ON THE ACTION OF MAGNETISM ON LIGHT. 369 



over, as these forces could only arise from the existence of a finite layer 

 of transition, so not only would their assumption be purely empirical, but 

 the present method of investigation of the prohlem of reflexion would 

 actually no longer apply : if there is to be a finite layer of transition, the 

 postulation of material continuity of the media across it by means of a 

 single set of surface conditions would be meaningless. 



29. In the light of these remarks it will be of interest to follow some- 

 what m detail KirchhoflP's discussion of the general problem of crystalline 

 reflexion and refraction, to find out how far his imposed surface forces 

 satisfy the conditions that we here demand of them, namely of being 

 deducible from a bodily energy-function. Kirchhoff' restricts himself to 

 an elastic solid setber ; three sets of waves will thus be possible with a 

 given front ; the restriction that the displacement for two of these waves 

 shall be m the plane of the front confines the energy-function to Green's 

 well-known form.' 



He then neglects the first term involving the compression, in Green's 

 forniula, on the ground that in the transverse waves the density of the 

 medium remains unaltered,^ so that such a term can have no influence on 

 the equations. If he had definitely omitted this term from the energy, 

 the analysis, as carried out by him without an introduced pressure, would 

 have shown that the function so modified belongs to a medium in which 

 a compressional wave is propagated with null velocity, in fact a medium 

 which (hke Lord Kelvin's foam) opposes no resistance to laminar com- 

 pression, though it does resist uniform compression with a finite volume- 

 elasticity. Green was not able to do away in this manner with the terms 

 producing a normal wave, because he thought his medium would be un- 

 stable ; and possibly the same idea suggested KirchhofE's cautious pro- 

 cedure. ^ 



This energy-function F, with the compression omitted, is easily ex- 

 pressed, in the notation of § 11, in the form 



F=U-2a„^(^-f . . . 4- 



-2a 



'23 



where 



I d(x, z) d{x, y) J 



2U=au/2 -I- a^^g^ + a^¥- + 2a^gh -f 2ajif+ 2ay,Jg, 



(/, g, h) being the curl of the displacement (?, r,, C) of the medium. By in- 

 tegration by parts, all the terms of the volume integral /Fdr except IJ are 

 clearly expressible as surface integrals ; while U, the remaining volume 

 distribution, is identical with the complete energy-function of MacCulWh's 

 medium The mterfacial part of the energy F, when thus expressed, is 

 (^m, n), being direction cosines, the diff-erence in value on the two sides 

 01 the interface of the expression 



\ax dy dzj 

 ' G. Green, Camlridge Phil. Trans., 1839. 

 _J^As^expUcitly recognised by MacCuUagh. See Sir G. G. Stokes' Report. 



B B 



