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

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

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

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 $\frac{\partial}{\partial x}\frac{\partial'}{\partial t}\int_{R}^{P}m_{x}dx$,|undefined

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)

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.