INVERSE METHOD OF DEFINITE INTEGRALS. 331 



Suppose 1- 1 substituted for t' in each term between the brackets, 

 then expanding each, the coefficient of t" in the whole will be 



n{n-'l)...{n-r + l) , ^y t-. , „ »-"^ , r(r-l) {n~m){n-m-l) , ,_ , 

 1.2...r "^ ^^ ^^^^l-m^ 1.2 ' {l-m){2-m) ^^^'^ 



«(«-!). ..(M-r+l).(-l)' (^„-„ d't'-'" d.f-'" </'->. r-" 



1.2...rx{l-tn) (2-m)...{r-m) '^ dt' dp ' dt'-' 



r.(r-\) d^ . t"-" d'-' . p-'" 

 ■^ 1.2 • dt^ '~dF^' ^^•^' 



when t is put equal to unity after the differentiations. 



But by the theorem of Leibnitz, the part within the latter 

 brackets is equivalent to 



fjr fr+n—2m 



— -^- — =(n-2m + l) (n-2m + 2)...{n — 2m + r).t"-'"", 



hence, the required coefficient of 



,_,_., «■(»- l)...{n-r+l) {n-2m+ 1) {n — 2m + 2)...{n — 2m + r) 

 ~^ '' 1.2...r ^ il-m) (2-m)...(r-m) 



Henpe, 



« _ (»-ffi)(n-m-l)...il-m) . . , n n-2m+l 

 7"- 1.2.3...n ^'~^' ^^~T' l-m '^ 



n(n-l) (n-2m-\- 1) (n - 2m + 2) ^ , , 

 "^ 1.2 • {l-'tn)i2-m) * > ^<^'h 



Again, by Art. 5., 



_ d"(ttT'" 



^"- l.2...ndf-^^^' 



(n + vi) (n + m—l)...(l+m) j^,„ n n + m ,„_, 



1.2...n *■ 1 1+m 





t'"-' t 



, n.jn-X) (n + m) (n + m -I) ^„.,^, \_, 

 "*■ 1.2 • (l+m)(2 + m) ' ^'^^.j, 



Vol. V. Part III. Xx 



