EXTINCTION ANGLES. 129 
Then 
,/T .\ . . . ,/. IT , r \ , T<f(V~ a') 
71 = cos 2 ( 8} -sin 26 sm 2 ( (-5J .sin 2 - 
7 2 = cos 2 (-+s) -sin 26 sin 2 fo d\ . sin 2 T ^ (T/ ~ a/) 
. ' .h-h = ah = K sin 20 {sin 2(0+3)-sin 2(0-5)} 
= 2K sin 20 cos 26 sin 26 
= AT sin 40 . sin 2 5 
This equation indicates that as decreases, d must increase; furthermore, 
that if the photometric sensibility were to remain constant, the minimum 
reading for would be sin 46= -. But the sensibility varies with the 
K. 
absolute intensity of illumination of the field, with the color used and with 
the condition of the eye of the observer. The effect of these factors is 
difficult to estimate accurately and to express in mathematical form. It 
is evident, however, that the limiting value of varies with the angle 5 and 
that under given conditions of illumination that angle 6 for which is a 
minimum is evidently the best. It is important, therefore, in practical 
work where the conditions of illumination and the sensibility of the eye are 
widely variable, that the half-shade apparatus be so constructed that 6 can 
be varied and the angle, for which the sensibility is a maximum, be thus 
readily obtained. In this arrangement, as in the method based on the 
rotation of the upper nicol, the abnormal interference colors, which result 
when white light is used and when the mineral plate is not precisely in the 
position of total extinction, change rapidly for a slight rotation of the crystal 
plate and are of great service in determining accurately the position of total 
extinction.* 
With a given color of monochromatic light extinction angles should be 
determined on plates of such a thickness that K is about i (the two emergent 
waves are a whole number of half wave-lengths apart). Thus, if sodium 
light be used, the plates should show in white light an interference color of 
about straw-yellow of the first order but not sensitive violet, since for this 
particular thickness the two waves are about 589 /z/z (555 nn) apart and the 
yellow waves are totally destroyed, with the result that the plate appears 
practically dark in all positions. It follows, furthermore, that a plate which 
is well adapted for determinations in one kind of monochromatic light may 
be useless for another color. Plates for which the path-difference of the 
emergent waves is between 200 and 300 w are best adapted for the measure- 
ment of extinction angles. 
It has been found that the insertion between the crossed nicols of specially 
cut plates and wedges of birefracting substances, as quartz and selenite, 
often serves to increase the accuracy of the measurement of the extinction 
angle on a given plate. The principle here involved is that of the super- 
position of birefracting plates, the action of which is to produce a resultant 
which differs from that of either component. It is possible to select a 
*A half-shade apparatus of variable sensibility was constructed by the writer in 1907 (Amer. Jour. Set. 
(4), 26, 375. 1908) but did not prove entirely satisfactory because of the depolarization of the transmitted 
waves by the total reflecting prisms used. 
