1894.] Optical Wave-Surfaces ly Homogeneous Strain. 31 



as extreme cases. It is almost a pity that magnetic eolotropy should 

 be insensible, because the investigation of the conditions regulating 

 plane waves in media possessing duplex eolotropy, and the wave- 

 surface associated therewith, possesses many points of interest. The 

 chief attraction lies in the perfectly symmetrical manner in which the 

 subject may be displayed, as regards the two eolotropies. This brings 

 out clearly properties which are not always easily visible in the case 

 of simplex eolotropy, when there is a one-sided and imperfect 

 development of the analysis concerned. 



In general, the fluxes displacement and induction, although in the 

 wave-front, are not coperpendicular. Corresponding to this, the two 

 forces electric and magnetic, which are always in the plane perpen- 

 dicular to the ray, or the flux of energy, are not coperpendicular. 

 Nor are the positions of the fluxes in the wave-front conditioned by 

 the effective components in that plane of the forces being made to 

 coincide with the fluxes. There are two waves with a given normal, 

 and it would be impossible to satisfy this requirement for both. But 

 there is a sort of balance of skewness, inasmuch as the positions of 

 the fluxes in the wave-front are such that the angle through which 

 the plane containing the normal and the displacement (in either 

 wave) must be turned, round the normal as axis, to reach the electric 

 force, is equal (though in the opposite sense) to the angle through 

 which the plane containing the normal and the induction must be 

 turned to reach the magnetic force. These are merely rudimentary 

 properties. I have investigated the wave-surface and associated 

 matters in my paper " On the Electromagnetic Wave-surface " 

 (' Phil. Mag.,' June, 1885 ; or ' Electrical Papers,' vol. 2, p. 1). 



Effects of straining a Duplex Wave-surface. 



3. The connexion between the simplex and duplex types of wave- 

 surface has been interestingly illustrated lately by Dr. J. Larmor in 

 his paper " On the Singularities of the Optical Wave-surface," 

 (' Proc. London Math. Soc.,' vol. 24, 1893). He points out, inci- 

 dentally, that a simplex wave-surface, when subjected to a particular 

 sort of homogeneous strain, becomes a duplex wave-surface of a special 

 kind. To more precisely state the connexion, let there be electric 

 eolotropy, say c, with magnetic isotropy. Then, if the strainer, or 

 strain operator, applied to the simplex wave-surface, be homologous 

 with c, given by c~* X constant, the result is to turn it into a duplex 

 wave-surface whose two eolotropies are also homologous with the 

 original c ; that is to say, the principal axes are parallel. This duplex 

 wave-surface is, of course, of a specially simplified kind, though not 

 the simplest. That occurs when the two eolotropies are not merely 

 homologous, but are in constant ratio. The wave-surface then re- 

 duces to a single ellipsoid. 



