MR. J. H. MICHELL OJi THE THEORY OF FREE STREAM LINES. 
395 
Let the distance between AB and EF be d-^ and that between EF and CD d^. 
Then integrating past the points w = — 1 , oj = 1 , we find 
(/j = — 77 
d,= 
A(l- c) 
77 ■ 
A a + c) 
which determine A and c, and, therefore, make the formula definite. 
Other examples will occur in the physical application. 
Peoblem II. 
The second transformation which we need may be stated most simply as an 
electrical problem. 
Let there be any number of infinitely long plane conductors, all in the same plane, 
and with parallel edges. 
It is required to find the potential at any point when these conductors are raised to 
given potentials. 
A P. C D E F 
Let AB, CD, EF, ... be the sections of the conductors by the plane (xy). 
Everything is symmetrical with regard to the line ABF which we take to be 
y — 0- 
Consider tbe specifications of the transformation-function V where i// is the potential 
and (f) are the lines of force. 
We must have dYjdxli = 0 over the conductors, since they are straight, and, there¬ 
fore, also dY/dy = 0. 
There will be infinite points at the edges A, B . . . of the conductors, and also at 
points in the field corresponding to branch points of \p (or <^). These last will be 
distributed symmetrically with respect to y = 0. 
From these conditions it is plain that the solution is that V is the potential of 
masses at the singular points in question, so that we may write 
V = log Il[{x — x,f -\-{y — + C, 
and, therefore, 
clz 
= An [z — zY'. 
dvj ' ' 
It remains to find the quantities iir- 
For a singular point in the field where m branches of xfj meet we have simply 
Ur— — m. 
3 E 2 
