MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
427 
Part II. 
We will now go on to consider problems in which the region of {x,y) is not simply 
connected, and consequently Schwarz’s transformation does not apply. 
First consider the area outside a closed polygon. 
We may state the problem electrically thus :— 
Problem III. 
To find the potential due to a polygonal prismatic conductor at a given potential 
which we may take to be zero, the field at infinity being at an infinite potential. 
Let y\i be the potential, (f) the lines of force, and let (f> increase by 2/ in going round 
the polygon. 
Then the area in the cj plane is a rectangle bounded byi//=0,r//=oo, ^= — Z, 
^ — 1. 
(p = — I 
(p — I 
^3 
- X 
The conditions which the transformation function V satisfy are 
dY 
{a) 
d \Jr 
= 0 over i// = 0. 
(b) V finite and continuous at all points within a finite distance in the rectangle. 
(c) V periodic in (f» so that 
V (<^ + 20 = V 
(d) V infinite at points .. . along x/; = 0. 
We can determine V from these specifications by means of W. Thomson’s method 
of images. For if we repeat the points ... at equal distances 2l along xp z= 0, and 
make Y the potential of these points, the conditions will clearly be satisfied. 
Hence 
V = A^, log'll {((/) — ({), — 2nlf + 
— CO 
and 
dz 
d w 
= An,.n„ {lu ~ —■ 2nl]‘‘ 
— CO 
{ TT 1 
sin {lu — cf),) ~ ^ , 
3 I 2 
