Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/419

328.] :$$r_1,\, r_2,$$ &c. the specific resistance referred to unit of volume.
 * $$K_1,\, K_2,$$ &c. the specific inductive capacity.
 * $$k_1,\, k_2,$$ &c. the reciprocal of the specific inductive capacity.
 * $$E$$ the electromotive force due to a voltaic battery, placed in the part of the circuit leading from the last stratum towards the first, which we shall suppose good conductors.
 * $$Q$$ the total quantity of electricity which has passed through this part of the circuit up to the time $$t$$.
 * $$R$$ the resistance of the battery with its connecting wires.
 * $$\sigma_{12}$$ the surface-density of electricity on the surface which separates the first and second strata.

Then in the first stratum we have, by Ohm's Law,

By the theory of electrical displacement,

By the definition of the total current, with similar equations for the other strata, in each of which the quantities have the suffix belonging to that stratum.

To determine the surface-density on any stratum, we have an equation of the form and to determine its variation we have

By differentiating (4) with respect to $$t,$$ and equating the result to (5), we obtain or, by taking account of (3), That is, the total current u is the same in all the strata, and is equal to the current through the wire and battery.

We have also, in virtue of equations (1) and (2), from which we may find $$X_l$$ by the inverse operation on $$u,$$