Page:Scientific Memoirs, Vol. 2 (1841).djvu/203

6.

If two points in space $P^{0}$, $P'$, be connected by an arbitrary line, of which d s represent an indeterminate element, and if, as before, $\theta$ signify the angle between d s and the direction of the magnetic force there existing, and $\psi$ its intensity, then

$\int \psi \cos \theta .d\,s=V'-V^{0}$

if we extend the integration through the whole line, and designate by $V^{0}$, $V'$, the values of $V$ at the extremities.

The following corollaries of this fruitful proposition deserve especial notice:—

I. The integral $\int \psi \cos .\theta .d\,s$ preserves the same value by whatever path we proceed from $P^{0}$ to $P'$.

II. The integral $\int \psi \cos \theta .d\,s$, extended through the whole length of any re-entering curve, is always $=0$.

III. In a re-entering curve, if $\theta$ is not throughout $=90^{\circ }$, a part of the values of $\theta$ must be greater and a part must be less than $90^{\circ }$.

7.

Those points of space in which $V$ has a value greater than $V^{0}$, are divided from those in which the value of $V$ is less than $V^{0}$, by a surface in all the points of which $V$ has one determinate value $=V^{0}$^[1].

It follows from the proposition in Art. V., that in each point of this surface the magnetic force has a direction perpendicular to the surface, and towards the side where the higher values of $V$ are found. Let $d\,s$ be an infinitely small line perpendicular to the surface, and $V^{0}+d\,V^{0}$ the value of $V$ at its other extremity; then the intensity of the magnetic force will be $={\frac {d\,V^{0}}{d\,s}}$. The series of points for which $V=V^{0}+d\,V^{0}$, form a second surface infinitely near to the first, and at different points in the whole intervening space the intensity of the magnetic force is in the inverse ratio of the distance apart of the two surfaces.

1. If the function $V$ could have any arbitrarily chosen form, then in particular cases a maximum or a minimum value of $V$ might correspond to an insulated point, or to an insulated line, around which only greater or only less values might be found, or it might correspond to a surface on both sides of which there might be greater or on both less values. But the conditions to which the function $V$ is subjected do not allow the occurrence of such excepted cases. A full development of this subject, as it is unnecessary for our present object, must be reserved for another occasion.