and the Conservation of Momentum. 627 



u The observer A, considering himself at rest, concludes that 

 the real change in velocity . . . ." (p. 518). If A considers 

 himself to be at rest, while he admits the principle of rela- 

 tivity, which states that he can have no rational ground for 

 such a belief, his pronouncements on any subject must be 

 received with considerable scepticism. When he proceeds to 

 calculate the "real change" without explaining what he 

 means by that term, I have a suspicion that he is using words 

 to which he cannot attach any significance whatsoever. 

 However, the argument can be expressed in a more satis- 

 factory form as follows. 



A and B, two observers furnished with instruments which, 

 when compared at relative rest, are similar, are moving past 

 each other with a relative velocity <£. Each projects a body, 

 a or b, appearing to him of mass m, towards the other. The 

 bodies collide and each observer determines the velocity of 

 each body relatively to his measuring instruments before and 

 after the collision. Let the components of these velocities as 

 determined by A be u a , v a , w a , u a ', v a ', w a \ u bi n, wv, u b \ v b , 



ii'b : and let those determined by B be U a W&'. Then 



Einstein's formulae for the composition of velocities give 

 relations between the w's and the IPs, etc. Thus, if the u and 

 U axis coincides in direction with <f>, 



_ cf>+U b _ (l-<ft 2 /c 2 ) 



etc. 



If the " law " of the conservation of momentum is true, 

 and m is the mass of b as it appears to A, we have 



m(u a — u a ') + m'(ii h — w 6 ') = 0, etc (A) 



But, by the first postulate of relativity, v a — nj =zJJ b —JJ b \ etc. 

 Hence, if all the u's and IPs are small compared to c, we have, 

 if the bodies move in the direction v or w, 



or, if they move in the direction u, 



^ = 1 (2) 



m 



On the other hand, if the v a etc., or U a etc., are not small, 

 the value of m'jm is a function of the u a etc.. and U fl etc. 



3. (1) represents Prof. Lewis and Dr. Tolman's result. It 

 appears of very much less generality than they imagined : 



