OF ELLIPTIC FUNCTIONS. 
435 
-|(QB— AS)=2{*v(i— £)' ■-•‘(s-l)} 
+ 
vr 
'M/’ 
viz. the left-hand side is 
= 2{_„<(1-«V) i + 2^(1 -«0 
+{sp(i-«‘)+lr (l-«V)-3(l- M *)}(i+ M >(»-2»*)); 
or say we have — ^ (QB— SA)=II, where 
n= 
+±.u%u- 20(1-*') 
. 4mV+w 4 (1 — 3 u s ) — 4wV+2w 4 
+ . — 2w 4 +.6vV(l— w 8 ) + w 4 (— 3+-5 w 8 ); 
wherefore the value of 2 y is ==^n^(t> 4 + 2«V— 2m— m 4 ). Similarly, writing 
rr= 
+jjl •»(»’— 2MX 1 -®*) 
. 4wV 4-^ 4 (3 — m 8 ) -{- 4wv — 2 mV 
+ ^(— 5 + 3zt 8 ) + 6m(l — m 8 ) + 2w 12 , 
we find 
2S =1 J IT (v 4 + 2 vht? - 2m - u 4 ) ; 
in verification whereof observe that this being so, the first equation gives the identity 
{(s- 1 ) (h+ 3 ) (^+^)](»‘+ 2 *v- 2 « ( -o+n-n'=o. 
50. The result is that, writing for the moment « 4 +2uV— 2m— u 4 = A, the values of 
the coefficients are 
a, 8 
7 > 
- 1 ’ 2 1 J’ 8 A 
J, — JL 0,7 
» 8 y A ’ 2 
/ I « 4 \ m 11 
v (sr-sv’ v 
and 
1 — y l—x/l—/3x + yx 1 — Sx 3 + ex 4 — 4 Z>5 \ 2 _ 
1 -j-y l+xyi+/3x+yx 2 + $x 3 + ex 4 +gx 5 J ’ 
the modular equation is known, and to complete the solution we require only an expres- 
sion for M in terms of u. v. 
