Page:Scientific Memoirs, Vol. 2 (1841).djvu/202

 190 contained in each of these elements by $$\operatorname{d}\!\,\mu$$, in which the southern fluid is always considered as negative; call $$\rho$$ the distance of $$\operatorname{d}\!\,\mu$$ from a point in space, the rectangular co-ordinates of which may be $$x$$, $$y$$, $$z$$; lastly, let $$V$$ denote the aggregate of $$\frac$$ comprehending with reversed signs the whole of the magnetic particles of the earth: or say Thus $$V$$ has in each point of space a determinate value, or it is a function of $$x$$, $$y$$, $$z$$, or of any other three variable magnitudes, whereby we may define points in space. We then obtain, by the following formulæ, the magnetic force $$\psi$$ in every point of space, and the components of $$\psi$$, parallel to the co-ordinate axes, which we shall call $$\xi$$, $$\eta$$, $$\zeta$$,

I shall first develope some general propositions which are independent of the form of the function $$V$$, and are worthy of attention from their simplicity and elegance.

The complete differential of $$V$$ becomes

If we designate by $$d\, s$$ the distance between the two points to which $$V$$ and $$V + d\, V$$ belong, and by $$\theta$$ the angle which the direction of the magnetic force $$\psi$$ makes with $$d\, s$$, we have because as $$\frac$$, $$\frac$$, $$\frac$$ are the cosines of the angles which the direction of $$\psi$$ makes with the co-ordinate axes, so $$\frac$$, $$\frac$$, $$\frac$$, are the cosines of the angles between $$d\, s$$ and the same axes.

Therefore $$\frac$$ is equal to the force resolved in the direction of $$d\,s$$; the same follows from the equation $$\frac=\xi$$ if we bear in mind that the co-ordinate axes may be arbitrarily chosen.