Electromagnetic Theory of Light, 233 



corresponding to this direction; but on account of the relation 

 pz=Yey, the senses will correspond to propagations of disturb- 

 ance in opposite directions.] When e and 7 are known, 8 and 

 /9 are known by Prop. II. 



Also, when e and 7 are known, p is known; since p = Yey, 

 and also a, since a = Y8j3. Thus when the direction of any 

 one of the four e, 7, 8, /3 is known, all six vectors are known. 



But Prop. IY. also suffices to determine all six when the 

 direction of p or of a is known, and gives us geometrical 

 constructions for the wave- and index-surfaces. 



Thus if the direction of p is given, the plane of e and 7 

 (which is perpendicular to p) is given. This plane cuts the 

 E-oid and H-oid in known ellipses, with regard to both of 

 which e and 7 are conjugate. Let (fig. 5) Hj E 2 , E x H 2 



be the common conjugate semi-diameters of these ellipses, 

 E 1? E 2 belonging to the E-oid, and H b H 2 to the 

 H-oid. e may be e ither + OE x or + OE 2 . If we take 

 e=OE 1 , then 7= + OH^, since 7 and e are conjugate. It 

 will thus be seen that p has any one of the four values given by 



p= + VOEi . OH; or + VOE 2 . OH 2 . 



By taking all possible directions for p, this gives a geo- 

 metrical construction for the wave-surface. From the con- 

 struction it is clear that the surface forms a double sheet with 

 centre at 0. Fresnefs construction is clearly what the pre- 

 sent construction degenerates into when one of the ellipsoids 

 is a sphere. 



Of course similar remarks apply to the index-surface as 

 depending on the D-oid and B-oid. 



Take e= 0E,, 7= 0H X . Then by Prop. II. -8~\ -/9" 1 

 are the perpendiculars from on the tangent-planes at E x 

 and Hi to the E-oid and H-oid. Since a = Y8J3 the wave- 

 front, supposed drawn through 0, contains both these normals, 



