OPTIC AXIAL ANGLE. 189 
the field of vision; that good results can be had from sections which show 
only one optic axis within the field, while for sections in which neither optic 
axis appears within the field the determination is uncertain and at best 
only a rough approximation. 
To summarize briefly the different methods best applicable to the four 
different possible cases cited above: 
(1) The optic axes are both within the field of vision and inclined between 
75 and 55 with the normal to the section. Determine the approximate posi- 
tions of the two optic axes by bringing each one, by means of HZ and V\, into 
the vertical position. 
Determine the position of each optic axis more accurately by means of 
optical curves in projection and check by means of extinction curves and 
exact location of principal planes of ellipsoid, especially the plane containing 
the optic axes. 
(2) The section is nearly normal to an optic axis; one optic axis A\ inclined 
less than 20 to section normal. Place the stage in the horizontal position 
(Hz and Ha in horizontal position and V\ normal to Vz}, turn H 3 and incline 
about Vz until the optic axis coincides with the axis of the microscope ; then 
rotate the section about V\ and turn HZ until darkness is attained, and thus 
determine plane of optic axes and /3. Incline Vz back to o position, rotate 
about HZ until the optic axis coincides with the plane normal to V\ and 
determine the extinction curve, the intersection of which with the plane of 
the optic axes in projection fixes the position of the second optic axis accu- 
rately. Check by determining a and 7 both from projection and observa- 
tion ; also by the extinction curve for the rotation about 7 2 . 
(3) One optic axis inclined 20 to 55 within the crystal to the normal of the 
section, the second entirely out of the field of vision. Determine visible optic 
axis by optical curves and second optic axis by means of extinction curves, 
both about V\ and V 2 . Verify results by determination of a, /3, and 7. 
(4) Both optic axes are entirely outside of the field of vision, i.e., are inclined 
at an angle of more than 65 in air with the normal to the section. In such 
instances the location of the optic axes is effected by means of extinction 
angles alone and the values obtained are not accurate, since an error of 
i in the determination of the extinction angle may affect the value of the 
optic axial angle up to 30. For accurate work, therefore, such sections are 
of little value at the present time for measuring the optic axial angle by 
the universal-stage methods; but in case the section be about normal to the 
obtuse bisectrix, the measurement of the optic axial angle is much more 
certain and satisfactory. 
As noted previously, experience has shown that the best and most rapid 
method of projection is that of Wulff, who uses an accurate projection plat 
as a base and tracing-paper on which to sketch the great circles and to exe- 
cute the actual measurements. The graduated hemisphere of Nikitin can 
also be used to advantage for the purpose. 
PROBABILITY OF ENCOUNTERING SECTIONS SUITABLE FOR OPTIC AXIAL 
ANGLE MEASUREMENTS. 
Since the accurate measurement of the optic axial angle can be accom- 
plished only on sections in which at least one optic axis is within the field 
