ON LINES OF INDUCTION IN A MAGNETIC FIELD. 321 
where y L and p, stand for the permeabilities of the two media, and n lt n, for the 
normals drawn from any point of the surface into the corresponding media. 
In dealing with elliptic cylinders and cylindrical shells, it is convenient to abandon 
the use of Cartesian rectangular co-ordinates, and to have recourse to the circular 
and hyperbolic functions. 
We shall take 
as our ellipse of reference. Then 
- + 7 , = 1 
cr Lr 
r 
■ (i) 
T- —— = I 
a- + lr + \ 
where X is a variable parameter such that X > — lr, represents the family of ellipses 
confocal with (I), while 
77 + yryr — • 
a - + v <r + v 
where v is a variable parameter such that — lr > v > — cr, represents the family of 
hyperbolas confocal with (I). 
If now we put 
v/ r -t X 
= cos 6, 
s/W + \ 
— sin 0, 
\/a* + 
= cosh u. 
y 
\/ — Qr + v ) 
sinli u, 
then u — constant is the equation to a certain ellipse, and 6 = constant to a hyper¬ 
bola, both curves being confocal with (1). 
It is obvious that u (which may vary from 0 to co ), and 0 (which may vary from 0 
to 27 t), completely and uniquely determine the position of a point in the plane of the 
ellipse (l). We shall use u and 0 as our co-ordinates. 
In order to arrive at the form of the induced potential function, we make use of 
the following considerations. Poisson has shown that the magnetisation of a solid 
ellipsoid placed in a uniform field is uniform. Let one of the axes of the ellipsoid be 
made infinite. Then we pass from the three-dimensional case of the ellipsoid to the 
two-dimensional one of the elliptic cylinder, and we see that in this case also the 
magnetisation is uniform. Such a magnetisation might be supposed to be produced 
by imagining two solid cylinders of the imaginary magnetic matter, of volume- 
density p and of opposite sign, originally coincident, to be displaced relatively to each 
other through a small distance in the direction of magnetisation, such that pSs is 
equal to the actual intensity of magnetisation of the cylinder. If then pY stand for 
the potential at any point due to one of the solid cylinders of the imaginary magnetic 
0y 
matter, the combined potential of the pair of displaced cylinders is p yy Ss. But 
(JS 
9 T 
VOL. CXCV. — A. 
