The Torsion Problem for Bodies of Revolution. 
178 
Superposing upon this the motion due to a uniform stream parallel to 
the 5^-axis, the resultant velocity-potential will be given by 
r \ r I ' a k'^ I 
From the above figure and a uniform stream we can get an approximate 
idea of the resulting stream-lines. If we take k > o and Vo > — ^- . 
then we shall get a body such as shown on Plate XVII, fig. 1 ; and if we take 
K < 0, and Vo > , then we get a body such as shown in Fig. 2. 
Within such bodies, then, the stress-components wall be given by 
T ' \ r I 
where the values of B^, Bo, Cj, are given by equations 23 and 24. 
The displacement tie will be given by 
( ^ . (A)^ I [2K - ^-±.,^^E] 
ar \ r I k" 
where the plane 2 = o is taken as the plane of zero displacement. 
If r^ is the radius of the body at the two distant ends, then the moment 
of the couples is 
TTVnr 
Section 12. 
Infinite Plate Source with Circular Hole. 
To find the velocity-potential due to a uniform distribution of sources 
over an infinite plate with a circular hole in it, we shall firstly find that due 
to a distribution of sources over an infinite plate, and then that due to a 
distribution of sinks of the same strength over a finite circular plate. The 
sum of these two potentials shall give us the velocity-potential of a uniform 
distribution of sources over an infinite plate with a circular hole in it. 
1. Infinite Plate. 
Let the plate pass through the origin, and let the 2^-axis be perpendicular 
to it. It is evident that if in this case we take the potential due to the 
plate source to vanish at infinity, then at finite distances the potential will 
be infinite. We shall, therefore, in this case take the potential to be zero 
in the plane of the plate. 
We have found that the velocity due to a circular ring- source at a 
point on its axis is given by 
