Page:BatemanElectrodynamical.djvu/33

Now, if

$ds'={\frac {ds}{r^{2}}},\ \rho ^{2}=r^{2}-t^{2},$ ${\begin{array}{r}{\frac {dx'}{ds'}}=\left(1-{\frac {2x^{2}}{\rho ^{2}}}\right){\frac {dx}{ds}}-{\frac {2xy}{\rho ^{2}}}{\frac {dy}{ds}}-{\frac {2xz}{\rho ^{2}}}{\frac {dz}{ds}}+{\frac {2xt}{\rho ^{2}}}{\frac {dt}{ds}},\\\\{\frac {d^{2}x'}{ds'^{2}}}=\left(\rho ^{2}-2x^{2}\right){\frac {d^{2}x}{ds^{2}}}-2xy{\frac {d^{2}y}{ds^{2}}}-2xz{\frac {d^{2}z}{ds^{2}}}+2xt{\frac {d^{2}t}{ds^{2}}}\\\\+2x-2{\frac {dx}{ds}}\rho {\frac {d\rho }{ds}}+4x\left({\frac {d\rho }{ds}}\right)^{2},\\\\{\frac {d^{3}x'}{ds'^{3}}}ds'=ds'\left[\left(\rho ^{2}-2x^{2}\right){\frac {d^{3}x}{ds^{3}}}-2xy{\frac {d^{3}y}{ds^{3}}}-2xz{\frac {d^{3}z}{ds^{3}}}+2xt{\frac {d^{3}t}{ds^{3}}}\right.\\\\\left.-4r{\frac {d^{2}r}{ds^{2}}}{\frac {dx}{ds}}+8x{\frac {dr}{ds}}{\frac {d^{2}r}{ds^{2}}}\right].\end{array}}$

Hence

$\left[{\frac {d^{3}x'}{ds'^{3}}}-{\frac {ds'}{ds'}}\left\{\left({\frac {d^{2}x'}{ds'^{2}}}\right)^{2}+\left({\frac {d^{2}y'}{ds'^{2}}}\right)^{2}+\left({\frac {d^{2}z'}{ds'^{2}}}\right)^{2}-\left({\frac {d^{2}t'}{ds'^{2}}}\right)^{2}\right\}\right]ds$ ${\begin{array}{r}=ds\left[\left(\rho ^{2}-2x^{2}\right)\left\{{\frac {d^{3}x}{ds^{3}}}-{\frac {dx}{ds}}\left[\left({\frac {d^{2}x}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}y}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}z}{ds^{2}}}\right)^{2}-\left({\frac {d^{2}t}{ds^{2}}}\right)^{2}\right]\right\}\right.\\\\-2xy\left\{{\frac {d^{3}y}{ds^{3}}}-{\frac {dy}{ds}}\left[\left({\frac {d^{2}x}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}y}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}z}{ds^{2}}}\right)^{2}-\left({\frac {d^{2}t}{ds^{2}}}\right)^{2}\right]\right\}\\\\-2xz\left\{{\frac {d^{3}z}{ds^{3}}}-{\frac {dz}{ds}}\left[\left({\frac {d^{2}x}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}y}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}z}{ds^{2}}}\right)^{2}-\left({\frac {d^{2}t}{ds^{2}}}\right)^{2}\right]\right\}\\\\\left.+2xt\left\{{\frac {d^{3}t}{ds^{3}}}-{\frac {dt}{ds}}\left[\left({\frac {d^{2}x}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}y}{ds^{2}}}\right)^{2}+\left({\frac {d^{2}z}{ds^{2}}}\right)^{2}-\left({\frac {d^{2}t}{ds^{2}}}\right)^{2}\right]\right\}\right].\end{array}}$

Now

${\frac {\partial x}{\partial x'}}=\rho ^{2}-2x^{2},\ {\frac {\partial y}{\partial x'}}=-2xy,\ \dots ;$

therefore

$\left(K'_{x}\delta x'+K'_{y}\delta y'+K'_{z}\delta z'-K'_{t}\delta t'\right)dt'=\left(K{}_{x}\delta x+K{}_{y}\delta y+K{}_{z}\delta z-K{}_{t}\delta t\right)dt.$

The Electrodynamical Equations for Ponderable Bodies.

We shall consider a system of electrodynamical equations for moving ponderable bodies of the type used by Minkowski,^[1] Einstein and Laub.^[2]

1. Göttinger Nachrichten (1908).
2. Annalen der Phys., Bd. 26 (1908).