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LXI. Investigation of the Transformation of certain Elliptic 

 Functions. By Arthur Cayley, Esq.^ B,A., F.C.P.S., 



Fellow of Trinity College, Cambridge. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



I should be obliged if you would do me the favour of insert- 

 ing the following investigation of the transformation of 

 elliptic functions, which appear to me completely to demon- 

 strate that no limitations are required in Jacobi's conclusions 

 (see Phil. Mag. S. 3. vol. xxii. p. 358 ; xxiii. p. 89). I should 

 not have resumed the subject had not the method employed 

 seemed to me to possess some independent interest. 



I remain, 



Your obedient Servant, 

 29 York Terrace, October 10, 1844. A. Cayley. 



The function sin amu (4> m for shortness) may be expressed 

 in the form 



*" = ""(' +27JktWk7-,) 1 



• V/^ 2wK + (2w' + l)K'JJ 

 where m, m' receive any integer, positive or negative, values 

 whatever, omitting only the combination m = 0, m' — in the 

 numerator (Abel, CEuvres, t. i. p. 212, but with modifications 

 to adapt it to Jacobi's notation ; also the positive and negative 

 values of W2, m' are not collected together as in Abel's formulae). 

 We deduce from this 



iMl) = n(.+ 



nO 



2mK + (2m'+l)K'i + 9. 

 Suppose now K = aH + fl'H'<, K' . = ^» H + Z»' H' ., «, 2», a', ^>' 

 integers, and aU — a' b a positive number (v). Also let 

 CO =/H +/' W;f,f' integers such that a/' - a'f bf - b'f 

 V, have not all three any common factor. Consider the ex- 

 pression 



_(pu.<p{7l + 2ca)...(p{u + 2{v—l)a)). 



<P{2co) ...4)(2(v— !)«;) 

 from which 



y = wn(^l +2mK + 2w'K'i + 2r9/ 

 "^ ^ V "^ 2»jK+(2m'+l)K'i + 2rfl/_ 



(3.) 



(4.) 



