Definite Integrals, with Physical Applications. 407 



multiplied by a constant quantity A,„ we may represent the combined 

 operation by A„^r", and the sum of any number of such operations as 



A m \p + A„ ^" + A, ^ + &c. 

 may be represented by the operative function F{$>). 



Suppose also that <p(x) the given function of x which is subjected 

 to the above operations, is by the inverse calculus reduced to the form 

 ftf(t) . f ; the result of the operation F(\ji) will be the sum of the 

 partial results of the same operation on each element f(t) . t" . 8 (t) ; 



but f" (t') = t' + '' h = ((")". (t*); 



.: Fty).t°=F(f).f; 



.: Fty).~<t>{x)=Jlf(f).F(t*).r (1.) 



Let the operation which is the inverse of Fty) be represented 

 by F' 1 (>//) so that the latter operation performed with the former neutral- 

 izes it, then 



F 1 ty) . { Ce'"'} = C.F 1 (£'»"). e" ; 



let the roots of the 



F-\(e n ") = be »«„ m„, m 3 &c; 



.-. F~ l ty) \A t e m >' + A % <•"•' + A, e'v &c. } = 0, where A„ A 2 , A 3 &c. 



are arbitrary constants ; 



.-. Fty) (0) = A t ^M«"+&c. ; 



which appendage must be added to the operation F(\js)(j>(x), since r/>(.<) 

 is analytically to be treated as <p (x) + 0. 



It is evident that the operation %//" is equivalent to multiplying the 

 given function by unity, and therefore ^ — ^" is equivalent to the opera- 

 tion A of finite differences; consequently ^ is equivalent to A + 1 ; Hence 



F{/\).<p(x) = f t F{P-\).f{t).t' (2); 



