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PROFESSOR C. NIVEN OR THE INDUCTION OF ELECTRIC 
Helmholtz, in an elaborate memoir on the “ Equations of Motion of Electricity,” 
in Crelle’s Journal (vols. 72, 78), has given an exhaustive analysis of the conditions 
which have to be satisfied in any problem regarding the movement of electricity, and 
has proved very clearly that the solution of any problem is unique; but he has not 
dealt with any special case of the problem of induced currents. 
Maxwell’s investigation remains up to the present, so far as I am aware, the only 
case in which the complete solution of any case of induction has been published. 
German writers on current electricity have usually adopted some form of the theory 
of action at a distance between the elements of different currents, and the free elec¬ 
tricity is conceived as a scalar quantity distributed with a certain density throughout 
the interior and over the surface of conductors. Maxwell’s theory, which is adopted 
in the present paper, though it leads generally to similar equations, differs notably 
from the other in both these respects. The energy is supposed to be seated every¬ 
where in the surrounding medium, and the free electricity is the convergence of 
a vector quantity termed the electric displacement. The total current, to which 
electro-magnetic phenomena are due, is compounded of the current of conduction and 
the time-variations of the electric displacement. Owing to this peculiarity of the 
theory, the conditions to be satisfied at the surface of separation of two substances 
will differ from those given by Helmholtz. I have therefore analysed them some¬ 
what fully : taking first, for the sake of generality and the simplicity which it gives, 
the most general case of two substances in which both the conductivity and specific 
inductive capacity are to be retained. We can then deduce the conditions at the 
common surface of two conductors, or of a conductor and a dielectric, which is the 
case with which we have to do. 
One special result of these conditions is that when the vector potential, at the 
surface, due to all the currents or magnets in the field is at each point perpendicular 
to the surface of the conductor, the electric potential will vanish everywhere, and 
there will be no free electricity present either in the conductor or on its surface. 
This happens in the case of an infinite plane plate of any thickness. The vector 
potential (or electro-magnetic momentum) is then everywhere parallel to the surface 
of the plate, and is derived by vector differentiation from a function P ; the current 
in it is also everywhere parallel to the surface and is derived from a single current 
function <t>. It- also appears that P is the potential of imaginary matter distributed 
with density <£> ; and, during the decay of the currents, P satisfies an equation of the 
same form as that which regulates the diffusion of heat throughout a solid. 
When the plate is infinitely thick there will be no reaction in the inducing system; 
but when it is very thin, the effect will be that given by Maxwell as already 
explained. The general formulae in this case reproduce his results. 
For a solid sphere or shell bounded by two concentric spherical surfaces, the vector 
potential and current are everywhere at right angles to the radius vector to the 
common centre, and their values may be derived from two functions, P and d>, which 
