﻿600 The late W. Gordon Brown on the 



Equation (12) is therefore by (17) and (20), 



V(d wV • q»)B + Vd m B - Vd m (q w v • B) 



-d m v.(E + Vq ra B) + V.d w (E + Yq ra B) = 0, . (21) 



d™ and q m being constant in the last term, and v operating 

 forwards only. 



In carrying out the simplifying transformations we may 

 drop for the moment the suffix m. 



From the last two terms we have, in part, 



— dy . E-f- v.. dE= + VdVyE 



= +VdcurlE. . . (22) 

 From the remainder we find 



YdB + V(d V . q)B + Vd(qy . B)-dy . VqB+ Vi . WqBi 

 = VdB + V(d V .q)B + V.d(qv.B) 



- V(dy . q)B - V . q(d V . B) + Vi • dVqB x 

 = VdB + V . d(q V .B)~V .q(dv . B) 



-V . d(qy . B) + V . q(dv • B) + Vdq . yB 

 = VdB + VdqdivB, (23) 



where the suffix restricts the action of v to the vector 

 carrying the same suffix. 



Equation (21) then reduces to 



Vd m (curlE + B+ divB). .... (24) 



Now d m will have different values according to the different 

 directions of the various sets of tubes ; hence (unless ail the 

 tubes are parallel) we may write 



curlE + B + q Wi divB = 0. . . . . (25) 



From this, since q OT is the velocity of any set of tubes, 

 unless all the sets have a common velocity, we must have 



divB = 0, ....... (26) 



and thus 



-curlE = B (27) 



We have now shown that the first four laws of the ordinary 

 theory of electromagnetism are consequences of the assump- 

 tions which have been made. It may be observed that 

 whereas, in the proof of the first two laws (3) and (4), no 

 departure of importance is made from the method of ' Recent 

 Researches/ the proof just given of the laws (26) and (27) is 

 quite different from that adopted in that work. This is 



