566 " Rings and Brushes " in a Spath Hemitrope. 



Let T, t be the thicknesses of the main crystal, and of the 

 thin twin layer respectively, and h x and 8 2 be the corre- 

 sponding path retardations. In the present case, if we 

 assume that the twin layer is also cut perpendicular to the 

 optic axis, then from the angle (30°) at which its plane cuts 

 the faces of the crystal, we must have ^ = 60° for the thin 

 layer. So that the expression for the retardation in terms 

 of the refractive indices takes the form, 



5 2 2 



— = \At 2 — sin 2 i 4- -^ — ^-s . \/3 . sin i . cos 

 t n ' Po - 3/V 



V d 



— 3/z e 2 — 4yu, sin 2 i cos 2 



fr 



2^(^ + 3^) 



For the rays which fall normally upon the face of the 

 crystal (i. e., the rays which make an angle of 30° with 

 the normal to the twin layer), 



8 2 = t {1-3186 + 0-05 cos + '0028 cos 2 0\ 



and 8 1 = T^ {f J i —/ J L e \. 



Moreover, 



I /*0 f*e J ' 



where i 1 ~ angle of incidence (very small). 



The thickness of the twin layer as seen under a high 

 power microscope and estimated with the help of a micro- 

 meter eyepiece is about 2\. So that 



S 2 = 2\(l-3186 + 0-05 cos + 0-0028 cos 2 0). 



Now as the azimuth angle increases from 0° to 90° the 

 value of 8 2 diminishes. On the other hand, 6\ goes on 

 increasing as the angle of incidence is increased, and is 

 constant for the same angle of incidence. 



If r be the radius- vector of the interference minima or 

 maxima, then we may put 



r 2 = k ^ 1 + 8 2 ,) 



where k is a constant. Hence we can expect the elliptic form 

 of the curves, the radius-vector for an azimuth of 90° being 

 the least. Moreover, at a distance from the centre where i 

 is great, S } will be very great in comparison to 8 2 , which 

 latter may then be safely neglected, the radius-vector in 

 those cases being given by r 2 = k8 lt 



