OF ELLIPTIC FUNCTIONS. 
441 
where the notations are as follows : 
(l+g 1+ «) . . is the infinite product (l+ 2 1 +B )(l+ 2 3 + ^)(l+ 2 ,S+B ") • • 5 
and 
„, 2 g ,*g ' 2 £ fi , 2 £ 
( 1+2 ») . . the infinite product (l+g “)(l+2 n ){l-\-q n ) • . ; 
and the like as to the expressions with exponents containing—^-. 
And then' attributing to g the values 1, 2, . .\{n— 1), and forming the symmetric 
functions of these expressions, we have the values of ; or a being put =1, say 
the values of (3, 7 , . . . a. 
It is easy to see, and I do not stop to prove, that if instead of &/=&/„ we have 
1 
cj. 2 . . . or <y rt _ 15 we simply multiply g n by an imaginary wth root of unity ; that is, 
\_ 
replace the real nth root q n by an imaginary nth root of q. 
In the case u— 0, that is g=0, we have — ^ — =0, and thence 3—0; and the like 
for the values.^, co . 2 , . . . a n _ x : the equation 2/3=^ — 1 £ives consequently for n values 
each = 1 , agreeing with the multiplier equation! 
61. We have for M„ the formula 
and, as before, 
1 , u fa _ n f sn2q;„ sn4a/„ , . . sn(ra— !)«/„ 
M„ ' ' [snc2a>„ snc4w„. .. snc(rc— l)w„ 
sn 2 gv n =f 2 (q) . 
2/r „.2 e , 
g»-i a-g '-)•• a-g*"* ).. 
2 ir 
hence 
2g OA-- g .,?? „ ?£ , % 
sn2 gu> n _ qn— 1 (1 — g «).. (1 + g ■):. (1— g »).. ( l+g ») .. 
me2ffa,n %» + l) (l+g S+ »)V. (i-g ,+ +.. (l+g 2 -»).. (1-g 1 --)./ 
and we thence derive the value of ; viz. observing that we have in the denominator 
(i 2 ) K ”-‘), =( — ) i(n_1) which destroys this factor in the expression of -L, this is 
( ?s , o+ 2g l+ 2? 
J j 1 -gn (l- g 2 + »)..(i + g 1 + ») 
— 11 1 M . ■>+?£ ,^A 
o “5" , 2 fi- 
;l -q »)••'(! +g ») 
[l+g* (l+g g J »)..(!+?■ -)..(l-g »)••] 
Now, giving to g its values, it is easy to see that we have 
nci-rx 1 -^). ,(i-* f “)..=^ 
(i -g"). . 
(i-g 2 )..’ 
- _ __ _ i 
where (1 — q n ). . denotes (\—q n )(l—q n )(l—q n ). • ? viz. if is the same function of qj 1 that 
