xlii 



evaluations or reductions of multiple definite integrals connected with 

 attractions and potentials in general, particularly his ' Memoir on Pre- 

 potentials,'* 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 inter- 

 vals, before drawing up his report about to be quoted, to the considera- 

 tion 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 quadra- 

 tures ; but the final solution of this case, in the most elegant form, is 

 due to Jacobi himself ; it involves single theta-f unctions. 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 memoirf 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 pro- 

 gress. 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 ' J 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 mathe- 

 matical topics, written by him for the ' Encyclopaedia Britannica.' 



Cayley also had a great enthusiasm for some of the branches of physi- 

 cal astronomy. Some idea of the value and importance of his labours in 

 this subject, particularly in connection 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. 



* ' Phil. Trans.,' 1875, pp. 675774. 

 t ' Mem. des Savants Etrang.,' t. 31 (1894), No. 1. 

 J ' C. M. P.,' Tol. 3, No. 195; ' Brit, Ass. Eeport' (1857), pp. 142. 

 ' C. M. P.,' vol. 4, No. 298; 'Brit. Ass. Report' (1862), pp. 184252. 

 || They are included, with very few exceptions, in the third and the seventh 

 volumes of the ' Collected Mathematical Papers.' 



