LXVII. On the Transformation of Elliptic Functions. By 

 Arthur Cayley, Esq.^ M.A., F.C.P.S., Fellow ofTrinit]) 

 College^ Cambridge^. 



IN ^ former paper I gave a proof of Jacobi's theorem, which 

 I suggested would lead to the resolution of the very im- 

 portant problem of finding the relation between the complete 

 functions. This is in fact effected by the formulae there given, 

 but there is an apparent indeterminateness in them, the cause 

 of which it is necessary to explain, and which I shall now 

 show to be inherent in the problem. For the sake of sup- 

 plying an omission, for the detection of which I am indebted 

 to Mr. Bronwin, I will first recapitulate the steps of the de- 

 monstration. '• 



If — CO, -^wf be the complete functions corresponding to 

 (p A", then this fpnction is expressible in the form 



^r=^n(i+; 



m CO + 



Letp be any prime number, j«., v integers not, diyisib.le jj»^^ 

 p, and 



"" ^ 

 The function j, 



— <p (:y + 2 fl) <p{a; + 4>Q) <p{x + 2 {p— })&) 



'Pi^-f^' ^ ^Tf~"' f{2{p-l}Q) 



is always reducible to the form 



:cu(i + , /^ , , ) - n /l+ ^ \ . 



\ m' co' 4- w' o7 I 1 1 j 



y m' + —uJ + n' + Y^/ 



rd 



Or (pj X is an inverse function, the complete functions of 



hich are — - co', -— v' 

 2 2 



CO, u by the equations 



which are — - co', -— u'. And where co', u' are connected with 



2 2 



co' = — (a CO + ^ y)) 

 u' = («' CO + f' u). 



mt'jH( 



a, I, «', S' being any integers subject to the conditions that 

 a, ^' are odd and «', ^ even ; also 



* Communicated by the Author. 



t Aualogous to the K, K' i of M. Jacobi. 



