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BELL SYSTEM TECHNICAL JOURNAL 



which are precisely the components of the momentum according to the 

 Newtonian theory. 



Finally, as v/c approaches zero, the differential equations of motion (1) 

 approach the forms^ 



- (moXn) = Xn, 



(10 



which are the Newtonian differential equations of motion. 



Thus we see that Newtonian dynamics is in effect a simplified approximate 

 form of relativistic dynamics which is valid when the speed of the particle 

 under consideration is sufficiently small compared with the speed of light. 



Let us carry out the indicated differentiations in equations (1), and then 

 solve the resulting equations for the quantities nioXi, m^xt, m^xz. The work 

 is straightforward, and need not be given here. We obtain the following set 

 of formulae: 



m^xi = (1 — V c 



2 -2\-l/2 



2 -2n-1/2 



niQX2 = {l — V c ) 



2 -2x-l/2 



(3) 



mo'Xs ^ {]. — v' c ') 



These equations are, of course, the differential equations of motion (1) 

 written in a new, but equivalent, form. 



If, at some particular instant, the particle is moving parallel to the .Vi-axis, 

 so that X2 = 2:3 = 0, the equations (3) reduce at that instant to the forms: 



moXi 



Xu 



moX2 



Xo 



moXs 



X3 



(1 - v'<r^y' ' (1 - v^'c-'-y^ ' (1 - v'-c-'-y- 



These equations show that a particle of rest-mass nio, moving with speed v, 

 responds to a force parallel to the velocity as would a Newtonian particle 

 of mass 



_ Wo 



^f ~ (1 -1,2^-2)3/2' 



* If this conclusion is not entirely evident, the reader is referred to equations (3), from 

 which the conclusion follows at once. 



* I.e. an ideal particle which obej's the laws of Newtonian dynamics. 



