1 62 On a Dynamical Theory of 



marked (i) in the corollary to Lemma IV., it follows that Oy' 

 (if we so call the direction of y'} is perpendicular to OQ, and 

 also that Oy' and OQ coincide with the axes of the elliptic sec- 

 tion made in this ellipsoid by the plane QOy', just as Oy' and 

 OR coincide with the axes of the section ROy' in the first ellip- 

 soid. The plane QOR is therefore perpendicular to Oy and to 

 the plane of the wave. Moreover, we have 



(OP) 2 = a 2 cos 2 a + b 2 cos 2 /3 + c* cos 3 ? = s 2 , 



so that OP is the reciprocal of OR, and is equal to the velocity 

 s with which the wave is propagated when its vibrations are 

 parallel to Oy' . 



Now let the figure TOSMloe equal in all respects to QOPR, 

 but in a position perpendicular to it, so that if QOPR were 

 turned round in its own plane through a right angle, the point 

 being fixed, the points Q, P, R would fall upon T, S, M re- 

 spectively ; and supposing the wave-plane ROy' to take various 

 positions passing through 0, imagine a construction similar to 

 the preceding one to be always made by means of the two ellip- 

 soids. Then while the points R and Q describe the ellipsoids, 

 the points M and T describe two biaxal* surfaces reciprocal to 

 each other, the latter surface being touchedf in the point Tby 

 a plane which cuts OM perpendicularly in S. But this plane 

 is parallel to the central wave-plane ROy', and distant from it 

 by an interval OS (= OP] which represents the velocity of the 

 wave ; and as the surface whose tangent planes possess this 

 property is, by definition, the wave-surface of the crystal, it is 

 obvious that the point T describes the wave-surface. The radius 

 OT, drawn to the point of contact, is then, by the theory of 

 waves, the direction of the ray which belongs to the wave JROy', 

 and the length OT represents the velocity of light along the 

 ray. As to the surface described by the point M, it is that 



* See Transactions of the Royal Irish Academy, VOL. xvn. p. 244 (supra, p. 24). 

 t Ibid. VOL. xvi. p. 6 (supra, p. 4). 



