readings 
of their tifs are taken on two horizontal 
scales. These scales are adjusted to be 
perpendicular to KL when the torsion 
heads read zero. If KqLq is the straight 
line through the zero positions of the tips 
and K, L are the tips when the bar has 
turned through d, Fig. 4 shows that 



Fig. 3. 



Lo 


— .^^^^^^^^"^ 


K 


L 


— ^;S===^ — 


H 


^ ^ 



Fig. 4. 



sm U = 



KH _ yi+j/a 



LK 



V 



•(2) 



where y-^ = KKq, y^ = LLq and p = LK, the whole length of the 
pointer system. 

The deflexion of the bar is best observed optically. A metal 
strip R is screwed to EF, packing pieces being interposed to allow 
the link N free movement, and a plane mirror is fixed to R. The 
deflexion can be observed by aid of a telescope and scale, or of a 
lamp and scale. It is, however, simpler to employ a goniometer 
such as those which have been in constant use at the Cavendish 
Laboratory for several years. A description of the instrument and 



64 Dr Searle, A bifilar method of measuring the rigidity of wires 

the method of using it for experiments of this type will be found in 
Proc. Catnb. Phil. Soc, xviii, p. 31, or in the author's Experimental 
Harmonic Motion, p. 35. The goniometer measures the tangents 

of angles. 

The motion of the suspended system, as so far described, being 
only slightly damped, it is consequently not easy to reduce the 
system to rest, and the vibrations of the building add to the diffi- 
culty. A simple damping device is therefore used. An annulus of 
thin sheet metal is carried by the bar GH, which is clamped to the 
rod Q. The annulus is immersed in motor lubricating oil or other 
highly viscous liquid contained in the annular trough U, which 
rests on the table. The rod Q passes through a hole in the table. 
By adjusting the height of GH, the annulus can be brought close 
to the bottom of the trough, and then the motion is so highly 
damped that tte system is practically immune to vibrations of 
the floor or the table. 

If the wires are overstrained by turning the heads through toO' 
large angles, the wires will no longer be vertical when the heads read 
zero, and it will be necessary to readjust the screws in the bar EF, 
To prevent overstrain, and at the same time to allow the heads to 
be turned through tt in either direction from their zeros, a movable 
safety device is used. A metal disk, about 1 cm. in diameter, 
can turn freely about its centre on a screw by which it is attached 
to the board XY (Fig. 2). A vertical pin is fixed excentrically in 
the disk, the greatest distance from the pin to the axis of the head 
being small enough to prevent the steel wire, which forms the 
index of the head, from passing the pin. The torsion head can then 
be turned only a little more than tt in either direction from zero. 

Care must be taken not to bend the wires near the soldered 
joints. A bend at 5 or Z) will alter the effective value of a^. If the 
wire AB is bent near A, the effect, when the torsion head is turned, 
will be the same as if the point A describes a small horizontal 
circle. This causes changes in a-^ as the head is turned, and, what 
is more serious, causes the bar EF to turn through angles which are 
by no means neghgible, in addition to the angles directly due to 
the torsion of the wires. For this reason, annealed wires are more 
suitable for the experiment than hard drawn wires, as they are 
more easily straightened. 

The torsion heads are read on circles divided at intervals of 45°, 
the dividing lines being scribed on the board XY. 

§ 4. Theory of the method. If each torsion head is turned from 
its zero through </> radians in either direction, the bar EF will turn in 
the same direction until the bifilar and torsional couples are equal. 
If EF turns through 6, the whole twist of each mre is (f> — 6. 

Let the radius and the length of each wire be r cm. and I cm.. 



Dr Searle, A bifilar method of measuring the rigidity of wires 65 

and the rigidity of the metal n dyne cm.~^. Since the wires are 
nearly vertical, the couple, due to torsion, exerted by the fair upon 
the bar is Trnr^ {(f) — 6) /I*, to a close approximation. 

The small couple due to the bending of the wires assists the 
bifilar couple; Kohlrauschf takes account of this small couple by 
writing in place of (1), 

G = ^^^^}^.Mg, (3) 

where V = I - r^ {iTrEjMgY, (4) 

and E is Young's modulus. 

Equating the torsional to the (corrected) bifilar couple, we have 

^md=C{4>-d), (5) 

^^^^^ ^ = ]^r ^^) 

Then n = ^^r.MC. (7) 

§ 5. Experimental details. The distances AC = 2^1, BD = 'la^ 
are measured. The diameters of the wires are taken at a number of 
points and the mean radius is found. 

The total mass, M grm., of the system carried by the wires is 
found. The masses of the screws are found before they are soldered 
to the wires. 

The torsion heads are first set to zero, and the scales on which 
the pointers K, L are read are adjusted to be perpendicular to KL. 
If a goniometer is used, it is set so that its arm is in the central 
position when the goniometer wire coincides with its own image. 

To eliminate errors due to slight bends in the wires, the readings 
must be taken over the range — tt to tt for ^ ; the theory assumes 
absence of hysteresis. But in experimental work in elasticity 
we must realise that hysteresis effects are unavoidable, when the 
strains are more than infinitesimal. To ensure that the effects of 
hysteresis shall be orderly and not irregular, the torsion heads are 
taken through a complete cycle from tt to — tt and back to it. To 
make the two readings for <^ = tt agree as closely as possible, a 
preliminary half cycle from — tt to tt is done. To make the conditions 
uniform throughout the cycle and a half, the readings for the pre- 
liminary settings are taken and recorded ; this will secure approxi- 
mately constant time intervals between successive readings. Thus 
the heads are set in succession at the following multiples of 7r/4 : 
-4,-3,-2,-1, 0, 1, 2, 3, 
4, 3, 2, 1, 0, - 1, - 2, - 3, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4. 

* G. F. C. Searle, Experimental Elasticity, § 39. 
•f- Kohlrausch, W jet?. ^rt,«., xvu, p. 737, 1882. 

VOL. XX. PART I. 5 



66 Dr Searle, A bifilar 7nethod of measuring the rigidity of wires 

The first 8 are the preliminary readings, and only the last 17 are 
used. 

If cf) goes through a complete cycle, and 6 is plotted against (j), 
a narrow hysteresis loop will be obtained. When readings are taken 
as above, there will be two values of 6 for each value of </> except 
(j) = — 7T. With careful work, the two values of d for <f) = tt will be 
exactly or very nearly identical; for the wires used in § 7, I have 
seldom found a difference between these two values as great as 
one minute. As a rough method of eliminating the effects of 
hysteresis, the mean of the two values of 6 for each value of (f> is 
taken as the value of 9 for that value of cf) . 

The effect of bends in the wires near their upper ends. A, C 
(Fig. 2), will be the same as if these points described small horizontal 
circles about the centres Aq, Cq, as in Fig. 5. Let </> be measured 




Fig. 5. 

from CqAqX, and let AAqX = (p + a, CCqX = cf) + y, while 
AqA = r, CqC = s, AqCq = 2ai. Then, if e is the small angle between 
AC and AqCq, 

r sin ((/) + a) — s sin (^ + y) 
2ai + r cos (^ + a) — s cos {<p + y)' 

When r and s are small compared with 2ai, tane may be replaced 
by 6 and the variable terms in the denominator may be neglected. 
Then, putting 

[r sin a — s sin 7)/2ai = P, {r cos a — s cos y)l'2ax = Q, 

we have 

6 = P cos(/> + ^ sin ^. (8) 

Here P and Q are the values of e when cf) = and ^ = Jtt. 

If the line BD makes an angle 9 with AqCq when the heads 
read </>, the angle between BD and ^C is ^ — 6. The wires will not 
be quite free from torsion when the heads read zero; let rj be the 
mean twist of the wires when </> = 0. We must thus write sin {9 — e) 
for sin 9 and cf) +rj — 9 for ^ — ^ in the equilibrium equation (5), 
which thus becomes 

sm{9-€) = C{cf)+r)-9). (9) 

To evade difficulties, 9 is kept small. Then, since e is also small, 
we may replace the sine by the angle in (9), and thus obtain 



I 



J 



Dr Searle, A bifilar method of measuring the rigidity of wires 67 

e = YTC i<f>+V)+^ = D{c/>+r))+e (10) 

If Oq, Cq correspond to ^ = 0, we have, since €q = P, 

e, = Drj+e,^Drj + P. (11) 

Thus 

6 - 6^ = Dcf>+e- P = D<f) + P cos (l> + Qsincf) - P (12) 

Since this equation is linear in 6, we may take 6^ as corresponding 
to any initial position of the bar which is yiear its ideal zero position. 
Thus, if /S is the angle at any time between the bar and some 
nearly ideal zero position, 

^ = Q-K (13) 

Since ^, though small — say less than 0*2 radian — is not infinitesimal, 
some correction should be made. An exact solution cannot be 
given, but accuracy is gained by writing sin ^ for ^, and then the 
final formula becomes 

sin^ = Z)(/. + P cos </. + Q?,\\\(j> - P (14) 

To eliminate P and Q, we combine the observations. Let /3^ 
correspond to (f> = imr/i. Then, putting cf) = tt and (f) = — tt, so that 
m = i and m = — 4, we have 

ttD = i (sin ^4 - sin ^_4). (15) 

A second value for ttD is found by giving m the values 3, — 3, 
1, - 1. Then 

7tD = sin ^3 — sin/3_3 — (sin^j — sin /3_i) (16) 

The two values of D are usually in good agreement, although, when 
P is plotted against <f), the curve differs considerably from a straight 
line. The mean value of ttD is used to find C. Thus 

ttD 

C' = -^. (17) 

Then n is found by (7). 

The actual values of P and Q are easily found. Thus 

P = - 1 (sin ^4 + sin ^_4), (18) 

Q = l (sin ^2 - sin ^.^ - ttD) (19) 

§ 6. Conversion table. A goniometer, such as those used at 
the Cavendish Laboratory, gives the tangent of the angle iJj through 
which the arm is turned from its zero. To find sin ip we subtract 
from tan j/» the small quantity s given in the table. 

5—2 



Q8 Dr Semie, A hifilar method of measuring the rigidity of wires 



tan\/^ 


s 


tan\//- 


s 


tan\//' 


s 


•01 


•00000 


•10 


■00050 


•19 


•00334 


•02 


•00000 


•11 


•00066 


•20 


•0C388 


•03 


•00001 


•12 


■00085 


•21 


•00448 


•04 


•00003 


•13 


■00108 


•22 


•00514 


•05 


•00006 


•14 


■00135 


•23 


•00585 


•06 


•00011 


•15 


•00166 


•24 


■00663 


•07 


•00017 


•16 


•00201 


•25 


■00746 


■08 


•00025 


■17 


•00240 






•09 


•00036 


•18 


•00285 







Simple interpolation, by "proportional parts," will give s with 
an error not exceeding unity in the fifth place of decimals. Thus, if 
tan xfs = •124, we find s = "00095, and then 

sin i/f = tan i// - 5 - •12305. 

§ 7. Practical example. The following results were obtained for 
a pair of soft brass wires. 

The distances AO, BD were each 6^00 cm. Hence aj = ag = 3 cm. 

Mean radius of wires = r = 0^0352 cm. 

Length of each wire = I = 47^30 cm. 

Mass of suspended system, excluding the weight W (Fig. 2) = 417^6 gm. 

The small correction for the buoyancy of the damper was neglected. 

The deflexions were observed by a goniometer. The distance from the 
centre of the pivot to the scale was 40^00 cm. The central, or zero, reading is 
10-00 cm. The following goniometer readings were obtained for the last 17 of 
the values of specified in § 5. 



<f> 


\ 

Reading Reading 


Mean 


X 


tan/3 


sin (^ 


sin /3 


radians 


cm. 


cm. 


reading 


cm. 


= tI40 


obsd. 


oalcd. 


IT 




14-10 


14-10 


14-100 


4-115 


-1029 


-1023 


-1033 


Itt 




12-96 1 13-04 


13-000 


3-015 


-0754 


-0752 


-0749 


i^ 


> 


, 11-85 , 11-98 


11-915 


1-930 


•0482 


•0482 


•0471 


iT 


10-80 


10-96 


10-880 


0-895 


-0224 


-0224 


•0220 





9-90 


10-07 


9-985 


0-000 


-0000 


•0000 


•0000 


-iTT . 


9-16 


t 9-33 


9-245 


-0-740 


--0185 


--0185 


- -0204 


— r/TT 


8-30 


8-43 


8-365 


-1-620 


- -0405 


- -0405 


--0415 


-|7r 


7-37 


1-U 


7-405 


- 2-580 


- -0645 


- -0644 


- -0653 


— TT 


6-33 


6-330 


-3-655 


--0914 


- -0910 


- -0921 



The value of x was found by subtracting from the mean reading, as given 
in column 4, the mean zero reading 9-985 cm. corresponding to (^ = 0. The 
differences between the readings in columns 2 and 3 are due to hysteresis. The 
seventh column shows that sin i3 is not proportional to <p. 



Dr Searle, A bifilar method of measuring the rigidity of wires 69 



m 



By (15), T^D = l (-1023 + -0910) = -09665, 

and by (16), nD - -0752 + -0644 - (-0224 + -0185) = -09870. 
Mean value of irD = 0-0977. 

Then, by (17) C= ""^ ^ = 0-03210. 

By (18), P= - i (-1023 - -0910) = - 0-0028, 

and by (19). Q=l (-0482 + -0405 - -0977) = - 0-0045. 

In the table, the column "sin/3 calcd." gives sin ^3 as calculated by (14), 
using the values of ttD, P and Q just found; there is fan agreement between the 
calculated and observed values of sin ,3. 

The total load M was 417-6 + 4999 = 5416-6 grm. 

Taking E = 10^^ dyne cm.--, we have r- {2TrE/Mg)^ = 1-35 cm., and hence, 
by (4), I' = 47-30 - 1-35 = 45-95 cm. 

Then, by (7), 

n = ?^!.MC = - ^'^J^~^l^, X 5416-6 x 0-03210 
TvrH' IT X 0-0352* x 45-95 



= 3-277 X 10" dyne cm 



-2 



A similar set of observations, m which ilf was 3417-lgrm., gave the following 

values of sin /3: 

•1510, -1123, -0743, -0361, -0000, - -0345, - -0735, - -1122, - -1540. 

Mean value of ivD = 0-1532. Hence C = 0-05127. 

Also I' = 47-30 - 1-70 = 45-60 cm. 

Then 

981 X 32 X 47-30 



TT X 003.52* X 45-60 



X 3417-1 X 005127 = 3-326 x 10" dyne cm." 



An independent determination of n was made by attaching a bar, of moment 
of inertia K = 4-766 x 10* grm. cm.^, to each of the two wires in turn; the mean 
periodic time of the torsional vibrations was T = 10-55 sec. Hence 

87r X 4-766 x 10* X 47-30 „ „. _ ,_., , .3 

10-55^ X 0-0352* = ^'^^^ ^ ^^ ^^^^ ^"^- " 



70 Mr Bennett, The Rotation of the Non-Spinning Gyrostat 



The Rotation of the Non-Sf inning Gyrostat. By G. T. Bennett, 
M.A., F.K.S., Emmanuel College, Cambridge. 

[Read 8 March 1920.] 

§ 1. The following extract is taken from an old examination 
paper* : 

"A symmetrical wheel free to rotate about its axle is 
moved from rest in any position by means of the axle and is 
finally restored to a position in which the axle again points 
in the same direction as formerly. Shew that the wheel, again 
at rest, will have rotated through a plane angle equal to the 
solid angle of the cone described by the varying directions of 
the axle." 

The proof of this result may be put briefly in a geometrical form. 
Translational and rotational movements being independent, the 
centroid of the wheel may be treated as stationary. As the gyrostat 
has no component rotation about its axis, the axis of rotation is at 
any moment some diameter of the wheel. This line has the central 
plane of the wheel as locus for the body-axode, and has a closed 
cone of arbitrary form as locus for the space-axode. The angular 
movement is therefore representable by the rolling of the plane 
on the cone. The angle of ultimate rotation of the wheel is thus 
(for cones of ordinary type) the excess of the four right angles of 
the plane surface above the total surface-angle of the cone. This 
difference is equal to the solid angle of the reciprocal cone described 
by the axis of the wheel. And hence follows the result quoted; 
namely, that the solid angle described by the axis of the wheel is 
equal to the circular measure of the plane angle of the resultant 
displacement of the wheel about its axis. Further, the sense of the 
displacement accords with the sense of circulation associated with 
the solid angle. 

§ 2. The result may be extended to the case in which the initial 
and final directions of the axis are different, say a and 6. For the 
axis may be restored to its original direction a by a subsequent 
movement in the plane ha; and this latter movement, which is a 
rotation about the normal to a and 6, leaves unaltered the angle 
that any diameter of the wheel makes with the plane ah. Hence 
the original movement, shifting the axis of the wheel from a to 6 

* Emmanuel and other Colleges, Second Year Problems, Wed. Jmie 8, 1898 
Question 11. 



Mr Bennett, The Rotation of the Non-Spinning Gyrostat 71 

by any conical movement, alters the angle between the plane ah 
and any diameter of the wheel by an angle equal to the solid angle 
enclosed by the cone formed by the conical surface ab together 
with the plane ha. 

§ 3. A geometrical integration of Euler's equation leads to the 
same result as § 1. The axis, with its direction given by spherical 
polar coordinates d and (/> (radial and azimuthal), generates a solid 
angle 

(7 = J(1 -COS^).f/(/.. (1) 

The equation of motion, being 

(^ COS ^ + i^ = 0, (2) 

with zero initial values for ^ and ^, has as its integral 

<f> + i/j = or. (3) 

If the axis of reference ^ = is supposed (conveniently) external 
to the cone then </> is zero finally as well as initially, and i/r is the 
angle of resultant displacement of the wheel and is equal to the 
solid angle a. 

If, more generally, the gyrostat has a constant spin Q. about its 
axis, the Euler equation becomes 

cf) cos 9 + i[i = Q (4) 

with 4> + ip = a + Qt (5) 

as its integral. And the final rotation of the gyrostat is then given 
by the solid angle of the cone described by the axis plus the time- 
integral of the spin. It may be noticed that the angle (f) + i/j, with 
a value independent of the choice of coordinates, gives in itself a 
natural measure of the total rotation of the wheel, as followed and 
estimated by projection on the plane 6 = tt/'I. For on that plane 
the circular disc shows as an ellipse, with ^ as the azimuth of the 
direction of the minor axis, and ^ as the eccentric angle, measured 
from the minor axis, of the projection of the revolving diameter of 
the wheel. A distant observer on the axis ^ = 0, able to distinguish 
the two faces of the wheel, would in this way precisely reckon the 
amount of rotation, whole turns and fractional. He does not give 
merely the ultimate position, by naming a plane angle to a modulus 
of four right angles, but assigns the multiple of the modulus neces- 
sary for a correct account of the movement intervening between 
the initial and final positions. 

A kinematic representation of the angle </> + «/' may be obtained 
by supposing the circular rim of the disc to have rolling contact 
with the rim of another equal disc whose plane keeps parallel to 
the plane 6 = 7r/2. The angle of rotation of this latter disc about 
its axis (which keeps the invariable direction ^ = 0) is then </> + ip. 



72 Mr Bennett, The Rotation of the Non-Spinning Gyrostat 

§ 4. For the special case in which d is constant, so that the 
axis of the gyrostat describes a circular cone, the rotation is stated 
by Sir George Greenhill* to be 27r — (conical angle described by 
the axle), as against the solid angle itself found above. The differ- 
ence of sign of the latter can be accounted for by a reverse sign- 
convention: but the term 27t is unnecessary if 27r is implied as a 
modulus, and it appears to be wrong if the precise angle of turning 
is intended. If, specially, the axis of the gyrostat described only 
a small cone, then the angle of consequent rotation is certainly a 
small angle, and not an angle nearly equal to four right angles. 

He adds the remark that the movement "can be shown ex- 
perimentally with a penholder held between the fingers and moved 
round in a cone by the tip of a finger appHed at the end." But the 
illustration is inapt; for the creep of the penholder occurs in the 
sense opposite to that of the conical movement. The body-axode 
is a circular cone and not a plane, and it rolls inside a shghtly 
larger circular cone as space-axode; and hence the reverse move- 
ment. 

§ 5. The movement of the non-spinning gyroscope here con- 
sidered is not yet among those that are familiarly recognised, though 
it has important practical applications and deserves to rank as a 
dynamical commonplace. Bodies suspended from a point on an 
axis of symmetry behave in the same way and for the same reason. 
No matter how the point of suspension may be moved about, and 
no matter what complicated conical movement is consequently 
executed by the axis, the applied forces have no moment about the 
axis, and the spin remains zero if originally zero. The resultant 
rotation is then given, as above, by the solid angle of the cone 
described by the axis. 

