Page:ComstockInertia.djvu/8

Since

$[E,\mathrm {\ curl} \ E]_{x}=-\left[E,\ {\frac {1}{V}}{\frac {\partial H}{\partial t}},\ H\right]_{x},$,

the last two terms in the bracket of (9) become together

$-{\frac {1}{V^{2}}}{\frac {\partial }{\partial t}}[EH]_{x}$,

which is minus the time rate of the density of momentum at the point. The time rate, however, refers to a point fixed in space, and to change to a point moving with the system we make use of the usual expression and write

$-{\frac {\partial }{\partial t}}m_{x}=-{\frac {\partial '}{\partial t}}m_{x}+v_{1}\left(l{\frac {\partial m_{x}}{\partial x}}+m{\frac {\partial m_{x}}{\partial y}}+n{\frac {\partial m_{x}}{\partial z}}\right)$,

(10)

where $m_{x}$ is the $x$-component of the density of momentum and ($l,m,n$) are, as formerly, the direction cosines of the constant velocity ($v_{1}$). The operator ${\frac {\partial '}{\partial t}}$ now refers to the rate of change at a point moving with the system.

Substituting (10) for the two last terms in (9) and noticing that

${\frac {\partial m_{x}}{\partial t}}$ may be written ${\frac {\partial }{\partial x}}{\frac {\partial '}{\partial t}}\int _{R}^{P}m_{x}dx$,

where the integration is to be taken from $R$ (meaning merely from a point outside the system where $m_{x}$ is zero) up to the point $P$ in question, account being taken of any discontinuity at the bounding surface, we have in place of (9)

${\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}=-\rho \Im _{x}$ $={\frac {\partial }{\partial x}}\left\{-{\frac {1}{8\pi }}(E_{x}^{2}-E_{y}^{2}-E_{z}^{2})-{\frac {1}{8\pi }}(H_{x}^{2}-H_{y}^{2}-H_{z}^{2})-v_{1}lm_{x}+{\frac {\partial '}{\partial t}}\int _{R}^{P}m_{x}dx\right\}$ $+{\frac {\partial }{\partial y}}\left\{-{\frac {1}{4\pi }}(E_{x}E_{y}+H_{x}H_{y})-v_{1}mm_{x}+{\frac {\partial '}{\partial t}}\int _{R}^{P}m_{x}dy\right\}$ $+{\frac {\partial }{\partial z}}\left\{-{\frac {1}{4\pi }}(E_{x}E_{z}+H_{x}H_{z})-v_{1}nm_{x}+{\frac {\partial '}{\partial t}}\int _{R}^{P}m_{x}dz\right\}$

(11)

This equation gives us what we were seeking, namely, the values of $X_{x},X_{y}$, and $X_{z}$ in terms of the electric and magnetic forces and the density of the momentum.