666 wilson: geometric potential 



to this tangent be R. If then 1 denotes the vector OQ, the 

 singular vector from the acting point to the point at which the 

 potential is desired, the potential 3 p is 



1 1 



p = p = ~ i~" (1) 



R 1-w 



The vector p satisfies the two fundamental conditions 



• p = o, • p = o, (2) 



of which the first is the wave equation needed to represent the 

 fact of the propagation of an electromagnetic disturbance, and the 

 second is the condition usually imposed upon the auxiliary func- 

 tions a and <p to make their determination complete. 4 



To validate the selection of (1) as definition of the potential, 

 a definition which thus far apparently has nothing but simplicity 

 in its favor, we may cite our proof that (1) is, apart from a 

 numerical multiplier, the only possible form for the potential p 

 which satisfies (2) and depends only on 1 and w, that is, on the 

 retarded position and velocity but not on the acceleration. 



2. I wish now to remove from the hypotheses the condition 

 • p = and to state the theorem : 



The only possible choice for the geometrical potential p, de- 

 pendent only on 1 and w but not on the derivatives of w, and 

 subject to satisfying the wave equation 0- Op = 0,is p = Al/R, 

 where A is a constant. 5 



When we have proved this theorem we have, from the point 

 of view of relativity, a completely rational basis for the theory 

 of the potential and field of the point charge, 6 and through it, 



3 This is the definition given by Minkowski in his Raum und Zeit, Gesam- 

 inelte Werke, vol. 2, p. 442. 



4 See, for example, Lorentz, los. cit., p. 239. 



5 Page, L., in Relativity and ether, Amer. J. Sci., 37: 169-187, 1914, apparently 

 reaches a similar conclusion in a totally different way; but it is difficult to com- 

 pare the arguments. 



r Even if we believe that electricity always occurs in continuous distribu- 

 tions, that is, that electrons are continuous surface or volume spreads of elec- 

 tricity with appropriate densities, it is convenient to have a rational theory of 

 the point charge for those investigations in which the size of the electron is 

 negligible, and particularly as the density within or upon the electron is un- 

 known. 



