of Definite Integrals. 441 



In like manner, if S' = 2„ . " L 



(1.2...n) (a-l .a-2...a-n) 

 we shall get .' ^ = —.x" .S'; 



it follows therefore that the complete solution of the equation 



J" = — .* .V, 

 ax a- ^ 



is y = cS -^ c'S'. 



Let <p (h) denote the value of the indefinite integral 



e".h"-fe-\h''-\ 



commencing from h = o, we evidently have the two equations 



- - + 1 





and <p (A) = 2„ . 



1.2 71. h"' 



h" 



a + l .a + 2....fl + « 



If we multiply together these two expressions, the part indei)endent 

 of A, evidently = S, and arranging the remaining terms according 

 to the powers of h, we get 



1 



<p(Ji).xe^' = S + T.h + U.h' + fV.h' + &c. 



+ t .h-^+u.h-- + w .h'^+ &c. 

 where T, U, W, &c. represent certain functions of x. 



ft h 



Multiplying both sides now by r, and integrating from h- — \ 

 to + 1, we get 



- 2t — ?fW — &C. 



3 l2 



