LEWIS AND TOLMAN. — THE PRINCIPLE OF RELATIVITY. 710 



which we are considering is the same in both systems, the observer A, 

 always using the units of his own system, concludes that the change in 



velocity of the ball b is smaller in the ratio — — ~- than the change in 



velocity of the ball a. The change in velocity of each ball multiplied 

 by its mass gives its change in momentum. Now, from the law of 

 conservation of momentum, A assumes that each ball experiences the 

 same change in momentum, and therefore since he has already decided 

 th at the ball b has experienced a smaller change of velocity in the ratio 



Vi — P 2 



, he must conclude that the mass of the ball in system b is 



1 



J2 • 



greater than that of his own in the ratio / 



VI — p 3 



In general, therefore, we must assume that the mass of a body in- 

 creases with its velocity. We must bear in mind, however, as in all 

 other cases, that the motion is determined with respect to some point 

 arbitrarily chosen as a point of rest. 



If m is the mass of a body in motion, and m its mass at rest, we 

 have 10 



m 1 



m 



o Vl - P 2 



The only opportunity of testing experimentally the change of a 

 body's mass with its velocity has been afforded by the experiments on 

 the mass of a moving electron, or (3 particle. The actual measurements 

 were indeed not of the mass of the electron, but of the ratio of charge 



to mass f - J . It has, however, been universally considered that the 



charge e is constant. In other words, that the force acting upon the 

 electron in a uniform electrostatic field is independent of its velocity 



relative to the field. Hence the observed change in — is attributed 



m 



solely to the change in mass. It might be well to subject this view to 



a more careful analysis than has hitherto been done. At present, 



however, we will adopt it without further scrutiny. 



The original experiments of Kaufmann u showed only a qualitative 



10 This equation and others developed in this section are identical with 

 those obtained through an entirely different course of reasoning by Lewis 

 (Phil. Mag.. 16, 705 (1908)). The equations were there obtained for systems 

 in motion w r ith respect to a point at absolute rest. We shall show here, how- 

 ever, that they are true, whatever arbitrary point is selected as a point of rest. 



11 See Lewis, loc. cit. 