Aeroplane compasses, in particular, are found to keep their 
cards practically parallel to the floor, under the combined action 
of gravity and lateral acceleration, during a banked turn of the 
aeroplane. Hence, from inertia alone, and apart from all other 
sources of control or disturbance, the compass-card would be 
rotated, as a consequence of the turn, through an angle equal to 
the solid angle described by the normal to the card. For an angle 
of banking a and a change of course ^ the solid angle is not much 
less than (1 — cos a)^ if the banking is taken and left quickly; and 
for very steep banking this angle is nearly equal to the change of 
course itself, and the card would almost appear to "stick." As 
compared with considerations of magnetic disturbance due to the 
vertical component of the earth's field, and of mechanical disturb- 
ance due to rotation of the bowl and liquid, the pure inertia efiect 

* Advisory Committee for Aeronautics. Reports and Memoranda, No. 146, 
Report on Gyroscopic Theory, p. 13, § 14. 



Mr Benneit, The Rotation of the Non-Spinning Gyrostat 73 

of the conical movement seems to need more emphasis than it has 
hitherto been awarded. It is here expKcitly isolated. 

The gyroscopic compass, Uke the magnetic compass, may at 
times suffer disturbance from this same source, if the compass- 
position in the ship and the run of the sea are such as to produce 
a circular or elliptical movement of the binnacle. 

§ 6. It would be hard to trace to its primitive source the know- 
ledge of the small piece of mechanics here discussed. It is really 
implicit in all treatises on Rigid Dynamics, but fails to emerge 
clearly amid the pressure of more important movements. Among 
empiricists it must be well-nigh prehistoric. The sailor in coihng a 
rope makes a winding motion of the feeding hand to remove the 
kinks from the overtwist of the piece which is to form the next 
turn of the coil. The circus clown, with the vertex of his conical 
cap resting on his finger-tip, or the end of a stick, easily makes it 
turn round and round; and the postman collecting his mail knows 
how to twist up the neck of his bag with a circular movement of 
the hand he holds it by. Later among empiricists are those who, 
accustomed to handle magnetic compasses, are very familiar with 
the rotation of the card produced so readily by giving the bowl 
a horizontal circular translational movement (without rotation). 
More lately still Mr S. G. Brown has noticed the conical motion 
and its effect. In the abstract of his lecture to the British Associa- 
tion* it is described as a "new phenomenon" and is stated as being 
"explainable mathematically." More fully in his lecture to the 
Royal Institutionf he states that in virtue of the "wobbling" 
{videlicet conical) motion, "the needles and. card would then have 
a force applied trying to carry the moving system round in the 
direction of the wobble." This mode of expression is of course 
entirely illegitimate. The rotational movement observed needs no 
"force" to explain it; the very essence of the inertia effect is that 
it occurs with no spin about the axis of rotation and no couple 
about that line either. Mr Brown announces also (but without 
demonstration) that if his compass-disc "is carried round in a 
horizontal circular path without any wobble the plate still goes 
round or tries to go round with the circular movement" and that 
this "should be of interest to mathematicians." It seems likely that 
the sheer paradox in angular momentum thus propounded will 
readily dissolve when all the relevant physical data are revealed: 
and meanwhile the interest is but that of a heresy resting on 
hearsay. 

* British Association, Bournemouth, 1919. Evening Lecture, Fr. Sept. 12, 
"The GjToscopic Compass." Abstract, 11. 9-14. 
t Nature, March 11, 1920, p. 45, col. 2. 



74 Mr Kienast, Proof of the equivalence of 



Proof of the equivalence of different mean values. By Alfred 
Kienast. (Communicated by Professor G. H. Hardy.) 

{Received 12 April: read 8 May 1920.] 

If Oi, ao, ... an, ••• denote the terms of a sequence of complex 
numbers, and 

8^^ = a, + ... + an, 



then lim 8^"^ i [ ) is called Cesaro's /cth mean* of the se- 



:.(0) 



quence 8\ 

Putting h^^^= ai + . . . + a^, 



A<:'=ii/,«;'+...+An, 



then lim A^^' is called Holder's /cth meanf of the sequence hj. 

In a paper "Extensions of Abel's Theorem and its converses |" 
I found it convenient to introduce the expressions 

5f=ai-F...+a,^ (?i=l, 2, ...), 

^'^-l[sT^-+sf_,] (.. = 2,3,...), 



and proved various theorems concerning the limits lim s^ . 

Several writers have proved 

Theorem 1. Whenever Cesaro's (Holders) Kth mean exists 
and is finite, then Holders (Cesaro's) /cth mean eooists too, and both 
have the same value. 

* Bromwich, Infinite series, p. 310. t Ibid., p. 313. 

t Proc. Cambridge Phil. Soc. , vol. xix, 1918, pp. 129-147. 



different mean values 75 

I propose to complete the researches of my above quoted paper 
by proving the theorem: 

Theorem 2. Whenever Holder s {and therefore Gesaro's) Kth 
mean exists and is finite, then lim s]f exists too, and both have the 

same value, ayid vice versa. 

The demonstration of both theorems is based upon relations 
between the mean values which it is possible to calculate com- 
pletely, as I have found, in a most simple manner. 

In §§ I to VI I determine the expression of s^^"^ by 

hf {X = l,% ...{n-K)), 

in § VII the expression of h\l^ by 

4'^ (X = l, 2, ...7i), 

in § VIII the expression of S^^^ by 

h^;:^ (\ = i, 2, ...7i), 

and finally in § IX I consider two more general mean values. 
I. From the definitions follow 

" A=i "■A=] ri 

,f= 1 S' ^J A<'',= i W hf+ 1 (2Af ' - ;,f >) + . . . 
^\=2 X ^^ w |_2 ^ 3 ^ ^ 

+»^j(„_2)e,-(„-3)esi 

Adding a term which is zero, 

(2)_ (n-l)(n-2) (2) _ 1 (1 , (2) , 1 . (2) J i-'^ j^2) \ 

etc. Now I suppose that, by continuing in this manner, I have 
arrived at the formula 

1 n—K , . 

''' A = l 

where c is a function of the indices n and k, and the coefiicients 



n, K 



<^a,k(X = 1, 2, ...) are, for each k, definite numbers which are the 
values of a function of X for \ = 1, 2, — 



76 Mr Kienast, Proof of the equivalence of 

Proceeding to build up the expression for s[l'^^\ applying the 
same transformations as above, we find 

"kAC^ - ") K'-"- (X - « - 1) Att'-M 



n-1 



„(«+!)_ 1 'V 



o(«+l)_. ^-'^-lz.('c+l) I'^v'^ 7,(«4-l) 



from which we conclude 



K't'l^-^ ^ d.,.+A''-'' (2); 



n K i- /ON 

(^n, K4-1 ^= Cn, K V"/) 

and a series of relations involving the numbers c?a,k and cIk^k+i- 

Equation (2) is of the same formation as (1), and therefore (1) 
gives the required expression of s^^''^ by the numbers h'-j^K 

II. Since Cn i = -^^ , (3) leads to 

n 

Cn, K = n~'' (n — K)(n-K+1) ... (n — 1), 

from which follows, for k = 2, Cn^= ~ ~ , which is in 

accordance with the expression for s^^^ above. 

III. c?A.,K may be determined in the following way. Putting I 

ai = a2= ...=aA-i = 0, a^ = l, aA+i=... = (4), 

we find 

s(0)= =,.(0) _A (0)_-, (0) _-, 

^(1>- - o(l)_A Jl) _ 1 „(1) _ 



^K ^' 6a4-1 — > , T> *A-4-S! 



+i~\+l' ^+2 X + 2' 



o(2). 






^+1 ' ^+2 (\ + l)(X + 2)' 






^+'' (\ + l)(X + 2)...(\ + /c)' 



different mean values 77 

and /,;*»=... =/t(VO, ^^=1, h['i=l, ..., 

n^ ... ^_i ^, /i^ ^, z^+i ^^-^, ..., 

Writing equation (1) for n = X + K, and substituting these special 
values, we obtain 

f \(\ + l)...(X + «-l) V^ ] 

rfx,.-(^ + «)| (\ + Kr (\+1){X+2)...(X + k)\' 

which is, for /c = 2, in accordance with the above expression for s^^\ 

IV. Lemma a. The coefficient c^a,k is a positive number for 
X = l, 2, ...,/c = 2, 3, .... 

It is easy to verify the inequalities 

^4^^—^'— (y^ = 0,l,2,...,.-l), 

X+ K X + K — fl 

from which results, by multiplication of all the left-hand and all 
the right-hand sides, 

X{X + 1)^. .(x± «-l) > '>^ 

(X + kY ' -{X + 1){X + 2)...(X + k)' 

which demonstrates the assertion. 

Lemma ^. The coefficient c?a,k satisfies the equation 

lini|<.-^(«-l)/c(« + l)i| = (5). 

To show this we expand c?a,k in the form 

and the proposition follows. 
A consequence of (5) is 

v. 

lim ]S f^A, K = 00 , 

)l-*-x A = 1 



78 Mr Kienast, Proof of the equivalence of 

and therefore the conditions of Stolz's theoreDi are satisfied. Thus 
we can state: 

Lemma 7. //' lim h^"^ = H exists and is finite, then 

lim 2 d,^X'^/ I d^,. = H. 

V. Let ai = l, ao = a3=---=0 (6); 

then , sl'^=l (n = l, 2, ...), 

^^=^^^ (n=2,3, ...); 

therefore lim s^^^^ = 1, and consequently 

lims;f=lims^^'>=... = l. 

Furthermore h^^^ = h^^^ = h'-^^ ^ . . . = 1 (n = l, 2, ...)• 

From (1) and (6) we obtain 

C ) \n-K 

Imi s\'^'= 1 = hm Cn,K - hm - S dx.K, 

n-^cc n-*-oo 7j->-Qo n 1 

1 n—K 

or lim - 1 dK,K = 0. 

VI. Now passing in equation (1) to the limit ?i-^qo , we find 

, , , , /I n-K \ (n-K , . ,n-K ) 

limsf = lim/i2«-lim (- S < J lim 2 d.,,/ii''7 S d,,. \, 

and this equation leads to the theorems: 

Theorem 3. Whenever lim h^^\^ exists and is finite, then 

lim s^"' exists too and has the same value. 

n 

More generally 

Theorem 4. When the function lf^^_^ oscillates between finite 
limits, then s^^ oscillates between the same limits. 

VII. The reverse propositions can be established in the same 
way. From the definitions follow 



n 



h^''=- 



J^^{(XH-2)4!>,-(X + l).f>J 



+ ^{(.+ 2).f>,-(. + 2)0} 



(n+^y J2) 1 I ^ + 2 (2) 
n(n + l) «+2 ^nKZiXiX + 1) ^+2" 



dijferent mean values 79 

Continuing this process we find 

A;:^=^n,«^-;i+' ^aai in 

In the same way as we determined Ch,k by (2) and (3), we 
obtain here 

n + K-\-l 



Thus, since 



6)1 K+] — 6j; 

n + 1 



_ (n + kY 

^"' " " n{n + l)...{n + K-l) ' 



Taking the values (4) for the numbers a^, we find from (7) 

f = X f (^ + l)(^+2)...(X + /c) { X+fcY ] 

■■''~ \ V X(\+l)...(\ + «-l)J ■ 

The considerations in § IV show that 

Expansion of /a_x in descending powers of X gives 
A. = g(«-1)«(« + 1)- + ^, + ...; 

n 

thus lim S /a, K = 00 . 

W-*.cc \ = 1 

Introducing in equation (7) the vahies (6) for the numbers a^, we 
find 

1 = lim e„^« + lim - 2 /a,« lim 2/a,«sJ;''L/ S/a^^ ; 

which gives, on account of Stolz's theorem, 

lim- i/A,. = 0. 

Thus equation (7) is completely determined and leads to 

Theorem 5. Whenever lim s^^'L exists and is finite, then 
lim h^"^ exists too and has the same value. 

n 

Theorem 6. Whenever the function s^^^^ oscillates finitely, 
then h^n oscillates between the same limits. 
Theorems 3 and 5 together constitute theorem 2. 



80 Mr Kienast, Proof of the equivalence of 

VIII. The relation connecting Cesaro's and Holder's means 
can be deduced in the same way. We have 






L\ = i 



n(n + l) n{n+l) 

= 2hf — -1—lxhf. 

« w(w + 1)a=i " 
Assuming therefore 

o(«) I n 
^ =C„ h'^:^ ^ r^ td, h['^ (8), 

we fand \ "[ K J ri+i,^K ~^^j^^^.«+i^ • 

Hence c„, „+i =(« + !) Cn+i, « , 

or Cn,K= k\. 

Starting from the numbers (4), we have 



sr'=... = sr_\=o. «r=i. -. *1"'=(""^+''). ■••; 



?"=... = si 

as is easily verified by the formula 

.ti ^ ™,roV « / V a: + i ;■ 

Writing (8) for 7r = X, we find 

d),^,= \(\+l) ... (X + «-l)-X«^0. 

Starting with the numbers (6), that is with the numbers (4) for 
X,= 1, we have 

and from formula (8) follows 

A = l V K 



different mean values 81 

so that finally 

^n + K-l\ " ^ -^7 

K J A = l 

Analogous considerations lead to 

n 
A<'')=1 ^!L_^ + fl_ M ^Zl ^ ...(10), 

\ K ) ,V'''\ K 

Formulae (9) and (10) prove theorem 1. 

IX. By similar considerations it is possible to arrive at a state- 
ment about the equivalence of two means of the kind examined 
in Part II of my above quoted paper. 

Let 6k , Ck denote the terms of two infinite sequences of positive 
real numbers, which have, when we write 

n n 

1 1 

the properties (i) lim Bn= 'X^ , lim Cn = go , 

.... 1 ^ /c6, 1 « KC^ 

(n) - 2 ^- , - 2 rT 

tend to limits or oscillate between finite limits. Then putting as 
before (o) ^ 

the means ^1'^^= w^K^T-i C^^)' 

t?=livt. (12), 

^n 2 

are connected by two analogous relations. From (11) and (12) follow 

b s^'\ = B s^'^-B J'\ (13), 

c s^'\=G f^-C J'\ (14). 

n n-l n n n-l n-l ^ ' 

VOL. XX. PART I. 6 



82 Mr Kienast, Proof of equivalence of different mean values 



Substituting (14) in (11) we find, on adding a term which is zero, 

„(!)_ 1 r^2^ ,(1) , ^sf^ .(1) p Ai)x , 



V ih_h±i] c t^^^ 



Since 



1 VCa Ca+1 



+ 



^n+1 ^n ,(1) 



-6. 



^n+i 



G,= i^~\C\-C\_,) + 



2 Ca 



= 5. 



we can write 



1 _ '"'+1 zl 









a, 






- i "+ 



Ca Ca+i/ I 



2 ( ^ — ^^^ 
,Ca Ca+1 



C, 



.(15). 



Now there may be distinguished two possibiHties : 
Theorem 7. If 



VCa Ca+t/ ^ ^ 



w /A 



Ca fA+] 



C. 



< K (fixed), 



0„ 6,1-1-1 



lim — = 1 

?i-*.Qo '^n+1 -'-'71 

and if ^JJ' a'pproaches a finite limit {or oscillates between finite 
limits), then s^^^ approaches the same limit (or oscillates between the 
savne limits). 

Theorem S. If 

" /&A &A+A p (1) 
■ i "-'a ''A 



Hm 



V /"A 

3 VCa 



= lim i^ 



^()a_6a+i\ ^ 

2 VCa Ca-|-i/ 
b G 
-~~^<K (fixed), 

and ift^^^^ approaches a limit, then s^^ approaches the same limit 

This is a known theorem *. 

The second relation results by substituting from (13) in (12) 
and proceeding in the same way.' The same formula is arrived at 
by interchauging in (15) b^ and c^, B^ and (7,^, ^'^ and s^^^ . From 
it we infer two theorems analogous to (7) and (8).' 

* Bromwich, Infinite Series, p. 386, Theorem V. 



Mr Harrison, Notes on the Theory of Vibrations 83 



Notes on the Theory of Vibrations. (1) Vibrations of Finite 
Amplitude. (2) A Theorem due to Routh. By W. J. Harrison, 
M.A., Fellow of Clare College. 

[Read 3 May 1920.] 

p^ (I) Lord Rayleigh in his Theory of Sound, Vol. i, has considered 

the effect of introducing terms depending on x^ and x^ into the 

d'^x 
simple equation of vibratory motion -j-^ + n^x = 0. He treats the 

added terms as small and employs the method of successive ap- 
proximatioD. The object of this note is to point out that exact 
integrals can be obtained in the form of the series of which 
Kayleigh determined the first two or three terms. The solutions 
now obtained are valid for any relative magnitude of the added 
terms subject to the motion remaining vibratory. 

(a) The Symmetric System. 

The equation of motion is 

d 00 

-^-^ + n^x T '2Bx^ = 0, 

dt'^ 

where /S is positive, and the upper sign is taken in the first instance. 
A first integral is 

where a is the amplitude of the vibration. 
We have 

<dx\^ (^2 _. ^2) („2 _ ^^,2 _ ^^2)^ 



fdxV 
\dt) 



or 



Jt 



where ax^^ =- x, k^ ^ ^a^Kn^ — ^a^). 

Hence* 

X = axi 

= a sn {{n^ - Ba^ft, h], [x =-0 when t = 0) 

_ 277a ^ g^+^ . (2m + I) tt (n^ - ^a^ft 

^ Kko 1 - q'""^' "''' 2E: 

* For the expansions of elliptic functions quoted in this paper see Whittaker 
and Watson, Modern Analysis, 1915, p. 504, and Example (5), p. 513; or Hancock, 
Theory of EUiptic Functions, Vol. i, 1910, pp. 486, 494, 495. 

6—2 



84 



Mr Harrison, Notes on the Theory of Vibrations 



Let the units of length and time be chosen so that a= l,n= 1. 

doc 

It is necessary that ^ < |, otherwise -^ vanishes first f or < x < 1. 

The effect of the term 2^x^ on the vibrations can be exhibited 
by the results of numerical calculation given in the following table: 



^ 


X 


V 


•01 


1-0006 sin />( + '0006 sin Hpt 


9924 


•1 


1-0074 sin pt + -0074 sin Spt 


•9214 


•3 


1-0335 sin pi -4- -0348 sin Spf + -0012 sin 5jjt 


-7309 


•4 


1-0632 sin pi + -0676 sin 3pi + -0046 sin 5pt 
+ -0003 sin Ipt 


-5997 


•45 


1-0928 sin pi + -1028 sin 3pt + -0108 sin 5pi 
+ -0011 sin 7 pi + -0001 sin 9pt 


•5063 


^ 


tanh{tl^^2) 





We proceed to consider the equation 
d?x 



dt^ 



+ n^x + 2^x^ = 0. 



If a is the amplitude of the motion as before, we have 

f^f = (a2 _ ^2) (^2 + ^^2 + ^^2)^ 

or r^j = {n^ + ^a^) (1 - V) (1 + l^V), 

where ax^^ = x, and [x^ = ^a^/{n'^ + ^a^). 

Write 1 — a;j2 = z^, so that 

^^y = (n2 + 2^a2) (1 _ ^2) (1 _ ^^2)^ 

where k^ = ij?/{l + jjl^) = ^a^j{n^ + 2^a^). 

Thus 2 = sn {{n^ + 2^a2)^^, A'}, (^ = when t = 0). 
Therefore 



dU CttJO-l 

= acn {{n^ + 2pa^)h, Jc} 

,m+i 12m + 1) 77 (%2 + 2i8a2)^^ 

cos r I 



^■na ~ q" 



v2m+l 



2Z 



X^ 1 + ?' 

In this case there is no limit to the value of ^, the motion 
remains vibratory, but the period of the gravest mode decreases 



Mr Harrison, Notes on the Theory of Vibrations 85 



as /3 increases. The results of calculation, with n -= 1, a = 1, are 
as follows: 



^ 


X 


P 


•1 


•995 cos 2)t + -005 cos 3pf 


1072 


•5 


•9818 cos pt + -0179 cos 3pt + -0003 cos 5pt 


1^318 


10 


•9742 cos pt + -0253 cos 3pt + •0006 cos 5pt 


1-569 


100 


•9582 cos 2it + -0402 cos 3pt + •OOIG cos 5pt 
+ -0001 COS 7pt 


3-975 


1000 


•9555 COS pt + -0427 cos 3pt + •OOIS cos 5pt 
+ •OOOl cos 7pt 


12-03 



n'^x + ^ax^ = 0, 



(b) The Asymmetric System. 
The equation of motion is 
d^x 

where a may be assumed to be positive, as changing the sign of a 
is equivalent to reversing the direction of the axis of x. 

Let the scale of time be such that n = 1, and the scale of 
length chosen so that the amplitude of the motion measured from 
aj ^ in the direction of x positive is unity. Then 

f , j =-■ {I — x) {I + a -\- x + ax -\- ax^) 
^{l-x){b + x) (c + x), 
where 6 = i {1 + « - (1 - 2a - ^a^f}la, 

c=\{l + a+{l--2a- 3a2)^}/a. 

