MULTIPLICATION OF ELLIPTIC FUNCTIONS. 



159 





yin „ „m— 1 



m ! , D"\sz"'-'^) 



{m—l)ll ^™+iD'^(s^"' 2) 



Hen 



ce 



(12.) P,=^m !! z"'+'\a-{)ii+\) TW^Yai 

 Aoain 



+ D"\s. A., + . . . + D"'sz"''-. A 



.] 



(13.) P,„+i = (-l)'"+^ 



},z, z^,... z'", s, sz, 



. sz 



l,'2z,... mz'"-"'^, Ds, Dsz, Dsz'"''^ 



m\,D"'s, D"\sz, ...D"'sz"''^ 

 B"^+h,D"'+hz,...D'''^hz"'^^ 



D-"'s, D'"'sz ...D-"'sz"'-'^ 



= (-1)'«+1 ;;i!! A . 



Suiwtituting the values of 1\ and P,„+i just found in the tirsf uf 



2in+l 



e(juation.s ((S.) and dividing by z ^ , we get 



(14.) 



V: = (-i)'«+V2 



_A--(/M+ 1)|/J''/— YAl + Z)'^^A,,+ ... + D''^s;^•™-2. A, 



A 



The expression f(K- A given in ([).) may be greatly sinipJitied. 

 J>y an easy reduction, we hud 



(15.) 

 A = 



D"'+\S {m+ \)D"'s, {,ic + l)D"'-i.s, {{)n+ \)iii. . .^ \D-s 



W'-^h, {m + 2)£>'«+\s, (VM + 2)u/i + 1 )D"\s, { {m + 2)(??i + 1) . . . 4 [ D=^s 



£»2^s, '2mD-"''h, 'Imi^m - 1 )L>=^"'--'i-, { 2m{2ni - 1) . . . (yy^ + 2) [ D"'+\s 



