1903.] On Skew Refraction tlhrough a Lens, etc. 519 



e 



Q 



30° 



$' 



180° 



136 c 10' 



60° | 90° 

 99° 35' I 68° 40' 



120° 

 43° 4' 



150° 

 20° 44' 



besides $ = 0' = 79° 10'. 



27. To trace the primary focal line, I have calculated, by formula? 

 (3), the equations of the rays (in terms of x y z) at these thirteen points, 

 and plotted on a large scale the sections of the rays by planes of 

 constant z, the smallest value of z being 5*304, which corresponds to 

 the intersection of the rays from 0° and 180°, the largest 6-05, which 

 corresponds to 79° 10', and the others being 5'4, 5*5, 5*6, 5*7, 5*8, 5*9, 6'0. 

 Some of these sections* are reproduced in Plate 10. The co-ordinates 

 x y of the points of self-cutting of the curves (drawn carefully by hand 

 through the plotted points) were adopted as the co-ordinates of points 

 of the primary line ; and the results are exhibited in Plate 10 in the 

 shape of three curves which are the projections of the primary line on 

 the co-ordinate planes of x (. These projections are on the same 

 scale as the curves in Plate 9. The scale of the intersecting curves is 

 five times as large. The projection on the plane of v i (the plane of 

 symmetry) is very nearly a straight line. The other two projections 

 show that the form of the primary line is approximately parabolic. 

 The length of the chord joining its ends is 0-952, and the distance of 

 this chord from the vertex, 0-911. The tangent of the slope of the 

 approximate plane of the curve is about - ^; and the ends of the 

 curve (which are its lowest points) are just above the plane of x £. 



28. A general view of the system as projected on the plane of 

 symmetry is given (on one-fourth the scale of Plate 9) at the foot 

 of Plate 10. The highest and lowest points of the annulus are 

 marked and 180, and the rays incident at these two points are 

 traced as far as their meeting with the secondary focal line. C is the 

 centre of curvature of the convex face of the lens, the radius of 

 curvature being ten times the radius of the annulus. CA, meeting 

 the plane of the annulus in A, is parallel to the rays within the lens, 

 and its production coincides with the secondary focal line. 



Cusps in the Sections. 



29. Every cross-section through an end of a focal line contains a 

 cusp, which is the transition from a small loop to a rounded-off angle 

 in sections taken near it. Three such sections, either of constant z or 

 of constant £ can be taken ; namely, one section through each end of 

 the secondary, and a single section through both ends of the primary. 



30. The values of dxjdO and dyjdd for constant z, and of dxjdO and 

 drj/dO for constant £, vanish at a cusp, and are very small in the region 



* The markings 21, 43, 69, 100, 136, 79, against the curves are abbreviations for 

 20° 44', 43° 4', 68 c 40', 99° 35', 136° 10', 79° 10'. 



