LEWIS AND TOLMAN. — THE PRINCIPLE OF RELATIVITY. 721 



Now let us assume that the whole system is in motion with velocity 

 v in the direction be. Obviously, merely by making such an assump- 

 tion we cannot cause the lever to turn, nevertheless we must now 



regard the length be as shortened in the ratio 



Vi-£ 2 



1 



while ab has 



the same length as at rest. We must therefore conclude that to main- 

 tain equilibrium the force at a must be less than the force at e in the 

 same ratio. We thus see that in a moving system unit force in the 

 longitudinal direction is smaller 

 than unit transverse force in the 



and therefore, by 



ratio 



Vl - P 2 



1 



b 



the preceding paragraph, smaller 

 than unit force at rest in the ratio 

 I-/? 2 



1 



It is interesting to point 



out, as Bumstead 13 has already 

 done, that the repulsion between 

 two like electrons, as calculated 

 from the electromagnetic theory, 



is diminished in the ratio 



VI - p 2 

 1 



Figure 3. 



if they are moving perpendicular 

 to the line joining them, and in 



1 — j8' 2 

 the ratio — - — if moving parallel 



to the line joining them. 



From the standpoint of the principle of relativity, one of the most 

 interesting quantities in mechanics is the so-called kinetic energy, 

 which is the increase in energy attributed to a body when it is set in 

 motion with respect to an arbitrarily chosen point of rest. Knowing 

 the change of the mass with velocity as given by equation I, the general 

 equation for kinetic energy, 14 E\ may readily be shown to be 



*=™ivT^- 1 ) 



II 



13 Bumstead, loc. cit. 



14 Consider a body moving with the velocity v subjected to a force / in the 

 line of its motion. Its momentum M and its kinetic energy E' will be changed 

 by the amounts dM = fdt, dE' = fdl = fvdt. Hence dE' — vdM, or substi- 

 tuting mv for M, dE' = mvdv + v 2 dm. Eliminating m between this equation 

 and equation I, and integrating, gives at once the above equation II. 



vol. xliv. — 46 



