128 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
If a be the percentage of the total monochromatic illumination which is 
necessary to produce a perceptible difference in intensity (o. i to 5 per cent 
or more), then in order that the intensity of the mineral plate be appreciably 
different from that of the field under the conditions of illumination 
/i /i = (7/2 = sin 26 sin 2(6 <f>) sin 2 
A 
but this expression reduces to 
=sin 2(6 <f>) 
K sin 26 
if we consider sin 2 = K, where K denotes the relative proportion 
A 
of the total monochromatic illumination which is emitted from the crystal 
plate, the remainder being destroyed by interference. From this equation 
it is evident that the sensitiveness of the method, so far as it is based on 
intensity relations alone, is largely dependent on the value of K. If the 
thickness of the plate be such that the intensity of the emergent light is a 
large fraction of the total illumination and the photometric sensibility (least 
perceptible increment of intensity), 0", be small (i to 2 per cent), then the 
least value which sin 26 can assume is - since the maximum value of the 
K 
right hand of the equation is unity. Under the most favorable conditions 
(7 = 0.015 and K=i or sin 20 = 0.015 or = 26'. If K were only a small 
fraction of the total intensity 7 2 , 6 becomes proportionately larger and the 
method less accurate. 
If white light be used, still another factor enters which tends to increase 
the sensibility of this method. On rotating the upper nicol, the intensity 
relations of the different colors throughout the spectrum are shifted by the 
addition of the light from the field, as indicated by the second term of the 
intensity equation, and this gives rise to a change in the interference color. 
Abnormal interference colors result, which for a slight rotation of the nicol 
to the right are noticeably different from those formed on rotating the nicol 
to the left. The eye is sensitive to this color change and, if the mineral 
shows brilliant interference colors between crossed nicols, it is possible to 
determine the position of total extinction of the plate by this method of 
nicol rotation with great accuracy (10' or less). If the interference colors 
of the plate are first-order grays, the change in color is hardly perceptible ; 
the field is relatively flooded with light and the method correspondingly 
less sensitive.* 
In case the half-shade principle be used to increase the accuracy of the 
determination of the positions of total extinction of a crystal plate, the 
comparison is between adjacent halves of the crystal plate rather than 
between the crystal plate and the field of the microscope alone. The eye 
recognizes differences in intensity between the halves of the analyzer field. 
Let <r be the photometric sensibility, 28 the angle between the halves of 
analyzer field, /i and 7 2 the observed intensities and I\ 7 2 = o7 2 . 
In nionorlinir mineral* dispersion of the bisectrices often give* rise to abnormal interference colon when 
viewed in white light under crossed nicols near the positions of total extinction. Abnormal interference 
colon may also arise from imperfect adjustment of the nicols and care should be taken not toconfound the 
two possible cause* dispersion of the bisectrices or imperfect adjustment of the microscope. 
