Page:Scientific Memoirs, Vol. 2 (1841).djvu/465

body $A$, $x'$ that of the body $B$, and $\left({\frac {du}{dx}}\right)$, $\left({\frac {du'}{dx}}\right)$ the particular values of ${\frac {du}{dx}}$, ${\frac {du'}{dx}}$ immediately belonging to them at the common surface, and in which it was assumed that the origin of the abscissæ was not taken on this common surface. The necessity of this last equation may easily be conceived; for were it otherwise, the two currents would not be of equal energy in the common surface, but there would be more conveyed from the one body to this surface than would be abstracted from it by the other; and if this difference were a finite portion of the entire current, the electroscopic force would increase at that very place, and indeed, considering the surprising fertility of the electric current, would arrive in the shortest time to an exceedingly high degree, as observation has long since demonstrated. Nor can a smaller quantity of electricity be imparted from the one body to the common surface than it is deprived of by the other, as this circumstance would be evinced by an infinitely high degree of negative electricity.

It is not absolutely requisite for the validity of the preceding determinations, that the two bodies in contact have the same base. The section in the one prismatic body may be different in size and form to that in the other, if this does not render the electroscopic force sensibly different at the various points of the same section, which, considering the great energy with which the electricity tends to equilibrium, will not be the case when the bodies are good conductors, whose length far surpasses their other dimensions. In this case everything remains as before, only that the section of the body $B$ must everywhere be distinguished from that of $A$; consequently the second conditional equation for the place where the two bodies are in contact changes into the following:—

$\chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\left({\frac {du'}{dx}}\right),$

where $\omega$ still represents the section of $A$, but $\omega '$ that of the body $B$, which at present differs from the former.

There may even exist in the prolongation of the body $A$ two prismatic bodies, $B$ and $C$, separated from each other, which are both situated immediately on the one surface of $A$. If in this case $\chi '\omega 'u'$ signifies for the body $B$, and $\chi ''\omega ''$, $u''$ for the body