152 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
or sin = , draw a circle with radius K (Plate 7, preferably in colored 
K 
ink); the intersection of the ordinate D with this circle determines then 
the angle E in degrees. To solve the equation sin = /3 sin V, or 
sin E sin V 
i j_ 
ft 
find the intersection of radius E with the circle for the given refrac- 
tive index and pass horizontally from this point to the intersection with the 
outer circle of Plate 7, which point indicates the angle V in degrees. The 
following are examples : 
(1) AT = 54.o D = 2i.i 
Intersection of ordinate D with ^-circle is at radius 23. 
(2) = 42 18=1.65 
Pass along the radius 42 to the intersection with the circle 16.5 =/3 and then 
horizontally to the outer circle and read ^ = 24. 
Having once determined, however, the angular equivalents of the micro- 
meter scale divisions for the particular lens system, it is simpler to plot these 
values once for all, as a curve from which the angle corresponding to any 
scale reading can be read off directly. Curves can also be drawn, showing 
directly the equivalent angle V for any angle E and refractive index /3, and 
vice versa. A convenient form for such curves is illustrated in Plate 8, in 
which the angles are the abscissae, the angles V the ordinates, and the 
curves the refractive indices.* But having once determined K or the 
equivalent angle curve, it is possible to prepare, once for all, a set of curves 
(with the angles V as abscissae, the readings D as ordinates, and the refrac- 
tive indices as curves derived from the equation sin E= =j8 . sin V or 
K 
D = K.0.sin V), from which the angle V can be read off directly, thus 
obviating one set of operations. Such a set of curves is then valid only for 
the particular microscope and lens system with which the observations 
were made. 
Mallard's method for measuring the optic axial angle is one of the most 
satisfactory of the microscopic methods and if sections showing the required 
phenomena are available Mallard's method should be adopted without 
question, especially if the measurements can be made with a double-screw 
micrometer ocular, or with a micrometer ocular with coordinate micrometer 
scale. The limits of error of measurements of 2 V by the Mallard method 
should not exceed i to 2 on clear interference figures. 
THE BECKE DRAWING-TABLE METHOD. 
In place of the single-screw micrometer ocular, which in itself is of very 
limited application, F. Beckef has substituted a graphical method in which 
the observed optical phenomena are projected by a camera lucida on a rotat- 
Another form of plot has been used by Dr. Mcrwin of this laboratory in connection with work on the 
alkali feldspar*; in hit plot the refractive indices were chosen as ordinates, and the angles E as abscissa; and 
the angles V at curves. For general work, however, the form of the curves in the arrangement of Plate 8 
teems more favorable. 
tP. Becke, Tschermak's Min. petr. Mittheil., 14, 563. 1894; I*, 180. 1896. 
