WILSON AND LEWIS. — RELATIVITY. 473 



dg = O/qxJSxl)*! = l,l<lq(lS, and dg ,IS = hldq. 



This vector is distrihuted uniformly over one of the hypercoiics and is 

 independent of the particuhir phmoid used in obtaining it. Then also 

 just as before 



-f = k^r=^l(ls, (90) 



dS Vl —v'^ 



where 4> is the angle between V, which passes through the centers of 

 the two spheres, and the line, from either center, to the chosen point 

 upon the surface. 



The Field of a Point Chart/r. 



48. ^luch of recent progress in the science of electricity has been 

 due to the introduction of the electron theory, in which electricity 

 is regarded not as a continuum but as an assemblage of discrete 

 particles. In Lorentz's development of this theory he has deemed it 

 necessary, however, to regard the electron itself as distributed over a 

 minute region of space known as the volume of the electron. This 

 deprives the theory of some of that simplicity which it would possess 

 if the charge of an electron could be regarded as in fact concentrated 

 at a single point. Whether the theory of the point charge can be 

 brougiit into accord with observed facts and with the laws of energy 

 cannot at present be decided. It seems, however, highly desirable 

 to de\elop this theory as far as possible. In our application of our 

 four dimensional geometry to electricity we shall therefore consider 

 first an electric charge as a collection of discrete charges or electrons, 

 each of which is concentrated at a single point. 



The locus of a point electron in time and space must be a (5)-curve. 

 If w is a unit tangent to such a curve, then we may consider at every 

 point the \ector ew, where e is the magnitude of the charge, negative 

 for a negatiAe electron, and positive for a positive electron (if such 

 there be). It is explicitly assumed that e is a constant. We shall 

 show that the geometric fields obtained from this x'ector by the 

 methods of § 43 give precisely the ec^uations which are of importance 

 in electromagnetic theory. 



The vector w determines at every point of our time-space manifold 



the \ector p = vr/R. Similarly the vector eW determines the vector 



field 



ew eV ek4 .„,s 



m = ep = = -~ + y- p— • (91) 



