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velocity. This is the fundamental equation of non-Newtonian mechanics^[1].

It has been shown from the principle of relativity^[2] that the mass of a moving body is given by the equation

$m={\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}},$

where $m_{0}$ is the mass of the body at rest and $c$ is the velocity of light. Substituting in equation (1) we obtain

${\mathsf {F}}={\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}{\mathsf {u}}\right)={\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}{\frac {d{\mathsf {u}}}{dt}}+{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right){\mathsf {u}}$

(2)

From an inspection of equations (1) and (2) it is evident that the force acting on a body is equal to the sum of two vectors, one of which is in the direction of the acceleration $d{\mathsf {u}}/dt$ and the other in the direction of the existing velocity u, so that in general the force and the acceleration it produces are not in the same direction. If the force which does produce acceleration in a given direction he resolved perpendicular and parallel to the acceleration, it may be shown that the two components are connected by a definite relation.

1. This definition of force was first used by Lewis (Phil. Mag. xvi. p. 705 (1908)). In Einstein‘s later treatment of the principle of relativity, Jahrbuch der Radioktivität, iv. p. 411 (1907), he defines force by the equations

   ${\mathsf {F}}_{x}={\frac {d}{dt}}\left({\frac {m_{0}u_{x}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right),\ {\mathsf {F}}_{y}={\frac {d}{dt}}\left({\frac {m_{0}u_{y}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right),\ {\mathsf {F}}_{z}={\frac {d}{dt}}\left({\frac {m_{0}u_{z}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right).$

   He there states that this definition has in general no physical meaning. We see, however, that these are merely the scalar equations corresponding to equation (2) above and hence derivable from equation (1), which is an obvious definition of force and has a physical meaning. In further support of this definition of force, it has recently been pointed out by the writer, Phil. Mag. xxi. p. 296 (1911), that, combined with the principle of relativity, it leads to a derivation of the fifth fundamental equation of electromagnetic theory in its exact form

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

   there being no necessity for distinguishing between longitudinal and transverse mass.

2. Lewis & Tolman, Proc. Amer. Acad. xliv. p. 711 (1909); Phil. Mag. xviii. p. 510 (1909).