782 PEOFESSOE BOOLE ON THE CO^HPAEISON OF TEANSCENHENTS, WITH 
Avhence, reducing the above expression for u, we find 
dx.x^~^ 1 F(^)r(n — i) 
r 
{a-\-hx-\-cx^Y c^{b -\-2 VacY'^' Tfn) 
and hence on changing x into -? we find 
I 
“ dx.x'^-i 
( 11 .) 
( 12 .) 
{a + bx + cx^Y V{n)a^[b + 2 A/flc)"-* 
The two last theorems were discovered independently and about the same time by 
Mr. Cayley, Professor Thomson* and ScHLOMiLcnf. It is to be noted that a. h, and c 
must be positive. 
The above examples have been selected in the fu’st instance because they relate to 
knoAvn results. But there is not one of the results amr’ed at which mav not be gene- 
ralized to an indefinite degree. 
Thus, since we have 

.... (13.) 
Ave have 
j2&-(^r=|r(i) 
(l^-) 
If w=2, this gLes 
whence 
ty 00 
f” 
(15.) 
agreeing with (5.). But let w=4, and we find 
(16.) 
and so on indefinitely without even proceeding to employ the more general forms of (2.). 
(!-•) 
34. Let us examine the definite integral 
x^’f(^x — 
By the general theorem (1.) we have 
u 
dx. 
= */(»)©[*”] »= (IS.) 
^ — X + - » 
X 
For, not being fractional, the interpretation of 0 is reduced simply to — Cj^. It is 
X 
Q^n + 1 
obviously desirable to express the term C i -i in a series consisting of powers of r. 
•' ^ —x‘^—vx—a ® ^ 
Now 
2n+\ 
X' 
x^^—vx—a 
—X 
,.2n + l 
VX 
x^ — a' [x^—df' of 
A ~2n+l 
-&c. =V- 
J — fl 
,2h + 2 .y>2« + s 
^-h&c. . (19. 
« rt)' [x^ — aY 
t Ceelle, vol. xxxii. 
* Cambridge aud Dublin Mathematical Joxu’nal, vol. ii. 
