Page:Scientific Memoirs, Vol. 2 (1841).djvu/466

$C$ what $\chi \omega u$ does for $A$, we obtain instead of the one conditional equation the two following:—

${\begin{aligned}&(u)-(u')=a,\\&(u)-(u'')=a',\end{aligned}}$

where $a$ represents the electric tension between the bodies $A$ and $B$, and $a'$ that between $A$ and $C$. In the same manner we now obtain instead of the second conditional equation the following:—

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

It is immediately apparent how these equations must change when a greater number of bodies are combined. We shall not enter further into these complications, as what has been stated suffices to throw sufficient light upon the changes which have in such a case to be performed on the equations.

14. To avoid misconception, I will, at the close of these general observations, once more accurately define the circle of application within which our formulæ have universal validity. Our whole inquiry is confined to the case where all the parts of the same section possess equal electroscopic force, and the magnitude of the section varies only from one body to the other. The nature of the subject, however, frequently gives rise to circumstances which render one or the other of these conditions superfluous, or at least diminishes their importance. Since the knowledge of such circumstances is not without use, I will here illustrate the most prominent by an example.

A circuit of copper, zinc, and an aqueous fluid, will wholly come under the above formula when the copper and zinc are prismatic and of equal section; when, further, the fluid is likewise prismatic and of the same or of smaller section, and its terminal surfaces everywhere in contact with the metals. Nay, when only these last conditions are fulfilled with respect to the fluid, the metals may possess equal sections or not, and touch one another with their full sections, or only at some points, and even their form may deviate considerably from the prismatic form, and nevertheless the circuit must constantly obey the laws deduced from our formulæ; for the motion of the electricity produced with such ease in the metals, is obstructed to such a considerable extent by the non-conductive nature of the fluid, that it gains sufficient time to diffuse itself thoroughly