Page:Scientific Memoirs, Vol. 2 (1841).djvu/201

the Intensitas Vis Magneticæ, &c., for our positive fundamental unity, that quantity of northern fluid which at the unit of distance exercises on an equal quantity of the same fluid a moving force equivalent to what we assume as unity.

When we speak of the magnetic force which in any point of space is produced by the action of the magnetic fluid elsewhere, we always mean to speak of the moving force which is there exercised on the unity of the positive magnetic fluid; therefore in this sense the supposed magnetic fluid $\mu$ concentrated in a point exercises at the distance $\rho$ the magnetic force ${\frac {\mu }{\rho ^{2}}}$, of either repulsion or attraction in the direction $\rho$, according as $\mu$ is positive or negative. Representing by $a$, $b$, $c$, the co-ordinates of $\mu$ in relation to three rectangular axes, and by $x$, $y$, and $z$, the co-ordinates of the point where the force is exercised, so that

$\rho ={\sqrt {}}([x-a)^{2}+(y-b)^{2}+(z-c]^{2})$;

and resolving the force in parallels to the co-ordinate axes, the components are

${\frac {\mu (x-a)}{\rho ^{3}}},\quad {\frac {\mu (y-b)}{\rho ^{3}}},\quad {\frac {\mu (z-c)}{\rho ^{3}}},$

which, as is easily seen, are equal to the partial differential co-efficients of $-{\frac {\mu }{\rho }}$ relatively to $x$, $y$, and $z$.

If besides $\mu$, there are also in operation other portions of the magnetic fluids $\mu '$, $\mu ''$, $\mu '''$, &c., concentrated in points, of which the distances from the spot where the force is exercised are $\rho '$, $\rho ''$, $\rho '''$, &c., then the components of the whole resulting magnetic force, parallel to the co-ordinate axes, are equal to the partial differential co-efficients of

$-\left({\frac {\mu }{\rho }}+{\frac {\mu '}{\rho '}}+{\frac {\mu ''}{\rho ''}}+{\frac {\mu '''}{\rho '''}}+{}{\&}{\mbox{c.}}\right)$,

relatively to $x$, $y$, and $z$.

4.

Hence may easily be shown what magnetic force is exercised in each point of space by the earth, however the magnetic fluids may be distributed therein. Imagine the whole volume of the earth, as far as it contains free magnetism (that is to say, separated magnetic fluids), to be divided into infinitely small elements; designate generally the quantity of free magnetic fluid