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

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 ${\frac {\operatorname {d} \!\,\mu }{\rho }}$ comprehending with reversed signs the whole of the magnetic particles of the earth: or say

$V=-\int {\frac {\operatorname {d} \!\,\mu }{\rho }}$.

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 $\xi$, $\eta$, $\zeta$,

$\xi ={\frac {d\,V}{d\,x}},\quad \eta ={\frac {d\,V}{d\,y}},\quad \zeta ={\frac {d\,V}{d\,z}},\quad \psi ={\sqrt {}}(\xi ^{2}+\eta ^{2}+\zeta ^{2})$.

5.

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

${\begin{aligned}d\,V&={\frac {d\,V}{d\,x}}.d\,x+{\frac {d\,V}{d\,y}}.d\,y+{\frac {d\,V}{d\,z}}.d\,z.\\&=\xi d\,x+\eta d\,y+\zeta d\,z\end{aligned}}$.

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

$d\,V=\psi \cos \theta .d\,s$,

because as ${\frac {\xi }{\psi }}$, ${\frac {\eta }{\psi }}$, ${\frac {\zeta }{\psi }}$ are the cosines of the angles which the direction of $\psi$ makes with the co-ordinate axes, so ${\frac {d\,x}{d\,s}}$, ${\frac {d\,y}{d\,s}}$, ${\frac {d\,z}{d\,s}}$, are the cosines of the angles between $d\,s$ and the same axes.

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