430 MR. J. H. MICHELL OR THE THEORY OF FREE STREAM LIRES. 
Let this point be 0 q, xJjq. Then we have 
dz _ IT,. {© [« {w — H [a {w — w,.)'] ^ 
dw {@ [« {w — i^o)] H [a (?fj — Y 
as the general expression for the potential of two polygonal prismatic conductors 
exterior to each other. 
Hollow Vortices. 
In conclusion 1 shall show how the methods of this paper may he applied to find 
the form of hollow vortices. 
Inside a vessel bounded by plane walls, let there be a hollow vortex in steady 
motion. 
Let \fj = 0 he the free steam line of the vortex, xfj = 2k the rigid boundary. 
In the IV plane the area is bounded hj iff = 0, \}j = 2k, (f) = — I, (f) = I, 2l being the 
circulation round the vortex. 
0 = 
0=Z 
IjV 0 
The function V satisfies the following conditions :— 
(a.) V = 0 over i// = 0, if the velocity along the free steam line be unity. 
(6.) clYIdxfj = 0 over xfj = 2k. 
(c.) V is periodic with respect to </>, so that V (</> + 2/) = V ((^). 
These conditions are to he satisfied by taking equal singular points at distances 2/ 
along i// = 2k, and then continually reflecting these points in the two planes ^ = 0, 
xjj = 2k, but in reflecting in xjj = 0 the image is of opposite sign to the object. 
Corresponding, then, to a point M at (^g, 2k) we have positive images at 
<^Q -j“ 277il, 2k -|“ 2ni . Ak, 
and negative images at 
+ 2ml, — 2k 2m . Ak. 
Therefore, corresponding to this point M, we have a factor 
(w — <^Q — 2ik) . dz 
— 0Q — 2ili) dw ’ 
where 
al = K\ 
Aak = K'j 
