640 TJIR GAMMA, OK l''A( 'l» )RI AT., FUNCTION. 



Hence (x, y) = ^^^ " (a-+ 1, r). 



Similarly (^-+1, y) = ~J^ (^^-+ 1, >'+ 1). 



therefore if (.v + i), (v+i) are positive, 



(^'>')= (.r+r)(.v+i) (-^+^'->>+i) . (11) 



Now in (f^'{i—dy'^'iiB all the elements are positive and less 



than unity, therefore the integral is finite. 



From (ii) it follows that (a-, v) is also finite, and we have 



JC + V4-2 



and {x+N, y) = J r+-(i - BydB = J (i - ^K ^^^^^ 



— /r _/lM+A'+IU' . III . 



~V ^ A- + A'+I 



which — > 



I ^// _ ny 

 /r v^ n(x+.v+i) n(ar+>>+A^+i) nv 



^■'■^^' Ha: ~^ (x + N-\-iy+' ■ n(A: + 



(A- + .V+iy+' n(A: + A^) 

 {x + NY+'Uy 



-^{x+N+iY^^ -^ ^y 



Ylx Hy 



•'• ^'^'■^'^ ^ ITC^^H^' ^^ ^^'^"^^^ ^"^ ^^^^^ ^"^^ positive. 



But if {y+ i) is a negative fraction, and a- has any value, (i) 

 shows that 



(a-, y+ I) - -^-pY (a-+ I, v) = 30 ; 



and if (a;+i) is positive, (a, v+i) is finite. 



.-. (x'i-i,y) is + CO. 



