WILSON: GEOMETRIC POTENTIAL 667 



by integration, for the theory of continuous distributions of 

 electricity. 



Before we can give the proof, we must make precise the mean- 

 ing of the statement that p is dependent only on 1 and w. Geo- 

 metrically 7 speaking, 1 and w determine a plane, and thus also a 

 second plane completely perpendicular to their plane, but they 

 determine no particular vector in this second plane or in their 

 own plane. 8 Hence, if a vector p is to depend on 1 and w alone, 

 it must lie in their plane. The scalar products of 1 and w by 

 themselves are 



M = 0, 1-w = — R, w-w = — 1. 



Hence the function p must take the form 



p = v(R) w+/(/2)l. (3) 



To show that p reduces to the form A l/R, we have merely to 

 substitute the general form (3) in the equation 0-0 p =0 and 

 see that the only possibilities are <p (R) =A/R, f (R) = 0. Now 

 if/ is a scalar and u, v are two vectors, 



0-0(/v) = (0-0/)v + 2 0/.0v+/0-0v 



0-0 (u.v) =v.(0-0u) + 2 0u:0v + u.(0-0v) 

 0-0f(R)=f"(R)0R-0R+f'(R)0-0R 



With the formulas that we have established (§44, loc. cit.), 

 namely 



A 1, /v 1 dc 

 w = — lc c = 



R Rds 



0l = / + Iiw 0fl=-w+ 1 +1 ' c l 



R R 



where c is the retarded curvature d\v/ds of the space-time locus 



7 We might discuss this question more in detail as H. Burkhardt does the cor- 

 responding general problem for three dimensional vector analysis in Ueber Func- 

 tionen von Vectorgrossen, welche selbst wieder Vectorgrossen sind. Eine Anwen- 

 dung invariantentheoretischer Methoden auf eine Frage der mathematischen Physik. 

 Math. Ann., 43: 197-215. 1893. For our present purposes this seems hardly- 

 necessary. 



8 The plane of 1 and w and the plane completely perpendicular to it cut the 

 singular cone in pairs of lines which are respectively real and imaginary, but 

 no vectors along these directions are determined. 



**; 



