292 
MR. Gr. T. WALKER ON REPULSION AND ROTATION 
might be thought that for some other surface an analogous property held, as that in 
an anchor ring the induced currents always lay in toroidal surfaces. 
The question may be stated as follows :— 
Given a system of orthogonal surfaces, 
a = constant, 
b = constant, 
c = constant, 
and a uniform conductor whose bounding surface has a constant, what is the condition 
that, whatever be the nature of the external field, the induced currents may lie in the 
a surfaces. 
If the length 85 of the line joining the consecutive points a, b, c, and a + Sa, 
6 + 86 , c + Sc be given by 
Ss 2 = A 2 8a 2 + B 2 S/> 2 + C 2 8c 2 , 
and if 
u, v, iv, 
ft, y, 
F, G, H, 
denote the components of electric current, of magnetic force, and of electromagnetic 
momentum along the normals to the three orthogonal surfaces through any point, 
then Maxwell’s equations of electric currents become 
47tBC .« = A (Cy) - | (B/3) 
4„CA.„ = !(A*)-A (Cy , 
4HAB. w = A (E/3) _ A (Aa), 
and if there be no magnetisable matter in the conductor, a, /3 , y are components of 
magnetic induction, and are given by 
bc« = A (C h)-A(bg), 
with two similar equations. 
If cr denote the specific resistance of the conductor, and xfj the electrostatic potential, 
then within the conductor we have (it being at rest) 
3F df 
<TU dt A 8 a’ 
with two similar equations. 
