Direction of Force and Acceleration, 459 



velocity. This is the fundamental equation of non-Newtonian 

 mechanics *. 



It has been shown from the principle of relativity f that 

 the mass of a moving body is given by the equation 



_ _ 77?0 



>A-£ 



where m is the mass of the body at rest and c is the velocity 

 of light. Substituting in equation (1) we obtain 



v _dl m \ m d*+d1 m \ (9) 



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 

 du/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 be resolved perpendicular 

 and parallel to the acceleration, it may be shown that the two 

 components are connected by a definite relation. 



* 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 Radioaktivitat, iv. p. 411 (1907), he defines force 

 bj' the equations 



mJI - 1 d \ — " V '' / I 



d 



dt\ , /!_!! I ' 



F = 



v^S 



F --= « 



m u 



..V' 



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 offeree, it has recently been pointed out by the 

 writer, Phil. Mag. xxi. p. 296 (1911), that, comhiued with the principle 

 of relativity, it leads to a derivation of the fifth fundamental equation of 

 electromagnetic theory in its exact form 



F=E+*vxH, 



there being no necessity for distinguishing between longitudinal and 

 transverse mass. 



t Lewis & Tolrnan, Proc. Amer. Acad.xliv. p. 711 (1909) ; Phil. Mag. 

 rviii. p. 510 (1909), 



