JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. VI DECEMBER 4, 1916 No. 20 



MATHEMATICS. — Note on relativity: The geometric potential. 

 Edwin Bidwell Wilson, Massachusetts Institute of Tech- 

 nology. (Communicated by Arthur L. Day.) 

 1. In our treatment 1 of the principle of relativity Lewis and I 

 introduced as fundamental the extended (i.e., four-dimensional) 

 vector m of which the space and time components, when once a 

 time-axis has been arbitrarily selected, are the ordinary re- 

 tarded vector potential a and the retarded scalar potential <p. 

 We called the vector m the extended vector potential, and by 

 its differentiation we obtained the electromagnetic field equa- 

 tions. This is the converse of the usual procedure, which is to 

 regard the field equations as fundamental and to introduce the 

 retarded potentials as "certain auxiliary functions on which the 

 electric and magnetic forces may be made to depend." 2 



We built up the potential for a distributed charge from that 

 for a point charge and reduced the potential of a point charge 

 to the product of the charge and a vector p, which may be called 

 a geometric potential because of its definition solely by geo- 

 metric means. To find the potential p at a point Q (of the 

 four-dimensional manifold) and due to a curve 8 which is the 

 space-time locus of a moving charge, the first step is to draw 

 the backward singular cone with vertex Q and determine its 

 intersection with the curve 5; then draw at the forward 

 unit tangent w to the curve, and let the perpendicular from Q 



1 Wilson, Edwin B., and Lewis, Gilbert N., The space-time manifold of 

 relativity; the non-euclidean geometry of mechanics and electromagnetics, Proc. 

 Amer. Acad. Arts ScL, 48: 389-507. 1912. 



2 See, for example, Lorentz, The Theory of Electrons, p. 19. What Lorentz 

 here calls electric and magnetic forces are what we call field intensities. 



665 



