TRANSACTIONS OF SECTION A. 4.83 



^ (2n-2)!(2n + 2)! . ^ ^ . 



"^2^ { (m - 1) ! (n + 1) ! r U-"i"^a.- + 1/ 



• •••••••• 



J (2w - 2r) ! (2w + 2r) ! . ^ ^ ^ 



?i — ?•) ! (n + r) ! } \x — r x + r) 



(4r^)! /I IN 



j (2m) ! V-r -n x+n) ^ ' 



Multiplying (1) and (2) throughout, by : — 



2-'» -j (?i - ?•) ! (w + r) 



1 



2-" s 



1- . 2* . . . «- and — - 



1- . 3- . . . (27i - 1)'^ 

 respectively, they become : — 



n / 1 , 1 \ 



V n! \ n) /I 1 \ 



( : _ ^) (I - y) . . . (:-^^) , 



1 ^ ^ 2^ /^— i_+ _i ^ 



+ J ^ J, J _ J_ li' - 1 .r + 1/ 



n 2n 



3 \ V.r - 2 .r + 2y 



1-1 



(-i)(-|) (-2l)(-|) 



+ &C. 



■which, when n is made infinite, give in the limit 



TT I I 1 1,1 



+ --^ + ~^. - &c. 



sin 7r.r .r x — 1 .f + 1 x — 2 x + 2 



1 1 r ^ 1 



« .1- - 1 "^ X + 1 X - 2 X + 2 



T cot 1TX = — + ^ + — — + ^ + + &c. 



I . . X f 



VIZ., on writing — lor .r, 



tr 



_^ = 1 _ _1 L^ , „V , _V - Sec. 



sin .r X X —IT X + rr x — ^tt x + M 



Il2 



