r.42 



Till': CAM MA, OK FACTOKIAL. FT'NiTloN. 



Then n^c Hy = dl . /^ j ciii flog - \ 



x+y+i 



(/+!) 



x+y-i-2 



= n(^+3'+i) 



dl . I' 



jo(/+ir+-^^' 



i™^)=p(/T^) • {M{Thy=s?'-''^'-'^' 



PROPOSITION VIII. 



Expansion of log Ylx and Ux. 

 From Gauss' definition : 



log n.v=I/ 



.V log .V + log nA' - ^ log (a- + /) 



2>-.;.U.v^2.^, 



JV 



n'o = Lt 



'°8'^'-S<r'-' 



jV 



j A' N — I 21 ^r~\ I 



= uY^\log'±'-\ynch<:^L 



■• "■<' 



iV+i 



— I 



and is therefore a definite quantity. 



It is known as ' Euler's Constant' (y),and y = —•5772156649... 

 ^lognA-= + > / , ■^, which < 



x-{-i 



^^„logn.r = (-r(,/-i)!^ L^^which <^( -)«(„- 2) 



I 



(^+1)"- 



