Page:ComstockInertia.djvu/9

Thus we may take

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

(12)

Similar values are of course to be found for the other six components of stress in the constraint $Y_{x},Y_{y},Y_{z},Z_{x},Z_{y},Z_{z}$.

8. The values for these nine stress-components are now to be substituted in equation (5). In doing this it is to be noticed that the last term in each of equations (12) will, after substitution, furnish a term of the type

$\int d\tau {\frac {\partial '}{\partial t}}\int _{R}^{P}v_{1}l^{2}m_{x}dx={\frac {\partial '}{\partial t}}\int d\tau \int _{R}^{P}v_{1}l^{2}m_{x}dx$,

(13)

and this, being a time derivative, gives an average value of zero when the time is allowed to increase indefinitely, since all quantities in the system remain finite. Also

$l^{2}+m^{2}+n^{2}=1\,$.

Making the substitution in (5) and simplifying, we have as the value for ($v_{1}W$)

$v_{1}W=V^{2}M-v_{1}\int \left\{{\frac {1}{8\pi }}(E_{x}^{2}+E_{y}^{2}+E_{z}^{2})+{\frac {1}{8\pi }}(H_{x}^{2}+H_{y}^{2}+H_{z}^{2})\right\}d\tau$ $+v_{1}{\frac {1}{4\pi }}\int \left\{(lE_{x}+mE_{y}+nE_{z})^{2}+(lH_{x}+mH_{y}+nH_{z})^{2}\right\}d\tau$ $v_{1}^{2}\int (lm_{x}+mm_{y}+nm_{z})d\tau$.

(14)

Now the first integral represents the total included energy, the two parts of the second integral represent the squares of the components of the electric and magnetic forces in the direction of the motion of the system, and the last integral represents the momentum in the direction of motion, which in this case is the whole momentum $M$, since we have assumed that $M$ and $v_{1}$ are in the same direction.

Calling

${\frac {1}{8\pi }}\int \left\{(lE_{x}+mE_{y}+nE_{z})^{2}+(lH_{x}+mH_{y}+nH_{z})^{2}\right\}d\tau =W_{L}$