154 BELL SYSTEM TECHNICAL JOURNAL 



that according to relativistic dynamics the mass of a five thousand volt 

 electron is about one per cent greater than the mass of an electron at rest. 

 From this we can infer that, while Newtonian dynamics may be adequate 

 for many purposes in our studies of electron motion, we do not have any 

 great amount of margin, and that it will be necessary to use relativistic 

 dynamics whenever we wish to obtain really good results concerning the 

 motion of even moderately high speed electrons. 



This article is purely expositor}^ Its purpose is to set forth the funda- 

 mental equations and theorems of relativistic particle dynamics in a clear 

 and concise form, unencumbered with any material relating to the theory of 

 relativity proper. Almost all of the material is to be regarded as already 

 known, but apparently it is only to be found in an inconvenient and scattered 

 form. The incomplete bibliography at the end of the paper gives references 

 to some of the more accessible sources of this and other related material. 



II. The Elementary Differential Equations of Motion 



Our discussion might be begun in any one of a number of ways, and no 

 doubt the different approaches would appeal unequally to different readers. 

 Considering the nature and purposes of this article, the author has deemed 

 it best to begin by writing down at once the differential equations of motion 

 of a particle (according to relativistic dynamics) in their most elementary 

 form. Then, for the purposes of this discussion, these equations will have 

 the status of a fundamental assumption. It need hardly be said that the 

 equations are not written down arbitrarily. On the contrary, they represent 

 the consensus of modern opinion as to the laws under which particles really 

 do move.^ The grounds, experimental and theoretical, for this opinion are 

 set forth in various of the works cited in the bibliography. 



For the time being, until the contrary is stated in the final section, we 

 employ a fixed rectangular coordinate system. Instead of denoting the 

 coordinates of the particle by x, y, and 2, as we have done provisionally in 

 the Introduction, we shall denote them by Xi, X2, and .V3. Then ±1, Xi, and 

 ^3 denote the components of the velocity of the particle. The components 

 of the force acting on the particle will be denoted by Xi, X2, and X3. For 

 the time being we need only note that the force may depend upon the 

 coordinates, the velocity, and the time; later on we shall introduce some 

 more explicit assumptions about the force. The symbol c will be used to 

 denote the speed of light in vacuo. 



^ The validity of these laws is not unrestricted. It is Umited on the one hand by the 

 quantum phenomena which become appreciable on the atomic scale, and on the other hand 

 by certain phenomena revealed by the general theory of relativity which become 

 appreciable on the cosmic scale. 



