INVERSE METHOD OF DEFINITE INTEGRALS. 145 



by Art. 12 ; and lastly, put for H,,^ its value 



(gA + 1) {qh + 1. +i)). . ..\ g^ + l+ { m-\).p\ . 



1.(1 +;?).. .|i+(»w-iy:jo"i 



we thus obtain 



+ »(^- g> L+a(i>±2) ^. (^ ^ (^ (^ 



(a: + l)(a;+jo + l)(a?+2^ + l) Sg'' '^ '^^ ^ /rv"/ / 



x(a; — 5')(a; — 2^) 



■*" {x + l)(a:+jt> + l)(ar + 2/> + l)(2 + 3ja + l) 



^ ^ 1^.2^^^ ^'^^^' + ^^^^^ +-^ + ^^^^^ + aja + 1) {qh) 

 + &c. when A is put = 0, 

 and where /> and q are perfectly arbitrary. 



Cor. 2. Put ^ = ^ = 0, and make 



where ^(0), 0'(O), 0"(O), and the values of ^{x) and its successive 

 differential coefficients when a; = 0, and the above expansion will become 



</>(x) = ^„.^ + ^,.^^-., + ^..^3 + &c. 



If, moreover, we put 



rr, , « , , , W.(»-l) (h. 1. O'' „ 



which is the same as A„ when we put f for ^ (a;), then it is easily 

 seen by the principles of the first Memoir, that jj Tj' = -, r-r—r , and 



/ r r , ji n (a; + l)"+'' 



since we have also fij'it) . /^ = {x), it follows that 



