430 Mr. Murphy on the General Properties 



For convenience sake, the definite integral of f{x), taken 

 with respect to x between stated limits, is denoted by D.f{x) ; 

 in the succeeding theorem the limits are 



X = a + h y/ — 11 

 X = a — h „J — \} 



which are capable of representing any whatever. 



Theorem I. 



n+l .«■ 



then shall 



D.f(,x) = ihj~^\.u„ 



n being made infinite in the value of m, . 



To prove this, let ^(x) be the indefinite integral of /(x), 



®"*^ '^* dT = "^ ("^' d^ =<t> {a)> &c. 



Suppose c,, c2...c„...Cj„ any series of quantities 2m in number, 

 and ai,a,..,.05„ formed from these by the following law: 



d.(p'(a) 



"i = ^' (,") — hci 



da 





"■in — "2»- 1 ~ "'^in- J J 



