Illustrate the Propagation of Light in Biaxals. 323 



ellipse in the principal section. This is not a mere coincidence, 

 but is a geometrical property of the wave-surface *. 



Section by Plane z = 3. 



Inner section is a point, the origin. 



Outer. 



SB. If. 



0-00 2-65 



0-50 2-52 



1-00 2-10 



1-09 2-00 



1-35 1-35 



1-39 1-00 



1-38 0-69 



1-37 0-50 



1-35 0-34 



1-32 0-13 



1-32 0-00 



Fig. 4 is a contour-map of the surface, the curves being 

 marked according to their distance from the principal plane 

 XOY. 



In fig. 5' we have the section of the surface by the ZOX 

 plane. 



The curves of section are the circle 



and the ellipse 



a 2 x 2 + c 2 z' 2 = a 2 c 2 . 



The radius of the circle is 3. 



* From the ease with which this relationship follows from the equation 

 to the surface it is improhable that it has not been noticed previously ; 

 but it is not, perhaps, well known. 



Writing the equation to the surface in the form : — 



(x 2 +f+z 2 ){a 2 x 2 +bY^c 2 z 2 )-a 2 (b 2 +c 2 )x 2 -b 2 (c 2 + a 2 )f 



— c 2 (cr + b 2 )z 2 -\-a 2 b 2 c 2 = 

 and substituting values z=b, and .r=0 we obtain 



{ y 2 +h 2 )(b 2 y 2 +c 2 b 2 ) -b\c 2 -\-ir)tf - b 2 c 2 (a 2 + b 2 ) + a 2 b 2 c 2 = 0, 

 or i/ 2 -\-b 2 — a 2 = 0, 



whence y— + V a 2 — b 2 , 



which proves the proposition. 



From the symmetry of the equation to the surface, it follows that the 

 two other analogous propositions are also true. 



