MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
431 
and 
M =-"-1. 
TT 
It is quite clear that all but the simplest cases will be of quite unmanageable 
complexity. 
One of the very simplest cases may be taken as an example. 
Example .—Hollow vortex between two parallel planes. 
Here for the two singular points a = 0. 
If for simplicity we put a = 1, and, therefore, Ah = K', the singular points are at 
(f) = 0, = K. 
Hence 
dw ^ / zKA „ / iW 
Riic 
H Uo - -h- K 
iK'\ / iK' 
S7l I IV -SJl Of — ^ — R 
= C" 
dn (10 — 
iK' 
iK'\ ( 'iK'\ 
sn 1 10 — ) cn ( 10 - - j 
This integrates at once, giving 
^ ai {w - ^ 
= C" log tn ( tv- ^iK') + O'" 
The equation to the vortex is, therefore. 
2 = 0" log tn (</) — 2 ^K') + O'". 
