MR. R. C. ROWE ON ABEL’S THEOREM. 
729 
III. c ‘ Abel’s Theorem .” 
As a third example, consider a problem analogous to that of Boole, art. 20 ; but 
more easily reduced by Abel’s theorem than by his. 
Let 
<K«D 
m 
{yjr(x)}n 
where is a rational integral or fractional function, is a rational integral 
function, while m and n are positive integers prime to one another. 
To this form any expression containing only a single term can be reduced. 
Let 
x=y"-t' 
while 
#=X 2 y— 
X L and X 2 being rational integral functions : also let 
Then, eliminating, 
and, in general, 
So 
so that 
Therefore 
e=xl— x/i y 
¥ 0 (x) = l. 
x= 
0i(g) n v u 
03 (% h( x )x'(y) 
fi{ x > y)=ny“~*<l>i( x ), 
M x ) =M X )- 
y 
© 
II 
s 
" 1 
02 0^0 
t log (X 2 y—X L )+C 
1 
= 0[0(x)]S-log (X^-XJ+C 
But, if 1, o) l; Wo, . . . <r n -i are the n th roots of unity the values of y are 
m m m 
0% 0™<y l5 . . . 0La,,-!- 
So the last expression becomes (putting co 0 for 1) 
f w— 1 
2 
0 
©[0(x)]0 n < 'Z OJ log Xo+ 2 CO log ( coxjj' 1 — 1 ) > -fC 
91—1 
$ 
o 
since 
m f 'ii —1 / m \ 
= ©[0(cc)]0 n j S co log (oo\Jj n -J> -f-C 
n—1 
S«J=0. 
0 
5 B 
MDCCCLXXXT. 
