Page:ComstockInertia.djvu/18

refers to a point fixed in space, and if we wish it to refer to a point moving with the system we must write as usual

${\frac {\partial w}{\partial t}}={\frac {\partial 'w}{\partial t}}-v_{1}{\frac {\partial w}{\partial x}}$,

(24)

where ${\tfrac {\partial 'w}{\partial t}}$ now means the rate of change measured from the moving point. Likewise, if we wish the velocities which enter into (23) to be expressed in terms of velocities relative to axes moving with the system, we must write

${\begin{cases}v_{x}=v_{1}+v_{2x}\\v_{y}=v{}_{2y}\\v_{z}=v_{2z}\end{cases}}$

(25)

where $v_{2x},v_{2y}$, and $v_{2z}$ are the components of these relative velocities.

Substituting (24) and (25) in (23), and remembering the simple proportionality between $S_{x},S_{y}$, and $S_{z}$ and the density of momentum $m_{x},m_{y}$, and $m_{z}$, we easily obtain

$V^{2}{\frac {\partial m_{x}}{\partial x}}-v_{1}{\frac {\partial w}{\partial x}}+v_{1}\rho \Im _{x}=-{\frac {\partial 'w}{\partial t}}-\left({\frac {\partial S_{y}}{\partial y}}+{\frac {\partial S_{z}}{\partial z}}\right)$ $-(v_{2x}\rho \Im _{x}+v_{2y}\rho \Im _{y}+v_{2z}\rho \Im _{z})$.

(26)

Now ($\rho \Im _{x}$) may be expressed in terms of the electric and magnetic force intensities, together with the density of the momentum. This involves only the fundamental equations of electromagnetic theory and has been done in paragraph 7, reference to which will show that with the present notation

$\rho \Im _{x}={\frac {\partial }{\partial x}}\left\{{\frac {1}{8\pi }}(X^{2}-Y^{2}-Z^{2})+{\frac {1}{8\pi }}(\alpha ^{2}-\beta ^{2}-\gamma ^{2})\right\}$ $+{\frac {\partial }{\partial y}}\left\{{\frac {1}{4\pi }}(XY+\alpha \beta )\right\}+{\frac {\partial }{\partial z}}\left\{{\frac {1}{4\pi }}(XZ+\alpha \gamma )\right\}$ $-{\frac {\partial 'm_{x}}{\partial t}}+v_{1}{\frac {\partial m_{x}}{\partial x}}$.

(27)

Substituting this for the ($\rho \Im _{x}$) which occurs on the lefthand side of (26), rearranging the latter, and putting

$w={\frac {1}{8\pi }}\{X^{2}+Y^{2}+Z^{2}+\alpha ^{2}+\beta ^{2}+\gamma ^{2}\}$

and

$w_{t}={\frac {1}{8\pi }}\{Y^{2}+Z^{2}+\beta ^{2}+\gamma ^{2}\}$,