OF ELLIPTIC FUNCTIONS. 
439 
In fact, n= 3, this is 
cw— - 
viz. the numerator is a — 2a 2 +l, = — 3a 2 , and the denominator is ( — a) 2 , =a 2 . 
So n= 5, the formula is 
p-1 « 4 — 1 V 
\a 2 + 1 ’ a 4 + 1 / 
= 5, that is^y|J=5; 
a 3 — 4a 2 + 6a 4 — 4«+ 1 _ 
a 3 + 2a 4 +l ~ b ’ 
viz. this is 5(l + a 3 +2a 4 ) — (1 — 4a — 4a 2 -fa 3 +6a 4 )=0, which is right; and so in other 
cases. 
We thus have 
i=(- )«-»«. E( ? ), 
which, on putting therein u= 0, that is g— 0, gives, as it should do, )& n ~ n n. 
9 0 
57. As regards the expression of R(g), observe that giving to g its different values, 
the factors 1 — a 2? g 2 and 1 — a”~^ 2 are a U the factors other than 1 —q~ of 1 — (f\ and 
so as to the other pairs of factors ; viz. we have 
viz. this is 
that is 
EM~ p-g 8 "-- 1 x +g 2 - 1 -g--V 
\l-q* .. 1 + g .. l+g 2 ».. l-g”. .) ’ 
_ / 1-g 2 ”.. 1+g” . A 2 ^_ / l— g 2 .. I+g.. \ 2 
— \l+g 2 ».“. l-g »." ) ~^l + g 2 .. 1-g../ ’ 
f(q n ) 
f(Q) ’ 
agreeing with a former result. 
58. We have of course the identity 2)3 0 =j^- — 1; that is, 
iq “ 8 + .»?)■• (!+«-»? )■• _, w_„ g(g) 
] +t ,:-s.T VI) (i + ««»}*) — ' > f (?) 
(#=1. V. ««-!)), which, putting therein ^=0, is an identity before referred to; a 
form perhaps more convenient is obtained by dividing each side by/ 2 (g). 
59. I notice further that we have 
v 0 =w re {snc 2 a 0 snc 4<y 0 . . . snc ( n — l)<y 0 } ; * 
the term in { } is 
l + J* ( l + «V)..(l+«”-^ g ) 3 .. 
11 2a? */ (l + a® 1 g ) . • (l + a ra - 2 ?g) . .’ 
where we have fl - +g = ( — )* (B ’ ! ~ 1) . For example, n= 3, the term is — — = — 1 ; 
3 M 
