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of the magnetic body, so as to be incapable of passing from one element to the next.

Suppose that an amount m of the positive magnetic fluid is located at a point (x, y, z); the components of the magnetic intensity, or force exerted on unit magnetic pole, at a point (ξ, η, ζ) will evidently be

$-m{\frac {\partial }{\partial \xi }}\left({\frac {1}{r}}\right)$, $-m{\frac {\partial }{\partial \eta }}\left({\frac {1}{r}}\right)$, $-m{\frac {\partial }{\partial \zeta }}\left({\frac {1}{r}}\right)$,

where r denotes |(ξ-x)² + (η-y)² + (ζ-z)²|^{${\frac {1}{2}}$}. Hence if we consider next a magnetic element in which equal quantities of the two magnetic fluids are displaced from each other parallel to the x-axis, the components of the magnetic intensity at (ξ, η, ζ) will be the negative derivates, with respect to ξ, η, ζ respectively, of the function

$A{\frac {\partial }{\partial x}}\left({\frac {1}{r}}\right)$,

where the quantity A, which does not involve (ξ, η, ζ), may be called the magnetic moment of the element: it may be measured by the couple required to maintain the element in equilibrium at a definite angular distance from the magnetic meridian.

If the displacement of the two fluids from each other in the element is not parallel to the axis of s, it is easily seen that the expression corresponding to the last is

$A{\frac {\partial }{\partial x}}\left({\frac {1}{r}}\right)+B{\frac {\partial }{\partial y}}\left({\frac {1}{r}}\right)+C{\frac {\partial }{\partial z}}\left({\frac {1}{r}}\right)$,

where the vector (A, B, C) now denotes the magnetic moment of the element.

Thus the magnetic intensity at an external point (ξ, η, ζ) due to any magnetic body has the components

$\left(-{\frac {\partial V}{\partial \xi }},-{\frac {\partial V}{\partial \eta }},-{\frac {\partial V}{\partial \zeta }}\right)$,

where

$V=\iiint \left(A{\frac {\partial }{\partial x}}+B{\frac {\partial }{\partial y}}+C{\frac {\partial }{\partial z}}\right)\left({\frac {1}{r}}\right)dxdydz$

integrated throughout the substance of the magnetic body, and