Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/84

where the vector (A, B, C) or I represents the magnetic moment per unit-volume, or, as it is generally called, the magnetization. The function V was afterwards named by Green the magnetic potential.

Poisson, by integrating by parts the preceding expression for the magnetic potential, obtained it in the form

$\iint (\mathbf {I.} d\mathbf {S} ){\frac {1}{r}}-\iiint {\frac {1}{r}}\mathrm {div} \mathbf {I} dxdydz$,^[1]

the first integral being taken over the surface S of the magnetic body, and the second integral being taken throughout its volume. This formula shows that the magnetic intensity produced by the body in external space is the same as would be produced by a fictitious distribution of magnetic fluid, consisting of a layer over its surface, of surface-charge (I.ds) per element dS, together with a volume-distribution of density — div I throughout its substance. These fictitious magnetizations are generally known as Poisson's equivalent surface- and volume-distributions of magnetism.

Poisson, moreover, perceived that at a point in a very small cavity excavated within the magnetic body, the magnetic potential has a limiting value which is independent of the shape of the cavity as the dimensions of the cavity tend to zero; but that this is not true of the magnetic intensity, which in such a small cavity depends on the shape of the cavity. Taking the cavity to be spherical, he showed that the magnetic intensity within it is

$\mathrm {grad} V+{\frac {4}{3}}\pi \mathbf {I}$,^[2]

where I denotes the magnetization at the place.

1. If the components of a vector a are denoted by (a_x, a_y, a_z), the quantity a_xb_x, a_yb_y, a_zb_z is called the scalar product of two vectors a and b, and is denoted by (a.b).

   The quantity ${\frac {\partial \alpha x}{\partial x}}+{\frac {\partial \alpha y}{\partial y}}+{\frac {\partial \alpha z}{\partial z}}$ is called the divergence of the vector a, and is denoted by div a.

2. The vector whose components are $-{\frac {\partial V}{\partial x}},-{\frac {\partial V}{\partial y}},-{\frac {\partial V}{\partial z}}$ is denoted by grad V.