786 PEOFESSOE BOOLE ON THE COMPAEISON OF TEANSCENDENTS. WITH 
Now 
0 
{x—hy 
1 
V — x-\ — 
j 
X X Aj 2 
V — h-\- 
+ ■ 
h — A| h Xn 
a / 
1 .2 ..{i — 1) J 
-|-Ci -Z T! 
^ix-hyfx-v-^..-^ ) 
\ X X } 
dji 
the former of the two terms in the second member being the coefficient of in the 
development of the function in ascending powers of .r — h. 
It is evident that the latter of the terms in the second member vanishes ; for the 
first term of the development in descending powers of x being there ^fill be no term 
A 
of the form — Hence we have merely to consider the term 
X ^ 
H standing for 
If^ = l, (8.) becomes 
Let ^=2, and (8.) becomes 
If ^=3, (8.) becomes 
Hence, generally, 
1.2. 
• b-i) ^ 
jhJ 
V— . 
h 
O-n 
A— Xi 
h- 
a 
dW 
dh 
{v- 
H)2' 
1 
r dm 
Q ( 
a 
did 
f 
ydh) 
1.21 
X 
1 
(v 
-H)3 
d y- 
' 1 
H, 
- _1_ 
( 8 .) 
H. 
H, 
(y— H) 
H„ Hg, . . H, being independent of v. The second member of the above equation, on 
addition of its several terms, becomes a rational fraction whose denominator is (r — H)\ 
and whose numerator is a rational and integral function of v of an order not higher than 
i — I. Hence the theorem is demonstrated. 
35. The conclusion to which these investigations lead is a remarkable one, and may 
be thus expressed. The evaluation of the definite integral 
r 
Po+PlX +P2'*'^. . +PiX' 
I - 00 9o + + S'a*® • • + 9/^ 
is redudhle to that of a definite integral of the form 
Pc + PlU + PgW*.. +P,V 
J_„ Qo + 
i'*'* n y ^ ^1 ^2 \ 
joid y a;— Xj X — Xj x — X,J 
dx 
