514 Prof. J. D. Everett. [Jan. 22, 



and the equations to an emergent ray are 



x — sin _ y - cos 6 _ z /a 

 V m' n 



When the sign of is reversed, the sign of /' is reversed, and there 

 is no change in w! or n r , 



17. For transforming to the axes oi x rj £ (see § 5), f being in the 

 direction of the original beam incident on the first face, the required 

 formulae are — 



l = l', m = (in - ri) n = (m + n) J^, 



I, m, n denoting the direction-cosines relative to the new axes. We 

 thus find, for the seven selected points — 





0°. 



30° 



60°. 



90°. 



120°. 



150°. 180°. 



-0-0778 

 0-997 

 -7071 



-0-0672 

 0-998 

 -6124 



-0-0403 

 0-998 

 -3536 



+ 0-00177 

 0-998 







+ -0495! + -0880 + -1040 



0-997 0-995 1 '995 

 -0 -3536 -0 -61241 -0 "7071 



i 1 





= £o 



and the equations of an emergent ray are reduced to 

 g - sin 6 = 17-7/0 = t-to 



where ^/o = to = Ji cos 0. 



These give the values of x and rj for a section at any distance f 

 from the origin. The curves in Plate 9 have been obtained in this 

 way, twelve points being plotted and a smooth curve drawn through 

 them. 



Description of Plate 9. (See § 11.) 



18. The sections are arranged in increasing order of the distance £ 

 from the centre of the annulus. 



The first row consists of sections nearer than the primary focal line. 

 The first of them is at distance 4 (radii of the annulus) and is much 

 flattened at the bottom owing to the large upward deviation of the 

 lowest rays. When we pass to the next section, at distance 6, the 

 lower side has become reentrant, and, as the distance increases, the 

 lower side becomes more concave, the upper becoming at the same 

 time less convex, till at £ = 7*25 they form sensibly parallel arcs. At 

 7*5 the middle has become the narrowest part. 



The second row consists of sections through the primary focal line. 



