Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/54

where $X$, $Y$, $Z$ are the components of $R$ parallel to $x$, $y$, $z$ respectively.

This quantity is, in general, different for different lines drawn between $A$ and $P$. When, however, within a certain region, the quantity

$X{dx}+Y{dy}+Z{dz}=-D\Psi \,\!$

that is, is an exact differential within that region, the value of $L$ becomes

$L=\Psi _{A}-\Psi _{P}$

and is the same for any two forms of the path between $A$ and $P$, provided the one form can be changed into the other by continuous motion without passing out of this region.

On Potentials.

The quantity $\Psi$ is a scalar function of the position of the point, and is therefore independent of the directions of reference. It is called the Potential Function, and the vector quantity whose components are $X$, $Y$, $Z$ is said to have a potential $\Psi$, if

$X=-({\frac {d\Psi }{dx}}),\quad Y=-({\frac {d\Psi }{dy}}),\quad Z=-({\frac {d\Psi }{dz}}).$

When a potential function exists, surfaces for which the potential is constant are called Equipotential surfaces. The direction of $R$ at any point of such a surface coincides with the normal to the surface, and if $n$ be a normal at the point $P$, then $R=-{\frac {d\Psi }{dn}}$.

The method of considering the components of a vector as the first derivatives of a certain function of the coordinates with respect to these coordinates was invented by Laplace^[1] in his treatment of the theory of attractions. The name of Potential was first given to this function by Green^[2] who made it the basis of his treatment of electricity. Green's essay was neglected by mathematicians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson^[3].

In the theory of gravitation the potential is taken with the opposite sign to that which is here used, and the resultant force in any direction is then measured by the rate of increase of the

1. Méc. Céleste, liv. iii.
2. Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828. Reprinted in Crelle's Journal, and in Mr. Ferrer's edition of Green's Works
3. Thomson and Tait, Natural Philosophy, $\S$ 483.