432 . Mr. Murphy on the General Properties 



Similarly, 



h ■iz., <l{a,.e'T-") ii5.„ rf(«,6'T-°) ll,,, di,p'a.e--h '') 

 — .f« . = -1.6/. , . . =i-e* 7"^;~T'~ 



2ir da 



(-1)" -^^^.a d"{(p'{a).e' !■'■") 

 Comparing tliis with the value of u, we have 



a„ = (-!)".«. 



Exactly a similar process applied to the remaining' quantities 



a,,.,, a„+ „....«„„, gives 



(-1)" .-;iTi.;.- ^^U,.e^") 



1.2....M 



d{e-7'")" 



^ . 6 —■" d"{u.eh-'') 



that IS, «.„ = i-;^;;;;- • ^ _. x« ' 



or a„„ = U, (2). 



Equate the values of «„„ in the equations (i) and (2), and we get 

 D .f{x) = 2h -J^l . M, . 



Theorem II. Supposing the limits of the integral now to be 

 .r = o, x=co, then .shall 



•' ^ I -f ^ ' 2' d(log. a) 3* rf(log.a)- 



For let f{a) = <f>(log.a); 

 therefore fi.ax') = (p (log. a + x log. x) 



= ^(log.a)+a:log.a;.0'(log.a)+ ■ . . .^"(log. «)+&c. 



