Page:Scientific Memoirs, Vol. 2 (1841).djvu/501

the repulsive force between these two elements is proportional to the time $dt$, to the product $uu'$, and, moreover, to a function dependent on the position, size, and form of the two particles, which we will represent by $\mathrm {F} '$; we accordingly obtain for the repulsive force between two particles the expression

$\mathrm {F} 'uu'dt$.

If we here proceed in the same manner as in §6, and signify by the moment of action $\chi '$ between two places, the product of $q'$, which expresses the force produced under perfectly determined circumstances between both, and its mean distance $s'$, so that

$\chi '=q'.s'$,

and determine $q'$ by putting $u=u'=1$ in the expression $\mathrm {F} 'uu'~dt$, and extending the action to the unit of time, we have

$\chi '=F's'$,

whence it follows that

$\mathrm {F} '={\frac {\chi '}{s'}}$.

Let us now imagine, as we did in §11, the prismatic circuit to be divided into equally large, infinitely thin discs, and call $\mathrm {M} '$, $\mathrm {M}$, $\mathrm {M} _{1}$ those immediately following one another, which belong to the abscissæ $x+dx$, $x$, $x-dx$; then, according to what has just been shown, the pressure which the disc $\mathrm {M} '$ exerts on the disc $\mathrm {M}$ is

$\mathrm {F} 'uu'~dt$;

and if we admit that the position, size, and form of the particles remain in all discs the same, the counter pressure, which the disc $\mathrm {M} _{1}$ exerts on the disc $\mathrm {M}$, is

$\mathrm {F} 'uu_{1}~dt$;

the difference between these two expressions, viz.

$\mathrm {F} 'u(u'-u_{1})~dt$,

gives accordingly the magnitude of the force, with which the disc $\mathrm {M}$ tends to move along the axis of the circuit. This force acts contrary to the direction of the abscissæ when its value is positive, and in the direction of the abscissæ when it is negative.

If we substitute for $u'-u_{1}$ its value proceeding from the developments given in §11 for $u'$ and $u_{1}$, the expression just found changes into the following: