OP ELLIPTIC FUNCTIONS. 
437 
viz. the cases a e , a n correspond to Jacobi’s first and second real transformations, and the 
others to the imaginary transformations. 
I remark that u=a 0 gives for snc 2 ga an expression which is rational as regards q, 
but a—a n an expression involving q n , the real nth root of q ; the other values, a t , a. 2 , . . 
I 2 
give the like expressions, involving uq n , a 2 q n , ... (a an imaginary nth root of unity), the 
imaginary wth roots of q. 
53. I consider first the expression 
1 1 dn 2 geo 0 
snc 2gco 0 ’ sn(K — 2_j/« 0 ) ’ ~ cn2gto 0 
2K£ 
Here, writing 2gco > 0 =— ^(£ for Jacobi’s x, as x is being used in a different sense), that is 
% 2K ' J ‘ n ’ n 
(and thence e^—e 11 =u s , e^=u? s , if a=e n , an imaginary nth root of unity), we have 
(Jacobi, p. 86) 
l , 2K? 2K£ 
— n — =dn- — - -f-cn — 
snc 2gu 0 ir tc 
C 2e*t (1 + qe 1 ^) . . (1 + qe~^) . . 
B ’ 1 + ' (I + ? V*) . . (1 +q*e~ 2i t). . 
that is 
(§=(§rfr^) =/%)) * 
1 . faf) . (l + «^g) .. (l+«"~^g).. 
snc 2gco 0 1 + o.2s J (14 -«‘^£ 2 ) . . (1 +a”“ 2 ^) . 
where, for shortness, I write (1 -\-qe 2i *). . . to denote the infinite product 
(1 +^)(1 +2 V^)( 1 +^)..., 
and similarly (l+g'V 1 ^) ... to denote the infinite product (1 + y V if )(l +2' 4 ^)(1 +# 6 ^ 2 ‘0 • • 
and the like for the terms in e ~ 2i f : the notation, accompanied by its explanation, is quite 
intelligible, and it would be difficult to make one which would be at the same time 
complete and not cumbrous ; and then attributing to g the values 1, 2 . . . \(n — 1), and 
forming the symmetric functions of these expressions, we have the values of -, £, &c., or 
a being put =1, say the values of 0, y, . . . <r. 
54. I stop to notice a verification afforded by the value of 0 O . Putting u— 0, that is 
q=t), we have 
2uS 
and thence 
snc 2gw 0 1 + 
0 [ « a 2 a 3 ] 
^ 0 = ^{l+« 2 "^l+« 4 'T' 1 + a 6 ' ' + 
MDCCCLXXIY. 
3 M 
