122 REPORT ON PHYSICAL OPTICS. 



bined with the laws of reflexion and refraction just alluded to, they 

 lead to a very elegant construction for the reflected or refracted 

 ray, which is, in most cases, more convenient than that of Huy- 

 gens. Thus, when a ray proceeds from air into any crystal, we 

 have only to construct the surfaces of wave-slowness belonging to the- 

 two media, and having their common centre at the point of inci- 

 . dence. Let the incident ray be then produced to meet the sphere, 

 which represents the normal slowness of the wave in air ; and 

 from the point of intersection let a perpendicular be drawn to the 

 reflecting or refracting surface. This will cut the surface of slow- 

 ness of the reflected or refracted waves in general in two points. 

 The lines connecting these points with the centre will represent 

 the direction and normal slowness of the waves ; while the perpen- 

 diculars from the centre on the tangent planes at the same points 

 will represent the direction and slowness of the rays themselves. 



This important curved surface presented itself also to M. Cau- 

 chy in his able researches on the propagation of waves in elastic 

 media, although he does not seem to have been aware of all its 

 properties. The properties of the same surface, and its use in 

 constructing the direction of a reflected or a refracted ray, were 

 also discovered, independently, by Mr. M'Cullagh, who has re- 

 cently applied them to the geometrical development of the theory 

 of double refraction.* 



The relations between the surface of wave-slowness, and that of 

 the wave, have led Professor Hamilton to the discovery of some 

 new geometrical properties of the latter. These properties are 

 demonstrated by means of certain transformations of the equation 

 of the wave-surface ; and it is shown that this surface has four 

 conoidal cusps, at the extremities of the lines of single ray-velocity, 

 at each of which the wave is touched (not by two planes as Fres- 

 nel supposed, but) by an infinite number forming a tangent cone 

 of the second degree ; while, at the extremities of the lines of single 

 wave-velocity, there axe four circles of plane contact, in every point of 

 each of which the wave-surface is touched by a single plane. 

 These singular properties have led Professor Hamilton to anticipate 

 two new laws of refraction called by him conical refraction, be- 

 cause in each case a single ray is refracted into an infinite number 



''Geometrical Propositions applied to the Wave-theory of Light," Transactions 

 of the Royal Irish Academy, vol. xvii. 



