Tlie Torsion Problem for Bodies of Revolution. 
177 
1'] 27r j-g 2'rr 
= {v^ — 47r2C J y . dr . de = {vo + ^tv^Ci) ^ r^dr . dO 
0 0 0 0 
2 ~ "2 ' 
where and ro are the radii of the body at 2^ = — a and or respectively. 
From this result we further note that 
r^* ~ -\- 4!Tr^c{ 
If another infinite plate source or sink with a circular hole in it, its 
plane passing through another point on the 2^-axis, be superposed upon 
the motion whose velocity-potential is given by <p^, then some other very 
interesting and practical cases can be obtained, and the problem seems to 
become ever more fascinating. 
APPENDIX. 
Volumes and Surfaces of ii-dimensional Spheres. 
The volume of an '^i- dimensional sphere of radius r will be given by 
V„ = J dXj dx.2 . . . dx,^ 
integrated over the region 
x^^-\- . . . -f x,2£r^-. 
We now make the 7i-dimensional polar transformation, 
x^ = r cos a^-i 
x.y — r sin a,,_i cos a„_2 
cCg = r sin a„_i sin a„_2 cos a„_3 
x-\u = r sm a„_i sm a„_2 . . . sm cos 
Xn = r sin a„_i sin a y?_2 . . . sin a.^ sin 
If Jn is the Jacobian of the transformation, then 
r TT n 2iT 
(r, a^, . . . c(.n-i)dr. da-n-x ■ . . dot^. 
r=o a = o <^2~o tt]=o 
n-l 
n-2 
But J„ = r"-i n siii^ a^ + i- 
27rr" r . 
.-. y„ — — n / sm?^ a^ + i C?a^ + i. 
n 
j9 = l a = 
