I'.i!; ^;\.\1MA. OK I A(.I<)KIAL, FUNCTION. 635 



Hence Lt vj^v+r — \/(-^) (known variously as Stirling's or 



J, A. Serret's formula). 

 We now have (if .r>o) 



e"n.r _\^4 l-^ _ V-< V~> I i-i 



11 = 1 1 = 2 



The right-hand expression can also be transformed by using 

 the relations : 





_ p (-i + Q)' _r'^i^ ^ p e^tit; 



00 



f>^ 



;i = o 



When X is big, this can be shown to lie between i/6(2.v+i) 

 and i/6(2.r) (using former Propositions for * simple ' functions). 



PROPOSITION IV. 

 Connection behveen Ux and n( — .r). 



nN.nN 



Ux . n( — .r) = by Gauss' Definition, Lt -, — , — c 7 — tta 



(-.v+i)...(-.v+A^) 



I T 



= LI ~, r ; r- = Lt 



■+7)-(>+i) (-■f:)-7:)--('-A^^ 



('-i)-('-^) 



but sin XI, = Ltxiv ( I - '^ [x - '^^j . . . (i - ^y 



.-. II.v. n(-,Y) = ■'^"" , or n.r. n(-.r-i) = 



Sin xt: ' ' sin x-K 



