390 Wright — Two Microscopic- Petrographical Methods. 



If c be acute bisectrix, then 



Y<45° 

 cos 2 V > i, and, therefore, 



pf>* ' (2) 



2 2 



a y 



Let 



(3-a=C 

 y —ft= A, or 

 a=/?-C 



y=/3 + A 



y — a = C-f-A 



The expression (2) may now be written 



2(/3-C) 2 A(2^ + A)>/3 2 (C + A) (20 + A-C). (3) 



On rearranging the quantities of (3), we obtain 



2^(A-C)(^ 2 -2CA) + i 8 2 (C-A) 2 + 2C 2 A 2 -6 ) 8 a CA>0 (4) 

 To prove that in the case of c acute bisectrix 



/3-a<y-/3 



C<A 



A-C>0. 



The value of the expression (4) is evidently dependent 

 on the value of A — C, for both A and C are such small 

 quantities (fractions compared to /3) that the members of (4) 

 /3 2 (C-A) 2 +2C 2 A 2 -4/3CA and usually 6/3 2 CA may be neg- 

 lected in comparison with 2/3* (A — C). The value of the 

 expression whether greater or less than zero, depends, there- 

 fore, generally on the sign of A — C as 2/3 3 is always -K Thus 

 when c is acute bisectrix, in general 



A— C > 0, or /3-a < y - (3. 



Hence, the acute bisectrix is, in general, direction of least 

 birefraction. An examination of the list of all biaxial rock- 

 forming minerals confirmed this statement. 



Application. — In measuring the extinction angle in any 

 monoclinic mineral in which b = b, the optical character of the 

 mineral can also be determined at the same time by the above 

 method. The birefringence (y — a) can likewise be ascertained 

 on the same plate if its thickness be known. The plate cut 

 perpendicular to the optic normal exhibits the strongest double 

 refraction (highest, brightest interference colors), and can be 

 picked out readily from a number of plates of the same min- 

 eral cut along various planes. Thus a plate of monoclinic 



