XXXIX 



where w = (ro, n) = mO-f nY, tho ratio Q : Y is not real, and the 

 product is taken for all positive and all negative integer values of m 

 and of n between positive and negative infinity, except simultaneous 

 zero values. He showed that such products can be used to obtain 

 Jacobi's elliptic functions by constructing fractions such as 



*OM- fa) -*(*)! 



and he also showed that the actual value of any product involves an 

 exponential factor e Biei , where the value of the constant B depends 

 npon the relation* between the infinities of m and of n. The results 

 were of definite importance at the time of their discovery, and they 

 still hold their place. But the form of the doubly-infinite product 

 has been modifiedf by Weierstrass, who takes 



}- 



a function the value of which is independent of any particular form 

 of relation between the infinities of m and of n. Owing to the latter 

 simplification, Cayley's results are, as he himself remarked,]! partly 

 superseded by those of Weierstrass. 



Cayley had great admiration for the works of both Abel and 

 Jacobi ; he had begun to read the latter's ' Fundamenta Nova ' imme- 

 diately after his degree. The prominent position occupied in that 

 work by the theory of transformation naturally attracted his interest ; 

 and, even as early as 1844 and 1846, he wrote short memoirs upon 

 the subject, obtaining in one of them a function, due to Abel and 

 now known as the octahedral function. Further memoirs of a similar 

 tenor appeared occasionally ; they deal chiefly with transformation as 

 concerned with the known differential relation of the form 



The contributions made to the transformation theory by Sohnko, 

 Joubert, and Herinite, as well as Jacobi's original investigations, all 

 depend upon the use of transcendental functions of the quantity 



* This is sometimes expressed differently, as follows. Points are taken having 

 m and n for their Cartesian co-ordinates ; those which occur for infinite values 

 of m and of n lie at infinity, and may be considered to lie upon a curve alto- 

 gether at infinity, the shape of which is determined by the relation between 

 the infinities of m and of n. 



The value of the constant B is said to depend upon the shape of this bound- 

 ing curve. 



t Weierstrass's investigations on infinite products are contained in his memoir 

 " Zur Theorie der eindeutigen analytischen Functionen" (' Abh. d. K. Akad. d. 

 Wiss. zu Berlin,' 1876) ; also in his book ' Abhandlungen aus der Functionenlehre,' 

 18S6. 



I ' C. M. P.,' vol. 1, p. 586. 



