490 PROCEEDINGS OF THE AMERICAN ACADEMT, 



expressing the conservation of electricity. We may write (141) in the 

 formO'M* = 0. Expressing M* as in (140), this equation becomes 



ah 



Similarly from (142) 



V^e + - = 0, 

 V-h = 0. 



de 



Vxh — ^ = 47rpv, 



V*e = 4 7rp. J 



These are the well known field equations. Finally (143) gives the 

 continuity equation 



It cannot be too strongly emphasized that all these equations follow 

 from the theorems of our four dimensional geometry without any 

 further assumption than that the geometrical vector potential field 

 derived from the locus of an electric charge is the extended electro- 

 magnetic vector potential. 



57. We have seen that the singular 2-vector field M' produced 

 by an accelerated electron determines a vector dg of four dimensional 

 significance involving quantities which may be identified with energy 

 and momentum in the radiation field. A search for similar vectors 

 due to the field M, which in general is not singular, proves, however, 

 to be unsuccessful. In the case of radiation we wrote 



dg = (f/S*-M')-M', 



or since it is readily shown (see footnote, § 62) that in this case 

 (c^S* • M') • M' = (f/S* • M'*) • M'* we could have obtained a more sym- 

 metrical form 



dg= i[(f/S*-M')-M'+ (cm*'M'*)'M'*l (144) 



In the case of the vector M we may write by analogy 



i [(dS* • M) . M + (f/S* • M*) • M*], (145) 



where c/S is the vector volume produced b}' intersecting a selected 

 portion of the four dimensional field by a planoid. However, this 

 cannot be made to give rise to a real vector in a four dimensional 

 sense, but will only have meaning for the particular planoid chosen. 