The limits of the vibration are x = \ and x = — b. It is 
necessary that a should be less than \, so that the greatest value 
of b is 2. 

Writing I — x = {b + I) y^, we, have 

|)'=i«(c+l)(l-?/2)(l 

where F= (6+ l)/(c + 1). 
Hence 

y= sn {|a^(c+ lft,l}, 

and x^l-{b-^l) sn^ {^a^ (c + l)^i, k} 

b+ I {, E 27r2 * mf " 



Fy^),. 



= 1 - 



/C2 



1 ^2772* 



2Z 



The results of calculation are as follows : 



86 



Mr Harrison, Notes on the Theory of Vibrations 



a 


b 


X 


P 


•1 
•2 
•3 


1-1125 
1-2680 
1-5657 


- -0838 + 1-0557 cos pt + 0284 cos 2pt 
+ -0006 cos 3pt 

- -2059 + 1-1306 cos pt + -0712 cos 2pt 
+ -0033 cos 3pt 

- -4634 + 1-2634 cos pt + -1783 cos 2pt 

+ -0190 cos Spt + -0018 cos 4p< + -0002 cos 5pt 


•9855 
-9477 
•8152 



The calculations have been performed for illustrative purposes 
only, and no special care has been taken to ensure the accuracy 
of the digits in the final decimal places. 

(c) The solution of the equation 



dt^ 



+ n^x + f 



ax^ 



in the form of a Fourier Series requires rather more elaboration 
of the algebra. 

The motion presents one novel feature which does not appear 
in the previous solutions. If ^ be positive, however small, the 
motion remains vibratory for any finite value of a, and if a and a/^ 
be great, the amplitude of the motion on one side is approximately 
a/jS times its amplitude on the other. 

(II) Kouth has shown (vide Advanced Rigid Dynamics, 1905, 
p. 56) that an increase in the inertia of any part of a vibrating 
system will increase all the periods in such a way that the modified 
periods are separated by the periods of the original system. This 
is true in general if the inertia of only one part of the system be 
increased, the definition of a single part being that the effect of 
increasing its inertia can be represented by a single term 

i (/^i^i +/2?2 + ■■■f 
in the expression for the kinetic energy, where g'j, q2, ... are the 
normal coordinates of the original system. For example, the 
theorem is applicable to the case of an additional mass attached 
at a single point of a stretched string, but not to the case of an 
increase of mass spread over a portion of the string, or to the case 
of two or more masses attached at different points. 

The theorem may be simply proved as follows. Let the modified 
kinetic energy be 

i iii + ?2^ +...) + ! (/ii?i + Mi + •••)^ 
and the potential energy be 

i(AiV + A2V +•••)• 
The equations of motion are typified by 

'ir + W + Mr (M*i + Mi +...) = 0. 



Mr Harrison, Notes on the Theory of Vibrations 87 



The determinantal equation for the periods is 
A2 (1 + ^j2) _ ;^^2^ Aii/^A2, /^i/^A2, .. 

flltl2^^, A2 (1 + IL^^) - Ag^, iUg.UgA^, ., 



= 0. 



;i) 



Let Aj2, Ag^, ... be arranged in ascending order of magnitude. 
If A^ = 0, the left-hand side of (1) is (— 1)" as regards sign. If 
A^ = Aj^, the left-hand side of (1) is equal to 



fh' 



V 














and this is (-- 1)"-^ as regards sign. 

Hence all the roots in A^ are decreased and they are separated 

The validity of this proof depends on (1) the non-equahty of 
any of the values of A^^, Xo^, ..., (2) the non-evanescence of any of 
the constants /Aj, /Xg, .... In case of (J) one period at least of the 
modified system is equal to a period of the original, but the theorem 
may be held to cover this case. 

In case of (2) the theorem does not remain true. Suppose 
the ju.'s are all zero except /x^, /Xg, f^t^ •••• Then only the periods 
corresponding to q,., qg, q^, ... are changed. The periods belonging 
to these coordinates will be increased and their new values will be 
separated by their old values. But these new periods bear no 
relation to the periods belonging to the remaining coordinates and 
can occupy any position in regard to them except as specified 
above. Hence the theorem does not seem to indicate where the 
modified periods must lie in regard to the complete system of 
periods of the undisturbed system. 

An example is afforded by the modification introduced into 
the periods of a stretched string by a load attached at a point 
dividing the string into two lengths which are commensurable. 
Rayleigh's argument (vide Theory of Sound, Vol. i, p. 122), which 
serves to maintain the validity of the theorem in this case, is 
acceptable owing to the strictly defined relations which exist be- 
tween the periods in both states. But in an ordinary dynamical 
problem the theorem must be held to break down in the excep- 
tional cases under consideration since it fails completely to indicate 
the position of the modified periods in relation to the original 
periods. 



88 Dr Searle, Experiments with a plane diffraction grating 



Experiments with a plane diffraction grating. By G. F. C. Searle, 
Sc.D., F.R.S., University Lecturer in Experimental Physics. 

[Read 3 May 1920.] 

Part I. Parallel Light. 

§ L Introduction^ . When a plane grating is employed in 
accurate measurements of wave length, the ruhngs are set per- 
pendicular to the direction of the incident beam of parallel hght. 
When these two directions are not at right angles, the diffracted 
beam is no longer parallel to a plane containing the directions of 
{a) the incident beam and (6) a hne intersecting the ruhngs at 
right angles. The formulae apphcable to this general case are 
obtained in §§ 4, 5, 7 ; they are tested by the experiment of §§ 8, 9 
for the restricted case in which the directions {a) and (6) are at 
right angles. 

§ 2. The grating axes. It is necessary to specify the three axes 
of a plane grating and the origin from which they start. 

For a transmission grating, the origin is a point on the centre 
hne of one of the openings. In a reflecting grating, would he 
on the centre line of one of the reflecting portions. 

The axes are 

(1) The normal ON to the plane of the grating. 

(2) The transverse axis OT, a line through cutting the 
ruhngs at right angles. 

(3) The longitudinal axis OL, a line parallel to the ruhngs. 
The grating interval, i.e. the common interval measured along 

OT from centre to centre of the openings, will be denoted by d. 

§ 3. Diffracted wave front and ray. At a distance of thousands 
of wave lengths from the grating, the wavelets due to the separate 
openings will merge into practically a single wave. For the mathe- 
matical purposes of this paper we shall speak of this wave as the 
diffracted wave front and of a normal to it as the diffracted ray. 
We may speak of the diffracted wave front passing through the 
origin 0, if we understand it to be a surface through cutting at 
right angles the normals to the distant wave fronts. The normal 
through may be called the diffracted ray through 0. 

In the case of reflexion or refraction at a pohshed surface, the 
time of passage from an incident wave front to a reflected or 

' * 1 have to thank Dr J. A. Wilcken of Christ's College, and Mr C. L. Wiseman, 
M.A. of Peterhouse. Dr Wilcken took the observations of § 12, Part T, and assisted 
in other ways. Mr Wiseman gave valuable help and criticism in the mathematicaV 
parts of the paper. 




Dr Searle, Experiments with a plane diffraction grating 89 

refracted front is independent of the particular ray. But, in the 
case of a grating, the time of passage from an incident to a diffracted 
front increases or diminishes by ir as the point of incidence of the 
"rav" is moved from the centre of one opening to the centre of 
the next. Here r is the periodic time of the vibration and i is a 
positive integer. 

§ 4. Diffraction of a plane wave; general case. Take the axes of 
X, y, z to coincide with the axes ON, OT, OL of the grating, as in 
Fig. 1. Let J? be a point on the centre hne of the 
qth. opening and let the coordinates of R be 
0, qd, h. 

Let the direction cosines of the forward di- 
rection OP^ of the incident beam be l-^, m^, n^, 
and let those of the forward direction OP^ of the 
diffracted beam of order i be l^, m.^^ n^. 

Through draw planes perpendicular to these 
two directions. The distance of R from the first 
plane, counted positive when the incident wave 
front reaches before it reaches R, is 7yi-^qd + n^h. The distance 
of R from the second plane, counted positive when the diffracted 
front leaves before it leaves R, is 7n^qd + nji. If Vq is the velocity 
of Ught and \q the wave length in a vacuum, and if yu^, /Xg are the 
refractive indices of the media on the two sides of the grating, the 
times corresponding to the two distances are 

/xj {mT<}d + nJi)lvQ and ijl^ {m.^qd + n^h^VQ, 
and these differ by qir. Thus, since tVq = Xq, we have 
1^2 {rn^qd + n^h) — /Aj {m-^qd + n^^h) = T qi^Q. 
This result must hold good for all positions of R on the grating, 
for which q is integral. We thus obtain 

fjL^m^ = [iiiniT iXo/d, (1) 

H2n2 = fJ-ini (2) 

These equations completely determine the directions of the 
diffracted beams of order i. 

Let the incident and diffracted beams make angles e^, 63 with 
OT and angles rj^, 172 with OL. Then 

cos ei = /%, cos 63 = W2, (3) 

cos r]■^^ = n^, cos ''72 = '^2 (^) 

Hence (1) and (2) may be written 

jLt2 cos €3 = fii cos ei =F iXJd, (5) 

1x2 cos rj2= [J-i cos t7j (6) 



90 Dr Searle, Experiments with a plane diffraction grating 

Since m^^ + Wg- cannot exceed unity when the direction cosines are 
real, the condition that a diffracted beam may exist is mg^ + n^^ 5 1, 
or cos^ 62 < sin^T^g- I^ ^2 ^^^ V2 li® between and In, this requires 
that ')72 + ^2 > i""- 

It is noteworthy that eg depends only upon e^ and iXJd, and 
that 772 depends only upon tj^. 

We shall not further consider the case in which fi^ and /Xg are 
unequal, but shall confine the work to the special case of /lij = /Xg- 
The reader will find no difficulty in making the necessary modifi- 
cations. 

§ 5. Diffraction of a plane wave; single tnediunfi. In practice 
each medium is air, of refractive index n, relative to a vacuum. 
If A is the wave length in air, Aq = /xA. We then obtain the simple 
equations 

m2 = m^ T iXjd, or cos e^ = cos e^ T iXjd, (7) 

'^2^*^15 O^ cos 172 = cos T^j (8) 

Since -q may be restricted to lie between and tt, we have 

'n2 = Vi = 'n (9) 

The direction of the diffracted fay is easily constructed on a 

spherical diagram. Let the axes of the grating intersect a sphere 

about as centre in N , T, L (Fig. 2), and let NON' be a diameter. 

Let the continuation of the incident ray 

_L through meet the sphere in P^. The 

^\ great circle arc TPj measures e^. Calculate e, 

. .-^^--^^N by (7) and take TQ = €^ on TP^. About f 

2 y\ \ and L as poles draw small circles through 

N'f^ /IH"^ ^andPj. Theni>Pi=-7;. If the small circles 

T / / do not intersect, there will be no diffracted 

beam either by transmission or by reflexion. 

If the small circles intersect in the points Pgj 

Fig. 2. P,', then OP2, OP2 will be the directions of 

the two diffracted beams. Of the arcs NP2, 

NP2' of the great circle NP^P.^N' , one is greater and one less 

than Itt. If NP2 is less than \tt, it corresponds to the transmitted 

beam, and then iV^P2' corresponds to the reflected beam. 

When ej and i are given, there are two values of eg, and hence 
there are two points Q_ and ^+ on TP^. Thus there will be two 
directions {OP2)- and (0P2)+ for the transmitted diffracted beam, 
and similarly for the reflected beam. 

It may happen that only one of the two beams {OP2)- and 
(0P2)+ exists. Unless P^ is on the great circle LN, there will be 
two distinct values of mg^, and the condition ^2^ + ^2^ ^ 1 may 
be satisfied by the smaller value of m^^ but not by the larger. 




Dr Searle, Experiments with a plane diffraction grating 91 

Since rj^ tas the constant value r], while m^ or cos eg depends 
upon A, it follows that, if white light is used, to each A there will 
correspond a position of Pg on the small circle through P^ with L 
for pole. 

§ 6. The deviation. If D is the angle (< n) between the forward 
directions of the incident and the transmitted diffracted beams, 
cos D = l-J.2 + ^1^2 + n-ffi^. If the plane ZOP^ (Fig. 1) cuts OXY 
in OHi, where XOH^ ^ ^i, then, since P^OZ = rj, 

l^ = sin 7] cos ^1, m^ = sin 17 sin <^i, Wj == cos rj, 

and similarly for Pg- Thus 

1 — 2 sin^ ID = cos D = sin^ 77 cos (^^ — </>2) + cos^ 77, 

and hence sin |Z) = siniy sin | (^1 ^ ^2)' (1^) 

as can also be shown from the isosceles spherical triangle P^LP^ 
in Fig. 2. 

In the case of the transmitted diffracted beam, l^^ is positive. 
Noting that n^ = n^^ cos 77, putting m-^= a + h, m.^ = a — b, and 
substituting for l^, l^, we find 
2 (sin2 17) - 62) _ sin^T^ - a^ _ 52 _ ^,^^^,^2^ _ ^2 _ ^2)2 _ ia%^f. 

Thus sin^ ID is greater than b^ except when a = 0, and then the 
two are equal. When a = 0, ni^ = — m^. If we take m^ positive, 
we see, by (7), that, since i is positive, m^ = — m2 = iA/2(^. Hence 
b = iXj2d. Thus, if Dq is the minimum deviation, 

sin iZ)o = iA/2«; (11) 

Since rj does not occur in (11), m^ = — m^ gives a minimum of D 
for any given value of rj — a minimum having the same value for 
all values of 77. 

§ 7. The sloped grating. For the experiment of §§ 8, 9 it is 
convenient to use axes differing from those of § 4, Now let OY 
(Fig. 3) coincide with OT and let OL make an angle 6 with OZ. 
Then the direction cosines of OL are sin 6, 0, 
cos 9, those of OT are 0, 1, 0, and those of ON 
are cos d, 0, — sin 6. 

Let ^1, m^, ^1 and I2, m^, n^ be the direction 
cosines of the forward directions OP^, OP^ of 
the incident and the transmitted diffracted beams. 
Let W2 = sin 0, so that the diffracted ray OP^ 
makes an angle i/f with the plane OXY, counted 
positive when P2OZ < In. Let the plane ZOP.^ 
cut OXY in OH^ and let XOH^ = m. Then, if -q is the common 
angle between OP^ or OP^ and OL, 




92 Dr Searle, Experiments with a plane diffraction grating 

?2 sin 6 + 712 cos 6 = cos 17 = l-^ sin 6 + n-^co&d. ...(12) 

We also have, by (7), if P^OT = e^, P^OT = e^, 

cos 62 = ^2 = m^ T i^^jd = cos e^ T iA/(? (13) 

Hence m^ is known at once. Using ^2^ = 1 — '^^^ — ^2^, we have, 
by (12), 

sin^ ^ (1 — m^ — ^2^) = (cos r) — n^ cos ^)2. 

Solving for n^ and taking the negative sign in the ambiguity, we 
have 

sin i/f = ^2 = cos 6 cos 7^ — sin ^ (sin^?] — m.^^)^. ..-(14) 

Using this value of Wg in (12), we find 

^2 = sin ^ cos ?7 + cos ^ (sin^ 17 — ^2^)* (15) 

Since cos NOP^ = I2 cos 6 — n^ sin 6, we find from (14) and (15) that 

cos iYOPa = (sin2 77 - mg^)* (16) 

Thus the negative sign has been correctly chosen in (14) for the 
transmitted beam, since for this cos NOP2 must be positive. If the 
positive sign is used in (14), cos NOP2 is negative, corresponding 
to the reflected diffracted beam. 

In terms of i/j and co, the direction cosines of OPo are cos i/j cos to, 
cos i/j sin 60, sin i/j. Hence Wg = cos j/f sin co, and thus 

sin to = mg/cos i/j = cos e2/cos i/f (17) 

The two angles ifs and co completely determine the direction of the 
diffracted ray. 

In the experiment of §§ 8, 9 the incident rays are parallel to 
the axis OX of Fig. 3. Hence ?j = 1, m^ = 0, n^ = 0. We then have 

cos 62 = W2 = T iX/d, (18) 

cos Tj = sin 9, (19) 

and thus, since sin^ 7] — cos^ e^ = sin^ eg — cos^ 77, 

sin ijj = n^ = sin 6 cos 6 — sin 6 (sin^ ^2 — sin^ 9Y, . . .(20) 

^2 = sin^ ^ + cos ^ (sin2 62 - sin2 ^)* (21) 

sin 60 = ma/cos ifj = cos eg/cos ijj (22) 

Since e^ and 9 may be restricted to be less than hr, we see that 
no diffracted beam will be formed if 9 exceeds its critical value e^. 

§ 8. Apparatus. The general arrangement is shown in Fig. 4, 
The grating G is attached to a horizontal shaft A, with its plane 
parallel to A and its ruhngs perpendicular to ^. A horizontal 
colhmator L has horizontal and vertical cross-wires intersecting 
at C in its focal plane ; these are illuminated by a sodium flame S. 
The straight Hue joining C to the appropriate nodal point of the 



Dr Searle, Experiments with a plane diffraction grating 93 




lens is the line of collimation, or axis, of the collimator. The parallel 
beam defined by C is parallel to this hne. After the light has 
passed through the grating, it is 

received by a goniometer K, and -^ S 

an image of the collimator wires r nC 

is formed in its focal plane. To 
fix the line of collimation, cross- 
wires are placed in the focal 
plane; they intersect in D. The u'^ 
goniometer is carried on a hori- 
zontal revolving shaft B, and its 
line of colhmation is perpen- 
dicular to the shaft. One cross- 
wire is parallel and the other 
perpendicular to the shaft; the E 
latter is also perpendicular to the U/i 
shortest distance from D to the 
axis of the shaft. The shafts are 
provided with divided circles E, Fig. 4. 

F, which are read by aid of the 

pointers U, U', V, V. A balance weight W is attached to the circle 
F. The point of intersection of the hne of colhmation of K with 
the axis of B should lie approximately on the centre of the grating, 
and the line of colhmation of L should pass through the same point. 
The angles 6 and i/j are measured by the circles E and F. 

§ 9. Experimental details. The shaft A is set horizontal by aid of 
a level. The colhmator is adjusted optically by an auto-collimating 
method. The plane of the grating is set horizontal by a level, and 
the shaft is then turned through 90°, as measured by the circle E, 
so that the plane of the grating is vertical. Light from a flame is 
then reflected by a plate of glass held at 45° past the cross-wires 
and through the lens on to the grating, and the colhmator, pre- 
viously set for "infinity," is adjusted so that the image of C, the 
intersection of the wires, coincides with C itself. If the coincidence 
is recovered when the grating shaft is turned through 180°, the 
plane of the grating is parallel to the shaft. The line of colhmation 
is then both horizontal and also perpendicular to the grating shaft. 

The line of collimation of the goniometer is set perpendicular 
to the goniometer shaft B by an optical method. An auxihary col- 
limator, set for "infinity," is placed so that it is approximately 
perpendicular to the shaft. A plate of plane parallel glass is attached 
to the circle F near its centre so that its faces are approximately 
parallel to the shaft. It is convenient to make the shaft vertical; 
the glass plate can then be supported on a small levelhng table 
resting on the circle. By adjusting both the colhmator and the 



94 Dr Searle, Experiments with a plane diffraction grating 

plate, the faces of the plate are made parallel to the shaft and the 
axis of the colUmator is made perpendicular to the shaft. In this 
case the image of the collimator wires can, by turning the circle, be 
made to coincide with those wires, when either side of the plate 
faces the colhmator. If the plate has been suitably placed, it will 
be possible, by turning the goniometer on its shaft, to receive the 
image of the collimator wires on the focal plane of the goniometer. 
The "vertical" cross-wire of the goniometer, i.e. the wire perpen- 
dicular to the shaft, is now adjusted so that it coincides with the 
image of the corresponding wire of the colhmator. The line of 
colhmation is then perpendicular to the shaft. The goniometer is 
then put into position and its shaft is levelled. 

The axes of the collimator, of the grating shaft and of the 
goniometer shaft are adjusted to be approximately in the same 
horizontal plane. The plane of the grating is made vertical, and 
the goniometer stand is adjusted in azimuth so that one of the 
diffracted images of the collimator wires can be made to coincide 
with the goniometer wires by turning the goniometer on its shaft. 

When the adjustments already described have been effected,, 
and when the plane of the grating G is vertical, the diffracted beams 
are parallel to the plane OTN. If OT is inclined at an angle 8 to 
the grating shaft, the direction of OT will be changed by 28 if 
G is turned through 180° about the axis of the shaft from Position 1 
to Position 2, when the plane is again vertical. The goniometer is 
turned to receive a diffracted beam when G is in Position 1. If, 
when G is turned into Position 2, the inchnation of the beam to 
a horizontal plane is changed, the grating must be turned in its 
own plane until the inclination is the same for both positions. 

