438 Mr. Murphy on the General Properties 



Again, '^''/(-^ = P + qv + rx'; &c.| 



+ Qx-'+ /2j-«, &c.| 



and therefore Z) jx''-'/(i)J = - P log-. ( -i) + ^ +~, 



& 



c. 



2Q 2R 



a li 



whence P = ^= D {^^ + .''-'./• ( -) 



Hitherto we have taken the limits of the integral independent 

 of .r, the same results hold in general when x is used for a ; 

 but it is to be observed that when the limits contain x, (as .\-h, 

 x + h,) the integral remains a function of x, and therefore is ca- 

 pable of being integrated again ; and the result may be called 

 the second definite integral of the given function, which being 

 integrated between the same limits, will give the third definite 

 integral, and so on: we shall denote by />', D", D", &c. the 

 successive definite integrals taken in this point of view : In 

 the following theorem which is similar to Theorem I, the limits 

 are x-h^-i and x + h^-i; it compri.ses that theorem as a 

 particular case, and admits of a similar proof. 



Theorem VI. 



Put U = '~^'^ X d-{f(x).e-r^) 



(1.2.3 nV" J ^L.A-"" 



1+1. »■ / ' \ 



then shall {2h^ -ly .u, be the value of the m"' definite integral 

 of /{■!)■, n being made infinite in the value of m,. 



