The Torsion ProUem for Bodies of Bevohition. 175 
In the neighbourhood of these points the series converge slowly. 
These expansions may also be written : 
(2 - A;2)K - 2E 
='i€(.i(V)*)---(E(^')*w)'- 
_ , f ^ r2 (2«. -1)2 - l y 2 (2» - 1) /2|) - 
Substituting this value of (2 - A;^) K — 2E and F = — — in the 
integral for <l>'\, we get : 
Since this series converges uniformly for all values of z and r excepting 
z — 0, r — a, therefore we may integrate term by term, and hence we have : 
^",= -.2., E22".r-^~^' n .C^) / 1,731 (A-^zl) 
» = 2 ^=1 ^ [^2 + (^_^^)2] 
The integral 
/ 
may immediately be solved by making the transformation a ~\r r — z tan |. 
This reduces the integral to 
tan -1^1. 
^(^T^y ^ ~ ^^"""^ • cos2"-^ I . dt 
^ , -1 >• 
tan — . 
It may also be solved by first of all reducing the integral to I^ by means 
of the reduction formula 