Since ijj is always small, cos i/j is nearly unity and hence, by 
(22), sin oj has a nearly constant value, for Wg is independent of 6. 
Hence, if the image of C, the intersection of the collimator wires, 
lies on D, the intersection of the goniometer wires, when the plane 
of G is vertical, the image of C will always lie very near the 
"vertical" cross-wire, and one setting of the goniometer stand will 
suffice for all values of 6. 

The plane of the grating is made horizontal and the index 
reading is taken. It is then turned through 90°; it is now vertical 
and in its zero position. The goniometer is next adjusted so that 
the image of C lies on the horizontal wire of K. The goniometer is 
then in its zero position. 

The grating is now turned through 10°, 20°, ... on sither side 
of the zero, and the goniometer is turned on its shaft to bring the 
image of C on to the horizontal wire of K in each case. If the 
grating circle has tivo indices, the grating is turned through 10°, 
20°, ... as indicated by one index. In reducing the observations the 
mean of the angles furnished by both indices is used. 



Dr Searle, Experiments with a plane diffraction grating 95 

Since e^ = ^tt, cos eg = Wg = iX/d. From the values of the 
interval d and the wave length A, cos eg is found and then the 
values of ip corresponding to the mean values of 6 are calculated 
by (20). These values are compared with the mean values of j/f 
given by the goniometer readings. 

§ 10. Distortion of the image. As 6, and consequently i/r, increases, 
the observer sees that the angle between the images of the col- 
limator wires undergoes great changes. When ^ = 0, the images 
are at right angles, but the angle diminishes rapidly as 6 reaches 
its critical value. The theory shows that they are actually tan- 
gential one to the other when 6 has its critical value, but, as no 
light is transmitted in the critical position, the phenomenon cannot 
be observed. If the collimator wires are stretched across a small 
circular opening, the image of the edge is distorted into an oval, 
which is practically an ellipse having the images of the wires as 
conjugate diameters. When, however, 6 approaches its critical 
value, the oval begins to deviate from an ellipse. 

In Fig. 5 let OX, OY or OT, OZ, OL, ON meet a sphere described 
about as centre in X, T, Z, L, N. Let OJ be the diffracted ray 
corresponding to the incident ray OX; 
the ray OX corresponds to the line of 
collimation of the collimator and OJ to 
that of the goniometer, when the image 
of C is brought to the intersection of the 
goniometer wires. Let OP^ be a ray nearly 
parallel to OX and let OP^ be the corre- 
sponding diffracted ray. Let the great 
circles through Z and P^, J, Pg cut the 
great circle TXS in H, K, M. Let *S be _ 

the pole of ZJK and let the great circle Fio- 5 

SJ meet ZP.^M in Q. Then ZJQ = ^tt. 

If the goniometer is mounted as described in § 8, and if its line 
of collimation coincides with OJ, its horizontal cross-wire will 
correspond to SJQ and its "vertical" wire to ZJK. The rays 
parallel to OP2 will come to a focus in the focal plane of the gonio- 
meter at D', whose coordinates referred to the horizontal and 
vertical wires through D (Fig. 4) are/ x angle QOJ and/ x angle 
P^OQ, where/ is the focal length of the lens. 

If points on a curve CC in the focal plane of the collimator 
give rise to diffracted rays whose directions are shown by points 
on the curve JPg o^ ^^^ sphere, and if the image of CC in the 
focal plane of the goniometer is DD' , the angle between the 
"vertical" cross- wire and the tangent to DD' at D is equal to /, 
the angle between the great circle JZ and the tangent at J to the 
curve JPg. 




96 Dr Searle, Experiments with a pJune diffraction grating 

If JK = ^, Pa^ = 'A'' ^^ = ^' -^'^^ = ^'' ^lien 

tanZ= limit of 75-^ = cosi/f f-y^ ) (23) 

If XH = a, PiH = y, the direction cosines of OP^, OP^ are 
?j = cos y cos «, m^ = cos y sin «, n-^ = sin y, 

?2 = cos if}' cos a>', m^ = cos j/(' sin to', ng = sin 0'. 
Since ZOL = d, the direction cosines of OL are sin 6, 0, cos ^. But 
P^OL = P2OL = 7], P-PT = ei, P^OT = 62, and thus the funda- 
mental equations (13) and (12) become 

cos iji' sin oj' = cos 7 sin a =F iXjd, >. . . (24) 

cos ijj' cos ii)' sin d + sin ip' cos ^ = cos y cos a sin ^ + sin y cos 6. 

(25) 

The vertical coHimator wire corresponds to the great circle ZX, 
and for this a = 0, but y varies. If ly is the inclination to JZ of 
the corresponding path described by P^, 

tan/, = cos^(|^y(|^)^. 

Differentiating (24) with respect to y and then putting y = 0, so 
that 0', co' become ifs, co, we have 

— sin i/f sin co {di/j'ldy)^ + cos ijj cos oj {dw'ldy)^ = 0. 
Hence tan /r.= sin i/j tan co. 

The horizontal colhmator wire corresponds to the great circle"' 
XT and for this y = 0, but a varies. Differentiating (25) with 
regard to a and then putting a = 0, we have 

— sin iJj cos CO sin 6 {di/j'/da)Q — cos i/j sin co sin ^ {da}'/da)Q 

+ cos i/f cos ^ (difj' lda)Q =--= 0. 
Hence, if Ih is the inchnation to JZ of the corresponding path 
ofP,, 



tan /^ = cos 

cos xjj cos ^ — sin ifj cos co sin 6 



.(26) 



sm 0L> sm u 

Multiplying numerator and denominator by cos a>, and replacing 
cos^ CO by 1 — sin^ to, we find 

T , T U cos d — Uo sin 6 .^„, 

tanZjj= tanZ^+ A ^ . „ 27) 

sm CO cos CO sm 

Since the direction cosines of ON are cos 6, 0, — sin 6, and I2, Wg 
in (27) refer to OJ, l^ cos 6 — n^ sin d = cos iVOJ. In the critical 
position, J lies on the great circle LT, which corresponds to the 



Dr Searle, Experiments with a plane diffraction grating 97 

plane of the grating. Then cos NOJ = 0, and the difference be- 
tween the tangents vanishes, i.e. the two curves touch at J . 

Since the distance of ZLX from T is constant, the curve 
corresponding to the vertical cross-wire is a small circle passing 
through J with T for pole. At J the small circle is perpendicular 
to the great circle TJ and the value of tan 1^ can be verified by 
spherical trigonometry. 

The horizontal cross-wire is represented by the great circle 
XT, but now both e^ and t] vary and no simple construction is 
available for the v:liole curve through J corresponding to this wire. 
The curve touches at ./ the small circle passing through J and X 
with L as pole, and cuts at right angles the great circle LJ . Hence 
tan III = cot LJZ, and then (26) can be verified by spherical 
trigonometry. If we find cos LJZ and sin LJZ and divide, we 
obtain the alternative form 

cos 6 — sintff sin 6 

tan iu = . : : 7P • 

cos i/f sm o) sm v 

If the angle between the two small circles which intersect in J 
is A, then A is the supplement of LJT. But LJ = ^tt — 6, 
JT = eg, LT = \tt, and hence 

cos A = tan 6 cot eg. 
In the experiment cos eg has the constant value =F iXjd, and thus 
cos A depends only upon d. The angle A will vanish when 
cos A = 1, and this occurs when d = eg, i.e. when 6 has its critical 
value. 

We can make visible a finite arc of the small circle w4th L for 
pole. If we illuminate with white light the small opening across 
which the wires are stretched, the position of J on this small circle 
will be different for the different colours. The short length of cross- 
wire will correspond for any colour to a small arc of a curve 
touching the small circle at practically its middle point. The 
envelope of these small arcs will be the small circle itself. The 
image of the horizontal wire will thus be a dark curved line running 
across the spectrum. 

§ 11. The critical values. The critical position of the grating 
is reached when 6 = eg, and we have, by (20), (22), the critical values 

cos e, 

sm ijjc = sm eg cos eg, sm oj^ = i , 

(1 — sin^ egcos^eg)^ 

cos e., ,^ J , cos^ eg 

tan a;. = — 9 , (tan i v)c == —■ " • 

^ sm^ eg sm eg 

For the grating used in § 12 and with * = 1, 

cos eg = 0-33568, sin eg = 0-94198, eg - ^tt - 19° 36' 49". 

Then ^1 - ]8°26', co^ - 20°43' 18" and (7^.)^ = 6° 49' 17". 

VOL. XX. PART I. 7 



98 Dr Searle, Experiments with a plane diffraction grating 

When ^ = 0, and therefore i/j = 0, 

sin o) — cos €2, oi —■ 19° 36' 49". 

Thus the maximum change in a> is only 1° 6' 29". 

§ 12. Practical exatnple. The following results were obtained 
by Dr J. A. Wilcken, using a grating having 14,468 lines per inch, 
' and, hence, an interval d = 1-7556 x 10"* cm. 

The adjustments described in § 9 were either effected or tested. The plane 
of the grating was not quite parallel to the grating shaft, but as both images 
of the first order were observ^ed, the mean results will be hardly affected. The 
calculated values of yjr were found on the assumption that the axis of the 
collimator is perpendicular to the transverse axis of the grating. Sodium light 
of mean wave length X = 5-893 x 10~^ cm. was used. Then, since the images 
were of the first order throughout, 
_ X _ 5-893 X 10- 5 
^^~ d~ 1-7556 X 10-* 
Thus e, = cos-i mg -= 70° 23' 11", and m^^ = 0-11268. 

Each of the observed values of yp' given in the table is the mean of four. 
Each of the first order images was observed, and for each image two values 
of d, one on either side of zero, were taken. The grating circle, which was 
printed on card, was a little eccentric relative to the shaft (it was a "home- 
made" affair), and, consequently, although the settings were made to integral 
degrees by one index, the other index did not always read integral degrees. 
Some of the mean values of 6 are, therefore, not integral degrees. The 
calculated values of -^ are those found from equation (20), which for con- 
venience is written 

sin t//' = ?i2 = J sin 26 - sin 6 [(cos 6 + m^) (cos 6 - m^)]^. 
For the sake of interest, the calculated values of w, /,- and I^ have been 
added. The last line in the table gives the critical values as found by calcu- 
lation. 



= 0-33568 = sin 19° 36' 49' 



d 


^ 


V' 


OJ 


Ir 


In 


mean obsd. 


mean obsd. 


calcd. 


calcd. 


/ // 


or'/ 






/ // 


a t n 











19 36 49 





90 


9 54 


36 


34 51 


19 36 52 


12 25 


86 38 28 


19 54 


1 12 45 


1 12 33 


19 37 7 


25 51 


83 1 5 


29 50 


155 30 


155 36 


19 37 34 


4113 


78 53 41 


39 51 


2 5145 


2 5018 


19 38 19 


1 46 


73 42 30 


49 48 


4 7 30 


4 7 25 


19 39 58 


128 20 


66 31 55 


59 49 


6 2615 


6 22 31 


19 44 27 


2 16 55 


54 29 53 


64 44 


8 3215 


8 29 8 


19 5024 


3 2 53 


44 114 


67 45 


10 54 


10 5113 


19 59 10 


3 55 4 


33 20 15 


69 44 


14 18 


14 10 3 


20 15 19 


5 9 40 


20 20 49 


70 2311 




18 26 


20 43 18 


6 4917 


6 4917 



The agreement between the observed and calculated values of ■^ is satis- 
factory. 



Dr Searle, Experiments with a plane diffraction grating 99 



Part II. Non-parallel Light. 

§ 1. Introduction. In the following experiments the incident 
light does not form a parallel beam. The diffraction now not 
merely changes the direction of the axial ray of the beam, but 
also, in general, introduces astigmatism into, or changes the astig- 
matism of, the incident beam. The exception is when the incident 
and diffracted axial rays are perpendicular to the rulings and the 
deviation is a minimum. The diffracted rays will, in general, pass 
through two focal lines when the aperture is small. If the aperture 
is increased, aberration will appear and all the rays will not pass 
accurately through the two lines. Aberration can be minimised 
by keeping the aperture small, but astigmatic effects are inseparable 
from the diffraction in the general case. 

The formulae for the general case are easily obtained, but are 
complicated. We shall, therefore, consider only the case in which 
the axial ray of the incident beam is perpendicular to the rulings. 

§ 2. Diffraction of an astigmatic hea^n. In Fig. 1 let OX, OY , OZ 
coincide with ON , OT, OL, the axes of the grating, as defined in 
Part I, § 2. For convenience, OZ will be taken as 
vertical. 

Let a beam, which started from a luminous 
point and therefore has a wave front, fall upon 
the grating near 0. Let OP^ be the continuation 
of the ray through 0, which has been restricted to 
lie in the plane OXY, and let OP^ be taken as the 
axial ray of the beam. Let P^OX = d^. Take OP^ 
as the axis of r^ in a new set of axes Or^, Os^, Ot^, 
of which Osi is in the plane OXY and Ot^ coincides with OZ. Let 
the equation to the incident wave front passing through be 

r, = i>SiV + W,s,t, + iT,t,^ (1) 

Let R he a point on the grating and let its coordinates referred 
to the grating axes be 0, qd, z, where d is the grating interval and 
q is an integer. Then the coordinates of R referred to the axes of 
the incident beam are 

ri = qd sindi, Si = qd cos dj^, t^ = z (2) 

If a line through R parallel to OP^ cuts the wave front O-^^ in F^, 
the second and third coordinates of F^ are qd cos ^j and z. By 
(1) and (2), the distance of F^ from the plane r^ = 0, which touches 
the wave front at 0, is ^S-^qH^ cos^ 6-^ + W^qdz cos ^^ + ^TjZ^, and 
the distance of R from the same plane is qd sin ^j. Hence 

F^^R - qd sin 6^ - ^S^qU^ cos^ d^ - W^qdz cos 6^ - \T^z^....{^) 

7—2 




100 Dr Searle, Experiments with a plane diffraction grating 



1 



When R and F^ approach 0, F^R becomes more and more nearly 
the normal at F^, and, for a small aperture, may be treated as the 
normal in the estimation of distances. Thus, ultimately, F^R is the 
ray distance from the wave front OF-^ to R. 

Let OP 2 be a diffracted ray of order i. By symmetry, OP2 is 
in the plane OXY, since OP^ is in that plane. Let P^X = 6^. 
Take the axial ray OP2 as the axis of ^2 in a set of axes Org, Os^, 
Ot2, of which OS2 is in the plane OXY and 0^2 coincides with OZ. 
Let the equation to the diffracted wave front passing through be 



_ 1 
"~ 2 



S2S2^+W2S2t2+hT2t2' (4) 



Then, if F2R, parallel to OP2, cuts the diffracted wave front OF^ 
in #2, the distance F2R is ultimately the ray distance from F2 to R. 
We then have 

F2R = qd sin ^2 - i>52?^^^ cos^ 62 - W2qdz cos 62 - lT2Z^....{b) 

The optical condition is that F2R differs from F^R by qiX, where i 
is a positive integer. We thus obtain 

F2R = FJl ± qiX. 

Since this holds for all values of z and all integral values of q, we 
have, by (3) and (5), 

sin 62 = sin ^^ ± iX/d, (6) 

S2 = k^S„ T4^2 = '^^i, T2=T„ ..(7) 

where k = cos ^^/cos $2. Since 6^ and $2 both lie between — ^tt 
and ^77 for a transmitted beam, k is positive. 

The direction of the axial ray of the diffracted beam is given 
by (6) and is independent of the constants S^, W^, T^. Equations (7) 
give the form of the diffracted wave front which passes through 0. 

If the deviation of the axial ray is a minimum, it follows from 
Part I, § 6, or otherwise, that sin ^^ = — sin ^j. Since d-^ and 9^ 
both lie between — ^tt and ^tt, cos ^2 = cos ^^, and thus k==\. 
Hence, in the case of minimum deviation, the form of the wave 
front is unchanged and the diffraction merely turns it through 2d 
about OZ. The restriction stated in § 2 must be noted. 

If the planes of the principal sections of the incident wave 
front are OXY and ZOP^, or, what is the same thing, the planes 
Or^Si, Orjtj^, then W^ = 0. It follows, from (7), that IF2 = 0, and 
thus the principal planes of the diffracted wave front are OXY 
and ZOP2. 

When TFj = 0, the section of the incident front by the hori- 
zontal plane ^^ = is r ^ = ^S^s^^, and the distance of the centre of 
curvature of this section from is Sj~'^. The vertical focal line of 
the beam passes through this centre of curvature. Similarly, the 
horizontal focal line is at a distance T^~^ from 0. The distances 



Dr Searle, Experiments with a plane diffraction grating 101 

from of the vertical and horizontal focal lines of the diffracted 
beam are S2~^ and ^3"^. 

If the incident beam is stigmatic, T^ ^ S^ and W^ = 0. Then 
S2 = k'^Sj^, Tg = S^. Hence S^ = k^T^, and so the diffracted beam 
is astigmatic, unless k=l, i.e. unless the deviation is a minimum. 

§ 3. The principal curvatures. The principal curvatures of the 
diffracted front can be found in terms of those of the incident front. 

Let the principal planes of the incident front intersect the 
tangent plane at in Orj^^, 0^-^ (Fig. 2). Take these, with Of^ 
along OPi, as axes for the front. Let the radii 
of curvature of the sections by 0^^171 and O^^^i 
be Bi~^ and C\~^. The equation to the incident 
front is then 




ii = hB,rj,^ + iC\l,^ (8) 

Let Orji make an angle tpi with Osi, as in Fig. 2. 
Then 

^1 "= ^1' Vi "" '^1 ^^^ "Ai + h ^i^ 'Ai' ^1 = ~ ^1 ^^^^ ^1 + ^1 ^°^ 'Ai' 

and hence (8) is equivalent to 

^1 = 2-^1 (^1 cos i/ji + ^1 sin 0i)2 + iCi (- Si sin tp^ + i, cos i/j^f. ... (9) 

Comparing (9) with (1), we find 

Si = i {B, + C,) + I {B, - C) cos 2<Ai, I 

If 1 = 1 (^1 - C'l) sin 2eAi, (10) 

T, = i(5i+Ci)-i(5i-Ci)cos2^,. I 

Then /Sg, Tf 2, ^2 can be found by (7). 

If the equation to the diffracted front referred to its principal 
axes is 

^2 = ^^27)2^ + ^02^', (H) 

and if T^gOsg = «/'25 ^^^'^ ^2, O2, ^2 ^^^ related to S2, T2, W2 by equa- 
tions similar to (10). Solving for B2, Cg, 'A2' we have 

B, 

C2 

tan 2^2 = 21f2/(.S2 - Tg) (13) 

The ambiguity in (12) has been settled so that, when i/j^ = 0, 
B2 = k^B^, C*2 = G^. Apart from mere reversals of direction, (13) 
gives two values of j/^a differing by hir, and corresponding to the 
axes 0-r]2, Ot,2- The arrangement of signs in (12) implies that when 
tjj^ = 0, ^2 = 0- Since, by (7), W^ and W2 vanish together, ijj^ 
and 02 niust reach \tt, tt, ^tt, ... together, and it follows that, for 
intermediate values, ^2 must lie in the same quadrant as ifj^. 



} = 1 (^2 + T2) ± [i (^2 - T2f+ W2^]\ (12) 



102 Dr Searle, Experiments with a 'plane diffraction grating 



^ 



Equations (12), (13) with (7) give B^, C^, i/fg in terms of >Si, T^, W^, 
which are given in terms of B^, C-^, i/j^ by (10). 

§ 4. ^ simple case. If we take C^ = 0, the incident wave front 
is cylindrical. We then obtain 

^2 = Pi {F + 1 + (P - 1) cos 2i/ri}, Cg = 0, ...(14) 

or tan i/(2 = A;~^ . tan ^1 (15) 

The maximum difierence between i/j^ and 1^2 occurs when 

tan i/j-^ = k^, and then sin (i/^i — iff.^) = {k — 1 )/{k +1). 

Since C*2 = 0, the diffracted wave front is cylindrical. There is 
therefore only one focal line at a finite distance from and this 
distance is -Bg"-^. If the principal planes of the incident front are 
turned round, i/j^ will change, so causing B2 to vary, and the 
distance of this focal line from will vary. 

