Page:Scientific Memoirs, Vol. 2 (1841).djvu/464

452 and in this equation positive values for $$S$$ show that the current takes place opposed to the direction of the abscissæ; negative, that it occurs in the direction of the abscissæ.

13. In the two preceding paragraphs we have constantly had in view a homogeneous prismatic body, and have inquired into the diffusion of the electricity in such a body, on the supposition that throughout the whole extent of each section, perpendicular to its length or axis, the same electroscopic force exists at any time whatsoever. We will now take into consideration the case where two prismatic bodies $$A$$ and $$B$$, of the same kind, but formed of different substances, are adjacent, and touch each other in a common surface. If we establish for both $$A$$ and $$B$$ the same origin of abscissæ, and designate the electroscopic force of $$A$$ by $$u$$, that of $$B$$ by $$u'$$, then both $$u$$ and $$u'$$ are determined by the equation (a) in paragraph 11, if $$\chi$$ only retain the value each time corresponding to the peculiar substance of each body: but $$u$$ represents a function of $$t$$ and $$x$$, which holds only so long as the abscissa $$x$$ corresponds to points in the body $$A$$; $$u$$ on the other hand denotes a function of $$t$$ and $$x$$, which holds only when the abscissa $$x$$ corresponds to the body B. But there are still some other conditions at this common surface, which we will now explain. If we denote for this purpose the separate values of $$u$$ and $$u'$$, which they first assume at the common surface, by enclosing the general ones between crotchets, we find according to the law advanced in § 10 the following equation between these separate values: where $$a$$ represents a constant magnitude otherwise dependent on the nature of the two bodies. Besides this condition, which relates to the electroscopic force, there is still a second, which has reference to the electric current. It consists in this, that the electric current in the common surface must in the first place possess equal magnitude and like direction in both bodies, or, if we retain the common factor $$\omega$$, where $$\chi$$ represents the actual power of conduction of the