On Skew Refraction through a Zens, etc. 



509 



" On Skew Eefraction through a Lens ; and on the Hollow Pencil 

 given by an Annulus of a very Obliquely Placed Lens." 

 By J. D. Everett, F.E.S. Pveceived January 22, — Eead 

 January 29, 1903. Keceived in revised form, April 20, 1903. 



[Plates 9 and 10.] 



Part I. — Outline. 



1. The investigation here described was undertaken with the view 

 of . explaining the curious curves obtained by receiving on a screen, 

 at certain distances, the hollow pencil which emerges from an annulus 

 of a lens placed at large obliquity (such as 30° or 45°) to the incident 

 beam.* 



2. The first requisite is a process for calculating the direction-cosines 

 of a ray refracted at a given surface, when those of the incident ray and 

 of the normal are given, with the relative index of refraction. In the 

 original computations for this paper, the method of procedure was, to 

 calculate first the direction-cosines of the tangent to the refracting 

 surface in the plane of incidence. A length unity along the refracted 

 ray was then projected on this tangent and on the normal ; and these 

 two projections were themselves projected on the axes of co-ordinates, 

 and added. It has subsequently been found better, instead of actually 

 calculating the direction-cosines of the tangent, to eliminate them 

 between two sets of equations,, one of which is obtained by projecting 

 the refracted ray in the manner above described, and the other by 

 similar projection of the incident ray. 



3. A single case of special simplicity is selected for the application of 

 the process ; the case of a narrow and thin annulus of a plano-convex 

 lens, with a parallel pencil incident at 45° on its plane face, the index 

 being 1*5. The annulus is supposed to be graduated, the division 3 

 being at the point furthest from the source, and the graduations being 

 from 0° to ± 180°. The plane containing the axis of the lens and the 

 axis of the incident pencil passes through the divisions 0° and 180°, 

 and is the only plane of symmetry of the system. For convenience of 

 description, we shall suppose this plane to be vertical, the point 0° being 

 the highest point of the annulus. 



4. Twelve points 30° apart, starting from 0°, are taken on the annulus, 

 as points of -incidence ; and the direction-cosines of the corresponding- 

 emergent rays are calculated to three or four places of decimals. The 



* See S. P. Thompson, "Experiments on Zonal Aberration," 'Archives Neer- 

 landaises,' 1901, and Fourth Traill Taylor Lecture, printed in the ' Almanac of the 

 Brit. Journ. of Phot.' for 1903 ; also Alice Everett, " Photos, of Sections of Hollo v.- 

 Pencils," 'Jour. Brit. Astron. Assoc.,' vol. 13, 1902, pp. 74-75. 



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