Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/95

an element of the shifted body whose quantity is –ρ δv at a distance –δx. The effect of these two elements is equivalent to that of a magnet of strength –r and length –δx. The intensity of magnetization is found by dividing the magnetic moment of an element by its volume. The result is δx.

Hence $-{\frac {dV}{dx}}\delta x$ is the magnetic potential of the body magnetized with the intensity δx in the direction of x, and $-{\frac {dV}{dx}}$ is that of the body magnetized with intensity ρ.

This potential may be also considered in another light. The body was shifted through the distance –δx and made of density –ρ. Throughout that part of space common to the body in its two positions the density is zero, for, as far as attraction is con cerned, the two equal and opposite densities annihilate each other. There remains therefore a shell of positive matter on one side and of negative matter on the other, and we may regard the resultant potential as due to these. The thickness of the shell at a point where the normal drawn outwards makes an angle e with the axis of x is δx cos ε and its density is ρ cos ε. The surface-density is therefore ρ δx cos ε, and, in the case in which the potential is $-{\frac {dV}{dx}}$, the surface-density is ρ cos ε.

In this way we can find the magnetic potential of any body uniformly magnetized parallel to a given direction. Now if this uniform magnetization is due to magnetic induction, the magnetizing force at all points within the body must also be uniform and parallel.

This force consists of two parts, one due to external causes, and the other due to the magnetization of the body. If therefore the external magnetic force is uniform and parallel, the magnetic force due to the magnetization must also be uniform and parallel for all points within the body.

Hence, in order that this method may lead to a solution of the problem of magnetic induction, $-{\frac {dV}{dx}}$ must be a linear function of the coordinates x, y, z within the body, and therefore V must be a quadratic function of the coordinates.

Now the only cases with which we are acquainted in which V is a quadratic function of the coordinates within the body are those in which the body is bounded by a complete surface of the second degree, and the only case in which such a body is of finite dimen-