﻿378 Prof. R. C. Tolman on 



Remembering that these were bodies which had the same- 

 mass when at rest, we see that the mass of a body is inversely 



proportional to \J 1 2 -, where u is its velocity, and have 



thus derived the desired relation, — 



V- 



Collision of Any Type. 



A treatment of the general case of any type of collision 

 between any two bodies elastic or otherwise is also possible, 

 and leads to the same conclusion as to the desirability of 



using the expression m a / 1— — for the mass of a moving- 

 body. 'V c 2 



For the mass m of a body moving with the velocity u let 

 us write the equation m = m f(u 2 ) where /( ) is the function 

 whose form we wish to determine. The mass is written as a 

 function of the square of the velocity, since from the homo- 

 geneity of space the mass will be independent of the direction 

 of the velocity, and the mass is made pr< portional to the mass 

 at rest since a moving body may evidently be divided into 

 parts without change in mass. 



Let us now consider two bodies having the masses m () and 

 » when at rest, moving with the velocities u and v before 

 collision and with the velocities U and V after a collision 

 has taken place. 



From the principle of the conservation of mass we have, — 



m f(u 2 + V + Uz 2 ) + n f(vj + v/ 4- v /) 



= > VW + U/ + ti,*) + n f(V x * + V/ + V), . (1) 

 and from the principle of the conservation of momentum, 

 i»of (yx + My 8 + u/)u x 4- n f(v/ + % 2 + v g *)v x 



= m f(u/ + iy + u/)u* + n f(v/ + v/ + v/) v„ (2) 



m f(u x 2 + u y 2 + u/)u y 4- n f(v x * + v 2 + v 2 ) v y 



= mofiUJ + U/ + U/)U, + n f(YJ + V/ + V, 2 ) V„ (3) 

 m f(u x 2 -f ii y 2 4- u z ")u z -f n f(v x 2 + r ?/ 2 + r/)v 



= ^ /(u r 2 +u/+u/)u,+ » /(v. r 2 +y/+ v/)v . (4) 



These velocities u x , u y , v z , v Xy ry, r~, U. T , &c, are measured 

 with respect to some definite system of " space time 

 coordinates. An observer moving past this system of co- 

 ordinates with the velocity <£ in the X direction would find 



