MUL'J'IPLICATIOX OF ELLIPTIC FUNCTIONS. ^ll 



dV 



a85) |l-a + AV- + ZV['^+{U^;?2._i)^,2_i),_2^,,2_i^y;,2. 



y3L 



' doc 



(il: 



We m:iv sujjpose tlie iiiimerator of sii //// to be aiTanged nccordinijf 

 'to the powers of /r nnd write 



(136) .r^Cr^) = 2] Pojr'". 



Then, denotino- the o])er:itor x^r- by 'V, we hnve 



dx 



d'P. 



(137) ^ 4- {»\^m + 1) - /y^jPa. = ( i^m - 3)»^ - -i/^^/y + â')P,,„^^ 



For 7^? = (). we have 



and, if we })iit 



then 



r/2p 

 a./'- 



Po = 2/3.-r^^+\ 



^9,(2r+l)2r = -^?^_,o,2_^2r-l)2) 



And, since ,3',, = «-, /I. = (— 1)'^ . , , -; thu.s we 



obtfiin the well-known series 



(138) P„ = 2 ,_ i,iH»!=iM-"=i?r^I)l ,.« . 



For 1]}:= 1, we have 



Since the lowest powcn- of .r in Pg i^^* 3, we write 



