1903.] 



On Skew Refraction through a Lens, etc. 



511 



downward slope of about 1 in 6. Each end of the primary line is the 

 intersection of two ultimately coincident rays from near 79°. 



After a long interval, the intersections on the secondary line begin, 

 the first being the intersection of rays from two points ultimately 

 coincident at 180°, and the last the intersection of rays ultimately 

 coincident at 0°. 



11. In the cross-sections, of which numerous specimens at gradually 

 increasing distances are given in Plate 9, every intersection of two 

 rays, or (what is the same thing) every intersection of a ray with one 

 of the two focal lines, appears as a double point, which is generally a 

 point of crossing of two branches, but is sometimes a cusp ; and in one 

 instance the two double points coincide in a point of contact of two 

 branches. 



For the purpose of identifying individual rays, the numbers 0, 30, 

 60, &c, are marked, indicating the positions, in each section, of the 

 rays which came from the points 0°, 30°, 60°, &c, in the right hand 

 half of the annulus. They facilitate the tracing of reversals of 

 position. 



12. It is by no means a general law for oblique refraction through 

 annuli that the primary crossings are completed before the secondary 

 begin. More usually there is a large region in which the two cross- 

 ings overlap — a circumstance indicated by the presence of three double 

 points in a section, the middle one of the three being in the secondary 

 and the two outer in the primary focal line. 



13. Sections of pencils from annuli of obliquely placed lenses have 

 been calculated by Steinheil* and by Finsterwalder,f the obliquity, 

 however, being only 0° 48' in Steinheil's calculations, and not exceed- 

 ing 6° in Finsterwalder's. In both cases, the method of computation 

 is that devised by Seidell based on spherical trigonometry ; and the 

 calculations are only for the positions 0°, ± 45°, ± 90°, ± 135°, 180°. 



Part II.— Details. 

 14. General Process for Skew Refraction. 



Let l h m h i\ be the direction-cosines of the normal, 

 U, ni 2 , n-2 those of the incident ray, 



then, calling the angle of incidence x? and the angle of refraction x , 

 we have 



cos x = hk + niim-2 + n^. 



Hence, knowing the index of refraction, we can deduce sin x' from 

 sin x . 



* ' Munich, Akad. Sitz. Ber.,' vol. 19 (1889). 



f 'Munich, Akad. Abhandl.,' vol. 17 (1892), p. 519. 



X 1 Munich, Akad. Sitz. Ber.,' 1866, p. 263. 



2 o 2 



