1894.] Optical Wave- Surf aces by Homogeneous Strain. 33 



corresponding fluxes, the displacement and induction, be D and B, 



then 



D = cE, B = fiH, (1) 



where c is the permittivity and yu, the inductivity, to be symmetrical 

 linear operators in general. We have also the circuital laws 



curl H = cE, curl E = /tfi. (2) 



Now, if we assume the existence of a plane wave, whose unit 

 normal is N, propagated at speed v without change of type, and apply 

 these equations, we find that D and B are in the wave-front, E and 

 H are out of it, and that there are two waves possible. We are led 

 directly to the velocity equation, a quadratic in v 2 , giving the two values 

 of v z belonging to a given N. Next, if we put B = N/u, then s is the 

 vector of the index-surface, and its equation is 



, (3) 



[yuT 1 ] (S/iS) [c- 1 ] (SCS) 



which are, of course, equivalent to the velocity equation (' El. Pa.,' 

 vol. 2, p. 11, equations (41)). Two forms are given, for a reason 

 that will appear later. I employ the vector algebra and notation of 

 the paper referred to, and others. Sufficient to say here that c" 1 and 

 fi~ l are the reciprocals of c and fi ; and that Scs means the scalar 

 product of S and cS ; for example, if referred to- the principal axes 

 of c, 



SCS G\S\ "| G^S^ ~T~ 303 9 \ / 



if Ci, Co, GS be the principal c's (positive scalars, to ensure positivity of 

 the energy), and Si, s 2 , s 3 be the components of s. Also, [c~'] denotes 

 the determinant* of c" 1 , that is, (c^Ca)" 1 . 



The operators in the denominators of (3) may be treated, for our 

 purpose, as linear operators themselves. But it is their reciprocals 

 that occur. For example, the first form of (3) may be written 



s = o, (5) 



asserting that the vectors s and [. . .] -1 S are perpendicular. The 

 expansion of (3) to Cartesian form may be done immediately if c and 

 ft are homologous, for then we may take the reference axes i, j, k 

 parallel to those of c and ^, and at once produce 



* It occurs to me in reading the proof that fhft use of [c] to denote the deter- 

 minant of c, which is plainer to read in combination with other symbols than |cl. 

 is in conflict with the ordinary use of square brackets, as in (5) and some equations 

 near the end. But there will be no confusion on this account in the present paper. 

 VOL. LV. D 



