INVERSE METHOD OF DEFINITE INTEGRALS. 127 



Now if we put for {p, q)^ the series assumed in Art. (12) aiid multi- 

 plying then by f, integrate from ^=0 to ^ = 1, we have 



ar + 1 a;+j9 + l a; + 2/> + l "*" x-\rnp + l 

 and actually adding these fractions, the denominator of the sum is 



{x + '\){x +p + l)(;r + 2jo + 1) {x + np + 1); 



and since the numerator is of n dimensions in x, and vanishes when 



x = 0, q, 2q....{n- I) . q, 

 it follows that the sum is of the form 



c .X . (x — q) (x—2q)....{x — {n — l).q] 



{x + l).{x+p + l)....{x + np + l) 



Multiply by ^ + 1 and then put x= —1; hence 



^^ c.{-l)\l.(q + l)(2q + l)....{{n-l).q + l} ^ 

 p . 2p . 3p....np 



whence deducing the value of c, and substituting in the above integral, 

 we obtain 



^'^^'^^''•^^^~P^'''l.{q + l){2q + l)....{{n-l).q+l\ 



^^ x.(x-q)(x-2q)....{x-{n-l).q} 

 {x+ l){x +p + l)....{x + np + l) ' 



hence y;(^, g)„ .^"^ = (-^y . ^ ^^^ ^| ; ^^^^^ ^^ ^y 



nq (nq - q) [nq — 2y) .... \nq - {n — 1) . q] 

 {nq + l){tiq+p + l)....(nq + np + l) 



from whence we obtain finally 



n" 1 . 2 . 3 . . . .?{ 



f^ ip, q)n {q, p)., = „(^ + ^) + i • i(^q^l)„..{{n-.l).q+l\ 



nq(n q — q) {nq — 2q)....{nq—{n — l). q} 



''~'Up + l){2p + l)....{{n-l).p + l} ■ 



R 2 



