Page:Scientific Memoirs, Vol. 2 (1841).djvu/502

$2F'u{\frac {du}{dx}}dx\,dt,$

and if we take, instead of the function $F'$ dependent on the nature of each single body, its value ${\frac {\chi '}{s'}}$, this expression, since $s'$ is evidently here $dx$, changes into

$2\chi 'u{\frac {du}{dx}}dt;$

or if we reduce the moment of action $\chi '$, referring to the magnitude of the section $\omega$, to the unit of surface, and at the same time extend the action to the unit of time, into

$2\chi '\omega u{\frac {du}{dx}},$

where the present $\chi '$ represents the magnitude of the moment of action reduced to the unit of surface. If we write this latter expression thus:

$2{\frac {\chi '}{\chi }}\chi \omega u{\frac {du}{dx}},$

in which $\chi$ denotes the absolute power of conduction of the circuit; and if we substitute for $\chi \omega {\frac {du}{dx}}$, by which, according to the equation (b) in § 12, the magnitude of the electric current is expressed, the sign $S$ chosen for it, and $i$ instead of ${\frac {\chi '}{\chi }}$, it is changed into

$2iuS.$

We hence perceive that the force, with which the individual discs in the circuit tend to move, is proportional, both to their innate electroscopic force, and to the magnitude of the current; and that this force alters its direction at that place of the circuit where the electricity passes from the one into the opposite state. And here occurs the circumstance which must not be overlooked, that this expression still holds, even when the electroscopic force $u$ of the element $M$ is changed in the moment of action, by any causes whatsoever, into any other abnormal $U$, while the electroscopic forces of the adjacent particles continue the same; only that in this case the value $U$ must be substituted for $u$ in the expression $2iuS$. It must also be observed, that the expression $2iuS$ which we have found refers to the whole extent of the section $\omega$, which belongs to that part of