204 FRESNEL'S THEORY OF DOUBLE REFRACTION. 



It may be shown that the direction of the vibratory move- 

 ment, at any point of the surface of the wave, coincides with 

 the projection of the radius vector upon the plane which 

 touches the surface at that point. Hence, if perpendiculars 

 be let fall from the centre, on the tangent planes to the two 

 sheets of the wave-surface, the lines connecting their feet with 

 the points of contact are the directions of the vibrations in 

 the two rays ; and therefore determine their planes of po- 

 larization. The perpendiculars themselves measure the 

 velocities of propagation of the waves, while the radii- vectores 

 represent those of the rays. 



(216) From the construction of the wave-surface, above 

 given, it follows that there are two directions, namely, the 

 normals to the two circular sections of the ellipsoid, in which 

 the two sheets of the wave-surface have a common radius 

 vector, and therefore the two rays a common velocity. If to 

 and a/ denote the angles which any line drawn from the centre 

 of the wave makes with these lines, v and v 1 the radii-vectores 

 in its direction terminating in the two sheets of the wave- 

 surface, the equation above given may be reduced to the fol- 

 lowing remarkable polar forms : 



tr* = (V 2 + ar 2 ) + J (c- 2 - <r 2 ) cos (w -i- a/), 

 t,'-* = 1 (<r + a' 2 ) + i (c- 2 - a' 2 ) cos (w - a/). 



Since the radius- vector of the wave-surface measures the 

 velocity of the ray in its direction, the velocities of the two 



slowness of the refracted waves, in general, in two points. The lines con- 

 necting these points with the centre will represent the direction and normal 

 slowness of the waves; while the perpendiculars from the centre on the 

 tangent planes at the same points will represent the direction and slowness 

 of the rays:' This construction was given by Sir William Hamilton and Pro- 

 fessor Mac Cullagh. 



