320 
PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
General problem of induction. 
§ 4. The problems to which we shall chiefly devote our attention in what follows are 
the induction of currents in a plate (i.) of finite and (ii.) of infinitesimal thickness, 
either at rest or revolving uniformly about an axis normal to its faces, and in a sphere 
or spherical shell either of finite or infinitesimal thickness under the same conditions. 
We may observe however, preliminarily, that any solution of any problem of induc¬ 
tion which satisfies all the conditions of the question is the only solution that can exist. 
For all the equations being linear the difference between the functions which express 
two solutions would also satisfy all the differential equations, and would correspond to 
the case in which the conductor is under no external inducing forces, and therefore no 
currents can be set up in it. 
This observation has been made by Maxwell and has been established more in 
detail by Helmholtz in his memoir. From the linearity of the equations we may also 
conclude that the effect of different systems acting either simultaneously or in succes¬ 
sion may be found by adding the effects due to each separately. 
In particular, when the conductor is at rest, its state at any time t may be considered 
as the aggregate effect produced by a continuous series of impulses during its 
previous history. The effect of each of these impulses, after being received, decays 
according to a certain law, and each contributes a certain amount to the total result at 
any time. The characteristic equation in these investigations is 
S v2p =l< p + p «).00 
where P 0 is due to the direct action of the external electro-magnets. 
Let an impulsive change take place in this external system, by which P 0 suddenly 
rises from 0 to P' 0 , and let P increase at the same time from 0 to P'; then, integrating 
the above equation during the impulse, we obtain 
F+P'o=0 
(26) 
The impulse having been administered, P decays according to a law peculiar to each 
system, and satisfies during its decay the equation 
a dl 3 
V J P = 
4tt dt 
(27) 
Let P 0 (i) be the value of P 0 at any time t : this value may be supposed to have 
been accumulated by impulses during the previous history of the system, and, if r be 
the time measured backwards from t, we may express this fact by the equation 
- n( ' T ) dr. 
dr 
