Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/215

and did not profess to have proved, that this process consists in the actual movement of a quasi-substance, electricity, from one plate of the condenser to the other, or of two quasi-substances, the resinous and vitreous electricities, in opposite directions. The process had been pictured in this way by many of his predecessors, notably by Volta; and it has since been so pictured by most of his successors: but from such assumptions Faraday himself carefully abstained.

What is common to all theories, and is universally conceded, is that the rate of increase in the total quantity of electrostatic charge within any volume-element is equal to the excess of the influx over the efflux of current from it. This statement may be represented by the equation

${\frac {\partial \rho }{\partial t}}+\mathrm {div} \ \mathbf {i} =0$,

where ρ denotes the volume-density of electrostatic charge, and i the current, at the place (x, y, z) at the time t. Volta's assumption is really one way of interpreting this equation physically: it presents itself when we compare equation (1) with the equation

${\frac {\partial \rho }{\partial t}}+\mathrm {div} (\rho \mathbf {v} )=0$,

which is the equation of continuity for a fluid of density ρ and velocity v: we may identify the two equations by supposing i to be of the same physical nature as the product ρv; and this is precisely what is done by those who accept Volta's assumption.

But other assumptions might be made which would equally well furnish physical interpretations to equation (1). For instance, if we suppose ρ to be the convergence of any vector of which i is the time-flux,^[1] equation (1) is satisfied automatically;

1. In symbols,

   $\mathrm {div} \ \mathbf {s} =-\rho$,

   ${\frac {\partial \mathbf {s} }{\partial t}}=i$,

   where s denotes the vector in question.

O 2