BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
xliii 
ns as on an inde- 
3 investigations are 
in order to secure 
efly with the cases 
results in this sub- 
>uble theta-functions 
if Weierstrass, deve- 
in his “ Memoir on 
lan by Rosenhain’s, 
die relations among 
atic relation among 
timer’s sixteen-nodal 
ssociated surface he 
in researches upon 
s on the porism of 
atic property of two 
lolygon (and, if one 
le and circumscribed 
ase when the conics 
elliptic functions for 
jussion of the most 
bainedli the relations, 
ueagon. And it may 
roceeding to general 
memoir** “On the 
ion as to the number 
pven curve or given 
given curves of any 
n completely in the 
; arise through the 
tie same curve. 
theoretical dynamics, 
as professor. Among 
lose on the attraction 
methods of Legendre, 
hi. (1828), pp. 376—389. 
:v. No. 267. 
Jacobi, Gauss, Laplace, and Rodrigues; and his evaluations or reductions of multiple 
definite integrals connected with attractions and potentials in general, particularly his 
“ Memoir on Prepotentials*,” in which he discusses the reduction of the most general 
integral of the type that can occur in dealing with the potential-problem related to 
hyperspace. He also frequently recurred at intervals, before drawing up his reports about 
to be quoted, to the consideration of the motion of rotation of a solid body about a 
fixed point under no forces. By introducing Rodrigues’s co-ordinates into the equations of 
motion he was able to reduce the solution of the problem to quadratures; but the 
final solution of this case, in the most elegant form, is due to Jacobi himself; it 
involves single theta-functions. It may be remarked that the next substantial advance 
made in the theory of motion of a body under the action of forces is due to the 
late Madame Sophie Kovalewsky, who, in a memoir+, to which the Bordin Prize of 
1888 was awarded by the Paris Academy of Sciences, has shown that the motion can, 
in a particular case, be determined in terms of double theta-functions when the body 
rotating round a fixed point is subject to the force of gravity. 
Sometimes, after reading widely upon a subject, Cayley would draw up a report 
recounting the chief researches in it made by the great writers. It occasionally happens 
in the development of a theory that periods come when the incorporation and the 
marshalling of created ideas seem almost necessary preliminaries to further progress. 
Cayley was admirably fitted for work of this kind, owing not only to his faculty of 
clear and concise exposition, but also to his wide and accurate knowledge. Among such 
reports, two are of particular importance; his “ Report on the recent progress of 
theoretical dynamics | ” and his “ Report on the progress of the solution of certain 
special problems of dynamics § ” have proved of signal service to other writers and 
to students. His knowledge and his power of summarising are shown also in some 
interesting articles on mathematical topics, written by him for the Encyclopedia 
Britannica. 
Cayley also had a great enthusiasm for some of the branches of physical astro 
nomy. Some idea of the value and importance of his labours in this subject, par 
ticularly in connexion with the development of the disturbing function in both the 
lunar theory and the planetary theory, and with the general developments of the 
functions that arise in elliptic motion, may be gathered by consulting the series of 
memoirs || which he communicated to the Royal Astronomical Society. 
Special reference should be made to one of Cayley’s astronomical papers. In 1853 
Adams had made a new investigation of the value of the secular acceleration of the 
moon’s mean motion, and, taking account of the variation in the eccentricity of the 
earth’s orbit, had obtained a value which differed from that given by Laplace. Unfor 
tunately, Adams’s result was disputed by some of the great school of French physical * * * § * 
* Phil. Trans. 1875, pp. 675—774. 
t Mem. des Sav. Etr., vol. xxxi. (1894), No. 1. 
+ G. M. P. vol. hi. No. 195 ; Brit. Assoc. Report (1857), pp. 1—42. 
§ C. M. P. vol. iv. No. 298; Brit. Assoc. Report (1862), pp. 184—252. 
|| They are included, with very few exceptions, in the third and the seventh volumes of the Collected Mathe 
matical Papers.