§ 5. Measurement of wave length. The results of § 2 can be 
applied in the determination of the wave length of sodium light 
by measurements made on an optical bench. On the bench slide 
three carriages Z), H, K, as shown in plan in Fig. 3; D carries a 




horizontal glass scale divided in mm., H carries the grating G*- 
(with vertical rulings), whose centre is 0, and K carries the con- 
verging lens system L, of focal length/. At the end of the bench 
is a vertical slit illuminated by a sodium flame F; to identify a 
point E on the slit, a wire may be stretched across the slit. The 
divided face of the scale faces the slit and this face and the plane 
of the grating are perpendicular to the bench. The line through 
the nodal points of L is parallel to the bench and passes through E. 
The scale is first placed in the position A^B^, at a distance 
from E exceeding 4/ by about 30 cm., and the lens is adjusted to 
form a real undiffracted image on the scale at C^. The axial ray 
of the incident beam is normal to the grating, and thus, if </> is 
the angle between the normal and the axial ray of either diffracted 
beam of order i, we have k"^ = sec^ <j). Since the incident beam 



Dr Searle, Experiments with a plane diffraction grating 103 

corresponding to the point E of the slit is stigmatic, the vertical 
focal lines of the diffracted beams of order i will be at a distance 
OCJk^ from 0, along lines OPj, 0^^, each making an angle cf> 
with 0C\. Sharp images of the slit will pass through P^, Q^ and can 
be focussed on the scale if it is moved to A\By' . With the grating 
used in § 9, the two sodium lines can easily be separated. Then 
sin (/) = ^PiQi/OPj^. Since OP^ is difficult to measure, we suppose 
OX-^ known, where Z^ is the mid-point of PiQi- If OX^ = x-^, 
P^Q^ = 2;yi, tan <^ = iPiQJOX, = yjx^. 

The glass plate protecting the grating prevents an accurate 
measurement of OX^. We therefore move the lens carriage along 
the bench so that the undiffracted image is focussed on the scale 
at C'g. If the scale is moved further towards 0, the difPracted 
images can be focussed at Pg? Q2- If OX2 = x^, ^2^2 ^ '^Ih' 
tan (j) = y^l^o,- Hence 

tan (j,^{y^- y,)l{x^ - x^) (16) 

Putting 6-1^ = 0, $2 = (f) in (6), we find 

X = d sin (jiji, (17) 

where i is the order of the image and d is the grating interval. 
From (16) and (17), A is determined. 

Since it is an angle we measure, small errors of focussing will 
be of little account, for, in spite of them, the point in which the 
axial ray cuts the scale in each case will be correctly estimated, 
and this is all that is necessary. 

§ 6. Test of laiv of obliquity. Let OCj = u^, OC^ = ^2' ^Pi = ^'i) 
OP2 ^ V2- Then v^ — v^ = [{x^ — x^)^ + (y^ — y^fY- But, since 
^2= sec^^, we have, by § 5, u^ = u-^ cos^ (J3, v^ = U2 cos^ ^, and 
thus i\ — V2= (w^ — Mo) cos^ cf). Since Mj — Wg is known from the 
bench readings, we can test the law for the vertical focal lines 
by comparing the two values of v^ — ?',• As we are now concerned 
with the positions of images the focussing must be accurate. 

If the slit is not too narrow, the diffracted images of order i 
of the horizontal wire stretched across it may be focussed on the 
scale. If these are at jOj, q^^ when OC = ^^ and at ^03, q2 when 
OC = ^2, and if p^q^^ = 2r]^, ^2^2 = ^772, then 

ViP2 - mi - Q' + (^1 - ^2)']^. 

Since, by § 2, 1^= T^, we have p^2 = '^h ~ ^'2- The two values 
of PxP2 are compared. 

It is difficult to obtain satisfactory readings for x and ^. This 
is largely due to the fact that the diffracted rays in the horizontal 
plane through do not meet in a point but touch a caustic of large 



104 Dr Searle, Experiments with a plane diffraction grating 

radius. If Q^Ri, drawn perpendicular to Q-fi in Fig. 3, cuts OC^ 
in i?i, the radius of curvature of the caustic at Q^ is ^QJH^. The 
length of the caustic between the points of contact of the tangents 
from Ml, iVi, where M^N^ is the width of grating actually used, is 
3 (^iMi ~ ^1^1 ) approximately. 

§ 7. Adjustment of the lens. The lens, a converging system, is 
adjusted optically. Let its focal length be / and the distance 
between its nodal points be t, where t is positive when the distance 
between the principal foci exceeds 2/; for a projection lens as shown 
in Fig. 3, t will be negative. When the distance of the luminous 
point E from the scale ACB exceeds 4/+ t, there are two positions 
of the lens for which an image of E is formed on the scale. Let 
M, N (Fig. 4) be the nodal points corresponding to the principal 



• 


iB 


^^ 


m' 


M 




1 




-— -3r^ 


~^3r^ 




-=:,,^^__ 




C 

A 


s' 


R' S 


R 


E 



Fig. 4. 

foci to the right and left of L. In Fig. 4 let EC be the horizontal 
line through E parallel to the bench and let the other Unes be pro- 
jections upon the horizontal plane through EC. Let MR, NS be 
the perpendiculars from M, N on EC. Let RE = p, SO = q. Then 
in the second position of the lens, R'E = q, S'C = p. When the 
angles are small, RS = R'S' = t. Let I, I' be the images of E in 
the two cases. 

Take CE as axis of a; and horizontal and vertical lines through C 
as axes of y and z. Let the second and third coordinates of M, N, 
I, r be y, z, 7), I, r, Z, Y' , Z' . Since /iV is parallel to ME, and 
I'W to M'E, 

7 = 7y + yq\p, Z = ^ + zqjp, T' = t^ + yp\q, Z' = ^ + zp\q. 

Since {p + q)\pq = II f, 

Y-Y' = y{q-p)lf, Z-Z' = z{q-p)lf. 

Hence, if the image has the same position for both cases, then 
y = 0, and 2 = 0, and thus M lies on EC. The emergent ray 
through N will then be parallel to EC for all positions of the lens 
carriage. 

The lens is best mounted so that it can turn about a vertical 
pivot whose axis passes through M. If the support to which the 
pivot is fixed is moved through the distance MR at right angles 
to the bench, M will be brought on to EC. Then by turning the 
lens about M, N can be brought on to EC. 



Dr Searle, Experiments with a plane diffraction grating 105 

To identify C, a pin is mounted on a carriage so that its tip 
coincides with E. The carriage is moved along the bench so that 
the tip touches the scale AB. The point of contact is C. If L is 
adjusted on its carriage so that / coincides with C for both cases, 
then M, N lie on EC. 

§ 8. Other experimental details. The scale ^ B is set perpendicular 
to the bench. A set square XYZ, with the right angle at X, is held 
with XY in contact with AB. A pin is held close to XZ. If, when 
the carriage D is moved along the bench, the distance from the 
pin to XZ is constant, AB is correctly placed on its carriage. The 
scale must be horizontal and the slit vertical. The plane of the 
grating can be set perpendicular to EC optically. The lens L is 
removed and a small triangle of white paper is fixed to AB so that 
a vertex coincides with C. The grating G is placed midway be- 
tween C and E and is adjusted on its carriage so that the image 
of C by reflexion at G coincides with E. To allow a close test of 
parallax, a few grains of lycopodium may be placed on AB when 
the image of the slit does not fall on a dividing line. 

§ 9. Practical example. Using a grating with d = 1-7526 x 
10~*cm., the following results were obtained: 

The image of first order was used; thus i = \. 

Bench reading 118-50 cm., glass scale readings 97-52, 73-84 cm. 
141-21 cm., „ „ „ 89-64, 82-18 cm. 

Hence 

ih = \ (97-52 - 73-84) = 11-84 cm., y^ = | (89-64 - 82-18) = 3-73 cm. 
Also x^- x^^ 141-21 - 118-50 = 22-71 cm. 

Hence tan cf) = {y^ - y2)l{x^ - x^) = 8-11/22-71 - tan 19' 39' 7". 
Then X = dsin (p/i = 5-894 x 10-^ cm. 

§ 10. Experiment with an astigmatic incident beam. The experi- 
mental test of the results of § 4 is a good exercise in optical manipu- 




lation. Fig. 5 is a plan of the apparatus. Two cross- wires, inter- 
secting in E, are fitted into a tube turning about a horizontal axis 



106 Dr Searle, Experiments with a plane diffraction grating 

in a hole in the board D. A circular scale is attached to D and P 
has a pointer J which indicates its angular position. Only one wire 
is used in the measurements, but the second wire is useful as 
identifying E. The wires are illuminated by the sodium flame F. 
A lantern projection lens L is placed so that E is at its principal 
focus; for the best results, that end of L should face E which faces 
the lantern slide. Beyond i is a cylindrical tube Q, resting in 
two F's, V, V, and against a stop C/, and thus having only one 
degree of freedom. A piano-cylindrical lens A is attached by its 
plane face to one end of Q. A lens of about + 2-5 dioptres, such 
as is used in spectacles, is suitable. The grating, centre 0, is 
placed at G. A ground glass screen H can slide on the main optical 
bench, which also carries G, Q, L; if possible, D should be carried 
on the bench. On a short auxiliary bench slides a second screen K; 
the angle between the benches is cf), where sin cf) = i\ld. The 
ground sides of the screens face 0. 

Suppose, for a moment, that E \^ & luminous point. Then E^ 
at the focus of L, gives rise to a parallel beam falling on the 
cylindrical lens A. This lens converts the plane wave front into 
a cylindrical front. If the "power" of ^ is + i^ dioptres, a "real'' 
focal line, parallel to the generators of A'b surface, will be formed' 
100/i'' cm. from A. This focal line can be received on the screen H, 
By § 4, the diffracted front is cylindrical and there is only one 
focal line at a finite distance. This focal line can be received on 
the screen K. If A is turned by turning Q on its axis, the focal 
line of the diffracted beam will turn about the axial ray and the 
distance of the focal hue from will change. The experiment tests 
the relation between the linear displacement of K and the angular 
displacement of A. 

When a wire is used instead of a luminous point, images of the 
wire will be formed on H and K when the generators of A are 
parallel to the wire. If the pointer J is set in any position, a sharp 
image can be obtained by turning Q. 

When the adjustments of § 11 have been made, H is set to 
receive the image of the wire, and HO is measured. As a correction 
we may add tj ^i, where t is the thickness and ix the index of the 
plate covering the grating. A line ruled on H is made vertical by 
aid of a set square and a level, and P and Q are adjusted so that 
the image is vertical and ^j = 0. Then K is set so that the diffracted 
image is in focus on it, and the reading of K on its bench is taken. 
Then P is turned by steps of 10° or 15°, Q is turned in response, 
and K is adjusted in each case so that the image is focussed. When 
P has been turned through 90°, so that ifj-^ = ^tt, the image is hori- 
zontal, and, by (14), since i/r^ = \tt, its distance from is equal to 
the measured distance OH. When the image is vertical, j^i = or it. 
Since B-^-^ = OH, B^-^ = OK, we have, by (14), 



Dr Searle, Experiments with a plane diffraction grating 107 

^^^ = F+l + (F-l)cos2<Ai' ^^^^ 

where k ■= sec cf) and sin <f) = iX/d. 

To compare theory with experiment, we may plot the value of 
OK given by (18) against the bench reading of K. If the zero of 
this bench is at the end nearest 0, the points will lie about a 
straight line equally inclined to both axes. An alternative method 
is used in § 12. 

§ 11. Experimental details. The cross- wires should be mounted 
so that E is as nearly as possible on the axis of P. The lines joining 
the nodal points oi L to E are made coincident and parallel to the 
bench by the method of § 7. The axis of Q is set approximately 
parallel to the bench; optical methods are available. The cylin- 
drical lens A is adjusted optically. For a given direction of the wire 
at E, there are two positions of Q, 180° apart, in which A forms 
a sharp image of the wire on H. If the positions of these images 
are not identical, the error can be corrected by moving A at right 
angles to its generators across the end of Q. 

To set the lens L so that E is at its focus, a plane mirror is 
substituted for H, Q and G are removed, the cross-wires are 
illuminated and L and the mirror are adjusted so that E coincides 
with its own image. The plane of G is made perpendicular to the 
bench by the same method, the plate covering the grating serving 
as the plane mirror. The bench on which K slides is adjusted 
optically. First set P and Q so that a vertical image of the wire 
is formed on K. Then slide L along the main bench and readjust 
K. If the position of the image relative to K is unchanged, the 
auxiliary bench is correctly placed. If a micrometer eyepiece is 
used in place of the screen K, two images will be seen except when 
the wire is horizontal, since sodium light has a double spectrum 
line. Unless the wire is very fine, the images will overlap. The 
doubling of the images causes no inconvenience. 

§ 12. Practical example. The following results were obtained 
with a grating of 14,493 lines per inch. 

For this grating, d = 1-7526 x 10~* cm. The wave length was 5-893 x 10~^ cm. 
The image of first order was used; thus i = 1. Hence 

sin (p =^ 0-33625, k" = sec^ </> = 1-1275, (/> - 19° 38' 55". 

A cylindrical lens of + 2-5 dioptre was used. The corrected value of OH 
was 38-91 cm. The angle ^|^^ was varied from 0° to 180° by steps of 15°. The 
bench readings in columns 2 and 4 arc theoretically identical, and their mean 
is given in column 5. The values of OK calculated by ( 18) are given in column 6. 
To facilitate comparison, the mean difference between columns 5 and 6 has 



108 Dr Searle, Experiments with a plane diffraction grating 

been added to column 5, as suggested by Dr Wilclcen, and the results are 
entered in column 7. 





Bench 




Bench 


Mean 


OK 


OK 


-Aa 


^1 


reading 


^1 


reading 


reading 


calcd. 


obsd. 




cm. 




cm. 


cm. 
18-52 


cm. 


cm. 







18-52 


180 


18-51 


34-51 


34-36 


o / 




15 


18-62 


165 


19-10 


18-86 


34-77 


34-70 


14 10 


30 


19-54 


150 


19-90 


19-72 


35-51 


35-56 


28 32 


45 


20-68 


135 


21-21 


20-94 


36-58 


36-78 


43 17 


60 


21-87 


120 


21-96 


21-92 


37-71 


37-76 


58 29 


75 


22-74 


105 


22-89 


22-82 


38-58 


38-66 


74 7 


90 


22-91 






22-91 


38-91 


38-75 


90 



The observed value of OK is a little low at 0° and 90° and a little high 
at 45°. Probably the incident wave front was not accurately cylindrical. 

The last column gives ^^ ^^ calculated from tan ^^ — ^~~^ ^^^ ^i- The 
difference between \//-2 and yjr^^ is too small to admit of measurement with simple 
apparatus. 



Mr Whiddington, The Shadow Electroscope lOQ' 



The Shadow Electroscope. By R. Whiddington, M.A., St John's 
College. 

[Received 15 June 1920,] 

A simple form of Electrostatic Voltmeter of low capacity is 
frequently useful in the laboratory. The instrument under descrip- 
tion is of the gold leaf type designed primarily for class instruction 
and while not capable Of the highest precision is yet sufficiently 
accurate for many purposes*. 

All leaf electroscopes with which I am familiar require some 
sort of optical system such as a microscope to view the leaf. 
Attempts have been made to use a scale placed near the leaf for 
measuring purposes, but when too near, disturbing electrostatic 
effects are encountered, placed too far away parallax errors become 
obtrusive. 

It occurred to me that the difficulties might be overcome by 
simply throwing a shadow of the leaf on a semitransparent scale 
some centimetres away, using a small 2-volt lamp as a source of 
Hght. 

The first instrument made on these lines consisted of a tin 
cigarette box with the lamp at one end, a transparent scale at the 
other end and the gold leaf system with its insulation in the middle. 
It was found as expected that quite a sharp shadow could be 
obtained when the lamp filament was nearly parallel to the leaf. 

The final design of electroscope is shown in section in the 
figure, the photographically reproduced scale, graduated in volts, 
being shown below. It will be seen that the scale is practically 
even from 100 to 500 above which the leaf becomes unstable. 

The quadrant shape of metal box was chosen as being most 
likely to give an even scale and a constant capacity over its 
working range. 

The tube (T) carries a well fitting sulphur plug fitted centrally 
with a quartz tube down which passes the rod (R) which carries 
the leaf within the case and a small cup at the top. 

The metal arm (A) is for clamping and tilting purposes and 
carries an earthing terminal (E). 

Just below (A), a short side tube is arranged carrying an ebonite 
block (B) in which a small lime coated spiral is fitted. When B 
is pushed home the spiral finds a place behind R. Its object is, 
when heated from a 2-volt cell, to provide a source of ions for 

* The original instrument, of which this one is the final form, was designed in 
1919 for the Naval officers under instruction in Physics at the Cavendish 
Laboratory. 



110 



Mr Whiddington, The Shadow Electroscope 



experiments on ionization, its position behind the leaf precluding 
the possibility of disturbing convection currents. 



Cup for Condenser 



To Earth. 




To £ l/bLTS. 



The voltage range of the instrument is from 100 to 500 volts 
and with a good leaf it is possible to estimate to 1 volt, an accuracy 
sufficient for most purposes. 



Mr Whiddington, The Shadow Electroscope 111 

The scale was graduated by applying known voltages from a 
small direct current generator*, measuring them by a standard 
Weston Voltmeter. 

I have found that with this instrument and the scale repro- 
duced above, it is sufficient, when no more than approximate 
results are required, to register the shadow of the leaf for two 
positions only — ^zero and one other, say 200 volts. To effect this 
it will generally be necessary to alter the sensitiveness somewhat 
by adjusting the height of the sulphur block in T. This is no 
doubt due to the non-uniform aluminium leaff available. 

Charging the Electroscope. 

After connecting E to earth, the leaf may be charged positively 
by induction from a rubbed ebonite rod. If a negative charge is 
required care should be taken not to overcharge the leaf. If an 
appreciable leak is observed a small piece of smooth silk rubbed 
lightly over Q will almost certainly cure it. 

Insulation troubles are nearly always traceable to hairs and 
dust particles attracted under the comparatively high voltages used. 
It is therefore best to conduct the experiments in a dust free room. 

The following are a few of the experiments which can be carried 
out with this instrument. 

Experiment 1. To determine the capacity (Cg) of the electroscope 
hy comparison with that of a sphere of radius r cm. 

Method. Charge the leaf to a voltage F^ as indicated by the 
scale reading (with the case earthed), and then share the charge 
on the J leaf with the insulated sphere thereby causing a drop in 
potential to Fg. 

Then since q = Cg F^ = (Cg + ?") Fg, ? being the original charge, 

r 

The following table shows a series of measurements taken on 

* Kindly lent by the Electric Construction Company, Wolverhampton. 

t Cut with scissors from leaf approximately -0004 cms. thick. 

X It is here assumed that the capacity of the sphere is equal to its radius. This 
is only true when the sphere is far removed from other conductors, a condition 
which can be approximately realised in practice if a long thin stiff vertical wire 
be inserted in the cup of the electroscope (or stalk of the condenser as the case may 
be) and the sphere touched to the top of the wire. If this precaution be neglected 
the results obtained will be too small. 

Further, it must be remembered that when bringing up the sphere to the 
electroscope for charge sharing, any charge on the insulating handle will affect the 
leaf by induction and spoil the results. This effect may be got rid of by passing 
the handle through a flame occasionally, merely touching the ebonite is often 
sufficient to produce a charge. 



112 



Mr WJiiddington, The Shadow Electroscope 



these lines using an insulated brass sphere of radius .3-25 cm.*; 
Fj and V^ are the scale readings in volts. 



Fi 


V, 


V^/V, 


490 
330 
235 
160 


330 
235 
160 

no 


1-485 
1-405 
1-470 
1-452 



From the above readings the mean value of Fj/Fg = 1'450, 
whence C^ = 7-2 cm. 

Experiment 2. To determine the capacity of a parallel plate air 
condenser by the method of Experiment 1 . 

The readings tabulated below were obtained with a specially 
designed circular plate air condenserf . The diameter of the central 
plate being 4-25 cm., and its distance from two outer earthed 
plates being 0-15 cm., the capacity C'a can be calculated from the 

formula for a parallel plate air condenser, viz. 2 — -^ cm. 

Inserting the proper values for the present case leads to the 
value 60-2 cm. 

The experimentally determined value may be expected, if 
anything, to be rather greater than this calculated value owing 
to the extra capacity of the edges of the central plate. 

The method is essentially the same as in Exp. 1 but in this 
case the formula is 



vjv. 



1 



Using the same sphere as in Exp. 1 the following results were 
obtained, the insulated central stalk of the condenser fitting in 
the electroscope cup (see figure) and the outer plates being con- 
nected to earth. 



* An ordinary bedstead knob mounted on an insulating ebonite rod. 

t The main point in the design is the protection of the central insulated plate 
from dust, small hairs etc., which under the comparatively high potentials em- 
ployed would be attracted to it with resulting insulation troubles. The central 
plate is therefore sandwiched between two outside parallel plates, one of which is 
provided with a peripheral spacing ring which in butting up against the other 
outer plate completely encloses the inner insulated one. Insulating grooved buttons 
of ebonite form the insulation. 



Mr Whiddington, The Shadow Electroscope 



113 



Fi 


V, 


vjv. 


350 


332 


1-054 


320 


306 


1-046 


500 


480 


1042 


460 


441 


1-043 


420 


402 


1-044 



From these readings the mean value of VJV2 = 1*045, whence 
C^ + Ce = 72-2 cm., and since C, = 7-2, C'^ = 65-0 cm. 

