1888.] On certain Definite Integrals. 313 



(x & 2 \ 



A + A l 23 - + A 233 + 



fl / 1\2 1 



But J (bg e -J^ = ^r(3) 



o 



Therefore we shall find 



x (x) = || X dv(log^J(A + K x vx + . . .) 



j 1 ^(log e i) 3 0(^) = ^. 



As xd> (x) or (x -— \ % (#) can have no constant term, all the terms of 

 V dx/ 



the expanded form of ( x x(£> (x) are suitable for the application of 

 \ ax/ 



the definite integral. 



Again let + 9*'g * U |) X(.) 



then #0) = (^ 8 J^ 8 £) XW 



so if X(«) = ^ ^ f"4^ * 3 • 



and a:0 (aj) = A a: + + . . .+ A„_i — as* + ... 



c j / \ A o* , -M 2 . A„_,a» 



we find x( x ) — i — 5 — e + o — a — a + — + ~7 — T~o\7 ; — T\ + ••• 



1 . . 5 I . i . b ii(n + 2)(n + 4) 



_ i JA n ri.* A^rq) a.,,^ 1 

 " 2" I r(4) "' r r(| + 3) - "' / 



_ l (A£ir(3) k-> A^rfFS i 

 = 1 {j o 1 ^(l-^ + A 1 ^+...) } 



r 1 ^ o / /- 24 xO) 



and so -7= (1 — v) 2 — — 



Jo Vv 



or if we please F cftt (1 - ^ 2 ) 2 (sew) = -^- ) . 



Jo x 



