INVERSE METHOD OF DEFINITE INTEGRALS 353 



The term which is independent of t' is 



but in general we have 



(l-A)(l+A)-(^"+^' = l-(2« + 3).A + (^" + ^^)(|^ + 5) ^._^^ 



and putting h = \, we find that the term independent of if is zero. 

 Again, multiplying the last equation by A"+'", we get 

 (*"+"■ - A"*""*') (1 +^)-<'"+'* = A"+'» - (2 w + 3) . A"-^""*' + (^"+yv^"+^) . ^.+".. >^ _ &c. 

 Now it is easily seen that when h = 1, we have 



-Tj^ (A"+'"-A"+"'+') = (w+w)(«+»?-l)...(«-/»+l)-(«+/» + l)(«+»»)...(«-w+2) 

 = — 2»w . (w + >w) (w + TW — l)....(w — m + 2), 



-(A"+'"-A"+'"+')= - (2m-l)(«+m) («+m-l)....(«-»w+3). 



&c. &e. 



and therefore when A is put =1 after differentiation, we have 



Jim 



_^^ {(A"+'»-A"+'"+>)(l+A)-'"+'} = -2-<""+''.2>«. («+»«) (w+»«-l)...(w-M + 2)x 



, , 2w+2 2OT-1 1 (2w + 2)(2w + 3) (2m - 1) (2?»-2) . . 



* ^* i ■«-»^^-2"^2^■ 1.2 ■(w-w + 2)(»-»» + 3) *^"» 



which series consists of only Im terms, and is equal to the infinite 

 series obtained by differentiating the other side of the equation, viz. 



w(« — !)....(« -OT+l)x (w + l)(« + 2)....(w + »«) 



- (2«+3. («+!)». ...(«-»» + 2) X (w + 2)(» + 3)....(» + »« + l) 



+ (^^+^)(^" + 5) (n + 2)(w + l)...■(w-»^+3)x(w+3)(w + 4)....(w + »? + 2) 



— &c. oe? infinitum. 



