1895.] in Geometrical Optics. 309 



from which Vi and r^ are measured being inverse points with 

 respect to it. 



Let then UiA U^ be the path of the central ray of an optical 

 filament refracted at A, at a spherical interface whose centre of 

 curvature is C. 



We can find the aplanatic points L^, L.. on the path of the 

 filament by inflecting the line GL^ to make the angle at L^ equal 

 to the angle of refraction ^a- These points will be exact conjugate 

 foci, no matter how wide the optical beam may. be. The easiest 

 construction for these points is to bisect by CL^ the angle between 

 the radii CJ.^ and GA^ drawn to the points in which the axes of 

 the filament again meet the spherical surface. 



As above, the point A is its own exact conjugate focus ; for any 

 pencil diverging from the point A continues to do so after re- 

 fraction. 



Further, we can find another pair of conjugate foci, not in 

 this case exact, by constructing a beam such that the angles of 

 incidence and refraction are the same for consecutive rays. For if 

 the ray incident at B is to have the same angle of refraction as the 

 ray incident at ^,the circle drawn through G, A, B must meet the 

 refracted ray A U.^ in the point where that ray intersects the ray 

 refracted from B. Hence, passing to the limit when B ultimately 

 coincides with A, the points in which the circle on JLC as diameter 

 meets an incident and refracted ray are conjugate foci ; and the 

 beam from one of these points diverges from the other, after re- 

 fraction, — at the same angle, since the deviation of each ray is the 

 same. This pair of conjugate foci are M-^, M^, the middle points of 

 the intercepts made by the spherical surface on the paths of the 

 central ray ; and it is noticeable that they are conjugate foci 

 whatever value the index of refraction may have. 



As then A is its own conjugate, and the relation between 

 conjugate foci is homographic, the line joining any pair of conju- 

 gates must pass through a fixed point. This is the pointy in 

 which the line connecting ifj and M^ meets GL^, which is the 

 perpendicular drawn to it from C. The march of conjugate foci 

 in the primary plane is now open to inspection. 



The primary focal lengths AF^, AF.^ of the refracting surface 

 for a pencil incident along the given direction U^A are the inter- 

 cepts made on each of the lines ATJ-^, Alio, by vectors from 

 parallel to the other one. 



The construction thus obtained for the primary focus after 

 oblique refraction has, I find, been already stated without proof by 

 Thomas Young \ " It approaches the nearest to Maclaurin's 

 construction ^ but is far more convenient." 



1 "On the Mechanism of the Eye," Phil. Trans., 1801. 

 ^ Treatise on Fluxions, § 413. 



