338 
PROFESSOR A. R, FORSYTH OH 
the summation on the Jeft-hand side being taken over all the roots of the equation 
E= 0 , which are assumed as the upper limits of the integrals, while on the right-hand 
side the summation is over all the roots of F^O, F c = 0 , . . . , F r _ x =0 considered as 
r — 1 simultaneous equations giving x . 2 , x 3 , . . . , x r in terms of x y 
9. Returning now to (iv) the conditions that the right-hand side should reduce to 
a constant are :— 
1°: That f{x) — \ , or be a factor of U ; 
and 2 °: 
which will be satisfied by 
Ci 2^ log 6=0 
c, T =°. 
Now J is of the order m-\~n—2 in x, and therefore the order of U may not be 
greater than to+ n— 4. In this case the number of terms in it will be 
m -\-n — l.m -h n — 2 ,m,+ n-~o 
1.2.3 
C X b 
But if the integrals \ —dx be formed they are not all independent for 
rv f 
I J-dx=< 
V m Y n 7 
--dx— 0 
where and Y„ are arbitrary functions of the orders to— 4 and u —4 respectively, 
and contain 
m — l.m — 2.m — 3 
1.2.3 
and 
n — l.n — 2.n— 3 
1.2.3 
terms. Hence the number of independent integrals in the case when the right-hand 
side reduces to a constant or to zero is 
m 
+ n — l.m + n — 2.m-\-n —3 m—l.m—2.m—3 n — l.n — 2.n — 3 
1.2.3 
1.2 3 
1.2.3 
— 4 ) +1 
This assumes that the surfaces F„, and F„ are the most general of the degrees 
to and n respectively and so possess no special singularities,. 
10. Abel’s theorem in the more simple case applies to the intersection of plane 
curves. There is a fixed curve given by 
x( x > y)=° 
