114 Prof. C. Mven. Induction of Electric Currents [Jan. 29, 



of the sphere or shell. The case of an infinitely thin plane sheet has 

 been made the subject of a paper by the late Professor Clerk Maxwell, 

 in the "Proceedings of the Royal Society," vol. xx (1872), and 

 Jochmann has also treated the case of a rotating sphere when the 

 inducing magnetism is symmetrical about the axis of rotation, and 

 that of a rotating plate when the mutual induction of the currents 

 is neglected. In the present paper the problems are dealt with 

 generally. 



Maxwell's equations of the field are adopted, and the peculiar view 

 which he takes of the electric current renders it necessary to enter 

 with some detail into the general boundary conditions. These are 

 given, first of all, for two substances at rest, which possess both con- 

 ductivity and specific dielectric capacity : we can then deduce the 

 conditions to be satisfied at the common surface of a conductor and a 

 dielectric. 



Let F, G, H be the components, at any point just inside the 

 surface of the conductor, of the vector potential due to all the currents 

 and magnets in the field, Jf the component along the normal (N) 

 to the surface, measured outward from the conductor, the electric 

 potential ; and let F', G', H' be the components of the vector potential 

 just outside, the boundary conditions are 



F = F' cW - dF ' cl ^-Q 

 "' dN dN'"' dt dN 



When df— 0, as in the cases treated in the paper, it is shown .that 

 must vanish everywhere, and there is no free electricity within the 

 conductor or on its surface. 



Infinite Plate. 



For the case of an infinite plate, the sealar potential (Q ) of the 

 external system and the vector components are given by 



The vector potential of the currents in the plate, and the currents 

 themselves are given by 



F=— G=~, H=0, 



dy dx 



u=— d ®, v = — , w=0, where 4sr<E*= — V 3 P- 



dy dx 



They are also expressed in semipolar co-ordinates p, 0, z, thus : — 



d¥ n d¥ ^ d$> d<t> A 



F=— — , G=— , H=0, u=--—, v=-—, iv=0, 

 pdcp dp pd(p dp 



where 4tt$= — V 2 P- 



