Page:TolmanFundamental.djvu/5

or in the case where $u_{x}=v,\ u_{y}=u_{z}=0$ :-

${\begin{array}{l}{\mathsf {F}}_{x}={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {du_{x}}{dt}}+u_{x}{\frac {d}{dt}}{\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {3}{2}}}}{\dot {u}}_{x},\\\\{\mathsf {F}}_{y}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}{\dot {u}}_{y},\\\\{\mathsf {F}}_{z}={\frac {m_{0}}{\left(1-\beta ^{2}\right)^{\frac {1}{2}}}}{\dot {u}}_{z},\end{array}}$

and by the further substitution of equations (19-20-21) we obtain

${\mathsf {F}}_{x}={\mathsf {F}}_{x}'$

(22)

${\mathsf {F}}_{y}=\left(1-\beta ^{2}\right){\mathsf {F}}_{y}'$

(23)

${\mathsf {F}}_{z}=\left(1-\beta ^{2}\right)F_{z}'$

(24)

which are the desired relations connecting measurements of force in the two systems.

The Fifth Equation.

Returning now to the consideration of an electron which is moving with the same velocity as the system S', we see that the transformation equations (13-14-15) together with the above equations lead to the relation

${\begin{array}{l}{\mathsf {F}}_{x}={\mathsf {F}}_{x}'={\mathsf {E}}_{x}'={\mathsf {E}}_{x}\\\\{\mathsf {F}}_{y}=\left(1-\beta ^{2}\right){\mathsf {F}}_{y}'={\sqrt {1-\beta ^{2}}}{\mathsf {E}}_{y}'=\left({\mathsf {E}}_{y}-{\frac {v}{c}}{\mathsf {H}}_{z}\right)\\\\{\mathsf {F}}_{z}=\left(1-\beta ^{2}\right){\mathsf {F}}_{z}'={\sqrt {1-\beta ^{2}}}{\mathsf {E}}_{z}'=\left({\mathsf {E}}_{z}-{\frac {v}{c}}{\mathsf {H}}_{y}\right),\end{array}}$

which is the desired equation:

${\mathsf {F}}={\mathsf {E}}+{\frac {1}{c}}{\mathsf {v}}\times {\mathsf {H}}$

This result agrees with that obtained by Einstein in his second treatment (Jahrbuch der Radioaktivität, iv. p. 411, 1907), where instead of defining force as equal to mass times acceleration, he defined it by the equations

${\mathsf {F}}_{x}={\frac {d}{dt}}{\frac {m_{0}u_{x}}{\sqrt {1-\beta ^{2}}}},\ {\mathsf {F}}_{y}={\frac {d}{dt}}{\frac {m_{0}u_{y}}{\sqrt {1-\beta ^{2}}}},\ {\mathsf {F}}_{z}={\frac {d}{dt}}{\frac {m_{0}u_{z}}{\sqrt {1-\beta ^{2}}}},$

which agree with our definition of force as equal to the rate of increase of momentum.