The Torsion Problem for Bodies of Revolution. 155 
If we agree to take an anti-clockwise direction as being positive, then the 
moment of the couples at the ends will be given by 
M„ = - / J^r- . Bz . dr . d9. 
0 r = o 
TTVo^rJ^ 
■ Therefore when a circular cylinder of radius is subjected to terminal 
couples of magnitude 
2~' 
then the stress -components will be given by 
r9 = 0 
QZ = VoT, 
and the displacement ue by 
Vorz 
lie — — • 
This agrees with the result that follows from Saint- Venaut's theory. 
Section 6. 
Source or Sink. 
Let a point source or sink be situated at the origin of the co-ordinate 
system. If the total flow in unit time across any closed surface surrounding 
the origin is Stt'c, then we shall call c the strength of the source or sink. 
Let (py be the velocity potential of the motion, and let a sphere of radius 
P with its centre at the origin be described. 
Now since the area of the surface of a sphere in space of fine-dimensions 
8 
is r-,:rV* (see Appendix), therefore we have 
o 
O ^ Op 
On integrating we get 
c 
'.'"A" " " " 
This is the only solution of (11) which is independent of d. 
For a source c is positive and for a sink c is negative. 
For the components of velocity we have 
_ OC 1 1 01^,^ 
^ ~ dp' ~~ p+ " ~ p^' sin^e ■ Jo 
_ _ 1 (501 _ _ . 1 1 ^''i^i 
~ p ' oO ~ ^ ~ ~^ p'""' sin'^ 0 iip ' 
where is the stream-function for a source or sink. 
(16) 
