ABEL’S THEOREM AND ABELIAN FUNCTION'S. 
347 
f*= 7 /V=7 \ 
2 E(w p ) —E( % uA 
/X=l V=i / 
expressed in terms of the functions of the u’s. 
The evaluation of the corresponding expression for the sum of the third elliptic 
integrals presents no difficulty. 
Section IJ. 
Abelian functions, after Weierstrass. 
13. The theory of these functions is detailed in a paper by Weierstrass in 
Crelle’s Journal, t. lii., hut such formulae as may be necessary in what follows will 
be proved. Let 
y 3 —P (x)=y 2 — (x—a 1 )(x—ci 2 ) . . . (x — a p ) = 0 
z 2 -Q(x)=z 2 -(x-a p+1 )(x-a p+2 ) . . . (x-rt 2p+1 ) = 0 
and 
#=My+Nz.(2) 
where M is of the degree p in x, N of p — 1; say 
M=x p +M 1 ai p . . . +M p _ 1 £c+M p "| 
N= . . . +N p _ 1 cc+N p J 
Then the equation for the roots x being 
My-N¥=0' 
is of the degree 3 p and involves 2 p arbitrary constants ; thus there must be p rela¬ 
tions among the roots. Let these roots be denoted by x v x 2 , . . x p ; . . ., f; 
Pi> P- 2 > • • -)P ? \ so that we may consider the p p's as given in terms of the £c’s and £’s 
by the p relations which might be exhibited in a determinantal form. Write 
dt(cc) = P(x)Q(#) 
and let 
u = Li P p_ V( ^ dx .(4) 
M — E(x) 
m which p, has in succession the values 1, 2, . . . , p as also in 
L P {x)dx 
a. bv — afy/E (x) * 
2 Y 2 
(5) 
