205 



rays are given by the preceding formulas. If we subtract 

 the latter from the former, we find (after a simple' trigono- 

 metrical reduction), 



r 2 - v'-* = (a-* - c~ 2 ) sin w sin a/. 



Hence the difference of the squares of the reciprocal veloci- 

 ties, in the two rays, is proportional to the product of the 

 sines of the angles made by their common direction with 

 the lines in which the two rays have a common velocity. 



The normal velocities of the waves are given by analo- 

 gous expressions 



w = 1 ( a * + c 2 ) + (a 3 - c 2 ) cos (o> + a/). 

 u' 2 = + <* + l a* - o 2 cos w - iK 



(D and a/ in this case denoting the angles which the common 

 normals to the two sheets of the wave-surface make with the 

 lines of single normal velocity, or with the true optic axes. 



Subtracting the latter of these expressions from the for- 

 mer, we find 



u - u /z = (a? -c 2 ) sin w sin a/. 



Thus the remarkable law of Brewster and Biot is a necessary 

 consequence of FresnePs theory. 



(217) The two sets of lines referred to in the preceding 

 article the lines of single ray -velocity, and single wave-velocity 

 are situated in the plane of the axes of greatest and least 

 elasticity, the lines of each pair making equal angles with 

 the axis of greatest elasticity on either side. The tangents 

 of these angles are respectively, 



