ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
343 
and substituting in the first we find 
From these we have at once 
^(A 2 +B*-FC 2 ) = PFV’ — d 6 & 3 c 8 (dropping subscripts) 
= -/(; /2 (i-3^V+2/tV) 
U 2 
/d 
AC = 6- 3 C? 3 , 
Hence 
^{(A 3 +B a -FC 2 ) 3 +4FA 2 C 3 }=(l-3WH-2^s 6 ) 2 +4Fs 6 (l-Ps 3 ) 3 
lb 
= 1 - 6&V+ 4 (P+fc 4 )s 6 - 3W 
and we therefore have to verify that 
8 WiP 
3E(w)—E(3w) = 
1 - 6/cV + 4(&®+£ 4 )s 6 - 3 k*s*' 
Now the ordinary addition formula for E is 
E(w) -j- E(/;) —- E {u-\-v )=^ 3 snwsnvsn(w -\-v) 
so 
that 
and hence 
But 
E(«) -f- E(2u) — E(3n) =/r 3 snusn2 usn 3 u 
2E(u)— E(2n) =& a sn a Msn2w 
3E (u) — E( 3 u) = Psnwsn2 !((sn u + sn3 a). 
3 s - (4 + 4/d).s 3 + 6 kh* - &V 
sn3w= 
1 - 6/,:V + 4(/d + k*)s G - 3 
or writing^ D for the denominator 
D(snw-(-sn3it) = 4s{ 1 — (1 ~\-k~)s' 2 -{-(k iJ rk i ')s i — ZAs 8 } 
= 4*(1-W)(1-^)(1-W) 
= 4s(l-fcV’)c s <Z 3 . 
Mot 
'eover 
sn 2 «= 
2 sal 
1 -las 
so that 
8/dsVv/ 3 
3E(u)-E(3w) = 
verifying the formula as required. 
