Crystalline Reflexion and Refraction. 



63 



sections in the plane of xy are a circle and ellipse with their common centre at 

 the origin, the radius of the circle being unity, and the semiaxes of the ellipse 

 being a and b, of which b is inclined at the angle X to the axis of ^; and there- 

 fore it is required to draw, parallel to the axis of x, a right line intersecting the 

 circle and ellipse, so that if tangents be applied to them at two points of inter- 

 section which lie on the same side of the axis of y, these tangents, when produced, 

 may cut each other on the axis of x. The angle which the tangent to the circle 

 makes with the axis of x is then the polarising angle zr^ ; and the solution of the 

 problem just stated leads directly and easily to the formula (68). From this way 

 of viewing the matter we see the reason why the polarising angle is the same in 

 the azimuths and 180°; for if tangents be applied at the two remaining points 

 where the parallel that we have spoken of intersects the circle and ellipse, it is 

 evident that these tangents also will cut each other on the axis of x ; since 

 tangents drawn at the extremities of any chord, either of a circle or an ellipse, 

 intersect the parallel diameter at equal distances from the centre. 



Let the reflecting surface of the crystal be in contact with a fluid medium 

 whose index of refraction out of vacuo is represented by n, and let b and a 

 respectively denote the ordinary and the principal extraordinary indices of 



refraction out of vacuo into the crystal 

 the preceding formula, and making 



L^ = A'sin'A -j- b' cos^X, 

 we readily deduce 



Hence we perceive that if l* = ab, that is, if 



tanX= v/-. 



Then putting - for a, and - for h, in 

 •^ ° A B 



-cJf 



..r^) 



(69) 





(in which case X will never be much above or below 45°,) the value of w, will be 

 always possible ; for then we shall have 



4. 2 *^ 



tanV,= ^- 



(70) 



But if X be diflFerent from this, and of course L'not equal to ab, the value of bt, 



/'^y^r^.^^'.^^ 



a 



~t 





% 



a 





x/^- 



. / 



■tv<< 



'n:i 



tU' 



u 



' l. 



J 



11 /u 



A 







/ 



V - 





