ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
345 
Moreover, if we clroose 
the equation X = 0 is of the form 
B=H=F=G=0 
—A=C=D=1 
and we can therefore take zero for the lower limit in all our integrals. Hence we 
shall have 
u \+%+ • • • —w 8 = 0 
where 
x=snu 
and the above relation will give sn(iq+ . . . +w 7 ) in terms of the elliptic functions of 
U^, . . ., Urj. 
Let us now find 
write 
S V ~T=& dx ’ 
Ba?4-C = w 3 ; Hr-{-0=1^; l+Fx + G x 2 ~iv z ; 
then the right-hand side of the equation is 
-C. 
z [w 3 — zwj— y(Az+w^'\ z ^ [w^+zw-^+yiAz— w 2 )l "j 
JJ ° \ w z- zw l + y{Az + w^)\ ~^~y & j w 3 + ziu 1 -y(Az - w z ) J J* 
On expansion, the ?r th term gives 
-Ci 
2n-l 
Cl 
zy 
2n — 1 x 
2w—2 
y 2 n Z(A 2 + W 2 ) 2 ” 1 y u 2 (A z — w^f n 1 
(Wg— 2 (it’g + 2W 1 ) 2ra_1 
_(w 3 2 — 
q{(Az+rt’ 2 ) 2 " 1 (m 3 +zry 1 ) 2w -{w 2 — As) 2 " l (w s —zw 1 ) !ln x ) 
So far as the result is concerned, the expression within the inner bracket is 
{w 2 w 3 —k 2 x 2 w 1 -^-z(A.w 3 ~\-w 1 w 2 )} 2n ~ 1 {w z w 3 —k 2 x 2 w 1 -z(Aw s -\-iv 1 w i )} 2 " _1 
= 2z{(2n—l)(io. 2 w 3 —Jc 2 x z w 1 ) 2>l ~ 2 (Aiv 3 J r w 1 Wc l ) J r . . . 
Now 
iv 2 iv 3 -Fxhv^x* BG - AH + (CG + B F - FI)) 
— ^v.3 
x 3 (Xi + -\ 3 ) say; 
MDCCCLXXXIII. 
