Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/139

96.]

CHARACTERISTICS OF THE POTENTIAL.

99

the density at that point are related to each other in a certain manner, and no relation is expressed between the value of ${\displaystyle V}$ at that point and the value of ${\displaystyle \rho }$ at any point at a sensible distance from it.

In the second expression, on the other hand, the distance between the point ${\displaystyle (x',\,y',\,z')}$ at which ${\displaystyle \rho }$ exists from the point ${\displaystyle (x,\,y,\,z)}$ at which ${\displaystyle V}$ exists is denoted by ${\displaystyle r}$, and is distinctly recognised in the expression to be integrated.

The integral, therefore, is the appropriate mathematical expression for a theory of action between particles at a distance, whereas the differential equation is the appropriate expression for a theory of action exerted between contiguous parts of a medium.

We have seen that the result of the integration satisfies the differential equation. We have now to shew that it is the only solution of that equation fulfilling certain conditions.

We shall in this way not only establish the mathematical equi valence of the two expressions, but prepare our minds to pass from the theory of direct action at a distance to that of action between contiguous parts of a medium.

Characteristics of the Potential Function.

96.] The potential function ${\displaystyle V}$, considered as derived by integration from a known distribution of electricity either in the substance of bodies with the volume-density ${\displaystyle \rho }$ or on certain surfaces with the surface-density ${\displaystyle \sigma ,\rho }$ and ${\displaystyle \sigma }$ being everywhere finite, has been shewn to have the following characteristics:—

(1) ${\displaystyle V}$ is finite and continuous throughout all space.

(2) ${\displaystyle V}$ vanishes at the infinite distance from the electrified system.

(3) The first derivatives of ${\displaystyle V}$ are finite throughout all space, and continuous except at the electrified surfaces.

(4) At every point of space, except on the electrified surfaces, the equation of Poisson

 ${\displaystyle {\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho =0}$

is satisfied. We shall refer to this equation as the General Characteristic equation.

At every point where there is no electrification this equation becomes the equation of Laplace,

 ${\displaystyle {\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}=0}$.