Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/246

590.] We are now able to determine the mode in which the quantity J depends on the direction of the element ds. For, by (4),

$J=F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}.$

(5)

This is the expression for the resolved part, in the direction of ds, of a vector, the components of which, resolved in the directions of the axes of x, y, and z are F, G and H respectively.

If this vector be denoted by ${\mathfrak {A}}$, and the vector from the origin to a point of the circuit by ρ, the element of the circuit will be dρ, and the quaternion expression for J will be

$-S{\mathfrak {A}}\,d\rho .$

We may now write equation (2) in the form

$p=\int \left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds,$

(6)

or

$p=-\int S{\mathfrak {A}}\,d\rho .$

(7)

The vector ${\mathfrak {A}}$ and its constituents F, G, H depend on the position of ds in the field, and not on the direction in which it is drawn. They are therefore functions of x, y, z, the coordinates of ds, and not of l, m, n, its direction-cosines.

The vector ${\mathfrak {A}}$ represents in direction and magnitude the time-integral of the electromotive force which a particle placed at the point (x, y, z) would experience if the primary current were suddenly stopped. We shall therefore call it the Electrokinetic Momentum at the point (x, y, z). It is identical with the quantity which we investigated in Art. 405 under the name of the vector-potential of magnetic induction.

The electrokinetic momentum of any finite line or circuit is the line-integral, extended along the line or circuit, of the resolved part of the electrokinetic momentum at each point of the same.

591.] Let us next determine the value of p for the elementary rectangle ABCD, of which the sides are dy and dz, the positive direction being from the direction of the axis of y to that of z.

Let the coordinates of 0, the centre of gravity of the element, be x₀, y₀, z₀, and let G₀, H₀, be the values of G and of H at this point.

The coordinates of A, the middle point of the first side of the