318 PROCEEDINGS OF THE AMERICAN ACADEMY, 



tan 2 p 1 + tan a tan /8 cos^ C" 



(6) 



tan 2 6" (1 + tan a tan {3) cos C 



1 — M cos C" 1 — w cos C" 



(7) tan a tan ^ = P^t — -^ = j^, ^-^ . 



^ ^ MOOS 6^— cos' C rcosC — cos^C 



tan 2 6' — cos C tan 2 p _ tan 2 9" — cos (7'' tan 2 p 

 ^ -^ tan 2 p cos C" — cos'^ C tan 2 6' ~ tan 2 p cos (7" — cos^ C" tan 2 6' 



Let tan 2 p = x, tan 2 6' = x' , tan 2 Q" = x' . We have 



x' — X cos C" x" — X cos C" 



^^^ a; cos C - x' cos'-' C ~ x cos (7" - x" cos^ C" ' 



Hence 



(10) X {x' cos C" sin^ C - x" cos C" sin^ C") = x' a:" (cos'^ C" - cos^ (7') . 

 Substituting the values of x, x\ and x", 



_ tan 2 g^ tan 2 &' (cos'' C^^ - cos' C) 



(11) tan 2 p - ^^^ ^ ^, ^^^ ^,„ ^.^, ^„ _ ^^ 2 ^„ ^^^ ^, ^.^^2 ^„ . 



Reducing to a form more convenient for the use of logarithms, we have 



' _ tan 2 & tan 2 &' sin ( C + C^'Q sin ( C - C") 



(12) tan 2 p _ ^an 2 6/' sin^ C cos C" - tan 2 6/" sin^ 6'" cos C ' 



We now have an expression by means of which the extinction can be 

 calculated even when the optical angle is not known. The theory of 

 the same problem was elaborated by Liebisch in his discussion * of the 

 general determination of the optical angle and of the direction of the 

 optic axes in biaxial minerals, using sections whose cry stall ographic 

 orientation is known, and on which the planes of vibration of the two 

 refracted rays are known. Of immediate interest to us is the case in 

 the monoclinic system where the optical plane is also the plane of sym- 

 metry, but the orientation of the two axes of elasticity is not determined. 

 He finds the problem soluble when two sections can be employed, and 

 this is what we practically have for optical purposes in the cleavage 

 piece placed at two positions with respect to the incident ray from the 

 polarizer. 



But it is evident that there must be some degree of error in the instru- 

 mental readings of 0' and 6"; we have next to inquire if equation (12) 

 is so sensitive to changes in 6' or 6" or to simultaneous changes in both 

 as to make any determinations of p by means of it valueless. By ac- 



* tJber die Bestimmung der optischen Axen durch Beobachtung der Schwings- 

 richtung ebener Wellen. Neues Jahrbucli fur Min., etc., 1886, Bd. I. p. 155. 



