EQUATIONS OF ELECTRON MOTION 155 



We assume that the particle moves, under the influence of the force (.Yi, 

 X2, X3), so that its coordinates satisfy the system of diflferential equations 



d moxr. ^^^^ (n = \,2,3), (1) 



dt Vi - (»Vc=') 



where Wo is a positive constant characteristic of the particle, and v- is an 

 abbreviation for the expression x{- + ±2^ + is'.* The positive value of the 

 square root is the significant one; and wherever square roots appear in the 

 subsequent work it will be understood, unless the contrary is stated, that 

 the positive values are intended. 



A few remarks may help bring out the significance of the foregoing assump- 

 tion and its relations to the corresponding fundamental assumption of 

 Newtonian dynamics. 



We call the constant Wo the rest-mass of the particle, and we assume (in 

 accordance with the experimental evidence) that mo is identical with the 

 mass of the particle which is used in Newtonian dynamics. In relativistic 

 dynamics the quantity ni defined by the equation 



mo 



Vl - (z'Vc^) 



is called the mass of the particle. We note that as v/c approaches zero the 

 mass approaches the rest-mass (whence the appropriateness of the latter 

 term), and that as v/c approaches unity the mass increases without limit. 

 Consider the vector having the components Pi, p2, ps defined by the 

 formulae 



VI - (v-/c-) 



We call this vector the momentum of the particle. The momentum is equal 

 to the velocity of the particle multiplied by the mass. 



Now equations (1) assert that the time-rate of change of the momentum 

 of the particle is equal to the applied force. 



We have already observed that as v/c approaches zero the relativistic mass 

 of a particle approaches the Newtonian mass. We now note that as v/c 

 approaches zero the components of the relativistic momentum approach 

 the values 



pn = mo Xn, (20 



* We might merel}' say that v is the speed of the particle. However, for our immediate 

 purposes, it is important not to lose sight of the fact that f is a certain particular function 

 of the components of velocity. 



