Page:TolmanEquations.djvu/4

acting on a body may be best defined as equal to the rate of increase of momentum^[1], i. e. by the equation

${\mathsf {F}}={\frac {d}{dt}}(m{\mathsf {q}})=m{\frac {d{\mathsf {q}}}{dt}}+{\frac {dm}{dt}}{\mathsf {q}},$

or

${\begin{array}{l}{\mathsf {F}}_{x}=m{\ddot {x}}+{\dot {m}}{\dot {x}},\\{\mathsf {F}}_{y}=m{\ddot {y}}+{\dot {m}}{\dot {y}},\\{\mathsf {F}}_{z}=m{\ddot {z}}+{\dot {m}}{\dot {z}}.\end{array}}$

By the substitution of the previous equations presented in this article we obtain:

${\mathsf {F}}_{x'}={\frac {{\mathsf {F}}_{x}-{\dot {m}}v}{1-{\frac {v{\dot {x}}}{c^{2}}}}}={\mathsf {F}}_{x}-{\frac {v{\dot {y}}}{c^{2}-v{\dot {x}}}}{\mathsf {F}}_{y}-{\frac {v{\dot {z}}}{c^{2}-v{\dot {x}}}}{\mathsf {F}}_{z}$

(15)

${\mathsf {F}}_{y'}={\frac {\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}}{\mathsf {F}}_{y},$

(16)

${\mathsf {F}}_{z'}={\frac {\kappa ^{-1}}{1-{\frac {v{\dot {x}}}{c^{2}}}}}{\mathsf {F}}_{z},$

(17)

which are the desired transformation equations of force. These equations, which have here been derived from the principles of non-Newtonian mechanics, are those which were chosen by Planck to agree with electromagnetic considerations^[2].

Field around a Moving Charge.

As an application of these transformation equations, we may calculate the force with which a point charge in uniform motion acts on any other point charge, merely assuming Coulomb’s

1. See Phil. Mag. xxii. p. 458 (1911).
2. In an article by Lewis and Tolman (loc. cit.) an attempt was made to deduce the transformation equation of force, which was unsuccessful, owing to the authors' assumption that the turning moment around a right-angled lever in uniform irrotational motion should be zero. This error and the interesting fact that in general, if we accept the relativity theory, the actual presence of a turning moment is necessary to produce a pure translatory motion in an elastically stressed body was pointed out by Laue, Verh. d. Deutsch. Phys. Ges. xiii. p. 513 (1911). For the particular case that the body on which the force is acting is stationary with respect to one of the systems, the transformation equations of force were correctly derived by the present author, Phil. Mag. xxi. p. 296 (1911).