156 
METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
page 1 50), each of these readings is reduced, as usual, to its corresponding 
angle value for the crystal by means of K and the average refractive index 
of the crystal, 
Having given the interference figure from a section of a biaxial mineral, 
cut so that one axial bar is visible, the course of procedure in measuring the 
optic axial by means of the double-screw micrometer ocular consists in: 
(a) rotating the microscope stage until the dark axial bar is parallel to the 
horizontal cross-hair of the ocular; (6) moving the horizontal cross-hair by 
means of the vertical micrometer screw V until it coincides precisely with 
the center of the dark axial line (Fig. 86, A\C\) ; (c) rotating the nicols (not 
the stage, as is the case with the Becke method) 
about a suitable known angle (usually 30 or 45), 
the exact position of the optic axis A\ then being 
the intersection of the axial bar with the hori- 
zontal cross-hair (Fig. 86, A\C\ with A\H\}\ (d) 
moving the vertical cross-hair by means of the 
horizontal micrometer screw until it coincides with 
this intersection and recording both vertical and 
horizontal micrometer-screw readings; (e) the 
stage is then rotated about an angle of 180, and 
similar readings for A\ taken in its new position, 
A' i. This last step is necessary in order to locate 
the center of the field (half the distances Ci C'i and A D'i) . The position of 
A! is thus fixed accurately and can be plotted directly after proper reduction to 
true angles within the crystal (sin r = for each coordinate angle)* in ste- 
n 
Two methods are available or locating a point // in the interference figure: (i) by means of its longi- 
tudinal angle o and its polar angle p. as in the Becke method; (2) by means of small circle coordinates, as in 
the method just described. Before plotting these 
angles of observation, they must first, be reduced 
(by use of the refractive index formula) to corre- 
sponding angles within the crystal plate. The ques- 
tion may then be asked. " Do the reduced angles of 
the two methods lead to the same point //' in the 
projection?" That they do is evident from the or- 
thographic projection (Fig. 87, projection of the in- 
terference figure as observed) in which O is the zen- 
ith, OX the first meridian, // the point observed, 
[.<>!! its longitudinal angle. O//( -sinp) its polar an- 
gle. OL (-sin a) and OM (-sin 0) its small circle 
coordinates. The reduction of the observed angles 
', P'. P'. to corresponding angles a, 0. p in the crys- 
tal plate is accomplished by the equations 
, sin a , sin sin p 
* W " T~' * m * " ~~iT~' * m p " ~1T 
But in right-angled triangle LOH. OH-OL+L// 
and the length L!l-O\f\ 
accordingly sin 1 p- sin* a -f sin* ft 
Therefore 
sin'p Rln* , sin'3 
p> ir o, 
u ' " 
In orthographic projection, accordingly , the points 
//' obtained by the two methods are identical in 
position and the reduction of the coordinate angle* 
M observed w therefore pertnissiMc. Sinre these 
coordinate angles define certain planes in projection, 
the stereoeraphic or any other projection may be used 
in place of the orthographic projection. In making the reduction it is assumed that the refractive index n 
is the same for the different directions //. /.. M: this is not strictly correct, but the error introduced thereby 
is practically negligible for minerals of ordinary birefringence. 
