MULTIPLICATION OF ELLIPTIC FUNCTIONS. 



185 



(70) ii/;= 



ii-r-zT+\ r-{i-r~zr, Ä-\i-Ä^.^r-s ... a-" 



0, 



]_ 



s '■ 



-1 ±d'- 



s ' lis' 



lis' ■!'. s' 



. 1 D»-I 



ml s 



1 D..-.1 ±D- 1 1 



(7;t— 1)! s ' m! s ' {iit+ i}\ 



D" 



1 _jr)2,„_L 



(li/yij: 



Ob.serve that the expression of sn mi ns giyen hy (f 7) and ((iS) is 

 that which has been indicated by Jacobi. It wiJJ, liowever, be more 

 convenient to write i)// as follows : 



(71) in; = 



ym+l 

 •' > 



0, 



J_ 



s ' 



— Z" 



..(-1)" 



-1 ^dI 



5 ' 1! s' 



1 ! s ' t2 ! s ' 



«il S 



■•(;y;+L)! 6' 





1 T.. 1 



C2;;i)! .s 



We may liere point ont that the comparison of the preceding 

 formula' with the corres])onding expressions of § 8, gives some elegant 

 theorems in determinants. Take for example il/ and M' . We find 

 easily 



where C denotes a constant wliich may be a function of m. To 

 determine C we may take some particular values of m. For ;// = 1, 

 we see at once that 



•2! 



Bh = -.s^ 



1, dK 



I) 



-s- ' '2 



-D- 



