The Torsion Prohleyn for Bodies of Revolution. 
157 
If now we take 
3c = K (finite), 
then we get for the velocity potential of a doublet 
K cos 6 
(18) 
(^2 + r2)f . 
gth of the doub 
Further, we have 
where ^ is the strength of the doublet. 
_ _ 4<K COS ^ _ 1 1 
dp' ~ ~ ^ ' sTn"^ * ^0 
_ I S(f>2 _ sin 0 _ 1 1 S^l^., 
p ' d9 ~ p'" ~~ p'^ ' sin'^ 0' dp ' 
where xj/., is the stream-function of a doublet. 
For the determination of j//^ therefore have : 
^i/zg 4>£ cos 0 sin' 0 
it ~ 'p 
Integrating, we get 
^^^K^m^ .... (19) 
P 
Let us now investigate the result obtained hy superposing the stream - 
function of a uniform stream parallel to the ^-axis upon that of a doublet. 
We may do this, since the equation for \p, equation 12, is linear. If is 
the velocity-potential, tp.^ the stream-function of such a motion, then 
K cos 6 
h — "Po -\- <p.2 = — Vop cos e — ^ - 
Vn i ■ A. f sin'*^^ 6 
= ^0 + h= —irP ^ + • 
^ P 
1. Case. 
If we put Vu = 4^, K = 1, then 
" \ P ' 
From this equation we see that 0 — 0 and — 1 are the, stream-lines 
