Page:On Faraday's Lines of Force.pdf/40

194 one another, and leave a single current running round the bounding line. So that the magnetic effect of a uniformly magnetized shell is equivalent to that of an electric current round the edge of the shell. If the direction of the current coincide with that of the apparent motion of the sun, then the direction of magnetization of the imaginary shell will be the same as that of the real magnetization of the earth.

The total intensity of magnetizing force in a closed curve passing through and embracing the closed current is constant, and may therefore be made a measure of the quantity of the current. As this intensity is independent of the form of the closed curve and depends only on the quantity of the current which passes through it, we may consider the elementary case of the current which ﬂows through the elementary area $$dydz$$.

Let the axis of $$x$$ point towards the west, $$z$$ towards the south, and $$y$$ upwards. Let $$x, y, z$$ be the coordinates of a point in the middle of the area $$dydz$$, then the total intensity measured round the four sides of the element is

$+\left(\beta_1+\frac{d\beta_1}{dz}\frac{dz}{2}\right)dy,$,

$-\left(\gamma_1+\frac{d\gamma_1}{dy}\frac{dy}{2}\right)dz,$,

$-\left(\beta_1-\frac{d\beta_1}{dz}\frac{dz}{2}\right)dy,$,

$+\left(\gamma_1-\frac{d\gamma_1}{dy}\frac{dy}{2}\right)dz,$,

Total intensity $$=\left(\frac{d\beta_1}{dz}-\frac{d\gamma_1}{dy}\right)dydz$$.

The quantity of electricity conducted through the elementary area $$dydz$$ is $$a_2dy/dz$$, and therefore if we deﬁne the measure of an electric current to be the total intensity of magnetizing force in a closed curve embracing it, we shall have

$a_2=\frac{d\beta_1}{dz}-\frac{d\gamma_1}{dy}$,

$b_2=\frac{d\gamma_1}{dx}-\frac{d\alpha_1}{dz}$,

$c_2=\frac{d\alpha_1}{dy}-\frac{d\beta_1}{dx}$.