1903.] 



On Skew Be fraction through a Lens, etc. 



513 



The Selected Case (see § 3). 



16. The parallel rays incident at 45° on the first face, which is 

 plane, are refracted into the lens so as to make with the normal (which 

 is the axis of z) an angle of 28° 7-|-\ The sine of this angle is 0*4714, 

 and its cosine 0*8819. The direction-cosines of the rays incident on the 

 second or convex face of the lens are therefore — 



l 2 = 0, m 2 = 0-4714, 7v 2 = 0*8819 ; 



and are the same at all points. 



Our calculations relate to a single narrow annulus of the convex 

 face,* the axis of this annulus being the same as that of the lens. The 

 radius of the annulus will be taken as the unit of length, and the 

 centre of the annulus as the origin of co-ordinates. 



The normals at all points are equally inclined to the axis of z, and 

 we assume the sine of this inclination to be 0-1000 ; in other words, 

 the radius of curvature is taken as ten times the radius of the annulus. 

 This makes the cosine 0*9950, and the angle itself about 5° 44J'. 



Let denote the angular distance of any point of the annulus from 

 the summit (which is the farthest point from the source). Then the 

 co-ordinates of the point are — 



.'•() = sin 0, y/ = cos z = 0, 



and the direction-cosines of the forward-drawn normal at the point a^e 



h = T V sin 9, nil = to cos ^ n i = 0*9950. 



From these we deduce, for the angle of incidence x> 



cosx = lik + m^ + n&z = 0*04714 cos 6 + 0*87749, 



and sin x' is § sin x \ hence cos x' is known, and we have all the data 

 for calculating the direction-cosines V m' n' of the emergent ray by 

 equations (3). The following values are thus found : — ■ 





0° 1 30°. 



60°. 



90°. 



120°. 



150°. 



180°. 



V ,. 



-0-0286 



-0-0513 



-0-0622 



-0-0568 



-0-0342 









650 0-658 



0-677 



0-707 



0*740 



-766 



0-777 



v! .. . 



-0*760 ; 0-753 



0-734 



0-704 



0-670 



0-641 



0-630 



* The annulus of the lens which corresponds to this annulus of the convex face 

 may have any thickness, but description is facilitated by supposing the thickness so 

 ^mall as to be negligible. 



