EQUATIONS OF ELECTRON MOTION 157 



and that the particle responds to a force perpendicular to the velocity as 

 would a Newtonian particle of mass 



•Wo 



For this reason, it was usual in the early work on relativistic dynamics to 

 ascribe two masses to a particle: the longitudinal mass m., and the transverse 

 mass nit. However, in general this procedure leads only to inconveniences, 

 and it has been almost entirely abandoned. 



This concludes our discussion of the elementary differential equations of 

 motion. Without any further general theory of relativistic dynamics it is 

 possible to solve many interesting and important problems. For instance, 

 it can be shown easily that the trajectory of a particle subjected to a force 

 which is constant in magnitude and direction is a catenary (rather than a 

 parabola, which is the curve predicted by Newtonian dynamics). In the 

 following sections we shall discuss some of the less elementary parts of the 

 subject. 



III. The Lagrangian Equations 



In the foregoing the components of the applied force have been any func- 

 tions of the coordinates, the components of the velocity, and the time. 

 However, in problems concerning the motion of electrons, and for that 

 matter in many other physical problems also, we are usually concerned with 

 forces of a somewhat special kind. Throughout the remainder of the article 

 we shall assume that the force belongs to this special class. 



We consider four given functions of the coordinates and time, namely 



V(Xi, X2, Xs, t), An(Xi, X2, X3, t) , (w = 1, 2, 3), 



and we assume that the components of the force are given by the formulae 



__dV _dAr 



A 1 — — - — —- i- X2 



dxi at 



dAi dA] 



dXi 8x2 



;] 



dX2 dt 1_ 8X2 



dV dAz , . VdAx dAi\ 

 dxz dt |_ dxz dxi J 



Let us suppose, for purposes of illustration, that we are considering the 

 motion of an electron. Then the physical interpretation of our assumption 



•5 L. A. MacColl, American Mathematical Monthly, Vol. 45 (1938), pp. 669-676. 



