BIREFRINGENCE. 105 
center or axial portion of the field, where the image is sharpest and the errors 
from the lens system are least serious. 
Suppose the extreme limits of error to be 0.003 mm. or 0.0015 mm. on 
either side of the correct value ; then an error of 5 per cent in the thickness 
determination may be considered probable. If this probable error be in- 
creased to 8 per cent to allow for multiplication by the refractive index and 
to introduce a safety factor, it can be safely assumed that the thickness of 
a favorable plate in the thin section, ranging from 0.03 to 0.05 mm. in 
thickness, can be ascertained readily to within 8 per cent of the correct value. 
For minerals in powder form, the thickness of the individual grains may 
be much greater and the thickness determination correspondingly more 
accurate. 
On the graduated wedge the scale divisions correspond to a path-differ- 
ence of low in the emergent light- waves and the error of the determination 
is not over one division on the scale (or o. i mm.), which is less than 2 
per cent. 
The total probable error of the determination of the birefringence of a 
mineral plate in the thin section may amount, therefore, to 10 per cent. 
As the birefringence of ordinary rock-making minerals ranges from about 
0.005 to 0.050, an error of 10 per cent is confined to the third decimal place.* 
In determining the birefringence (y a) or (y /3) or (/3 o) of a mineral, 
the optical position of the mineral plate under examination is ascertained 
by observations in convergent polarized light. In actual work it is not 
always easy to find a plate cut precisely perpendicular either to the optic 
normal or to the bisectrices, and it is of interest to know the percentage 
error caused by using sections inclined at low angles to the correct directions. 
For a given plate the birefringence can be calculated approximately from 
the usual formula, 
- = sin 0'.sin0 
7-a 
in which 6 and 0' are the angles which the normal to the plate makes with 
the two optic axes (optic binormals) respectively. A graphical solution 
of the equation is given in Plate 5, in which the abscissae and ordinates 
represent the angles 8 and 6' respectively; the curves indicate the pre- 
centage ratio . From this plate the values of 6 and 0' for any given 
7-a 
birefringence ratio can be found directly with sufficient accuracy for 
practical purposes.f 
In Figs. 63-68 these relations are shown graphically in stereographic 
projection. In each figure the angular distance between any two successive 
concentric circles is 10. Thus in Fig. 63 the positions of the sections are 
indicated whose birefringence is 2 per cent less than the maximum bire- 
fringence (y a) exhibited by a plate exactly perpendicular to the optic 
normal. The position of these lines of equal birefringence is different for 
different optic axial angles, as indicated by the lines for 2 F = o ,45 and 90; 
*An average of the birefringences of the 1 18 minerals listed under birefringence on pp. 292-295 of Rosen- 
busch-Wulnng gives 0.040 as the mean value, while the value of the members midway between the two 
limits is 0.020 to 0.025. This value represents more nearly the mean value of the birefringence of rock- 
making minerals than the arithmetical mean 0.040. 
tCompare also Duparc and Pearcc, Traitl, etc., 229, 1007. 
