63^ TIItL GAMMA, OR FACTORIAI,, FUNCTION. 



Hence also n(x+i) . ll(-x-i-i) = ■ 7^ ^ - (-)' • T^ 

 ^ ' ' \ ' sin (.v+/)'r sin . 



and in particular, putting .r = — ^, 



{n(-i)}^=7r, II(-|)=V^. 



Introducing negative values of .v in Proposition III, 

 log -J — 77 : becomes log -7—77 ^^ — 777 ^: • -■ 



^ \ e-''n{ — x) 2 sm xttJ 



t;-^n( — .1") V~\ 6^ 



.-. log _^ i 7--~\ — log (I COSeC.VTr) — > Al' , - , .- ., 



PROPOSITION V 



n- ux . n(x-l) . iif^-^V . . n(x-'^ 



To prove that .^ "^ ^) /'^ ^ ^ (.-/.-. 



■^ Il{nx) ^ 



■ ■ C27rV"~'"' 



// IS a positive integer) js independent of x and = ^ — ^ — . 



\l n 



For I{{nx) = Lt ' 



{nx+i) (//.r+2) . . . {nx-\-Nn) 



n(;/.v+ nN) 



= It 



""-^ lx+l\(x+l-\-i\ . . . (x+'^+N- I 



x + -](x+i + ^-\ ... Lr + r + .V-i 

 nj\ n V n 



{x+i){x+2)...{x + N) 



which. Lf """+"^' —^ " I .^- » I ... "M 



«"-v 



n(.+^^-i+.v) n(.-^-i+iv) nG^+^^) 



