Wright — Optical Character of Birefr acting Minerals. 289 



ized light ; A^A^ the projection of the optic axes, and P that of 

 any ray in the achromatic hyperbola. Fresnel's law states that 

 the planes of polarization of rays traveling in any direction P 

 are the bisectors of the angles between the planes A X P and A 2 P. 

 For small angles of incidence, the traces of the planes of polari- 

 zation of the rays will approximately coincide with the bisectors 

 of the angle AjPA,. Since P is a point of the achromatic curve, 

 the bisector of the angle A X PA 2 must be parallel to one of the 

 principal planes of the nicols. The triangle FPA 2 is then isos- 

 celes, and the triangles PFD and PDA 2 are similar. Therefore 



X JL JL -\~ JL , , 



- 1 = \ or (1) 



y—Vx y+y, 



x y = x ,y l ( 2 ) 



the equation of an equilateral hyperbola. In order that this 

 curve be tangent to a circle, its tangent must be perpendicular to 

 the radius the equation for which is 



dx 

 By substituting the value of -j- from (2), (3) becomes 



x = y (4) 



which shows that the hyperbolic curves are tangent to the circles 

 along the diagonals of the nicols. For these points (2) reads 



x 2 = x 1 y 1 (5) 



Transforming (5) to polar coordinates, we find 



p a = r 2 sin 2<f> (6) 



From Mallard's method above, it is evident that 



r = K sin E 

 and p = K sin O 



rp, n ^ Si" O 



.therefore, sm & = — = ^ 



V sin 2<f> (7) 



where sine O is the constant of the circle used and to be deter- 

 mined once for all by the Mallard method. For any given angle 

 of revolution (<£) the corresponding E can be found by finding 

 in fig. lb the intersection of the horizontal line at the distance 

 sine O from the base line with that arc which corresponds to the 

 angle <j>. 2E can then be reduced to 2V by fig. la, if the medium 

 index of refraction be known. 



Owing to the width of the achromatic curves, the results 

 attained by this method are only approximate but of sufficient 

 accuracy to be useful in many instances. The angles <f> can also 

 be figured for sections not exactly perpendicular to the bisectrix ; 

 they possess, however, only slight practical value. 



