Page:Scientific Memoirs, Vol. 2 (1841).djvu/468

The electroscopic force, at any place of such a prismatic body, may be deduced from the differential equation (a) found in § 11. For this purpose we have only to integrate it, and to determine, in accordance with the other conditions of the problem, the arbitrary functions or constants entering into the integral. This matter is, however, generally very much facilitated, with respect to our subject, by omitting one or even two members, according to the nature of the subject, from the equation (a). Thus nearly all galvanic actions are such that the phænomena are permanent and invariable immediately at their origin. In this case, therefore, the electroscopic force is independent of time, consequently the equation (a) passes into

$0=\chi {\frac {d^{2}u}{dx^{2}}}-{\frac {bc}{\omega }}u.$

Moreover, the surrounding atmosphere has (as we have already noticed in § 9.) in most cases no influence on the electric nature of the galvanic circuit; then $b=0$, by which the last equation is converted into

$0={\frac {d^{2}u}{dx^{2}}}.$

But the integral of this last equation is

$u=fx+c,$

(c)

where $f$ and $c$ represent any constants remaining to be determined. The equation (c) consequently expresses the law of electrical diffusion, in a homogeneous prismatic conductor, in all cases where the abduction by the air is insensible, and the action no longer varies with time. As these circumstances in reality most frequently accompany the galvanic circuit, we shall on that account dwell longest upon them.

We are enabled to determine one of the constants by the tension occurring at the extremities of the conductor, which has to be regarded as invariable and given in each case. If, for instance, we imagine the origin of the abscissæ anywhere in the axis of the body, and designate the abscissa belonging to one of its ends by $x_{1}$, then the electroscopic force there situated is, according to the equation (c),

$fx_{1}+c\colon$

in the same way we obtain for the electroscopic force of the other extremity, when we represent its abscissa by $x_{2}$,