280 Mr. Gr. H. Livens : Influence of Density on Position 



state of the sether. This latter current is shown to be ex- 

 pressed by the time rate of change of the electric force 

 intensity, viz. E, and therefore the total current is 



U=P + E = (1 + 4tt<7)E 

 = KE = mKE, 



and this is the relation that is really required. It is like a 

 generalized Ohm's law, and depends essentially on the con- 

 stitution of the substance under consideration. 



With the two fundamental electrodynamic equations and 

 this constitutive relation we have a sufficient number of 

 equations to enable us to proceed to a complete analysis of 

 the electrodynamic field in the body when the given wave- 

 disturbance passes through it. 



§ 4. Mathematical formulce : their reduction to the optical 

 case. — We have the following mathematical relations between 

 the vectors involved in the theory: — 



((Exdx + Hydy + -Rdz) = 1 (j (ZU* + rotJy + fiU*)dS, 

 \(E x da; + Kjdy + ^dz) = - l -j t (j\zH* + mH y + nH*)dS, 

 giving in the usual vector notation 



-U=CurlH, --H = CurlE, 

 c c 



with U=mKE = KE. 



We shall now examine the propagation of the light in a 

 medium where these equations are satisfied. In order to 

 simplify the general equations we adopt a standard conven- 

 tion, to which we shall always adhere. We consider the 

 propagation of plane homogeneous waves taking place in 

 the direction of the axis Os, so that the components of E, H, 

 and U involve the coordinates of space and time by the 



exponential factor e m "~ qs \ where q is in general a complex 

 quantity, a function of n the frequency of the light dis- 

 turbance. 



Now since all differentials with respect to oc and y vanish, 

 the general equations reduce to 



inqK y =- -U, <?Ez= -H r 



