54 



Mr. Mac Cullagh on the Laws of 



We come now to apply our theory to the case of uniaxal crystals ; and, in 

 doing so, we shall take the crystal to be of the negative kind, like Iceland spar, 

 so that the ordinary refraction will be more powerful than the extraordinary. 

 On the sphere described with the centre O and radius OS, let XY be a great 

 circle in the plane of incidence, the radii OX, OY being the positive directions 

 of the coordinate axes of ^ and_y. 'Suppose the right lines iO and Oi', inter- 

 secting the sphere in i and i', to be the incident and reflected rays; let the ordi- 

 nary refracted ray and the extraordinary wave normal be produced backwards 

 from O to meet the sphere, at the side of the incident light, in the points o and 

 e respectively; let the right line OA, cutting the sphere in A, be the direction 

 of the axis of the crystal ; and draw the great circles Ao, Ae, AY. The points 

 i, e, 0, i' are all on the circle XY. The point E, where the extraordinary ray 

 OE produced backwards meets the sphere, wlUbe on the circle Ae; and if, as 



in the figure, the arc Ae be less than a quadrant, the 

 point e will lie between A and E. The polar plane 

 of the ordinary ray is obviously the plane of the 

 circle Ao ; but the polar plane of the other ray 

 must be found by a construction. On the arc AeE 

 take the portion ef, so that the point e may lie between the points E and/, and 

 so that the tangent of e/" may be to the tangent of Ee as the square of the sine 

 of the arc eY is to the difference between the squares of the sines of iY and e Y. 

 Through f draw the great circle /if perpendicular to the circle AeE ; and it is 

 manifest from (29) that the plane of/ is the polar plane of the extraordinary 

 ray. On each circumference Ao and/, the points which are distant 90° from i 

 and i', the distances being measured by arcs of great circles, are the points where 

 the uniradial transversals, prolonged from the centre, intersect the sphere. Let 

 Ao and/ intersect each other in t, and let ti' be an arc of a great circle connect- 

 ing the point t with the point ^'. 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 O^ ; the angle of incidence is then the polarising 

 angle ; the plane oi ti' is the plane of polarisation of the reflected ray ; and the 

 angle ti'Y is 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. 



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