774 PEOFESSOE BOOLE OX THE COMPAEISOX OE TEAXSCEXHEXTS. -WITH 
The first term vanishes by Art. 11. The second may be reduced as follows. e have 
\po 
- 20 ly,S!i= - e [ J 2y3f = e 
Now 0 
is!)=6r 
L 
^Po 
(9.) 
This is evident if we collect the different parts of the intei’pretation of 0 from the terms 
in each member, observing that in all cases it is upon the same function that 0 operates. 
Now the first term in the second member vanishes by Art. 11. Hence 
0 
Po 
^v=Q 
<p{x) 
VnV ■ ^ 
Pi 
Po 
Attaching now the symbol of integration to the second member, and integrating, since 
f and 0 are now transposable, we have 
K) 
Writing in the first term of this expression — for — and adding the result to (8.), we 
Po 
have 
e[_(p{wj] { ^y, log {u—yrv) — log — 0 
<p(x) 
Pi 
Po 
logv. 
or 
Q\(p{x)]1y^og{^-y,J-Q <p{x)^^ 
logv^). 
i/C * 
Hence replacing ~ by and adding the constant of integration, we have 
^^(p(x)ydx=e[<p(xj]^y,. log 0 log v+C, 
( 10 .) 
X) 
the expression required. It will be observed that as <p{x) and ^ are always rational 
functions, the operation 0 may always be performed on each term of the second member. 
As the summation Ijy^ log — ?/,.) can only be effected by connecting the several terms 
included under S by the sign of addition, it will be most convenient to express the 
solution in the following form. 
’Po 
log^j+C. 
( 11 .) 
'^)i(p{x)ydx= Xz^Q\yp{x)']y, log (x-^J- © 
28. If^j = 0, we have 
'l^<p{x)ydx=X="^ \_<P{x)'\yAog (x-2/r) + C (12.) 
This includes as a particular case the theorem of Art. 22. For if y=-4/”, we have 
of which any root y^ will be given by the formula 
