I3O METHODS OP PETROGRAPHIC-MICROSCOPIC RESEARCH. 
wedge or plate of such a character that the interference phenomena pro- 
duced by it alone are extremely sensitive to the slight changes produced by a 
second crystal plate when it is not precisely in the position of total extinction. 
From the mathematical standpoint, the insertion of a second plate in- 
volves a new set of conditions for the vibrating ether elements and the 
equations for the resultant are correspondingly more complex. Their 
derivation, however, is exactly the counterpart of that for the intensity 
of a single crystal plate and the final result, only, need be given here. If 
the nicols be crossed and 0i be the angle which y'\ of the crystal plate of 
thickness d\ includes with the principal plane of the polarizer, and 2 the 
angle between 7^ of the inserted plate or wedge of thickness </ 2 and the 
polarizer plane, and d\ (y'i of\) = Ai and d 2 (y'z a'z) =A 2) then the relative 
intensity is given by the standard equation 
/i = sin 2(0 2 0i) sin 261 cos 20 2 sin 2 - 
A 
+sin 2(0 2 0i) cos 20i sin 20 2 sin 2 -A 2 
A 
+cos 2 (0 2 -00 sin 20i sin 20 2 sin 2 - (Ai+A 2 ) 
A 
sin 2 (02 00 sin 20i sin 20 2 sin 2 - (Ai Ao) 
A 
From this formula the relative intensity can be calculated for any given 
values of 0i, 2 , Ai and A 2 . 
if 
In case the crystal plate is of such a thickness that sin 2 AI = i and at 
A 
the same time the inserted plate is also of a thickness that sin 2 - = i , this 
A 
equation reduces to 
/! = sin 2 2(0 2 0i) 
an expression for a curve similar in every respect to those of Figs. 74-78 
but which is zero for 2 = 0i and also for 2 = - + 0i, and reaches its maximum 
2 
of i at 0j = - +0i. The intensity from the second plate alone is 7 2 = sin 2 20 S . 
4 
If this intensity be compared with that of the two superimposed plates, the 
difference in intensity between the two fields is 
1 1 7j = sin 2 2(0j 0i) sin 2 20, 
If this difference be just perceptible 
/i h = <rh = sin 2 2 (0$ 0i) sin 1 20i 
From this equation it is evident that if the angle 0i is to be small, 2 must 
also be small, because <r/2 is a small quantity. In other words, the sensi- 
bility of the method is greatest when the ellipsoidal axes of the inserted test 
plate include only a small angle with the principal planes of the nicols. 
If both test plate and crystal plate are not strongly birefracting, abnormal 
interference colors result under these conditions and aid in determining the 
position of total extinction accurately. 
