154 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
mately normal to an optic axis by this method, the axial bar is first drawn 
when in a position parallel to the horizontal cross-hair (Fig. 84), the straight 
line A \ C in this position being the trace of the plane of the optic axes ; the 
microscope stage and drawing-table are then 
rotated in the same direction through some con- 
venient angle 30 or 45 and the axial bar drawn in 
the new position (A\ U of Fig. 84). These draw- 
ings are repeated after rotation of the microscope 
stage or drawing- table alone through 180 (A'\C' 
and A'\H'}\ the center O of the projection then 
bisects the distance A\A'\. 
The point H in the projection is any convenient 
point on the achromatic brush or zero isogyre, and 
is therefore a direction in the crystal along which 
light-waves are propagated whose plane of vibration on emergence coincides 
with the extinguishing plane of the analyzer. The plane of vibration for 
the point H is thus known, and the law of Biot-Fresnel can be applied 
directly to find by construction the second optic axis A 2 . 
By means of the Mallard formula the polar angular values p equivalent 
to the distances OA\ and OH in the interference figure are first deduced, and 
these in turn are reduced to true angles within the crystal by means of the 
refractive index formula 
sin i 
smr= 
where i is the angle observed in air, r is the angle within crystal desired, and 
ft is the mean refractive index of the crystal. The error committed in using 
ft for the reduction of the angle equivalent to OH instead of the actual value 
is not great and can be neglected, since the latter does not differ appreciably 
from ft in minerals of ordinary birefringence. 
The form of graphical construction used by Becke in applying this rule 
is shown in Fig. 85. The observed points H and A\ are first plotted (by 
means of their observed longitudinal angles < and reduced polar angles p) 
on tracing paper above the stereographic projection plat of Wulf (Plate 4) ; 
the tracing paper is then rotated about the center, (tracing paper held in 
place by a needle-point through O) until H coincides with the vertical 
diameter of the underlying plat and the great circle PK, whose intersection 
with the vertical diameter is 90 from // (polar circle to H), is then sketched. 
Similarly, the great circle DE, containing A\ and the extremities of the 
horizontal diameter, is located and drawn. The great circle HT, which 
indicates the plane of vibration of //, is then determined by Becke as the 
one tangent at // to the straight line LQ parallel to the trace FOI (in the 
projection) of the principal section of the lower nicol.* The great circle 
HA\, passing through U and A \ is then sketched and its intersection A '\ with 
the polar circle PK is accurately determined. The projection of the second 
optic axis .rl'iis found by making A\T = A'\T (Biot-Fresnel's law). The 
intersection of the great circle HA'i with the plane of the optic axes A\A* 
determines then the position At, and the angle A\A\ in projection is 2 V, the 
angle between the optic axes. 
Recently Professor Becke (T. M. P M.. 28, 393. 1909) has suggested a simpler method of construction 
which leads to the same result (see footnote, page 160). 
