The Torsion Problem for Bodies of Revolution. 161 
The displacement ue is given by 
^. / ^{2! -\- a) k {z — a) \ 
ue — — \ voZ H ■ ■ r + 1- . 
The moment of the terminal couple will again be + — ^ where is 
the radius of the body at ^ = ±: oc. 
Section 9. 
Line Source. 
Let X be the linear intensity of a uniform and continuous distribution of 
sources along the 2;-axis from z = — a to z = + a. 
P {z, r) 
Q t-H 1 z 
z = — a 
The velocity -potential due to an element Sz-^ is 
X. 
{(z-z,r^r-}i 
at the point (z^r). 
The velocity-potential due to the whole distribution is therefore 
+ a 
clz. 
— a 
Putting z — Zy~ y tan ^, we get 
-I z + a 
tan 
COS 'C . al 
+ tan -^'-^ 
\ ( z -]- a z — a 
