OPTIC AXIAL ANGLE. 
149 
Conversely if the angle E be known and the distance D be measured, the 
constant K can be calculated from the above equation and this in turn 
applied to the measurement of optic axial angles whose values are not known. 
This method, however, does not furnish a check on the value of K thus 
obtained, and its validity can not be considered verified for other angles E, 
unless many similar sections of different biaxial minerals be taken and the 
K of the microscope for each angle E be ascertained. The Mallard equation 
above assumes that the loci of the focal points of waves entering in all 
directions lie on a perfect spherical surface, an assumption which actual 
microscopic objective lens systems do not fulfill in the strict sense of the word. 
The validity of this formula has been tested by the writer by several 
different methods* and also by E. A. Wulfingf on four mineral plates of 
known optic axial angle. The observations show that the accuracy of 
the Mallard equation is not the same for all objectives, certain objectives 
furnishing values which agree closely with the values of calculation, while 
marked differences between observation and calculation occur in others. 
As a rule, the agreement is close and not unsatisfactory. 
TABLE 6. 
E 
Degrees. 
D 
Horizontal 
scale 
D 
Vertical 
scale. 
Sin= 
K-40.4. 
5 
a 5 
D 
IO 
I ; 
7-0 
IO.4 
7.0 
7.0 
20 
25 
13 9 
17.2 
I 3 .8 
13.8 
30 
ac 
20.4 
2} .4 
20.2 
20.2 
40 
45 
26.2 
28 8 
26.O 
26.O 
50 
El 
31.2 
33 3 
31.0 
31.0 
60 
65 
35 2 
36.6 
35-0 
35 o 
70 
75 
37-9 
38.8 
37.6 
38.0 
g 
39-5 
39 .0 
39 .8 
The simplest and most convenient method for testing the Mallard 
equation and for calibrating a microscope for all possible angles is that of 
Flett, who uses an Abbe-Zeiss apertometer for the purpose. The micro- 
meter scale is introduced as usual in the focal plane of the ocular and the 
angle corresponding to any number of divisions of the scale is read directly 
on the apertometer. In this manner an objective can be calibrated rapidly 
for all possible angles within the field of vision and an empirical, correct 
table prepared which is independent of the Mallard constant, thus obviating 
all errors due to the fundamental assumption on which the Mallard formula 
is based. 
*Amer. Jour. Sci. (4). 14, 327. 1907. 
tRosenbuscb- WQlfing, Mikroskop. Physiogr. 1, i. 330. 1904. 
