178 L. A. COTTON AND M. M. PEART. 



The roots n and r 2 of equation (2) are given by the theory 

 of equations and may be obtained by solving the two follow- 

 ing equations : — 



and 





1 



r 2 2 " 

 'l '2 



I 2 a 2 



+ m 2 P 2 + 

 a 2 /3 2 f , 



n 2 f 



"- ( 



1 



i 



1 



l 2 + m< 



f 



' l 2 + n 2 

 + P 2 + 



m 2 + 

 a 2 



n 2 





(3) 



1 



Now it is known from the properties of the optical indi- 

 catrix that the two refractive indices of a mineral plate 

 are represented in magnitude by the major and minor axes 

 of the central section of the indicatrix which is parallel to 

 the plane of section of the mineral plate. 



If therefore, the values a, P, y are the axes of the indi- 

 catrix and Z, m, and n are the direction cosines of the 

 normal to the plane of the mineral section, then the refrac- 

 tive indices y l and a 1 of this particular section are given by 

 the values of n and r 2 derived from equations (3) and (4). 



Now these formulae are not nicely adapted for numerical 

 computation and in any case they involve the solution of 

 equations (3) and (4). 



As an alternative to this rather tedious method the 

 author proposes the following method which is largely 

 graphical in character. 



Let a, p, y as before be the principal refractive indices 

 of the mineral and therefore the principal axes of the 

 indicatrix. 



The equation of the indicatrix is therefore as before 



x 2 y 2 z 2 

 li 2 + ~P 2+ f = L 



Let Vi and r 2 be the refractive indices of a mineral plate 

 the direction cosines of whose normal are I, w, and n. 



