272 Dr. L. Treii chard More on the Coincidence of 



Solution for a Biaxal Crystal. — The calculations are for 

 aragonite, as it has an elliptic section of considerable eccen- 

 tricity. 



1st Case. — The ellipse and circle intersect (fig. 2). Upon 



substituting the values z=d- — Ti , x=—d—. — „ in the equa- 



° sin 6 sin 6 i 



tion (a) of the locus, we obtain 



f/ 8 sin 4 a cos a \« sura cos 4 <x rt°snracos «/ \d cos 4 a 



r/ 4 sin 2 a cos 2 a d sin ctj \d? cos^ a d- snr «/ 



Using carbon bisulphide as a bounding medium, l/d = 

 1*63034. For aragonite, Rudberg gives the following indices 

 of refraction:— l/a = l*530i3, 1/6 = 1*68157, l/c = 168589, for 

 the D line. 



Let the surface of the crystal intersect a conical point ; 

 from Rud berg's values 2=81° 4' 47 ;/ . Here it was possible 

 to confirm the correctness of equation (a) ; as the coordinates 

 of a conical point are readily found, they may be introduced 

 into the equation of the locus which should then vanish. 

 This was done, and the correctness of the equation verified. 

 To return from this digression, the values of the constants 

 were substituted, giving 



1*45067 sin 8 (9 — 3*23411 sin 6 6>+ 1*93912 sin 4 6 



-0*15038 sin 2 <9 + 0-000032 1 = 0. 



Two roots are coincident, and the ray in the crystal grazes 

 the surface ; as l/b>]/d, it cannot pass into the bounding 

 medium, and the roots are imaginary. The other root is 

 1<°30' 15". 



Bounding medium again carbon bisulphide ; a = S6°. 



34-06891 sin 8 0-74-87614 sin G + 41*80691 sin 4 6 



-0 72988 sin 2 0+0-0000321 = 0. 



6" = 1° 42' 30". Other roots are imaginary. 



As it was found to be very difficult to choose a value for 

 a that would give three real roots, a somewhat more indirect 

 method was adopted. If we assume a medium with a re- 

 fractive index exactly equal to that of the circular section of 

 the crystal, then the coincident rays will not deviate in 



