Page:Scientific Memoirs, Vol. 2 (1841).djvu/463

change in their electroscopic force by the same quantity of electricity, then there would still remain to be added a coefficient $\gamma$ corresponding to this property of the various bodies. Experience has not yet decided respecting this supposition borrowed from the relation of heat to bodies.

If we assume the two expressions just found for the entire change in the quantity of electricity in the disk $M$ during the moment of time $dt$ to be equal, and divide all the members of the equation by $\omega \,dx\,dt$, we obtain

$\gamma {\frac {du}{dt}}=\chi {\frac {d^{2}u}{dx^{2}}}-{\frac {bc}{\omega }}u,$

(a)

from which the electroscopic force $u$ has to be determined as a function of $x$ and $t$.

12. We have in the preceding paragraph found for the change in the quantity of electricity occurring between the disks $M'$ and $M$ during the time $dt$

${\frac {\chi (u'-u)\,dt}{dx}},$

and have seen that the direction of the passage is opposed to the course of the abscissæ when the expression is positive; on the contrary, it proceeds in the direction of the abscissæ when it is negative. In the same way the magnitude of the transition between the disks $M_{1}$ and $M$, when we retain the same relation to its direction, is

${\frac {\chi (u_{1}-u)\,dt}{dx}}.$

If we substitute in these two expressions for $u_{1}$ and $u'$ the transformations given in the same paragraph, and at the same time $\chi \omega$ for $\chi$, i. e. the absolute power of conduction for the relative, we obtain in both cases

$\chi \omega {\frac {du}{dx}}\,dt,$

whence it results that the same quantity of electricity which enters from the one side into the disk $M$ during the element of time $dt$, is again in the same time expelled from it towards the other side. If we imagine this transmission of the electricity, occurring at the time $t$ in the disk belonging to the abscissa $x$, of invariable energy reduced to the unity of time, call it the electric current, and designate the magnitude of this current by $S$, then