xl 



q (= e K ) : yet the results are such that they ought to be deducible 

 by ordinary algebraical processes. It was Cayley's wish to deal with 

 this theory by pure algebra ; two simple cases had already thus been 

 discussed by Jacobi, but the extension to the less simple cases proved 

 difficult. Cayley's ' Memoir on the transformation of elliptic 

 functions,'* carries on the algebraical theory and places it in a 

 clearer light than before. But though he made a distinct advance in 

 dealing with particular cases, he still found it necessary to use the 

 ^/-transcendents for making any definite advance in the general case. 

 And the same compulsion occurs in the chapters of his 'Treatise' 

 (dated 1876), where transformation is discussed at considerable 

 length. 



He resumed his investigations in 1886, still dealing with the 

 algebraical method, but applying it to a simplified form of elliptic 

 integral due to Brioschi. Though the problem is not solvedt com- 

 pletely for the general case, he has devised a method which is 

 effective at least in part ; it easily leads to- new results connected with 

 the modular equations in the known simpler cases previously solved. 

 The theta-functions are the subject of several of his papers. He 

 began J with a direct establishment of Jacobi's relation 



obtained in the ' Fundamenta Nova ' by a long and cumbrous process ; 

 and he proceeded to the construction of the linear differential equa- 

 tions satisfied by the theta-functions. Except, however, in so far as 

 they arise in the transformation theory, they do not appear to have 

 occupied him until about 1877. In that year and in, the succeeding 

 years he wrote a number of papers dealing with the theta-functions 

 HS on an independent basis and not as a detail in elliptic functions. 

 Though the investigations are concerned with p-tuple functions, yet, 

 partly for simplicity, and partly in order to secure the greater 

 detailed development of the theory, the papers deal chiefly with the 

 cases p = 1, p = 2. 



Previous to Cayley's investigations, the most valuable algebraical 

 results in this subject were those of Rosenhain and Gopel, || which 

 had connected the double theta-functions with the theory of the 



* ' C. M. P.,' vol. 9, No. 577 ; ' Phil. Trans.,' 1874, pp. 397456. 



f The memoirs of this period belonging to the transformation of elliptic 

 functions were published in the ' American Journal of Mathematics,' vol. 9 

 (1887), pp. 193224; vol. 10 (1888), pp. 7193. 



J " On the Theory of Elliptic Functions," ' C. M. P.,' vol. 1, No. 45 ; ' Camb. 

 and Dubl. Math. Journal,' vol. 2 (1847), pp. 256266. 



'Mem. des Savans Etrangers,' t. 11 (1851), pp. 361 468; the paper is dated 

 1846. 



|| ' Crelle,' vol. 35 (1847), pp. 277312. 



