250 Mr. W- Williams on the Relation of Dimensions 



It might be more convenient to put X = iL, Y=jL, Z = &L, 

 then the dimensions of the energy of the medium become 

 ML 2 T -2 . There is thus nothing to indicate how or with 

 reference to what dynamical reactions the expression is 

 derived. In what follows, however, Cartesian expressions 

 will still be used though more cumbrous and involving more 

 explanation. They may be immediately converted into the 

 above. The energy of the medium will be expressed in terms 

 of an instantaneous linear displacement R upon which our 

 conceptions of the electromagnetic displacements at a point, 

 whatever their nature may be, must ultimately depend. By 

 keeping R, at first, separate from X, Y, and Z, the formulae 

 gain in generality, and by retaining R, X, Y, Z instead of 

 their equivalent vector forms rh, ih, jh, JcL, the dynamical 

 connexion between the various quantities is more clearly 

 expressed. 



The quantity it enters prominently into electromagnetic 

 relations, and it becomes necessary at the outset to determine 

 in what manner it is to be dealt with. This subject has been 

 discussed by Mr. Oliver Heaviside (Elec. October 16th and 

 30th, 1891), and his conclusions may be briefly summarized 

 as follows : — 



If m and g be point sources of induction and displacement 

 respectively, the measure of the induction and displacement 

 at a distance r from the source (if the fluxes emanate isotro- 

 pically) is 



B 



= (4^) D =(w} 



where B and D are the densities of the fluxes over spherical 

 surfaces enclosing the sources. And, similarly, the density 

 of any radial flux at a point should be estimated by the total 

 flux through a spherical surface having its centre at the 

 source and passing through the given point divided by the 

 surface. Writing B = /xH, and D = AE, where /jl and k ex- 

 press physical properties of the medium, we have 



Now, H and E express the strengths of the fields produced 

 by the fluxes m and q at distances r from the source. Hence 



Tn o 



— and \ express the strengths of the sources. Again, multi- 



plying the above by m and q respectively we get 



