﻿of the Optical Constants of a Crystal. 33 



Substituting this value for p 2 in (8), and remembering 

 the relation in (7), we deduce 



Imn |^(^-c 2 ) + -(c 2 -a 2 ) + ^(a 2 -.6 2 )} =0, 



whence we see that the critical values are 



I = 0, m =s 0, n - 0, . . . . (12) 



^(/> 2 -c 2 )+-( C 2 -a 2 )+^(a 2 -6 2 )=0. . (13) 



If I = 0, we have from (11) p = a ; and similarly, if m or 

 n = 0, we obtain p = b or c. 



Combining (13) and (7), we get 



I m 



X { fi\a 2 - b 2 ) - v\c 2 - a 2 ) \ y, \ v\b 2 - c 2 ) - \%a 2 - b 2 ) \ 



v\X 2 {c 2 -a 2 )-fjL 2 (b 2 -c 2 )\' 



and substituting in (11) after some reduction we finally 

 obtain 



/ = a 2 \ 2 + &V 2 + cV. 



It will be noticed that this value of p is the inverse of the 

 radius of the ellipsoid which is parallel to the edge of the 

 prism. 



We might arrive at the same result in another way. The 

 directions of the axes of any central-section, whose equation 

 is lx-\-my + nz = 0, is its intersection with the cone 



-(b 2 -c 2 )+ ^(c 2 -a 2 )+-(a 2 -2> 2 ) = 0. 



x y s 



Comparing these equations with (13) and (7) we see at 

 once that the fourth critical value is the inverse of the semi- 

 diameter parallel to the edge of the prism. 



Let us consider which of the critical values are maxima or 

 minima. We must differentiate equations (7), (8), and (9) 

 twice, regarding one of the variables, let us say ???, as inde- 

 pendent. We have 



dl mv — nfjb dn lft — 7n\ 



dm~ n\ — lv ' dm nX — Iv ' 



d 2 l v d*n -X 



dm 2 ~ (ftX-/v) 3 ' dm 2 ~ {n\-lv) z ' 



Phil Mag. S. 6. Vol. 12. No. 67. July 1006, D 



