ON LINES OF INDUCTION IN A MAGNETIC FIELD. 
32 3 
The second gives 
0(Y o +Y0 0(Y o + Y,) dV 0 3V e 0V ; 
* = "“aT“» or (/x_ 0 cn = dn “'Sv 
dn 
where /x is the permeability, and n is the outward-drawn normal at any point of the 
ellipse (1). 
Now ~ so that the last equation becomes 
on on an 1 
, _ x sv 0 av, 
^" 1 )a^= ftT 
0V; 
^ dll • 
Carrying out the differentiations, and then putting n = tanh 1 we get for the 
second condition 
[ihA -f- (a — 0)B = (p — 1)0H.(3). 
Solving (2) and (3) for A and B, we find 
A = ^ . i>H 
B 
fib + a 
fM — 1 ab 
fib -J- a ' a — b 
H. 
We thus have 
v. + v, = - . H.r 
fib -f a 
Y„ -f- Y e = - ^ ^ . Ujib' — a 2 ) cosh u cos 6 — (a — 1) ab sinh u cos 6} 
(fib + a) (a — b) 
and the equipotential lines may at once be plotted from these equations. 
The equation to the lines of induction inside the cylinder is obviously y — constant. 
In order to find the equation to the external lines, we have to determine the 
function which is conjugate to Y 0 4- Y e . Since the function which is conjugate 
to cosh u cos 6 is sinh u sin 9, and that conjugate to sinh u cos 6 is cosh u sin 6, 
we have 
K cosec 9 = cosh u — -———^ sinh u 
(fi — 1) ab 
for the equation to the lines of the external field, K .being a constant which varies 
from one line to another. 
If we next assume that the direction of the impressed field is along the minor 
axis of the ellipse, we similarly find for the equation to the lines of the external field 
K sec 9 — cosh n — ( ^— —-sinh a, 
fiar — rr 
K being as before a parameter which varies from one line to another. 
The treatment given above for the case of a solid elliptic cylinder may be extended 
