Definite Integrals, with Physical Applications. 363 



The coefficient therefore of 



1 . <b(a) A B C 



- in -*— — = — + — + -3 + &c. 

 a x — a x x* x 



= <p(x). 



nrt 



Substituting this in equation (2) we have f—.t* = <f>(x) (3) 



T 



which shews that f{t) = — as announced above. 



6. The following examples are intended to shew the manner, 

 in which this theorem may be applied. 



Ex. 1. Given ftf(t).t x = — - — , m being essentially positive, 



*C ~t~ i ft 



(vide Art. 2), to find/(/). 



We must find the coefficient of- in /"'.-— — and divide it bv t. 



x x + m J 



t— 



Now since 



x + m 



= {l-xh.l.(/) + ^l.(h.l..^- f ^ rf .(h.l.0 3 + &c.} 

 fl m m" g ] 



the coefficient of 



= r, 

 therefore the required function /(() = (""'■ 



Ex. 2. Given f,f(t).t'=- to find/(/) 3 (n being integer). 



' " ' (x + my " ' ° 



Vol. IV. Part III. 3 A 



