Theory of a Contractile JEtlier to Optical Problems, 529 



Take the case in which A = 0, and let 



\' = \/a 2 , fJL' = jn/b 2 , v' = v/c 2 . 



Then (35), (24), and (29) become 



V 2 = a 2 V 2 + &V 2 + cV 2 , (36) 



l\' + miM' + m/ = 0, (37) 



J_ (62 _ c2)+ ^ (c2 _ a2)+ Zi( a2 _^ ) = 0. . (38) 



Thus, draw a plane normal to the direction of vibration to 

 touch the ellipsoid. The quantities A/, ///, v' will be the direc- 

 tion-cosines of the radius vector to the point of contact, and 

 this radius vector, by (37), lies in the wave-front. 



Moreover, the velocity of propagation is given by the length 

 of the radius vector in the direction V, ///, v' ; and this radius 

 vector is an axis of the section of the ellipsoid by the wave- 

 front. This is, of course, Fresnel's construction for the 

 velocity. If Fresnel's construction were completely fulfilled, 

 A', fJ ', i/ would give the direction of vibration. As it is, that 

 direction is given by a 2 X/, b 2 /jb f , cV ; and these are the 

 direction-cosines of the perpendicular on the tangent-plane to 

 the ellipsoid at the point where it is met by V, yu/, V . 



Moreover, this perpendicular clearly lies in the plane which 

 contains the wave-normal, and the axis of the section of the 

 ellipsoid by the wave-front ; and since, according to Fresnel, 

 the axis is the projection of the ray on the wave-front, the ray 

 lies in this same plane. 



Thus, in fig. 1, let N represent the wave normal and R 



Fig. 1. 



the ray. Take a section of the ellipse aV + ^y-fc¥=l by 

 the plane NOR. Let P be a radius of that section per- 

 pendicular to N \ P is in the wave-front, and is the axis 



