484 PROCEEDINGS OF THE AMERICAN ACADEMY. 



ent result when we obtain b from a resultant M (no longer necessarily 

 a singular vector) and when we add the b's obtained from the original 

 ]yi"s. All the classic ideas of electromagnetic energy assume that 

 it is the vectors M that are additive at a point. 



The Field of Coniinuous Distributions of Electricity. 



54. Since the locus of an electric charge is not a singular line, we 

 may regard the charge as distributed continuously over a given region 

 or regions rather than as concentrated at one or more discrete points. 

 Thus instead of a single vector representing the locus of an electron, 

 we may consider a vector field. Let a small (5)-tube be parallel to 

 and comprise ti electron- loci each of charge e. Then we may replace 

 these on the one hand by a single vector neVf, and on the other hand 

 by a vector field q such that, if c/S is the volume of any portion of the 

 tube cut off by a planoid perpendicular to w. 



/ 



qd^ = new. 



Or if dS is the vector volume cut off by any planoid whatever, then as 

 in §45, 



/ (f/Sxq)*w = new. (127) 



If now we write 



q = Pow, (128) 



Po evidently represents the density of electricity as it appears to an 

 observer stationary with respect to the charge. To an obser\'er with 

 respect to whom the charge appears to be moving with the velocity V 

 the density appears to be different. For we may write (127) in the 

 form 



/ 



((/S*-q) w; 



and if dS is the volume cut off by the planoid perpendicular to the 

 chosen time-axis k4, dS* = d2>^i; then writing 



w= (v+k4)/Vl -r, 

 we have 



/ 7 d<c = n 



J Vl— 1-2 



6W. (129) 



