39G 
MR. J. H. MICHELL ON THE THEORY OF FREE STREAM LINES. 
For an edge of the conductor we may proceed thus :— 
then 
z — = 
(z — 2^)" = R" (cos nO + i sin nO), 
so that as 6 goes from — tt to tt, {z — z^^’‘ goes from 
R" [cos (— nir) -h i sin (— ?r7r)] 
to 
PF'[cos (wtt) + i sin {nTr)\ 
and, therefore, the amplitude of dzlclw increases by 2?i7r. Now the amplitude goes 
from 0 to TT, therefore, n = \. 
ffence 
dio ~ 
dz {z — Xg)^ ’ 
where ?• refers to a point in the field and s to the edge of a conductor. 
If one of the conductors reduce to a line, we have two of the equal, say 
Xg = Xs + i, and there is a factor 2 — x^ in the denominator, and so for any number of 
line conductors. 
There are many other cases for which a formula like the above applies ; it is not 
always necessary that the conductors should be in the same plane. 
It is very easy to perceive sucli cases by considering whether the equation 
(lY/chi = 0 is satisfied over the conductors where chi is an element of a normal to a 
conductor. 
Ry combining this transformation with that of Schwarz and Christoffel we get 
a general solution for the free-stream-line problem, as I shall presently show. 
It is necessary first to deduce some special formulm, which will be continually used 
hereafter. 
(a) Take first the case of one conductor and one line— 
■ —-—X. .-X. 
X = — b X = b X = a 
Let the conductor extend from x = — h to x = h and the line distribution 
be at a; = a. 
