232 Prof. A. McAulay on the 



relations of p, a, 8, and e and the corresponding surfaces, as 

 given by Prop. II. An inspection of the figure shows that 

 8 is parallel to the projection of the ray on the wave-front, 



Tlfoe 



judex surf. 



that e is parallel to the projection of a on the tangent-plane 

 of the index-surface, and that H and B are perpendicular to 

 all the other vectors. 



By Prop. II. it will be noticed that not more than three of 

 the six surfaces, wave-, D-oid, &c, are independent. As a 

 matter of fact two, and but two, are. These may be taken 

 as the E-oid and H-oid (or as the D-oid and B-oid); for when 

 these are given, the D-oid and B-oid are given by Prop. II., 

 and the wave-surface and index-surface by Prop. IV. 



Prop. IV. e and 7 are, in direction, conjugate to one another, 

 with regard to both the "E-oid and H-oid. 



8 and /3 are, in direction, conjugate to one another, with regard 

 to both the D-oid and B-oid. 



This proposition (when the E-oid and H-oid are given) 

 suffices to determine all of the four e, 7, 8, y3 when the 

 direction of any one is given. For the ellipsoids serve by 

 Prop. I. to give the magnitude of each when its direction is 

 known, so that it is only necessary to determine the direction 

 of each. Suppose the direction of e given : since 7 is 

 conjugate to e with regard to the E-oid, 7 is confined to one 

 plane ; since it is also conjugate with regard to the H-oid, 

 it is confined to another plane. Its direction therefore is 

 determinate. [Its sense may have either of the two values 



