So 
METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
the dark hyperbolas of the biaxial interference figures perpendicular to the 
bisectrices, the chief difference between the two being one of intensity and 
rapidity of motion. The hyperbolas are very weak and require close obser- 
vation to be noticed at all. 
Since the ordinary approximate methods for calculating the positions of 
the zero isogyres for different angles of rotation of this section are involved 
and difficult, a graphical method with the stereographic projection plat as 
base can be used to advantage. Having given the positions of the optic 
axes, the directions of vibration of waves propagated along any given direc- 
tion can be found directly by use of the Biot-Fresnel rule. In this method 
the slight rotatory effects of the boundary surfaces of the crystal plate on 
the plane of polarization of the transmitted waves are disregarded and the 
average refractive index of the 
mineral is used in place of the 
correct indices of refraction for 
the different directions. These 
assumptions render the results 
obtained by use of the graphi- 
cal method slightly inaccurate, 
but the general relations are 
not seriously affected thereby. 
To find the lines of vibration 
for any given direction in the 
stereographic plat, the entire 
projection plat is rotated first 
on its horizontal and then on 
its vertical axis until the given 
direction coincides with the 
pole or center of the projection FlG - 5 - 
plat. The wave-front is then the plane of the plat and the directions of 
vibration bisect the angles of the projections of the optic axes on the wave- 
front. Fig. 50 was constructed in this manner and shows that the recession 
of the dark achromatic lines from the center of the field of view for the optic 
axial angles 2V = o, 10, 80 and 90, after a rotation of the stage through 
i, is very marked and that except in the limiting case of 2 V 90, the dark 
hyperbolas pass out of the field most slowly in the direction of the acute 
bisectrix. For 2^ = 90, the hyperbolas in all quadrants recede from the 
center with equal rapidity. In Fig. 50 the lines between the outer and 
inner marginal circles represent the actual positions of the bisectrices and 
the optic axes under the conditions stated. 
Owing to the fact that for this section the angles of extinction are very 
low for all waves whose angle of incidence is small, the intensity of the 
waves adjacent to those of the achromatic curves is also low, since it varies 
with the square of the sine of the angle p between the planes of polarization 
of the nicols and that of the section according to the standard formula 
,TA 
7 = sin* 2p sin* - 
The black curves are, therefore, indistinct and require careful scrutiny to 
be observed at all. 
