Reflexion and Refraction. 



It deserves to be remarked, that, at any angle of incidence, 

 if the incident and reflected wave planes be intersected by a 

 plane drawn through the two refracted transversals, the inter- 

 sections will be corresponding transversal directions ; that is to 

 say, if the incident transversal coincide with one intersection, 

 the reflected transversal will coincide with the other. For it is 

 evident, from our fourth hypothesis, that if three of the trans- 

 versals be in one plane, the fourth transversal must be in the 

 same plane. 



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 and radius OS, let XY be a great 

 circle in the plane of incidence, the radii OX, OY being the po- 

 sitive directions of the co-ordinate axes of x and y. Suppose the 

 right lines iO and Oi', intersecting the sphere in i and /, to be 

 the incident and reflected rays; let the ordinary refracted ray and 

 the extraordinary wave normal be produced backwards from 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, A Y. The points i, e, o, i' are all on the circle 

 XY. The point E, where the extraordinary ray OE produced 

 backwards meets the sphere, will be 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 ob- 

 viously 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 tf/may be to the tan- 

 gent of Ee as the square of the sine of the arc eYis to the dif- 



i 



Fig. 19. 



