of Relativity of the Fifth Fundamental Equation. 299 

 The substitutions 



are an obvious consequence of the relations between the 

 units of length and time used in the two systems. For 

 example, if a body has an acceleration in the Y direction, o£ 

 magnitude u when measured in the system S, evidently its 

 acceleration uj as measured in the system S' will be greater 

 because the units of time used in that system are "lengthened" 

 in the ratio 1 : \J I— /3 2 . Remembering that the units of 

 length in the Y direction are the same in both systems, and 

 noticing the time enters to the second power in the expression. 



for acceleration, the relation uJ — j^ — v ~ 7 ^k is evident. The 



y (1—00 

 other relations may be obtained in a similar way. 



If now we define force as the increase in momentum per 

 second we shall have, as has already been pointed out by 

 Lewis *, 



t^ d , N ■ du dm 

 r = — ( mu) =m-: — \-u -5— , 

 dt K J dt dt ' 



where a possible change in mass as well as a change in 



velocity is allowed for. It has, moreover, been shown by 



Professor Lewis and the writer j-, that the two postulates of 



relativity, themselves, combined simply with the principle 



of the conservation of momentum are sufficient for a proof 



that the mass of a body is increased when set in motion in 



the ratio 1 : y/1 — ft 2 , so that in general the mass of moving 



m 

 body m— . R2 . Substituting in the equation above, we 



have 



n m du d w n 



£ = , — — -77 + u 



V'-r v 



* Lewis, Phil. Mag. xvi. p. 705 (1908). 

 t Lewis and Tolman, loc. cit. 



X2 



