234 Prof. A. McAulay on the 



i. e. it is the plane perpendicular to the line of intersection of the 

 tangent planes. Further, since 8 = Yyo-, /3= We [Prop. III.] 

 the directions of 8 and ft are given by saying that they are in 

 the wave-front perpendicular to the projections of 7 and e 

 respectively on the wave-front. Thus the directions of all six 

 vectors are given by fig. 5, and the line of intersection of the 

 tangent planes at E, and Hj. 



If the two ellipses of fig. 5 are similar and similarly situated, 

 the directions of e and y are those of any two conjugate dia- 

 meters of either ellipse. Since the area of the circumscribing 

 parallelogram of either ellipse is constant we get but one 

 value of p( = Ve<y), including of course its opposite. 



The above construction for the wave-front still holds; and 

 we see that the front must, in this case, touch a cone. It is a 

 simple problem, directly from the geometrical construction, 

 to show that this cone is of the second degree. That there is 

 but one pair of directions (and their opposites) of p giving 

 this case, is most easily seen by orthogonally projecting the 

 H-oid into a sphere, when we fall on the usual case, as 

 depending on the circular sections of the ellipsoid into which 

 the E-oid has been projected. The geometrical properties 

 connected with conical refraction can thus be treated by the 

 present method. This brief indication of the fact must 

 suffice. 



It might be thought that these generalizations of the par- 

 ticular form of theory usually adopted would necessitate a 

 complex proof. On the contrary, it will be found, I think, 

 that the following proofs are simpler than those ordinarily 

 given for the particular form. 



From Maxwell's 'Theory of Electromagnetism' we take 

 the following; omitting the 47r's, as recommended by Mr. 

 Heaviside, in such work as the present, 



D = VVH, B=-WE, . . . . (1) 

 D=cE, B=/xH, (2) 



where c, p, are self-conjugate linear vector-functions of a 

 vector (permittivity and permeability), and 



ifSDE + SBH) = -w, (3) 



where w is the energy per unit volume, i. e. the " intensity." 

 The only kind of wave w T e contemplate is a plane wave in 



which the quantities have constant value over the wave-front. 

 Take the axis of z as normal to the front and k as the unit 



vector in that direction. Denote partial differentiation by 



D* &c. Thus if a wave with velocity v can be propagated in 



