1 1 4 On the Laws of Crystalline 



ference between the squares of the sines of iYa.nd.cY. Through 

 /draw the great circle ft perpendicular to the circle AcE ; and 

 it is manifest from (29) that the plane of ft is the polar plane of 

 the extraordinary ray. On each circumference Ao and ft, the 

 points which are distant 90 from i and ', the distances being 

 measured by arcs of great circles, are the points where the unira- 

 dial transversals, prolonged from the centre, intersect the sphere. 

 Let Ao and ft intersect each other in t, and let ti' be an arc of a 

 great circle connecting the point t with the point i'. When the 

 connecting arc ti' is a quadrant, the two uniradial transversals, 

 belonging to the reflected ray, coincide with each other and with 

 the right line Ot ; the angle of incidence is then the polarizing 

 angle ; the plane of ti' is the plane of polarization of the reflected 

 ray ; and the angle ti'Yis the deviation. 



To find the equations appropriate to uniaxal crystals, we may 

 suppose formulae (34) to belong to the ordinary, and formulas 

 (35) to the extraordinary ray. Then will e = 0, and E' = the arc 

 Ee. Putting B and B' for the spherical angles Aoi and Aei, we 

 shall easily see that 2 = B + 180, and 0' 2 = 0' + 90, if we con- 

 ceive the point A and the positive axis of z to be both on the 

 upper side of the plane XOY. And if a/ denote the arc Ae, 

 while b and a respectively express the reciprocals of the prin- 

 cipal indices, ordinary and extraordinary, the law of Huyghens, 

 for the double refraction of uniaxal crystals, will give us 



a~-b z 



tan e'= - sin a/ cos a/, (36) 



s 



where 



> Sin I 2 ,o / , jn\ o / /O"y\ 



2 = -r- = 6 2 + (a 2 - b z ) sm 2 w . (of) 



Observing these relations, we have, from (34), 



tan 0j = cos (ti - i z ) tan 0, 



(38) 

 tan 3 = -cos (i + i 2 ) tan0, ) 



for the ordinary ray ; and from (35) we get 



