MULTIPLICATION OP ELLIPTIC FUXCTIOXS. 201 



or, ]uitting for a moment Wn — ir + ^j — '/'ii > 



ÇV2+%''n+^ . = 0. 



Observing </'i=-^, <r's=—'ö'' we find 



ç''„= coef. of ,«" in -^ ^y:j-^ "" ^~^^ ^ "2 



Hence, 



9?2 »til ,, ^,2 »2zl 71 



(114) ;,-| + (_l)2|._l, ^,,^_|-_(_1)2|., 



that is. 



or 



1 1 



'^« = VC" + 1)-1 ' /A, = ^"("--l). 

 according as -—^ — is odd or even. 



This premised, we now h^ubstitnte D=^^H,/£'' in (99), whirli may 

 1)e written in the form 



(i.^'-^r=:r — dr — 1 )«ç -j^ + 2ura-—- = ( 1 + ç" -y^— 'J(7r — 1 )?'-î^ 



+ «\^r-l),-^F+8n^|^, 

 and obtnin 



(115) ç^-^^0f—\)^^^+2n'qH,=^O, 



(116) ç^ '^^^ - Or- 1 ),-'^ + 2u\q - l)7f,_i=(l + ç^)^^ - 2Cn^- 1 )ç^'^' 



+7?,v-i;ç^H , 



