720 
MR. R. C. ROWE ON ABEL’S THEOREM. 
Assuming the truth of this theorem we may proceed with the investigation as 
follows :— 
Since f(x, y)^6y is a rational integral function of x, y and may therefore be 
expressed in the form NP r y' . P,- being a rational integral function of x and r a positive 
integer not greater than n —1, while y is a root of the equation y(y) = 0, we have, by a 
known theorem of partial fractions,* 
E, 
M x >y)-s7*Qy 
2 --_p 
x\y) 
n —1 
We have then, by art. 7, 
By Boole’s theorem this 
v 
r -u-! 
>■ 
U-1 
- 
cl log F 
— rJ 
L/ 8 (*)f 0 (*)F(*)J 
clx 
p«-i 
F'(aj) 
— 
_M x Wo( x ) F '( x ). 
F(U 
i 
— 
_fi ( a; ) F o( a ')_ 
Y(x) 
For since P„_ 1 is an integral function it contributes nothing to the interpretation of 
© by being within the square bracket : and, if we assume that ~F'(x) and F(a') have no 
common factor (which is also the case for F'(x)—which contains the as —and F u (a;) and 
F 2 (£c)—which do not), we shall have in the expansion of 
«-i 
f 2 (x)Y 0 (x)T(x) 
no term involvint 
the reciprocal of a linear factor of F'(.r), which therefore may also be brought out of 
the square bracket. 
The expression last obtained 
= 0 
1 
E y) My 
FO) x'(y) 6l J 
= © 
q F„(x).: 
y;(r, y) My 
x(y) °y 
Under this form the sum is immediately integrable, for the new variables (of which 
alone this is now a function) occur only in the factor 
Integrating we find 
i[ fix, y)dx — t V }, dx =© 
J v Ud x )x(y) L/alUU 
( x )_ 
By' 
F 0 (x)S^log^+C. 
* See also note on art. 10, (i.). 
