224 Mr Sharpe , Liquid Motion from a Single Source 
Inside the circle, that is for r < a put 
pa cos 6 Ci (a m r m ' 
u x = c + ^ + S [~- m cos rnd 
r V a j 
aa sin 6 ~ 
Uy = >S 
a r 
a m r m . A 
— zr- sin m6 
\ a m ) 
(1) , 
( 2 ) . 
This makes u x 4 - iu y a function of x — iy, and therefore repre- 
sents irrotational motion in two dimensions, m is integral and S 
represents the sum of a number of terms like that written down, 
but with different values of m. 
Outside the circle, that is for r> a put 
pa cos e „ (a m a m \ 
“* = « + - — + S[— COSmO) (3), 
pa sin e 
+ S 
sin md) 
which make u x + iu y a function of x — iy. 
(4) 
