CERTAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRALS. 753 
in ascending powers of z. The development is 
up I p , ^ 
z‘‘\ b — a ' do 0 — a '1.2 db^ b — a ' 
in which the coefficient of - is 
z 
d W) 
db b — a 
(5.) 
Lastly, the coefficient of - in the development of the function 
{x — a){oc — bY 
in descending powers of x being represented according to a familiar notation by 
C, 
we have, on adding (4.) and (5.), and subtracting (6.) from the sum, 
aj{a) d bj{b) x%x) 
(6.) 
( 7 .) 
nf ^ I O' oj\o) p ^71^) 
\_{x — a){x — bYy^ ^ {a — bY~^ db b — a ^{x—a)[x — bY 
As a particular illustration, lety’(.r)=log and let us seek the value of the last 
term in the above expression. Now 
a + 2b 
See. 
{x—a){x — bY x' x‘^ 
on developing in descending powers of a:, and 
log (o+i) = log o+i - &o. 
Multiplying these together, the coefficient of ^ in the result is log c. Whence 
^\_{x — a)[x — bY\ {a — bY b — a 
■log c, 
( 8 .) 
in which it only remains to perform the differentiation in the second term. 
Formulae applicable to the determination of the result of the operation 0 in any case, 
may readily be found by the aid of Taylok’s theorem. Thus we should have, f(x) not 
becoming infinite when x=a or x=b, 
ef I 1 fu) /(ffL 
L(a7-ar(a:-6)”J-7^ ^“1.2...(m-l)V«y {a-b)” 
d\”-^ f{b) ^ fix) 
1 fib) 
’^l.2...in-l)\dbj (b-a-r T 
(9.) 
ib-a'r Ti^-a)^ix-bY 
an expression in which the general law of such formulae is manifest. 
If there be n distinct simple factors in the denominator of the rational fraction within 
the brackets, the result of the operation 0 will consist of w + 1 terms, the first n of which 
5 F 
MDCCCLVII. 
