TJie Torsion Problem for Bodies of Revolution. 
165 
We now apply the transformation, 
x\) ■= a cos ap 
a?'3 = a sin ttj cos a^. 
x'^ — a sin sin cos a^. 
x'.^ — a sin sin sin a.^, 
which we may call a four-dimensional polar transformation. 
The integral is then transformed into 
^ (a; g, x'^, x\, x'r^) 
^ (a, aj, ag, ttg) 
da . da^, da^y da.^ 
{^2 _|. ^,2 ^ _ 2ar cos a^jt 
where r = Xcy, and the Jacobian 
^ (^X 2 J 3> 4> '^5) 
d (a, aj, 03, 
CX^o ^X'o ^^^'9 
8a 
= d-^ sm^ 01 sm <7o. 
The limits of the above integral may immediately be got by considering 
that the transformation must be a single-valued one. If we take a to range 
from 0 to di, then ranges from 0 to tt, a., from 0 to tt, from 0 to 27r. 
We therefore have 
i* /" /* ^^^^^ ^1 ^^^^ ^'^ ' ' ^^^^ ^^*^3 
V y y ^{^2 _|_ «3 4. _ 2ar cos ajl~' 
a = o a\ = o a.2 = o d^ = o 
— 47rc /* /* ^^^'^^ • da . da^ ^ 
~ '^^V / { ^2 + ^2 ^ ^3 r 2ar cos aj}!* 
a = o aj = o 
Now the rate of variation of this potential with respect to the radius 
will be given by 
^^6 ^ 4^^3c / Sin^ ai . da^ 
Set V {^2 4. ^2 _|_ ^2 _ 2ar cos ajl 
and therefore the potential of a ring element of radius a and breadth Sa 
will be given by 