Experiment 3. To determine the Specific Inductive Capacity of 
Ebonite. 

This can be readily carried out by using a second condenser 
exactly similar to the one used in the previous experiment but 
with circular plates of ebonite separating the plates instead of air. 

Then if this condenser (capacity C'j,) is placed on the electro- 
scope in the manner of the previous experiment, and charged to 
a potential (F^), and the sphere is used in the manner previously 
described, the resulting collapse of the leaf will be so small as to 
be hardly readable owing to the large capacity of the ebonite 
condenser. It is therefore more convenient to use the air con- 
denser of measured capacity Ca in place of the sphere. It is suffi- 
cient to hold Ca by its outer case for earthing purposes, touching 
its central plate momentarily to the corresponding plate of the 
condenser Cj, on the electroscope. The potential resulting from this 
sharing of charge (Fg) is noted. 

We then have that 

Ca+C, + Ce _ Fi 



whence 



C, + C, 



Fo 



a 



Fi/F,-1' 



in which both C^ and Cg have been previously determined by 



Fa 


V, 


vjv. 


330 


250 


1-320 


340 


262 


1-296 


258 


192 


1-.341 


480 


362 


1-336 


350 


265 


1-322 


260 


195 


1-318 



VOL. XX. PART I. 



114 



Mr Whiddington, The Shadow Electroscope 



experiment. The above table gives the results of an experiment. 
From which the mean value of V1/V2 comes out to be 1'322. 

By calculation from this value Cj, = 194-7 cm. 

Assuming the identical dimensions of the two condensers* the 
Specific Inductive Capacity of Ebonite is just the ratio 

fi iri _ 194-7 

^' '^ 65-0 

= 2-98. 

A value not far removed from the accepted value which ac 
cording to the table of Kaye and Laby will usually lie between 
2-7 and 2-9. 

Experiment 4. The comparison of two capacities by the ioniza 
tion leak method. 

It is convenient to illustrate this method by giving as an 
example the results of an experiment using the same two con 
densers as the preceding experiment. 

Method. If when the lime coated spiral is glowing steadily the 
slow leak of the electroscope be observed firstly with C'^ in position 
and then with Cj, in position, the capacities can at once be compared, 
for if Ta and T^ be these times it can be shown that 

^g + ^e _ Tg 
Cb + ^e Tj, 

The following table gives some results obtained with this 
method. In order to eliminate as far as possible any variations 
in the amount of ionization (which depends very greatly on the 
temperature of the filament and therefore on the e.m.f. of the 
power supplying cell) the readings for T^ and Tj, were taken 
alternately and as quickly as possible. It will be seen that under 



Times in seconds j 


■^ a 


T, 


9-2 
9-2 
9-6 
9-4 


25-0 
24-6 
25-8 
25-4 


Mean 9-35 


25-2 



* This can easily be tested experimentally. 



Mr Whiddington, The Shadow Electroscojpe 115 

the conditions of this experiment, in which a well charged 2-volt 
lead accumulator was used, there is very fair concordance between 
the various readings. 

Leak observed from 400 volts to 200 volts. 

If now in the above-mentioned expression we assume the pre- 
viously determined values of C^ and Cg, viz. 65-0 cm. and 7-2 cm. 
respectively, the value of C^ comes out to 188-0 cm. leading to a 
value for the specific inductive capacity of ebonite of 2-90. This 
value is in as good agreement as is to be expected with the deter- 
mination of Exp. 3. 



8—2 



116 Professor Baker, On the Hart circle of a spherical triangle 



On the Hart circle of a spherical triangle. By Professor 
H. F. Baker. 

[Read 9 February 1920]. 

This note is concerned with the problem, given three arbitrary 
plane sections of any quadric, of finding a fourth section which 
shall be tangent to four of the tangent planes of the three given 
sections. If the three given sections are concurrent on the quadric 
they have only four tangent sections, and the fourth section is 
unique, the projection of the figure on to a plane (from the point 
of concurrence) giving rise to Feuerbach's theorem of the nine- 
point circle. In general the three given sections have eight common 
tangent planes; in fact any two of these sections lie on two quadric 
cones, and the six vertices of the cones so obtainable lie by threes 
on four coplanar lines; the three cones whose vertices are on any 
one of these lines have a pair of common tangent planes, which 
thus touch the three sections. The eight tangent planes of these 
are thus accounted for. There are now fourteen ways of selecting, 
from these eight tangent planes, four which all touch another 
section ; six of these ways, in which the four tangent planes selected 
are tangent to a fourth section passing through the point of con- 
currence of the three given sections, are easy to recognise, and do 
not need further consideration. There are however eight ways of 
choosing four from the tangent planes which shall all touch another 
section lying in a plane w forming with the planes of the three given 
sections a finite tetrahedron. 

§ 1. We are thus lead to the problem of the condition necessary 
and sufficient in order that the sections of a quadric by the four 
faces of a tetrahedron should have four common tangent planes; 
and the main object of this note is to state this condition in a 
form which in fact leads to great simplification of what is generally 
presented as a somewhat intricate theory, and to point out several 
results, apparently new, which follow from this. Let the tetra- 
hedron be 0, X, Y, Z; denote the intersections of the quadric 
with OX hy A, A', those with OY by B, B' and those with OZ 
by C, C"; similarly denote the intersections with YZ by V, U', 
those with ZX by F, 7' and those with XY by W, W. In general, 
if each edge of the tetrahedron be joined by j)lanes to the two 
points in which the quadric meets the opposite edge, the twelve 
planes so obtained touch another quadric. But it may happen that 
this new quadric degenerates into two points, say S and S' ; then, 
with a proper choice of notation, the four lines AU, BV, CW are 
concurrent in a point S, and the four lines A'U', B'V, CW con- 






Professor Baker, On the Hart circle of a spherical triangle 117 

current in another point S'. That this should be so is a necessary 
and sufficient condition that the four sections of the quadric by 
the faces of the tetrahedron should have four common tangent 
planes. The condition may be stated in another form; take on 
the edge OX, the point Ai separated harmonically from A by 
and X, and the point Ai separated harmonically from A' by 
and X; in the same way take on each edge of the tetrahedron 
the harmonic conjugates, with regard to the vertices of the tetra- 
hedron lying on that edge, respectively of the intersections of the 
quadric with that edge. The twelve new points so obtained lie on 
another quadric, which we may describe as the harmonic conjugate 
of the original in regard to the tetrahedron. The condition in 
question then is that the harmonically conjugate quadric should 
break up into two planes, say a and a'; these will be the polar 
planes of >S and S' in regard to the original quadric. 

We may illustrate this condition by applying it to the (Feuer- 
bach) case of three sections of the quadric which are concurrent 
on the quadric, say in 0. The fourth section of the quadric touched 
by the four common tangent planes of the three given sections 
OYZ, OZX, OXY is then constructed as follows: on the plane YOZ 
take the line p through 0, harmonically conjugate with respect to 
OY, OZ, to the line in which the plane YOZ is met by the tangent 
plane of the quadric at 0; let this line p meet the quadric again 
in P; obtain the points Q, R of the sections ZOX, XOY in a similar 
way. The plane PQR is the fourth plane required. In this case 
one of the planes a, a' is the tangent plane at 0. 

§ 2. We may obtain a direct verification of the sufficiency of 
the condition in general by using it to obtain any one of the eight 
(Hart) sections m which can be associated with three given sections 
YOZ, ZOX, XOY, so as to form four sections with four common 
tangent planes. Let the quadric, referred to YOZ, ZOX, XOY and 
the polar plane of 0, have the equation 

ax'^ + by^ + cz^ + ^fyz + 2gzx + 2hxy = t^ ; 

with an arbitrary choice of the signs of Va, Vb, Vc, take 

u = h (/ -f Vb Vc), V = -| {g + Vc Va), w = | {h + Va Vb), 

and then I, m, n so that 

mn = u, nl = v, hn = w; 

the eight planes required are then expressed by 

Ix + my + nz — t^ = 0. 

It is at once seen that this follows from the condition stated above. 
If we introduce A, fi, v so that 

/ = Vb Vc cos X, g ^ Vc Va cos /x, h = Va Vb cos v. 



118 Professor Baker, On the Hart circle of a spherical triangle 

a plane of this latter form is 



/- cos la cos |v /r 

Va -.s + y V6 

cos |A 



- cos l-v cos ^A 



A cos |-jLt 

/-cos iA cos ia ^ ^ ,TT\. 

+ Z-VC — — -2r_ t = 0, (H) ; 

cos |v 

on the other hand a common tangent plane of the three given 
sections in x = 0, ^ = 0, 2 = is at once found to be 

X Va cos {s- X) + yVh cos (s — /x) + 2 Vc cos {s — v) — t-^ = 0, (I) 

where s = ^ (A + yu, + v) ; and it is easy to see that the section (I) 
touches the section (H) at the point of the plane (H) which lies on 

X Va : y^/h : zVc = p (q— rf : q (r — p)^ : r (p — qf, 

where, for brevity, p, q, r stand respectively for 

sin {s — A), sin (s — fj,), sin {s — v). I 

The four planes (I) which touch the section (H), as well as the 
original sections inx = 0,y = 0,z = 0, are obtained from the above 
equation by replacing A, yu,, v by ± A, ± /x, ± v, respectively. 

The eight sections (H) are obtainable from that above by re- 
placing Va, Vb, Vc, A, /x, V respectively by 1 

{Va, Vb, Vc, A, fji, v), (— Va, Vb, Vc, A, 77 + ix,7t -\-v), 

{Va, — Vb, Vc, X + TT, ix,v + tt), {Va, Vb, — Vc, X + tt, jx + tt,v) 
together with those obtainable from these by changing the sign 
of ^1. 

§ 3. The following result gives a construction for the position, 
upon the section, i, of the quadric by the plane (I), of the point 
in which this section is touched by the plane (H). Upon i we have 
three points, its contacts with the sections in a? = 0, ?/ = 0, s = 0; 
we also have two points, namely those in which i is met by the 
plane from to the intersection of the planes ABC, A'B'C , which 
plane is at once found to have the equation 

x Va + y Vb + s Vc = 0. 
The point to be constructed is the apolar complement of the two 
latter points in regard to the three former points. This result may 
be made clearer perhaps by stating it for a sphere in Euclidian 
geometry: If D, E, F be the mid-points respectively of the sides 
BC, CA, AB of a spherical triangle, the planes of the great circle 
arcs EF, BC give a diameter, and the three diameters so obtained 
are coplanar; let I, J denote the intersections of their plane with 
the inscribed circle of the triangle ABC; let P, Q, R be the points 
of contact of this inscribed circle with the sides BC, CA, AB. Then, 



Professor Baker, On the Hart circle of a spherical triangle 119 

upon this inscribed circle, the point of contact with the Hart circle, 
which touches this and certain other three tangent circles of the 
sides of the triangle, is the apolar complement of I, J in regard to 
P, Q, R. For the particular case of the nine point circle of a plane 
triangle the result has been remarked by Prof. F. Morley, as was 
pointed out to the writer by Mr J. H. Grace, Bulletin of the American 
Math. Soc, I, 1895, 116-124 ("Apolar triangles on a conic"). 

§ 4. Another result may also be stated here. To introduce it 
and render its meaning clearer we state it first for the Hart circle 
of a spherical triangle in Euclidian geometry. If this circle meet 
the sides of the spherical triangle ABC respectively in U, U' on BC, 
V, V on CA, W, W on AB, then, with proper choice of notation, 
the arcs AU, BV, CW are concurrent, say in S, and the arcs 
A'U', B'V, CW are concurrent, say in S' . The result in question 
is that S, S' are the centres of similitude of the circumscribed 
circle of the triangle ABC and the Hart circle. It is a direct 
generalisation of the corresponding familiar fact for the nine point 
circle of a plane triangle. 

Stated in the more general way here adopted, which is also the 
more precise way, the theorem is that the lines OS, OS' are each 
the intersection of two planes through which touch both the 
section m and the section by the plane ABC. If PQR, P'Q'R' be 
two sets of three points lying respectively on two plane sections 
of a quadric, such that PP' , QQ' , RR' are concurrent, the sections 
lie on a quadric cone having this point of concurrence for vertex; 
thus a plane through touching the section [x by the plane ABC 
equally touches the section /x' by the plane A'B'C . Now S, the 
point of concurrence of AU, BV, CW, is the vertex of one cone 
containing the sections //,, xd; and S' is similarly the vertex of one 
cone containing the sections jx' , rn. The line OS', joining the vertex 
of one cone containing the sections fx', m to the vertex of one cone 
containing the sections fj., fj,', passes through one of the vertices 
of the two cones containing the sections, /jl, m; as OSS' are not 
collinear, this line OS' passes through the vertex other than S of 
a cone containing /x and w. The two cones containing fx and m 
thus have their vertices on OS and OS'. Now to each of these 
cones there can be drawn from two tangent planes, which 
intersect in the line joining to the vertex of the cone; the four 
planes so obtained touch the sections fi and w, and thus are the 
four common tangent planes of the cones with vertex standing 
on the sections /x, ta. Two of these planes therefore intersect in 
OS and two in OS'; which is the result we desired to obtain. 

There are as we have said eight sections m each touched by four 
of the common tangent planes of the sections in YOZ, ZOX, XOY. 
These tall into four pairs, the planes of a pair intersecting on the 



120 Professor Baker, On the Hart circle of a spherical triangle 

polar plane of 0, being harmonic conjugates in regard to this plane 
and 0; for the pair associated as above with the two planes ABC, 
A'B'C the lines OS, OS' are the same. There is another pair 
associated similarly with the planes AB'C , A'BC, a third pair 
with the planes BC'A', B'CA and a fourth pair associated with 
the planes CA'B', CAB. And it may be remarked that the sections 
by the planes ABC, AB'C, BC'A', CA'B' are all touched by four 
planes, as follows at once from the fact that AA', BB', CC are 
concurrent; so also the sections by the planes A'B'C, A'BC, B'CA, 
CAB are all touched by four planes. 

§ 5. Another remark may be made, relating to a property which 
appears in Euclidian geometry as Salmon's theorem that the 
tangent of the radius of the circumcircle of a spherical triangle is 
twice the tangent of the radius of the Hart circle. 

Let P be the pole of any plane section of a quadric, upon which 
any point A is taken, and be any other point; denote by p the 
Cayley separation of the lines OP, OA in regard to the quadric, 
and by S the Cayley separation of P from 0. It can then be shown 
that p is independent of the position of A upon the section, and is 
indeed symmetrical in regard to P and 0, being connected with 8 
by an equation sin S sin p = ± 1. Calling p the radius of the section 
in regard to the point 0, it can be shown that if p, p' be the radii 
of any two sections a, a' whose planes intersect in a line I, and the 
planes joining I to and to the vertex of one of the two cones 
containing a and a' be respectively denoted by co and y, then 
tanp/tanp' is equal to the homography (y, a>; a, a') or to the 
negative of this. In particular when the planes y, a are harmonically 
separated by to and a', this leads to tanp = 2 tan p'. In our figure 
the plane a, which is the polar of S in regard to the quadric, passes 
through the line of intersection of the planes ABC and w, since S 
is the vertex of one of the cones containing the section by ABC 
and the Hart section zu, and this plane a also contains the vertex 
of the other cone containing these sections; it can easily be proved 
that the plane co which joins to the line of intersection of the 'planes 
ABC and m is harmonically separated from m by the planes ABC 
and a; thus the planes a, ABC, oj, w have the relation of the 
respective planes y, a, w, a in the general description just given. 
It follows tbafc if p, R be the radii of the sections m, ABC, we have 
tan -R = 2 tan p ; which is what we wished to prove. 

§ 6. A last remark may be added bringing into relief the con- 
nexion between the present point of view and that of the Euclidian 
geometry. As hitherto, let OXYZ be a tetrahedron whose faces 
meet a quadric in sections having four common tangent planes. 
Denote by ia, i^, iy the Cayley separations OX, OY, OZ in regard 



Professor Baker, On the Hart circle of a spherical triangle 121 

to the quadric; by ia, i^' , iy' the Cayley separations YZ, ZX, XY] 
by (A), (B), (C) the Cayley separations of the pairs of planes 
meeting respectively in OX, OY, OZ; and by (A'), (B'), (C) the 
Cayley separation of the plane XYZ respectively from the planes 
YOZ, ZOX, XOY. Each of these separations is ambiguous in sign 
and by additive multiples of it, unless we enter into further detail. 
There are however equations by which all of them are deducible 
from a, P,y; and these equations may be represented, when proper 
regard is paid to the ambiguities, by 

a' = ITT + ^ — y, ^' = iiT + y — a, y' = iV + a — /3, 

... sinha , , .,. sinh (^ - y) 

tan (A) = — 7~ r , tan (A) = — , ' , q ^—. 

^ ' cosh (e + a) ^ ' cosh (e + ^ + y) 

where e is such that 

2 tanh e = tanh a tanh ^ tanh y — tanh a — tanh ^ — tanh y. 

And these lead to 

(A') = {B) - (C), {B') = (C) - (A), (C) = (A) - (B), 

which may be used to define the Hart section. 

§ 7. In what has preceded we have stated a sufficient condition 
for the Hart section, namely that AU, BV, CW are concurrent. 
It can however be proved that this is also a necessary consequence 
of the existence of the four sections of the quadric all touched by 
four other planes, provided we exclude certain particular possi- 
bilities which are easily stated. Precisely, given three arbitrary 
plane sections of a quadric, no one of which degenerates into two 
straight lines, so that the equation of the quadric referred to these 
and the polar plane of their point of intersection is of the form 
{abcfgh^xyzf = t-^, in order that these with a fourth section (also 
not two straight lines) should form a set of four sections all touched 
by four planes, if no relations are assumed to hold among the 
coefficients a, h, c, /, g, h, it is necessary that the condition in 
question (that AU, BV, CW are concurrent) should hold. 

In order that the sections by a; = 0, ?/ = 0, z = 0, ^ = of the 
quadric {abcdfghuviv\xyztf- = should have four common tangent 
planes, the cones enveloping the quadric along these sections must 
be concurrent; if A be the four-rowed determinantal discriminant, 
and ^4, 5, ... the minors therein, it follows that the necessary and 
sufficient condition for this is that the equation 

{ahcdfghuvivjVA, VB, VC, VDf = A 

should be satisfied for four choices of the signs of VA, VB, VC, vD. 
It proves to be possible to examine all the ways in which this can 
happen, and the result is as stated. 



122 Professor Baker, On a projperty of focal conies 



On a property of focal conies and of bicircular quartics. By 
Professor H. F. Baker, 

[Read 9 February 1920.] 

The property of focal conies referred to in the title is the well- 
known one that if P, R be any two points of the focal hyperbola 
of a system of confocal quadrics, and Q, S be any two points of the 
focal ellipse, then the distances PQ, PS have the same difference 
as the distances RQ, RS. The theorem remains true if every one 
of the distances be replaced by the Cayley separation of its end 
points in regard to an arbitrary quadric of the confocal system, 
and the original theorem is then obtainable by making the para- 
meter of this arbitrary quadric increase without limit. It is shown 
that the generalised theorem is equivalent to the geometrical 
theorem that two enveloping cones of the arbitrary confocal exist, 
each of which touches the four lines PQ, QR, RS, SP. The theorem 
that the sum of the two focal distances of a point of an ellipse is 
constant may similarly be replaced by the theorem that the sum 
of the Cayley separations of a point of the ellipse from the foci is 
constant, in regard to an arbitrary confocal conic; a theorem is 
obtained which includes both this last result and the former. It 
is unnecessary to point out that this last result is equivalent \^'ith 
Chasles's theorem that a variable tangent plane of a quadric cone 
makes angles with the planes of circular section whose sum is 
constant (Chasles, Geom. Super., 1880, § 812, p. 517). 

The property of bicircular quartics referred to is that the angles 
which a variable bitangent circle of one mode of generation makes 
with two fixed bitangent circles of another mode of generation, have 
a constant sum (Jessop, Quart. Journ., xxiii, 1889, 375). This is 
shown to be equivalent to the former theorem. 

There exist much more general theorems in regard to the 
generation of a quadric with the help of a thread of constant 
length, whose systematic investigation is in connexion with the 
theory of hyperelliptic functions (Chasles, Liouville, xi, 1846, 15; 
Darboux, Theorie des surfaces, Livre iv, Ch. xiv, 296-312; Staude, 
Math. Ann., xx and xxii, 1883; Finsterwalder, Math Ann., xxvi, 
1886; Maxwell, Works, ii, 156 or Quart. Journ., 1867). I have added 
some lines in regard to this general point of view. 

