1884.] equations of the electromagnetic field. 129 



Now it is from this equation modified by the suppositions that 



ft = <&, and that -r, ( ~4~ + i — I" -r - ) = 0, that Helmholtz obtains 

 dt\dx dy dz J 



his normal wave. 



For the equation expressed in terms of J then becomes 



ofJ 

 fafike -=-j = (1 + 47re) yV. (25). 



Hence J and therefore <I>, since J= — /juk -j- , travel with a velocity 



v 7 



dt 

 (1 + 4tt6) 



This agrees with Helmholtz if we remember that in a magnetic 

 medium of magnetic inductive capacity fx, he replaces k by kj/x. 

 Thus the existence of a longitudinal wave travelling with this 

 velocity depends on the fact that Helmholtz assumes that <E> is the 

 potential arising in the dielectric from the polarization ; if we do 

 not make this supposition then. equation (24) gives us, omitting as 

 before, the terms in X, Y and Z, 



^' fi =4 vV -? < 26 >. 



V^M^-^V* (27). 



If we adopt Maxwell's view as to the current we must distin- 

 guish thus between <E> and XI, for <3> is then the potential due to 



matter of density — Vj + j + ~r\ while O is that due to matter 



of density _/*+* + *). 



J \dx dy dz) 



We arrive at Maxwell's case by putting 



<£ = and -fxk~r-= J, 

 and then the relation between 12 and J becomes 



y a =-w < 28 >- 



This corresponds to Maxwell's equation 8 (Electricity and Maqnet- 



dX 

 ism, II. 783), for since — — h . . . = 0, 



y 2 ft = y 2 ^* (29). 



* In Maxwell y- has the opposite sign to that with which it is used here. 

 VOL. V. PT. II. 9 



