﻿32 Mr. G. F. H. Smith on the General Determination 



touches the wave-surface at r ; and further, that for rays with 

 the same direction of wave-normal, the corresponding 



values of the wave-normal, pi — - V are the inverses of the 



semi-axes of the perpendicular central-section. Whenever 

 the wave-normal is at right angles to a principal axis, one 

 of the two values ohtained is half the inverse of that axis. 

 Hence, if the axes are a, /3, 7, supposed to be in ascending 

 order of magnitude, it is obvious from geometrical consider- 

 ations that the maximum and minimum values in a complete 

 semi-revolution will give us 7 and a. Since each curve has a 

 maximum and a minimum value, the question remains what 

 do the lower maximum and the higher minimum values give; 

 one must clearly be /3. 



Let a 2 # 2 + &y + cV = l be the equation to the indicatrix, 



where a = — , b = &i c =— ; and let Xav be the direction 



a ' p 7 



cosines of the central line parallel to the edge of the prism. 

 Since the wave-normal is always perpendicular to this line, 

 the central section turns round it. Let the equation of this 

 section be Iz + my + nz = 0. 



Then Ik + mix + nv = 0. . . . 



Now the semi-axes of the section are given by 



I 2 



nr 



b*-p 



+ 



■F 



= 0, 



(7) 



(8) 



where p is the inverse of the semi-axis and therefore the 

 value of the corresponding wave-normal. 



F ur ther we have I 2 + m 2 + n 2 = 1 . 



(!») 



For critical values of p, dp = 0. Hence we differentiate 

 the above three equations in the usual way with regard to 

 /, m, and n and eliminate the differentials, and arrive at the 

 determinant 



m 



n 



a* — p x 

 X 

 I 



•F 



c*—p 



V 



n 







(10) 



which reduces to 

 X 



T 



(6«-u»)(a«-^) + * (c«-a f )(6 f -p«) + -(a f -6 f )(c , - j p f )=0. (11) 