§ 1. Ii P, Q, R, S be four coplanar points of a quadric, and 
through the lines SP, PQ, QR, RS be drawn four arbitrary planes, 
respectively, a, ^, y, 8, the lines a^, ^y, yh, 8a meeting the quadric 




and of bicircular quartics 123 

again respectively in P', Q', R', S', then (1) the points P', Q', R', S' 
are equally on a plane, (2) if by the angle between the sections of 
the quadric by the planes a, ^ be understood the Cayley separation 
of these planes, measured by the homography of these planes in 
regard to the two tangent planes to the quadric drawn from their 
line of intersection, then the sum of the angles at P, R, determined 
respectively by the sections a, ^ and y, 8, is equal to the sum of 
the angles at Q, S, determined respectively by the sections /3, y 
and 8, a. 

That P', Q', R', S' lie in a plane follows from the fact that the 
four quadrics consisting of (i) the original quadric, (ii) the planes a, y, 
(iii) the planes ^, 8, (iv) the planes PQR, P'Q'R', have seven, and 
therefore eight points in common. For the relation between the 
angles, denote by 6 the section by the plane PQRS, and in general 
by (a, B) the angle between the sections (a, ^). Then we have 

77 ^ (d, a) + (a, ^) + (^, d) = {d, y) + (y, 8) -^ (8, d), 

and therefore 

(a, /3) + (y, 8) = 277 - {6, a) - (9, ^) - {d, y) - {d, 8), 

which is also the value of (^, y) + (8, a), the ambiguities of inter 
pretation being properly settled in each case. 

In a plane we have the theorem that if P, Q, R, S be concyclic 
points through which pass pairs of four circles a, ^, y, 8, namely 
a, B through P, /?, y through Q, y, 8 through R and 8, a through S, 
then the two angles (a, /3), (y, 8) have the same sum as the two 
angles (^, y), (8, a); and this, not depending on the Axiom of 
parallels, may well be regarded as a fundamental theorem. Further 
if P' be the other intersection of a and ^, etc., the points P', Q', R', S' 
are concyclic. The connexion of this result with the theorem of the 
angles is incidentally remarked by Prof. W. McF. Orr, Trans. Camb. 
Phil. Soc, XVI, 1897, 95. 

§ 2. Regard the bicircular quartic in question as the projection 
on to an arbitrary plane of the section of a quadric by a quadric 
cone of general position, the centre of projection being an arbitrary 
point of the quadric. An arbitrary tangent plane of the cone cuts 
the quadric in a section projecting into a conic having two points 
of contact with the bicircular quartic, and this conic, passing 
through the nodes of the quartic, is for us a bitangent circle, of 
one mode of generation. The other three modes are obtained by 
considering the other three quadric cones through the intersection 
of the quadric and the first cone. Take then two bitangent 
circles of the bicircular quartic of the first mode of generation, say 
a and y; their points of contact will be on another circle, say/?, 
as appears from the three dimensional figure. Take also two 



124 Professor Baker, On a property of focal conies 

bitangent circles of a second mode of generation, say ^ and S, 
with points of contact on a circle, a. We shall prove that the 
eight points of intersection of the pairs of circles (a, ^), (/S, y), 
(y, §), (8, a) lie on two circles 6, 6', four on each. These circles 6, 6' 
pass through the two intersections of the circles p, a, and separate 
these circles harmonically; the circle p is orthogonal to the principal 
circle to which the bitangent circles of the first mode are all ortho- 
gonal, with a similar statement for cr. For the proof, let a = 0, 
y = be the equations of any two tangent planes of a quadric 
cone, whose generators of contact lie on a plane p = 0, so that 
the cone has the equation ay — p^ = 0. Let ^S — o-^ = be another 
quadric cone, whereof ^ = 0, 8 = are tangent planes touching 
the cone on ct = 0. Then a quadric E = through the curve of 
intersection of the two cones has an equation of the form 

E = ay -p^-m^ (^S - a^) = 0, 

so that the four lines a = 0, j3 = 0; ^ = 0, y=0; y = 0, 8 = 0; 
8 = 0, a = 0, in which the two first planes a, y meet the two latter 
planes ^, 8, intersect the quadric ^ = in eight points lying in 
the two planes p + ma = 0, p — ma = 0. 

We have then a proof of Jessop's theorem in regard to the 
bicircular quartic curve*. 

§ 3. Eeciprocally let any two conies be taken in space, not 
intersecting one another. Consider a quadric touched by the 
common tangent planes of these two conies. Then if A, C be any 
two points of the first conic, and B, D any two points of the second 
conic, it follows from § 2 that the pairs of tangent planes to this 
quadric from the lines AB, BC, CD, DA touch two enveloping 
cones of the quadric, say F and G. Or, as a line lying in a tangent 
plane of a cone is a tangent line of the cone, there are two en- 
veloping cones of the quadric which touch the lines AB, BC, CD, 
DA. And, comparing the equations of § 2, the vertices of these 
cones lie on the line joining the points R, S, in the planes of the 
conies, which are the poles respectively of AC, BD in regard to 
these conies, and separate R, S harmonically; the positions of the 
vertices depend on the quadric taken to touch the common 
tangent planes of the conies. Moreover, as the reciprocal of the 

* The direct analytical proof is, of course, simple. Let the fundamental quadric 
be x^ + y~ + z^ + I" = 0, and bitangent circles of two modes be obtained by pro- 
jection of the polar sections respectively of the two points 

[{a-d)x, (b-d)y, {c-d)z, 0], [{a-c)^, {b-c)v, 0, (c^-c)t]. 

Then the angle between these circles, being the Cayley separation of these points, 
is the angle, in rectangular Cartesian coordinates, between the two lines 

X/px = Y/qy, X/p^ = Y/qr], where p^ = {a- d) [a - c), q^=(b- d) (b - c). 

This generalises at once to the Cyclide; cf. Jessop, Quartic surfaces, 1916, p. 106. 



and of bicircular quartics 125 

theorem in regard to the angles, if we consider the homography 
of A,B in regard to the quadric, say a, and take the corresponding 
homographies for the pairs B, G; C, D; D,A respectively, say b, c, d, 
we have ac = bd, or a/d = b/c. In words, the difference of the 
Cayley separations of A from 5 and D, in regard to the quadric, 
is the same, for unaltered positions of B, D on the second conic, 
when A is replaced by any other point C of the first conic. 

This result includes the particular case of the focal conies of 
a confocal system, for which we may also consider the further 
particular case of actual Euclidian distances between the points. 
(Cf. § 10 below, where the relation between the separation and the 
distance is given.) 

§ 4. If we assume that the sides of the skew quadrilateral 
ABCD in § 3 touch an enveloping cone of the quadric, we can 
deduce the relation between the Cayley separations in another 
way. In fact if the sides of a skew quadrilateral touch any quadric 
having ring contact with a given quadric, the sum of the Cayley 
separations belonging to the sides of the quadrilateral, each taken 
in proper sense, is zero, the separations being measured by the 
latter quadric. For if ^T be a tangent to a quadric V, which has 
ring contact with a quadric U, drawn from a point A, the Cayley 
separation AT in regard to U is independent of T. If A be 
(^, 77, ^, r), T be (x, y, z, t), so that, with usual notation, V^. = 0, 
7^^ = 0, and Z7 be 7+ P^^ 0, then TJ^ = PJ^, U^^ = PJ"^, and hence 

which is independent of x, y, z, t; and U^^KUJJ^Y is the cosine of 
the separation in question. Therefore, if the ^iAqsAB, BC, CD, DA 
of the skew quadrilateral touch 7 respectively at L, M, X, Y, 
we have the following relations among the separations 

(AB) = (AL) - (BL), (BC) = (BM) - (CM), 

(CD) = {CX) - (DX), (DA) = {DY)- (AY), 

{AY) = iAL), {BL) = {BM), {CM) = {CX), {DX) = {DY), 

leading to 

{AB) + {BC) + {CD) + {DA) = 0, 

or {AB) - {AD) = {CB) - {CD). 

In the application of this result above, 7 was a cone. 

§ 5. We may however make an application in which C/ is a 
cone, and 7 not a cone, U being an enveloping cone of 7. Namely, 
if the sides of a skew quadrilateral touch a quadric, the sum of 
the four Cayley separations of the vertices, each in proper sense, 
in regard to any enveloping cone of the quadric, is zero. The 
reciprocal theorem, is that if two plane sections a, y of a quadric 



126 Professor Baker, On a property of focal conies 

be both touched by each of two other sections ^, S — and if, taking 
a fifth arbitrary section, co, of the quadric, we measure the angle 
between the planes of two sections a, 13, which touch one another, 
in the usual way, by considering the homography of these planes 
in reoard to the tangent planes drawn from their line of intersection 
to the section co — ^then, with proper sense of measurement, [a, /3] 
denoting the angle between these planes, we have 

[a, P] + IP, y] + [y, 8] + [S, «] = 0. 

Now take one of the two quadric cones containing the sections 
a, y, and regard this cone, and the section co, as fundamental; 
speak of a, y as circular sections of this cone, of opposite systems 
because each has two points common with the other and with a». 
Then we have ChasJes's theorem that a variable tangent plane of 
a quadric cone makes angles of constant sum with two planes of 
circular section of the cone, of opposite systems. 

§ 6. The reciprocal theorem is that a generator of a quadric 
cone makes angles of constant sum with two conjugate focal lines 
of the cone, that is, considering the conic in which the plane of w 
cuts the cone, and the quadrilateral formed by the common 
tangents of this conic and oj, makes angles of constant sum with 
the lines joining the vertex of the cone, to an opposite pair of 
intersections of two of these common tangents (Chasles, loc. cit., 
§ 827, p. 528). Projecting on to an arbitrary plane we have the 
theorem that if P be a variable point of one of two conies having 
S, H as common foci, the Cayley separations PS, PH in regard to 
the other conic have a constant sum. An elementary proof can 
be given depending on the fact that if PS meet the other conic 
in >Si, S2, and PH meet the other conic in Hj^, H^, then, with proper 
notation, each of S^H.,, S^H^ passes through a fixed point of the 
line SH. 

§ 7, This theorem for conies is a particular case of the following : 
Two conies V, W, have both double contact with a conic U, and 
also both have double contact with another conic K. From a 
point P oi K a tangent PX is drawn to F, and also a tangent P Y 
to W; then the Cayley separations PX, PY, taken in regard to U, 
have a constant sum (or difference) as P varies on K. Two tangents 
are possible from the point P to the conic F; but the separation PX 
is the same for both. 

If F degenerate into the pair of tangents to U from a focus S 
of U, and W into the pair of tangents to U from the conjugate 
focus H, then the conic K, touching these four tangents, will be 
confocal with U, and the tangents PX, PY will become the lines 
PS, PH. Thus the theorem includes that of § 6. 



and of bicircular quartics 127 

§ 8. The proof of the general theorem of § 7 is, analytically, 
identical with that of the following theorem, of three dimensions, 
which leads, in § 9, to the theorem of § 4, and may thus be regarded 
as summarising all the analogous theorems here obtained: — If two 
quadrics V, W both have ring contact with a quadric U, and also 
both have ring contact with a quadric K, and PX, PF be tangents 
respectively to V and W from a point P of K, the sum (or difference) 
of the Cayley separations PX, PY, in regard to U, is independent 
of the position of P upon K. When V and W coincide the difference 
of the separations is zero for all positions of P and the quadric K 
is unnecessary. 

The theorem is easy to prove. In order that two quadrics 
V = 0,W = should both have ring contact with another quadric 
U = 0, they must, if P = 0, Q = be suitable planes, be capable 
of the forms V = U - P^, W ^ U - Q^ and thus F, W must have 
two points of contact, there being an identity of the form 

V-W = pq, 

where p = 0, q = are two planes. Any quadric having ring 
contact with both V and W is then capable of either of the 
identical forms 

V +1 {a-'^p - aqf =0, W +1 {a-'^p + aqf = 0, 

wherein a is a constant, and two such quadrics can be drawn 
through an arbitrary point. We. may then suppose 

U=V + l {a-^p - aq)\ K=V + 1 (b'^p - hq)\ 

where ii = is the quadric of the enunciation, and 6 is a constant. 
Thus we have the identity 

U - K = l (a-2 - 6-2) (^2 _ a^j^Y), 

involving in particular that U, K have two points of contact on 
the line joining the points of contact of V and W. Putting 

P = I [a-^p — aq), ^ == | {a-'^p + aq), 
this is the same as 

(1 - a2) (C/ - A') = P^+Q^+ 2aPQ, 

where a = {a^ + b^)/{a'^ — 62). This again, if U is not zero, is the 
same as 

(P2- U) (Q2 - U)- {PQ f aUf = (1 - (t2) UK. 

We remarked however above (§ 4), that if 6, (f> be the Cayley 
separations PX, P Y, taken in regard to V, 

P Q 

cos 8 = — 7 , cos (b = —- 



128 Professor Baker, On a pro;perty of focal conies 

where the coordinates in IJ, P, Q are those of the point P. If this 
point be on the quadric K ^ 0, but not on U = 0, we thus get 

cos 6 cos <f) + a ^ ± sin ^ sin (f> 

showing that 9 ± (f) ^ constant, as was stated. 

§ 9. Now suppose a skew quadrilateral ABCD of which the 
sides AB, BG both touch the quadric V, say in X and Y, respec- 
tively, while the sides CD, DA both touch the quadric W, say in 
Z and T respectively. The quadrics 7, W are supposed to have 
two points of contact, so that quadrics can be drawn having ring 
contact with both. Let TJ be one such; let K be another such 
passing through C, and let A be on K. Then, considering Cayley 
separations in regard to V, we have {BX), {BY) equal because F 
has ring contact with U, and also {DZ), (DT) equal because W 
■ has ring contact with U. By § 8 we also have (AX) — (AT) equal 
to (GY) — {GZ), if a proper sense be assigned to the separations 
involved. 

We infer therefore that 

(AB) - (AD) = (AX) + (XB) - [(AT) + (TD)] = (AX) - (AT) 

+ (XB) - (TD) = (GY) - (GZ) + e{YB)-l (ZD), 

where e, ^ are each ± 1. Without making the proper detailed 
examination, we shall put both e and ^ equal to 1, so obtaining 

(AB)- {AD) = {GB)- (GD). 

This is verified (§ 4) in the particular case where the quadrics F, W 
coincide, there being then no need for the condition that A, G 
lie on the same quadric K having ring contact with F and W. 

§ 10. A line joining a point of one focal conic to a point of 
another focal conic of a confocal system of quadrics is a particular 
case of a line touching two confocals of the system. And such a 
line is part of a continuous curve which on either of these two 
confocals may consist partly of arcs of the line of curvature which 
is the intersection of these two fundamental confocals, and partly 
of arcs of geodesies touching this line of curvature. As was recog- 
nised by Chasles this continuous curve has everywhere the geo- 
metrical property that if we take two other confocals of the 
system, the homography of the tangent planes drawn to one of 
these from a tangent line of the curve, in respect of the tangent 
planes drawn to the other, is the same for every point of the curv^e. 
That the analytic formulation may equally be regarded as uniform 
for all parts of the curve seems often to be unnoticed; it is recog- 
nised however by Staude in the papers above referred to. Let us 
consider the system of confocals 



and of bicircular quartics 129 



x^ y^ z" 



+ A-.+ ^^ = h 



a + X b + X c + X 

where a > b > c, using A for ellipsoids, ju. for hyperboloids of one 
sheet, V for hyperboloids of two sheets. Suppose that the straight 
portion of the curve touches the confocals for which X = 'p,X=^ q, 
and denote by dw the Cayley separation of two consecutive points 
of the curve taken in regard to the confocal of parameter A = ^. 
Putting 

F {x) = ^{x + a) {x ■i-b){x + c) {x - p) {x - q), 

L^=f{X), 3P=f{iJi), N^=f{v), 

the curve is such that 

(A -p)dX ^ r (/x - p) d^i ^ ( {v - p) dv ^ ^^ 



L ' J M ' ] N 

(A -q)dX [ ifx - q) dix ( {v - q) dv _ 

L +j — li ' J iT^-^' 

while, with S^ = F (6), 

[{X -p){X- q) dX ^ f (/z - ?)) ( ^ - q ) dfx , ^{v -p ){v- q) dv 



J {X-d)L } {yi-6)M ' j {v-e)N 

e 

where tv = jdw. By supposing 6 to increase indefinitely, and re- 
placing 6^dw by ds, we have the corresponding result when 
Euclidian distance is used. 

In the notation of hyperelliptic functions (see Multiply -periodic 
functions, Cambridge, 1907, pp. 35, 36, using the p, q as a^, ag ^re 
there used), we have 

p {x — p) {x — q) dx 



2{d-p){d-q)]^^ x-e y 

y , e,x,s, x,Xo,y /,,^. =o\ ,, x,xo, 1 1 „ J (w ' + A;) 

where ((f)) denotes the place conjugate to (6), Ic is such that '^ {h) 
vanishes identically, and when d is large the significant terms of 

the functions l,-^, l,^ ^^^ d~^pq and d~^ {p + q). 

Along a straight portion of the curve, joining, suppose, the 

points (Aq, /Xq, Vq), (A, IX, v), the places (A), (/i), {v) of the hyperelliptic 

construct are coresidual with the places (Aq), {ixq), [vq), and we can 

satisfy the identities 

F {x) - [0 {x)f = 4 (X - A) {X - ix){x- v) ix ~f,) (x -/a), 
F (x) - [ipo {x)f =4:{x- Ao) {x - (Xq) {x - Vq) (x -/J (x -f^), 

VOL. XX. PART I. 9 



130 Professor Baker, On a property of focal conies, etc. 



where 

ip (x) = 2 {Ix^ + mx+ n), iJjq (x) = 2 {l^x'^ + m^x + n^). 

With this notation we find, for the Cayley separation of the two 
extreme points, 

„ = tanh-(^)-ta„h-(^^), 

leading in particular, if r be the distance of these points, to 
r = I — Iq, and to 



r^{d-X,){d~f.,){e-v,) 



2 tanh w 



Q + ipQ (d) tanh w ' 

The character of the symmetric functions of the places 
(A), (fx,), (v), regarded as functions of w, along any portion of the 
curve, seems eminently worthy of investigation. 

And it appears that the total value of tv, along any closed 
portion of the continuous curve, is expressible by an aggregate of 
the periods of the integral 

Q [ {x-p){x-q ) 

2{d-v){d-q)] {x-d)y ^' 

where y^ ^ F {x), with integer coefficients; these will then be un 
altered by any continuous small deformation of the arc of the 
curve. This remark appears to lead to all the known results. 

In conclusion I should like to refer the reader to a most inte 
resting note by Mr A. L. Dixon, Messenger of Mathematics, xxxii, 
1903, 177. 



Professor Baker, Ninth point of two plane cubics 131 



On the construction of the ninth point of intersection of two plane 
cubic curves of which eight points are given. By Professor H. F. 
Bakee. 

[Read 3 May 1920.] 

Cayley has collected, in a paper reprinted in Vol, iv of his 
Papers, pp. 495-504 {Quart. J., v, 1862), the various solutions given 
of this problem, regarded as a problem of plane geometry, by 
Pliicker, Weddle, Chasles and Hart, depending for the most part 
on the generation of a plane cubic curve (two points at a time) 
by the intersection of a pencil of lines and a homographic pencil 
of conies. So far as I have been able to notice, geometrical con- 
ceptions present themselves to an unbiassed child in the first 
instance as three dimensional, and he feels it to be an abstraction 
to regard plane geometry as self-contained; the discussion of the 
most natural Axioms of geometry seems also to point in this direc- 
tion; and the most valuable part of a training in geometry would 
seem to lie in the cultivation of a faculty for visualisation of 
relations in space. However these things may be, it appears to 
me always to be an interesting extension when a property of space 
is shown to follow from a property in space of higher dimensions, 
this being generally accompanied by the removal of some artifi- 
ciality. Thus, I regard the very simple example which now follows 
as being logically at least as fundamental as a proof in the plane. 

Let A, B, C, M, N and P, Q, R be the eight given coplanar 
points. Take a point D outside the plane of these. There are qo ^ 
quadric surfaces containing A, B, C and the lines DM, DN; let Q 
be one of these (other than that consisting of the planes ABC, 
DMN). Let DP, DQ, DR meet this quadric again in P^, Q^, Rj^. 
A definite twisted cubic curve can be drawn through D, A, Pj, Q^, R^ 
to have BC as a chord (see below). This cubic curve, meeting D. 
in D, A, Pj, Qi, Pj, meets O in a further point, say 0^. If DO^ 
meet the original plane in 0, this is the ninth point required. 

For the space cubic is the intersection of two quadric surfaces 
drawn through D, A, P^, Q^, Pj, both having the line BC as a 
generator; denote these by U and V. The quartic space curve of 
intersection of U with O contains D, A, B, C, P^, Qi, R^, and 
meets the generators DM, DN of Q; this curve then projects 
from D on to the original plane into a cubic curve containing the 
eight given points A, B, C, P, Q, R, M, N. The curve of inter- 
section of V with Q. projects from D into another cubic through 
these eight points. The point 0^, on the space cubic, lies on U 

9—2 



132 Professor Baker, Ninth point of two plane cubics 

and V, and on Q, and so projects from D into a point common to 
the two plane cubics. This justifies the statement. 

