I9O METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
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 120, and the area 
on the surface of the unit sphere thus covered for a biaxial crystal is evi- 
dently 
5 = 4* . 2(1 cos<)=4r . 4sin 2 - 
2 
2< being the angle of vision of the field reduced to the true value within the 
crystal; if the observed angle 2^ be used, the average refractive index of 
the mineral /3 and that of the glass segments n should be introduced into 
the formula 
The probability, PI, that a section will show an optic axis is evidently meas- 
ured by the relative surfaces s to S, the surface of the sphere itself: 
4T . 4 sin 
2 
P 5 2 . </> 
P = =- =4Sin 2 -=: 
S 4* 2 
In case the areas covered by the two optic axes overlap, the formula 
should be changed, as Ce"saro has shown,* to 
_j /sin_F\ 
. 
= 4 sin- -- cos l - cos < cos 
in V\ 
- ) 
in</ 
2 irl_ \sin</ \tan< /J 
in which 2 V denotes the angle between the optic axes. The probability 
of encountering a proper section with minerals of average refractive index 
1.65 and with glass hemispheres of refractive index 1.52 ranges from 2 to 
5 in uniaxial crystals to 4 to 5 in biaxial crystals for which the fields for the 
optic axes do not overlap. The degree of probability is high, and one 
should be able to find suitable sections in every slide for measuring the 
optic axial angle of each mineral present. 
METHODS BASED ON THE RELATIVE BIREFRINGENCE OF^KNOWN SECTIONS. 
In 1883 Michel-I^vyt suggested that the optic axial angle could be 
determined approximately by measuring the path-differences on two 
oriented sections perpendicular to one of the two bisectrices or to the optic 
normal in the same slide. The path-difference is dependent both on the 
birefringence (7' a') and on the thickness of the section, but if two sections 
be chosen from a uniformly thick slide the ratio of their path differences in 
monochromatic light can be used directly in the calculation of the optic axial 
angle from any one of the following standard approximate equations : 
( 
Ab d(y a.) y a y a yp 
in which V t is the angle between the least ellipsoidal axis C and one of the 
*G. Clsaro. Mem. de l'Ac*d. Roy d. Sci. d. Belgique. 54, 1895. 
tBull. Soc. Min. Pr., 6, 147, 1883; Les Minlraux des Roches, 53, 1888. 
