MULTIPLICATION OF ELLIPTIC FUNCTIONS. 215 



and, generally, 



(145) (2w+l)(2wi + 2)Z)2,„+2,2r = -M{m-r+l)ir-m'\D,„,^,,_, 



- 4(;vr - vi')D,„^ ,, - («^ - 2m + 1 ){n' - 2m + 2)D„„_,^ ,^-2 , 



the la«t coefïicient being given by 



(2w+l)(2m + 2)Z)2,„+2.m+i = -8|^^^^i^-i';r|A..,m-i 



or 



(2;;i+lX2w+2)D2„+2,^ = -2\{m + 2W-2m'\D,„,^,„_, 



-^{^n'-m^D„,^„,-{n'-2m+l)i)i'-2m+2)D,^ 



accordino- as m is odd or even. 

 By means of (145), we find 



(146) 



2>n-'2, w-2 > 



6! 



7^V-l)(7^'-4) 

 8! 



A = - ''^''' tl ^ ^{^Oi'-'àW + k'j + ain'-^m^, 



^^_ 7^;^^ 1)(M^^^4)(^/^ '')) .^2{4{n'-l6W + k')+ l5(M^-4)(^^ + A«) [ , 

 Dj2= _ »'(^''^^X»^-^)(»'-9) 3,g4(,„2_i(3)(,,2_25)(^2^^.io) 



+ .S(?i2- 16X47»2_ 185)(Ä;H Ä;«)+ I5(45?i*-5G9M2+ 1544)Ä''[ , 

 Dj^= »'0^'-n^^'-4)(^'''-^)32|ö4(M^-l6)(7^^-25)(;^^-36)(A.-HA:^^) 



+ 1(3(^^2- 16)(34yi^-982;r + 3300)(Ä,'»+ A-^") 

 + (1549;^'-43925?iH 357196m-- 8l5040)(Ä;''+A-8)}. 



