358 F. E. Wright — Measurement of the Optic Axial Angle 



value of optic axial angle up to 30°. For accurate work, there- 

 fore, such sections are of little value in general at the present 

 time for measuring the optic axial angle by the universal 

 stage methods. In case, however, the section be about normal 

 to the obtuse bisectrix, the measurement of the optic axial 

 angle is much more certain and satisfactory. 



As noted previous^, experience has shown that the best 

 and most rapid method of projection is that of Wulff, who uses 

 an accurate stereographic or orthographic plat as a base and 

 tracing paper on which to sketch the great circles and to 

 execute the actual measurements. 



Since the accurate measurement of the optic axial angle can 

 be accomplished only on sections in which at least one optic 

 axis is within the field of vision, it is of interest to note the 

 probable relative frequency of occurrence of such sections in a 

 rock section. The microscopic field of the universal stage 

 fitted with glass segments includes an angle of about 60°, and 

 the area on the surface of the unit sphere thus covered for a 



biaxial crystal is evidently s — 4:7r.2 (1 — cos (f>) — 4:irA sin 2 — ;, 



2(f> being the angle of vision of the field reduced to the true 

 value within the crystal ; if the observed angle 2-^r be used, 

 the average refractive index of the mineral j3 and that of 

 the glass segments n should be introduced into the formula 



s = 4:7r.2— - sin 2 — . The probability, P 1? that a section show- 



P . . 

 ing an optic axis is evidently measured by the relative surfaces 



s to S, the surface of the sphere itself : 



d> 



47T.4 sm ! — , • , 



s 9 <p n, . „ w 



P = — = =4 sin 2 - = 4^ sin 2 ^- 



S 4,r 2/3 2 



In case the areas covered by the two optic axes overlap, the 

 formula should be changed, as Cesaro has shown,* to 



P = 4 



<£ 2 / /sin V\ , /tan V\ \ 



( arc cos ( — : — cos <£ arc cos ( I ) 



2 tv \ \sm $ I ^ \ tan <f> / / 



in which 2V denotes the angle between the optic axes. 



Assuming an average refractive index of P65 for ordinary 

 biaxial minerals, and 1*52 for the glass hemispheres, the prob- 

 ability of encountering a proper section ranges under these 

 conditions from 4 to 10, in unaxial crystals, to 8 to 10 in 

 biaxial crystals for which the fields for the optic axes do not 

 overlap. The degree of probabilitj^ is high and one should be 

 able to find suitable sections in every slide for the measure- 

 ment of the optic axial of each mineral present. 



* G-. Cesaro, Mem. de l'Acad. Eoy. d. Sci. d. Belgique, liv, 1895. 



