138 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
upper nicol. A comparison of this curve with the other curves of Fig. 71 
shows clearly that the distribution of colors for a quartz plate of this thick- 
ness and cut normal to the axis is analogous to that resulting from a sensitive- 
tint plate of quartz or of selenite. The sensitive-tint is not precisely the 
same in both cases, nor is the resultant total visual intensity, but for prac- 
tical purposes the quartz plate normal to the axis and about 3.55 7 mm. (rota- 
tion = 90 for light of wave-length 547 nn) thick serves well in place of the 
sensitive-tint plate. Its color depends on the kind of illumination used, and 
it is not satisfactory when used with deeply colored plates. To increase 
its sensitiveness, Bertrand combined two plates (2 mm. in thickness) of 
right-handed quartz with two plates of left-handed quartz, so that each 
right-handed plate is adjacent to a left-handed plate. This plate is in- 
serted in the focal plane of the ocular; the sharp junction lines serve as 
cross-hairs. The Bertrand plate can be used in monochromatic light, 
provided that for the particular wave-length used its angle of rotation is 
not a multiple of v, in which case darkness ensues and the observed effect 
is nil. By rotating the upper nicol it is possible in white light to bring out 
the sensitive interference tint over the entire field covered by the Bertrand 
plate, and in such a position that a very slight turn of an intervening crystal 
from its position of total extinction is sufficient to disturb this equality of 
interference color and to divide the field into four quadrants, the opposite 
sections of which are similarly colored, while adjacent sections are differently 
colored. Even with colored mineral sections it is possible with the Klein 
or Bertrand plates to obtain, by rotating the upper nicol, a tint which is 
sensitive, under the conditions of observation, to slight changes in path- 
difference and thus to the extinction direction of the mineral. 
The Bertrand plate is best adapted for use in white light; but it may 
also be used in monochromatic light, provided its thickness be correct for 
the particular wave-length of light employed. 
In the case of quartz plates cut normal to the axis the conditions for 
maximum sensibility under given conditions of observation are readily 
derived from the standard equations above. With crossed nicols the effect 
of the inserted quartz plate in monochromatic light is to rotate the plane 
of vibration of the waves from the lower nicol through an angle 5. For two 
adjacent quartz plates, the one right-handed and the second left-handed, 
the intensity of the field for the first plate becomes I\ = / sin 2 b\ ; and for the 
second plate 7 Z = / sin 1 &. Both fields will appear equally lighted if 81 = fe. 
For small values of the angle 8, the intensity varies with 5*. 
If now a crystal plate be inserted below the quartz plate, it can readily 
be shown that the intensity of the field for monochromatic light will be 
7i = sin 2 5+sin 26 sin 2(0+*) sin 
in which is the angle included between the polarizer and the ellipsoidal 
axis y' of the section. For the second half of the quartz plate (left-handed 
circular polarization) the intensity equation is 
20 sin 2(6-8) sin* * d ^' ~ *"* 
