ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
357 
which is satisfied by the 2p values of x, viz.: x 1} x 2 , . . ., x p , . 
therefore 
, £; and 
N, 
/y* P 1/y /y» P 2-p 
/i/p, . 
• . 2h 
+ 
x i P 2/n 
x x p . 
• .2/i 
zy> P l/*v /y> p 2 --p 
^ 2 ? * 
• . 2/a 
x 3 p y 3 , 
x 2 p ~\, . 
• . 2/3 
zy* P“"“l /y P 2-, 
^p *^p ^p 5 • 
• . y P 
x P p y P ’ 
n— 0 
x/ h p , . 
•.2/p 
• . t?i 
£iVi> 
£i P ~ 3 £i. ■ 
• . 
fr'L • 
•. ^ 
&v P > 
•. ^p 
= 0 . 
20. As an example of (22) and (23) consider the elliptic functions, i.e., the case in 
which p= 1 ; then 
= 0 
(dropping suffixes), or 
that is 
N x 
z ?/ 
+ 
xy 
y 
£ 
€y 
y 
and 
n i( 2 V —y 2 £r)={£—x)yy{zy+yi) 
N x { (x-aJig-aJ - (a L - a 2 )(a 1 - a 3 )} = -y v (z v +yQ ; 
dx 
U =T> 
h \/ X ~ a \' X — ^.X — ff 3 
jp \/x—a 1 dx 
Let 
then 
~L, lV / x — % 
x - a Q 
x=a 1 J r (ao — ct^t 2 , F= 
<x 0 — a-. 
a 3 — a x 
U- 
1 f* 
cu — a , 
<:/£ 
p £ 2 ffe 
« 3 — ffj o-v/1- A1 — & 2 / 2 ' 
Let s, c, c/ denote elliptic functions of u\/a 3 — a Y 
S,C,D ,, ,, v\Za 3 —a Y 
(T 
Then 
and therefore 
sn {{u-\-v)\/a 3 —a l } i.e., -smo/ 
S~=t 9 ^- 
x — a. 
