﻿376 Prof. R. 0. Tolman on 



it has been believed by Professor Lewis and the writer, that 

 in general, without respect to direction, the expression 

 w?o/ y/ 1 — w 2 / 6 ' 2 is best suited for the mass of a moving body. 

 They have already shown {Joe. cit.\ from the theory of 

 relativity and the principles of non- Newtonian mechanics 

 outlined above, that the consideration of a " transverse 

 collision" between two moving bodies does lead to this 

 expression for the mass of a moving body; and the purpose 

 of the present article is to show that the consideration of 

 any type of collision also leads to the same expression. 



The immediate occasion of the present article is a recent 

 attempt made by Mr. Nor man Campbell * to show that the 

 consideration of a " longitudinal collision " does not lead 

 to the expression m /^/l — u' 2 lc 2 for the mass of a moving 

 body. There appears, however, to be an obvious error in 

 his reasoning. Mr. Campbell wishes to find a relation between 

 the mass of a body and its velocity and yet assumes that 

 the mass of each of his bodies is the same after collision as 

 before, although the velocities of course have changed (see 

 equation (A) p. 627). Thus, although endeavouring to 

 determine how the mass of a body depends on the velocity, 

 he assumes in formulating his fundamental equation that it 

 does not depend on the velocity at all f . 



Longitudinal Collision. 



Consider a system of ('artesian coordinates and two bodies 

 moving in the X direction with the velocities +u and — u in 

 such a way that ;i "longitudinal collision" will take place. 

 Suppose the bodies are elastic and perfectly similar, each 

 having the mass m when at rest. On collision the bodies 

 will evidently come gradually to rest, and then under the 

 action of the elastic forces developed start up and move 

 back on their original paths with the respective velocities 

 — u and -f u of the same magnitude as before. 



Let us now consider how the collision will appear to an 

 observer w T ho is moving past the above system of coordinates 

 with the velocity v in the X direction. Let u Y and u 2 be the 

 velocities of the two bodies as they appear before collision 

 to this new observer. From Einstein's formulae for the 



* Phil. Mag. xxi. p, 626 (1911). 



t In the same article, Mr. Campbell has also criticised the writer for 

 referring the acceleration of a body under consideration to moving axes 

 which have at the moment in question the same velocity as the body 

 itself. As this is a procedure which has long been familiar to students of 

 theoretical mechanics, has not in the past led to erroneous results, and in 

 the cases under consideration leads to self-consistent conclusions, the 

 writer cannot agree with Mr. Campbell's criticism. 



