Non-Newtonian Mechanics. 155 



These equations give the force acting on e 1 at the time t. 



v 

 From equation (4) we have t= -^x since ^ = 0. At this time, 



the charge e which is moving with the uniform velocity v 

 along the X axis will evidently have the position 



v 2 

 x e — — 2 x, ye = 0, z e = 0. 

 c 



For convenience we may now refer our results to a system 

 of coordinates whose origin coincides with the position of the 

 charge e at the instant under consideration. If X, Y, and Z 

 are the coordinates of e x with respect to this new system, we 

 evidently have the relations 



t? 2 ...... 



X=# — g #=tf~?#9 Y=y, Z = z, X — x, Y=y, Z=z. 



Substituting into (21), (22), and (23) we may obtain: — 

 F*=f(l-/3 2 ){x+^(YY + ZZ)}, . . (24) 



F,= ^(l-/32)(l-^)Y, (25) 



F s =^(l-^)(l-^) Z , (26) 



where for simplicity we have placed /3= -, and 



c 



s= vx*+(i-/32 )(Y 2+z 2 y. 



These same equations could also be obtained by substituting 

 the well-known formulae for the strength of the electric and 

 magnetic field around a moving point charge into the 

 fifth fundamental equation of the Maxwell-Lorentz theory 

 F=E + l/cvxH. It is interesting to see that they can be 

 obtained so directly, merely from Coulomb's law. 



If we consider the particular case that the charge e ± is 

 stationary (i. e. X = Y = Z = 0) and equal to unity, equations 

 (24), (25) and (26) should give us the strength of the electric 

 field produced by the moving point charge e, and in fact they 

 do reduce as expected to the known expression 



F=E=i( 1 -/ 32 >> 



S 6 



where r = Xi + Yj -h Zk. 



