/ 
ABEL’S THEOREM AND ABELIAN FUNCTIONS. 337 
X = A F 4-A F -J-A F 
— 4 F 
— -^p 1 - p 
and therefore substituting this in the above which will replace the right-hand side of 
(iii) and inserting the value of dw the equation becomes 
x—a. 
u 
Sib 
-C ,2 
X 
' U 1 8F ; , 
F, 
x-u j/IL, FA f p 
— 
X = a 
1 
i 
r 1 1 
u 
X — a 
j/F m , FA 
F, 
- \ y,« ) 
— 
by the use of Boole’s symbol © as before. The summation on the left-hand side is of 
course over the mnp roots x ; on the right-hand side it is over the mn roots y and z in 
terms of x of the equations F m = 0 and F >< = 0 . We may obviously integrate as before ; 
and using the distributive property of 0 we obtain as our result 
p.=mnprxp. {J y x 
Ji J M 
f(x) 
■C=© 
A X ). 
Ii. = mn f TT , , 
• ^ 1 /F F\ lo S ¥ pf • • • H’ 
m=i 
_FA 
l' ) { y> z ) 
8 . The general theorem will proceed on lines not widely different from the above, 
and may be enunciated as follows, Let 
fi 
& 
TT 
o 
Fo 
(x x , x z , . 
T 
o 
F,_ 
-l(*Fu x Z} • • 
4* 
T 
o 
be r —1 equations, of degrees vi x , m 2 , . . . , m r ^ x respectively, giving x. 2 , . . . , x r in 
terms of x x ; and let 
F r {x 2 , x 3 , . . , , x r ) 
be a function of these dependent variables, the coefficients of which are functions of x x 
containing any number of arbitrary constants. Form the eliminant E of all the F’s 
so that we shall obtain the set of roots x x by equating E to zero ; and denote by U any 
algebraical rational integral function of x x , x. 2 , . . . x r . Then 
:f - 
J/W 
dx x 
) t/ *i’ b 3 , .. JA-i 
Xo 
, x r 
:© 
J( x l)_, 
2 x 
_ U log F, _] , . 
j/ W F 3 ,.. ■, L-A j +A 
\ X -2> X 3> • • ■ > x r J J 
MDCCCLXXXI1I. 
