758 PEOFESSOE BOOLE 02s THE COZVIPAEISOX OF TEA2sSCE^^)EyTS, WITH 
d 1 d 
Now log E = — coefficient of^in the development of «a;''^logE by the defi- 
nition of 0, since, as the function within the brackets is enth'e, there will be no ascend- 
ing developments. Hence 
— e[fia']^logE (4.) 
Consider, secondly, the expression 
[x—pY 
Now expressing (3.) in the form 
1 I 1 
dx^ ® x—p — {X]^—p)'x — p — [xc^—p)'''x—p — {x^—p)' 
and developing the several terms in the second member in ascending powers of x—jp, 
the aggregate coefficient of wiU be 
1 . I 
{x-pY^{x^-pY" ^{x^ 
1 
'{x-pY 
Hence = — coefficient of ffin the development of^logE in ascending 
powers of x — p, 
= --coefficient of in the development of ^los E in ascendins: 
x—p V (^x—pY dx ° ® 
powers of x—p ; 
2 — — coefficient of in the development of log E in ascending 
powers of x—p, 
Hot 0 ^ log E = coefficient of in the development of ^^x—p)' ^ ^ 
ascending powers of x—p. 
For 1st, the only distinct simple factor in the denominator of the expression within 
A 
the brackets is x—p ; 2ndly, there will be no term of the form — in the development 
a d 
of i^x—pY ~dx ^ descending powers of the first term of that development being 
in 
na 
evidently -fpf. Hence 
’ [x — pY 
•0 
“ r.li'ogE- 
(5.) 
.{po—pYj dx 
It appears from the above, that when we decompose the rational fraction (p{x) into a 
series of terms (Pi{x), -{■(p 2 (x),, which are individually either of the form ax' or 
of the form (^x—p)' '> have 
2 p,(x) + - e [f . - e 