Incidentally any two cubic curves in a plane are shown to be 
the projections of two quartic curves in space lying on the same 
quadric; and the plane problem is put in connexion with the space 1 
problem of finding the remaining eighth intersection of three 1 
quadrics with seven common points. 

To construct a twisted cubic curve with five given points 
D, A, Pj, ^1, Ri to have a given line BC as chord, we may for 
instance first construct a quadric surface by the intersection of 
corresponding planes of two homographic axial pencils with DA, 
BC as axes, three pairs of corresponding planes being those con- 
taining Pj, Qi, Ri, and then construct a quadric surface by the 
intersection of corresponding planes of two homographic axial . 
pencils with DP^^, BC as axes, three pairs of corresponding planes 
being those containing A, Qj^, Rj^. These quadric surfaces intersect 
in the cubic curve required. 

It is seen that analytically each step requires only the solution 
x)f linear equations. Indeed, if the conic through A, B, C, M, N 
be written (referred to ^, P, C, D) as Ayz + Bzx + Cxy = 0, the 
line 31N being x + y -\- z = 0, we may take for Q, the quadric 
t {x + y + z) = Ayz + Bzx + Cxy. The general plane cubic curve 
through the five points A, B, C, M, N may be taken to be 

{Ayz + Bzx + Cxy) {Ix + my + nz) + {x + y + z) x [qy + rz) = 0, 

and two cubics through these and P, Q, R may be found by solving 
for the ratios of I, ni, n, q, r in the three equations obtained by 
substituting the coordinates of P, Q, R. Corresponding to two sets 
of ratios l-^ : m^ : n^ : q^ : rj, and l^ '. m^ : % • % '■ ^2 ^^ chosen, 
there are two quadric surfaces 

t {l-^x + nijy + fijz) + X {q^y + r-f^z) =-- 0, 

t {I2X + m^y + n^z) + x (q^y + r^z) = 0, 

which intersect in a cubic curve containing D, A, P^, Qj^, Pj and 
having BC for chord. The combination of these with the equation 
of O will lead to a linear equation for Oj, from which is found. 
Or the solution may be stated, naturally enough, without reference 
to three dimensions. 



Professor Baker, On a proof of the theorem of a double six, etc. 133 



On a 'proof of the theorem, of a double six of lines by projection 
from four dimensions. By Professor H. F. Bakek. 

[Read 9 February 1920.] 

The theorem in question is that if five lines in three dimensions, 
of which no two intersect, say a, b, c, d, e, have a common trans- 
versal, say / ', and we take the five transversals other than / ' of 
every four of these five given lines, the five new lines so obtained 
have also a common transversal. Namely if a' be the transversal, 
beside/', of b, c, d, e, and b' be the transversal, beside/', of 
a, c, d, e, and so on, so that we have the scheme 

a b c d e 

a' b' c' d' e' f 

in which every line intersects those not occurring in the same row 
or column with itself, but not the others, in general, then there is 
a transversal/ of a' , b', c' , d' , e' . 

We see that the theorem is that if we take eight lines a, b, c, d 
and a', b', c' , d' , so related that a' meets b, c, d, while b' meets 
a, c, d, and c' meets a, b, d and d' meets a, b, c, and if e',/' be 
the two transversals of a, b, c, d and e, f be the two transversals 
of a', b', c', d', then the meeting of one of the two former, / ', 
with one of the two latter, e, involves the meeting of the other, e', 
of the two former, with the remaining one, /, of the two latter. 
But the original relation of the eight lines a, b, c, d, a', b', c', d' 
has a certain artificiality; the object of the present note is to show 
that there is a simple figure in four dimensions, possessing perfect 
naturalness, being determinate when four arbitrary lines of that 
space are given, from which the figure in three dimensions may 
be derived by projection ; and that the condition for this derivation 
is precisely the intersection of the two transversals e and/'. The 
naturalness of this figure lies in the fact that three lines in four 
dimensions have just one transversal. 

§ 1. In order to show this, it is necessary to enter into some 
detail in regard to the elements of the geometry of four dimensions; 
this appears worth while for its own sake; and in order not to 
over-emphasize the importance of the theorem in three dimensions 
which is here made the excuse for this, we first give an elementary 
proof of this theorem, employing only three dimensions (Proc. 
Roy. Soc. A, Lxxxiv, 1911, 597), 



134 Professor Baher, On a proof of the theorem of a double six 

With the notation above, denote the respective intersections 
(6', c), (6, c'), {c', a), {c, a'), {a', h), {a, h'), (a,/'), {bj'), {c,f') by 
4, A', B, B', C, C, W, r, W. Let/ be the transversal other 
than e of a', b', c', d', which we may represent by/ = {a', b', c', d')le, 




'f V, 



Pig. 1 



and denote the points (a'J), {b',f), (c'J) respectively bv U, V, W. 
Similarly let /^ be the transversal other than d of (a'," 6' c' e') 
which we may denote by/^ = {a', bi, c', e')/d', and let'the points 
(«',/i), (6',/i), (c',/i), be U^, Fi, W,. ^ 

Now take the lines 

a, b, c,f; e 
a',b',c'J'-d''' 

The two quadric surfaces defined respectively as containing {b, c e) 
and {b', c', d'), have, both of them, the two generators e and 'd' 
which are intersecting lines. The other common points of these 
two quadrics are then coplanar. Such points are A and A' respec- 
tively {b\ G) and {b, c'), and U' or (a,/') and U or (a'J). Thus U 
les on the plane A, A', V). So, by considering the quadrics 
(c, a, e), [c , a , d ), we find that V lies on the plane (B B' V) 
and by considering the quadrics (a, b, e), (a', b' , d'), that W lies 
on the plane {C, C , W). By taking the hnes ■ 

^, b, c,f^\ d\ 
<&',c',/';4' 
and considering the pairs of quadrics 

{b, c, d), {b\ c', e'); (c, a, d), {c' , a', e'); {a, b, d), {a', b', e') 
we similarly show that V, V, W„ lie respectively on the planes 
{A, A, U), [B, B', V), {C, C, W), and therefore coincide 



of lines by projection from four dimensions 135 

respectively with U, V, W, being the intersections of these 
planes respectively with the lines a', b' , c' . Thus/^ =/is a common 
transversal of the lines a' , b' , c' , d' , e' ; as was to be shown. 

§ 2. Now take four arbitrary lines a, b, c, d in four dimensions, 
of which no two intersect. Two of these lines, determined by four 
points, two on each, determine a threefold space, defined by the 
four points, and this meets a third line in the four dimensional 
space in a point. From this point, in the threefold space, can be 
drawn an unique transversal to the two lines spoken of. Thus three 
lines in four dimensions, of which no two intersect, have an unique 
transversal. Let then a' be the transversal of b, c, d, and similarly 
6', c', d' the transversals respectively of c, a, d; a, b, d and a, b, c. 
Denote the points (6', c), (6, c'), (c', a), (c, a'), (a', 6), (a, b') 
respectively by A, B, C, A', B', C' and the points (a, d'), (6, d'), 
(c, d'), {a', d), (6', d), (c', d) respectively by P, Q, R, P', Q\ R'. 




In general use the word plane for the planar twofold space 
which is determined by three points, and the word space, or 
threefold for the planar threefold space determined by four 
points; as above remarked two lines determine a space, each 
line being determined by two points; reciprocally two spaces, in 
the most general case, intersect in a plane, there being a duahty 
of properties in four dimensions wherein a space is reciprocal to 
a point and a plane to a line. The points A, A', being respectively 
on the lines C'Q', BR', are in the space {a, d), and evidently are 
in the space (6, c); the points P, P', being on the lines QR, B'C 
respectively, are in the space (6, c), and are evidently in the space 
{a, d). Thus the four points A, A', P, P' lie in a plane, which we 



136 Professor Baker, On a proof of the theorem of a double six 

may denote by a, namely that common to the two spaces (6, c) 
and {a, d). We see how much more naturally this arises than the 
statement, to which it is evidently analogous, in the three dimen- 
sional figure considered in § 1. It follows that the lines A A' and PP' 
intersect one another, say in L. Similarly the plane, ^, of inter- 
section of the spaces (c, a), {b, d), contains the lines BB' and QQ', 
which then intersect, say in M; and the plane, y, of intersection of 
the spaces {a, b), {c, d), contains the lines CC and RR' , intersecting, 
say, in N. The points L, M, N are however all in each of the spaces 
[a, a'), {b, b'), (c, c'), and so in a line, the intersection of these 
spaces. For instance the line A A' joins a point {A') of the line b, 
to a point (A) of the line b', and so is in (6, b') ; and joins a point (A) 
of the line c, to a point (A') of the line c', and so is in (c, c'); 
thus L, on the line AA', is in the spaces {b, b'), (c, c'). But the 
line PP' joins a point (P) of the line a to a point (P') of the line a'; 
thus L is equally in the space {a, a'). Similarly both M and N 
are in the line of intersection of the spaces {a, a'), {b, b') and (c, c'). 
Thus the space {d, d') passes through the line of intersection of 
the spaces {a, a'), (6, b'), (c, c'); for we similarly show that each of 
L, M, N is in the space {d, d'). We denote this line by e; evidently 
its relation to the lines a', b', c', d' is exactly similar with its relation 
to the lines a, b, c, d; the plane, a, for example, defined as that 
common to the spaces (6, c), (a, d), is equally the plane common 
to the spaces {¥, c'), {a', d'); and so on. It is usual to speak of e 
as the line associated with a, b, c, d; examination of the figure of 
fifteen lines and fifteen points which we have constructed will show 
that there is entire symmetry of mutual relation, and that we may 
speak equally well of any one of the five lines a, b, c, d, e as being 
associated with the other four; further e is also associated with 
a', b', c', d'; and indeed, taking any line of the figure, the eight 
lines of the figure which do not intersect it, consist of a set of 
four skew lines and their transversals, and the line in question is 
associated with either of these two sets of four. There are then 
15'2 ^5 = 6 ways of regarding the figure as depending upon a set 
of five associated lines. 

§ 3. Consider now what planes exist meeting the lines a, b, c, d. 
In four dimensions an arbitrary plane does not meet an arbitrary 
line; two such elements which meet lie in a threefold space. It 
can be shown that a plane meeting a, b, c, d can be drawn through 
two arbitrary points, one on each of any two of these four lines, 
so that there are oo ^ such planes. Further that every such plane 
also meets the associated line e. Further that two planes meeting 
a, b, c, d can be drawn through an arbitrary point of the four 
dimensional space, and, for instance, an infinity of such planes 
can be drawn through any point of the Une e. Also, if the two 



of lines by projection from four dimensions 137 

planes through an arbitrary point 0, to meet a, b, e, d, meet the 
line e in T and U, then the two planes which can similarly be 
drawn through to meet the lines a', b', c', d', meet the line e 
in the same two points T and U. In general two planes in four 
dimensions have only one point in common; when they have two 
points in common, the join of these points lies in both the planes 
which then both lie in the same threefold space. By what we have 
said there is a plane through OT intersecting a, b, c, d and also 
a plane through OT intersecting a', b', c', d', with a similar state- 
ment for planes through OU. Namely considering the two planes 
through which meet a, b, c, d and also the two planes through 
which meet a', 6', c', d' either one of the former meets one of the 
latter in a line. 

To prove these statements we may proceed as follows. The 
joining line of two points arbitrarily taken respectively, say, on the 
lines b and c, will meet the space {a, d) in a point, from which, in 
this space, a transversal can be drawn to a and d. Then the plane 
of the original join and this transversal is a plane, say w, meeting 
the four lines a, b, c, d. The point of intersection of these two lines 
determining this plane m is evidently on the plane, a, common to 
the spaces (6, c) and {a, d). Similarly the point of intersection of 
the plane m with the plane, ^, common to the spaces (c, a) and 
(6, d), is a point from which two transversals can be drawn respec- 
tively to the pairs of lines c, a and b, d; and the plane of these 
transversals is a plane through this point meeting the four lines 
a, b, c, d; conversely the join of the two points where the plane C7 
meets the lines c and a lies in the space (c, a), and so intersects the 
plane /S, namely in the supposed unique point common to w and j8; 
this join is thus identical with the transversal drawn from the 
point (m, ^) to the lines c, a. There is thus an unique plane m, 
meeting a, b, c, d, passing through any general point of the plane «, 
beside the plane a itself. It will follow from the general result 
enunciated above, to be proved below, that the plane w', drawn 
through the same point of the plane a to meet the lines a', b' , c' , d' , 
meets C7 in a line intersecting the line e. 

Take now any general point 0, and a varying point P of the 
line d; a plane can be drawn through OP to meet the lines a, b, 
this being the plane containing OP and the common transversal 
of OP, a and b. Let this plane meet a, b respectively in P^ and P^. 
Thereby any position of P, on the line d, determines the position 
of Pi on the line a. Conversely given and Pj, a plane can be 
drawn through OP^ to meet 6 and d, which, being unique, coincides 
with the former. Thus any position of Pj on a determines the 
position of P on d. The correspondence being algebraic, it follows 
that Pj, P describe homographic ranges respectively on a and d. 
Using the line c instead of b, we obtain another range (P') on d. 




138 Professor Baker, On a proof of the theorem of a double six 

also homographic with (Pj). Thence the ranges {P'), (P), on d, are 
homographic ; and, if not coincident, they will have two common 
points, which may coalesce. When P has a position in which it 
coincides with P', there is a single plane containing 0, Pj, P2, P3, P, 
where P3 is the point of c on the plane OP, P' . Thus through the 

point can be drawn, either an 
infinity of planes meeting all of 
a, b, c, d, or else two, which may 
however coincide. 

When is on the line e, the 
plane Od' meets a, b, c, and it 
also meets d because, as we have 
shown, e, d, d' are in a three 
dimensioned space. Equally 
the planes Oa', Ob', Oc' meet 
a, b, c, d. As there are thus 
more than two planes through 
meeting a, b, c, d, it follows, 
by what we have shown, that 
there is an infinity; this is 
when is anywhere on the line e. The aggregate of planes so 
obtained, by taking to be every point of e, is identical with the 
aggregate of all planes meeting a, b, c, d, namely any plane meeting 
a, b, c, d can be identified with one of these; for taking on e, 
and P on d, this P determines Pj, P^, respectively on a, b, when 
regarded as belonging to one of the coincident ranges on e, and 
determines Pj, Pg, respectively on a, c, when regarded as belonging 
to the other range on e. Thus every plane meeting a, b, c, d also 
meets e, or more generally five associated lines are such that every 
plane meeting four of them also meets thefifth"^. 

In general, as we have seen, from any point on a plane meeting 
a, b, c, d (and e), there can be drawn another such plane. If the 
point be on the conic through the five points in which the first 

* The reader may compare the proofs of this result given by vSegre, Circolo 
Mat., Palermo, n, 1888, 45, Alcune considerazioni....The elementary theorems 
here given for the geometry of four dimensions are of course well known; but I 
have thought that it was necessary for the purpose of this Note to supply demon- 
strations. The reader may consult Bertini, Introduzione alia geometria projettiva 
degli iperspazi, Pisa, 1907, a volume of 400 pages, p. 177. In English there is 
Mr Richmond's paper On the figure of six points in four dimensions, Quart. Journ., 
XXXI, 1899; Math. Annal., Lm, 1900 (see also Trans. Camb. Phil. Soc, xv, 1894, 
267), which deals with a diagram intimately related with that of the text, and 
CooMdge, A treatise on the Circle and Sphere, Oxford, 1916, p. 482, etc., where the 
lines of four dimensions are replaced by spheres. The origin of the five associated 
lines seems to be a result given by Stephanos, Compt. rendus, xcm, 1881, p. 578. 
I have not seen it formally remarked that the property of the double six follows 
from the geometry of four dimensions ; indeed the argument given in § 1 was invented 
in ignorance of this. The fifteen points and lines of our figure (Fig. 2) are the 
diagonal points and transversal lines of the figure considered by Mr Richmond. 
See also Hudson, Kummer's Quartic Surface (1905), Chap. xii. 



of lines by projection from four dimensions 139 

plane meets a, b, c, d, e, the second plane coincides with the first. 
It is not necessary for our purpose to prove this, 

§ 4. The theorem that two planes can be drawn from an 
arbitrary point to meet the lines a, b, c, d is obvious from the 
theorem in three dimensions that four skew lines have two trans- 
versals, the proof of which also depends on the fact that two homo- 
graphic ranges on a line have two common points. For, if we project 
a, b, c, d from 0, on to an arbitrary threefold space E, the planes 
joining to the two transversals of the four lines of S so ob- 
tained, all meet a, b, c, d. And, we now see, e projects into a fifth line 
meeting these two transversals. When is on e, the projections 
in 2 of a, b, c, d are all met by the projections in S of a', b', c', d' ; 
for the plane Od' , for example, meets a, b, c, and meets d because 
e, d, d' are in the same three dimensional space; thus the pro- 
jections in S of a, b, c, d are four generators of the same system 
of a quadric surface of which the projections of a', b', c', d' are 
generators of the other system. The planes from each meeting 
a, b, c, d intersect the space S in lines all meeting the projections 
of a, b, c, d; that is, in lines which are generators of this quadric 
of the same system as a', b', c', d'. The planes from meeting 
a', b', c', d' similarly give rise to generators of the system (a, b, c, d). 
Thus any plane from the point meeting a, b, c, d meets any 
plane from drawn to meet a', b', c', d' in a line through 0; and 
every line drawn from in a plane of the former system is the 
intersection with this plane of a plane of the second system. If 
be a point of e lying on a plane drawn from a point H, not on e, 
to meet a, b, c, d (which therefore also meets e), the line HO lies 
in a definite plane meeting a', b' , c' , d' . Thus either of the two planes 
of the first system, those meeting a, b, c, d, drawn from a point H, 
not on e, meets one of the two planes of the second system, those 
meeting a', b' , c' , d' , drawn from H, in a line; and the two lines so 
arising intersect the line e. 

§ 5. Hence we can obtain from the four dimensional figure a 
figure in three dimensions with the characteristics of that used in 
proving the double six theorem. 

If, in the four dimensions, p, a be the planes drawn from an 
arbitrary point to meet a' , b' , c' , d' , and p , a' those meeting 
a, 6, c, d, and if p and a' meet in a line, as also p and a; and if 
we consider the intersections with an arbitrary threefold space S, 
of these four planes, and also of the planes joining to a, b, c, d, 
a', b', c', d' , denoting these twelve lines respectively by (p), ...,{a), ..., 
then, arranged as follows: 

{a) (b) (c) (d) (p) (a) 

{a') (6') (c') id') ip') {a'), 



140 Professor Baker, On a proof of the theorem of a double six 

these form a double six, any one of the lines meeting the five which 
do not lie in the same row or column with itself. 

§ 6. Conversely we now proceed to show that if 

a^ h^ Cj d^ 

a{ h{ c-l d^ 

be eight lines in three dimensions such that no two of a^, h-^, c^, d^ 
intersect, while d-l intersects a^, 6^, c^, a{ intersects h-^, Cj, d^, etc., 
and if one of the two transversals, say I, of a^', hy, c{, d^, intersects 
one of the two transversals, say m' , of a^, \, Cj, d-^, then these lines 
may be obtained by projection from four dimensions; namely 
«j, &!, Cj, d-^, a{, bj', Cj', c?j' are projections of four lines a, b, c, d 
in space of four dimensions and of the transversals a', b', c', d' of 
threes of these, respectively, while I and m' are the intersections 
w^ith the original three dimensional space of planes in four dimen- 
sions meeting respectively the set a', b' , c' , d' and the set a, b, c, d. 

We give an analytical proof of this. And for this purpose first 
explain an analytical view of the theorems which have been given 
in §§ 2, 3, 4, which indeed renders these very obvious. 

It is fundamental that a point may be represented by a single 
symbol, say P, the same point being equally represented by any 
numerical multiple of this, say mP, where m is an ordinary 
number. Then a space of r dimensions is one in which every y + 2 
points, Pi, P^, ..., Pr+2^ are connected by a sy2ygy, 

w^Pi + m^P^ + ... + m^+2P^+2 = 0, 

where m^, ..., m^+2 are ordinary numbers; thus the space is deter- 
mined by any r + 1 points of it, themselves not lying in a space 
of less than r dimensions ; and, in terms of such r -f- 1 points, say 
ylj, ..., ^y+i, every other point of the space may be represented by 
a symbol Xj^A-^^ + x^A^ + ... -f x^+i^^+j, where x-^, x^, ..., x^+i are 
ordinary numbers ; whose ratios may be called the coordinates of 
this point, relatively to Aj^, ...,Ar^j^. Thus any point of a line 
determined by two points A, B, is representable by a symbol 
mA + nB, in which m, n are numbers; and any point of a plane 
determined