Medium, according to the Principles of Fresnel. 1 1 



will be in the sub-duplicate ratio of the elasticities in the direc- 

 tion of their respective vibrations, and therefore (by No. 3) in- 

 versely as the semiaxes of the section to which those vibrations 

 are parallel. 



If a wave be propagated in all directions from an origin 

 within a crystal, its surface will at each instant be touched by 

 the simultaneous position of a plane wave which passed through 

 at the instant when the former began to be propagated 

 (Memoir, p. 127). Hence, to find the surface of the double 

 wave in a crystal, let the above-mentioned ellipsoid be cut by 

 any plane through its centre 0, and imagine two other planes 

 parallel to this section, and at distances from it which are third 

 proportionals to its semiaxes, OR, Or, and any given line k : the 

 double surface which touches these planes in all their positions 

 will be the surface of the wave. 



Now conceive another concentric ellipsoid, having the direc- 

 tions of its semiaxes the same, but their lengths a, b, c, inversely 

 proportional to those of the former, so that the rectangle under 

 any coinciding pair of semiaxes is equal to k* : then if a plane, 

 touching this second ellipsoid in Q, cut OR perpendicularly in 

 P, the line OP will be a third pro- 

 portional to OJ?and k (Lem. 2) ; and 

 if Or intersect the second ellipsoid 

 in q, the semiaxes of the section 

 QOq will be OQ and Oq (Lem. 4). 

 Draw OT perpendicular to QOq 

 and equal to OQ, and conceive the 

 surface which is the locus of the 

 point T to be described, and a tan- Fi s- 5 - 



gent plane, to which the line OS is drawn perpendicular, to be 

 applied at the point T. Then OS will be perpendicular to the 

 plane ROr and equal to OP (Lem. 5) : and hence the point T 

 always lies in the surface of the wave. Similar things may be 

 proved with respect to the other semiaxis Or of the ellipse ROr. 

 Hence we deduce the following construction for the surface of 

 the wave : 



