270 Dr. L. Trenchard Mors on the Coincidence of 



A solution of these equations gives 



7 cos a _ sin a 



z=d-^—-w, x=—d-^— a . 

 sin u sin u 



These values substituted in the equation of the envelope of 

 the uniaxal crystal, 



give 



>-V) |l^-(l/^ + ^H2M^^)}=0 ; 



w Vsin 2 ^ m *Mao#a \° V Wo » sin 2 « - sin 2 <9/ J ~ U; 



where m = d/b and m e = d/a. 



The first factor sin# = +??? sin a is the equation which 

 satisfies the condition of coincidence of rays along the optic 

 axis. As # and a are on opposite sides of the normal to the 

 surface, the minus sign is the one to be used. The second 

 factor, of the sixth degree, can only be solved approximately. 



In order that this phenomenon in refraction may be ob- 

 served experimentally, it is advisable to solve these equations 

 for definite crystals. As a type of uniaxal crystal Iceland 

 spar was chosen. The minimum value for a (the surface is 

 then tangent to the envelope) is found from the equation 



tan 2 « = ^ 2 = ^(l/a 2 + l//> 2 ) + 2 



ab V w 2 -b 2 



The indices of refraction for spar given by Mascart are 

 l/a = 1-48651 and 1/6= 1-65846 for the D line. The mini- 

 mum value of a=69° 32' 30"; the corresponding coordinates 

 of the point on the locus are x= +0*695 and z = +0*2593. 



The equation of a tangent to the circle from the above 

 point is 



^cos h + z sin B = b, 



where $ is the angle between the ray in the crystal and the 

 optic axis X. Substituting the values for the remaining 

 terms of the expression, 8 = 56° 4'. The angle made by the 

 normal to the surface and the ray, that is the angle of re- 

 fraction, equals c/> = (90-«) + (90-8) =54° 23' 30". From 

 those angles, a and <£, the minimum and maximum values for 

 the refractive index of a medium that will permit this ray to 

 pass through it, assuming the ray to pass the bounding 

 medium from the crystal, may be found. For, let the angle 



