PROFESSOR HELE-SHAW AND MR. ALFRED HAY 
to any number of hollow confocal elliptic cylinders of different permeabilities. For 
if (taking the direction of H to be along the major axis of the ellipse) we assume 
that the induced potential V; m the innermost space is of the form 
V; = A y/a 2 — L~ . cosh u cos 9. 
that the induced potential V„ in the substance of any one of the hollow confocal 
elliptic cylinders is 
V, t = y/cr — b 2 ( A„ cosh u + B„ sinh u) cos 9. 
and that finally the induced potential in external space is 
Y e — A c x /a~ — !:r . e " cos 6. 
then all the functions Yi ... Y lt ... Y e .. . satisfy Laplace’s two-dimensional 
equation, and the last function V, vanishes at an infinite distance. If, therefore, 
we can determine the various constants A, . . . A,„ . . . A e , so as to satisfy (1) 
continuity of potential, and (2) continuity of normal induction, the only possible 
solution of the problem will have been obtained. 
Let us suppose that there are m hollow cylinders. The number of constants in 
the assumed expressions for the potential will in that case he 2 m + 2. There are 
m + 1 hounding surfaces, and at every such surface the two conditions of continuity 
of potential and continuity of normal induction must he fulfilled, thus giving two 
conditional equations for each boundary. There will therefore he 2(m-J- I) equations, 
and these will completely determine the values of the 2 m + 2 constants. 
By way of further illustration, we shall work out the case of a hollow elliptic 
cylinder of iron placed in a uniform field. Let the internal hounding surface be the 
ellipse 
* 7 + yW = U 
or u — tanh “ 1 (b/a), and the external bounding surface the ellipse 
or 
Let the impressed potential Y 0 he 
Y„ = — Hy/ck — b~ . cosh u cos 6, 
corresponding to a direction of the field along the major axis of the cylinder. 
We assume for the induced magnetic potential functions :— 
V, == A \/ a 2 — b 2 . cosh u cos 6, 
V, = A \/a 2 — h 2 (A, cosh v + B, sinh u) cos 6 
Y e = A e \/o 2 — b' 2 . e~ ri cos 9. 
and 
