230 Prof. A. McAulay on the 



Terminology and Notation. — Intensity is understood to mean 

 energy per unit volume. D, E, B, H each-r- (intensity) \ will be 

 denoted by 6*. e, ft, 7 respectively. If it is thought desirable 

 to give names to these vectors, they may be called the quasi- 

 unit displacement, the quasi-unit E.M.F., &c. The terms 

 D-oid. E-oid, &c. used in Prop. I. below may be regarded as 

 convenient abbreviations of displacement-ellipsoid, E.M.F.- 

 ellipsoidj &c. The vector of ray-velocity and the index-vector 

 will be denoted by p, cr respectively: that is to say. the mag- 

 nitude of p is the velocity of propagation of a ray parallel to p. 

 and the magnitude of cr is the reciprocal of the corresponding 

 wave-front velocity, cr itself being perpendicular to the wave- 

 front and in the direction of its motion. The locus of the 

 extremities of p and cr, supposed drawn from a given origin 0, 

 are, as usual, called the wave- and index-surfaces respectively. 

 To fix the ideas, we may suppo-e the unit of length to be 

 1 cm., and the unit of time to be 1 (3 x 10 10 ) of a second. Thus 

 in vacuo both wave- and index-surfaces will be spheres each 

 of about 1 cm. radius. 



Prop. I. The loci of the extremities of 8. e, ft, 7, supposed 

 all drawn from 0, are ellipsoids with common centre 0. These 

 ellipsoids will be called the D-oid, E-oid. B-oid. and H-oid 

 respectively. 



Prop. II. The B-oid and E-oid are reciprocal polars; and 

 8 and e are corresponding vectors of them. [S£e=— 1, 

 0=SSde=SedoV] 



The B-oid and HL-oid are reciprocal polars ; and ft and 7 are 

 corresponding vectors of them. [Sy@7= — 1 . = $ftdy = $ydft.~\ 



The wave-surface and index-surface are reciprocal polars; 

 and p and a are corresponding vectors of them. [Spcr=— 1, 

 = Sp da= So dp.~] [See fig. 4 below.] 



Fig. 1. 



The last of these three statements results of course from the 

 definitions of the vector of ray-velocity and of the index- 

 vector. It is given here merely to bring out the similarities 

 between the six vectors. 



