Page:Scientific Memoirs, Vol. 2 (1841).djvu/460

448 11. This being established, we will now proceed to the subject, and in the first place consider the motion of the electricity in a homogeneous cylindric or prismatic body, in which all points throughout the whole extent of each section, perpendicular to its axis, possess contemporaneously equal electroscopic force, so that the motion of the electricity can only take place in the direction of its axis. If we imagine this body divided by a number of such sections into disks of infinitely small thickness, and so that in the whole circumference of each disk the electroscopic force does not vary sensibly for each pair of such disks, the expression ♂ given in § 6 can be applied to determine the quantity of electricity passing from one disk to the other; but by the limitation of the distance of action to only infinitely small distances mentioned in the preceding paragraph, its nature is so modified that it disappears as soon as the divisor ceases to be infinitely small.

If we now choose one of the infinite number of sections invariably for the origin of the abscissæ, and imagine anywhere a second, whose distance from the first we denote by $$x$$, then $$d\,x$$ represents the thickness of the disk there situated, which we will designate by $$M$$. If we conceive this thickness of the disk to be of like magnitude at all places, and term $$u$$ the electroscopic force present at the time $$t$$ in the disk $$M$$, whose abscissa is $$x$$, so that therefore $$u$$ in general will be a function of $$t$$ and $$x$$; if we further suppose $$u'$$ and $$u_1$$ to be the values of $$u$$ when $$x + d\,x$$ and $$x - dx$$ are substituted respectively for $$x$$, then $$u'$$ and $$u_1$$ evidently express the electroscopic forces of the disks situated next the two sides of the disk $$M$$, of which we will denote the one belonging to the abscissa $$x + d\,x$$ by $$M'$$, and that belonging to the abscissa $$x + d\,x$$ by $$M_1$$; and it is clearly evident that the distance of the centre of each of the disks $$M'$$ and $$M_1$$ from the centre of the disk $$M$$ is $$d\,x$$. Consequently, by virtue of the expression (♂) given in § 6, if $$\chi$$ represents the conducting power of the disk $$M'$$ to $$M$$, is the quantity of electricity which is transferred during the interval of time $$d\,t$$ from the disk $$M'$$ to the disk $$M$$, or from the latter to the former, according as $$u' - u $$ is positive or negative. In the same manner, when we admit the same power of conduction between $$M_1$$ and $$M$$,