The same property may be shown by similar reason ing to belong to a concave lens, whose first surface :MKN, Fig. 13. or that which receives the parallel rays AB, is part of a prolate 'spheroid ; but the second sur face OR must bc spherically convex, and have II for its .:entre.
If a lens has its posterior surface part of a hyperbo loid, formed by the revolution of a hyperbola, Whose greater axis is to the distance between the foci, as the sine of incidence is to the sine of refraction ; and if the other surface is plane, a pencil of lays parallel to the axis of the spheroid will be refracted so as to converge accurately to the farther focus of the spheroid.
Let PDQ, Fig. 14. be the lens, WIIOSC posterior sur face PDQ is part of a hyperboloid, (generated by the revolution of the hyperbola PBDQ, about its-major aNis DK, having Fl and I for its foci') and whose other surface PQ is plane, and perpendicular to DK, the pa 1 allel rays All, MD, will be refracted to its farther focus I. Having joined 1113, IB, then by constructing the diagram as in Prop. XIX. it may be shown•by almost the same steps that the ray AB, which suffers no re fraction at all, by falling at right angles upon the plane Furface PQ, will be retracted by the hyperboloidal sur face to the focus I.
The same property ntay be shown to belong to a piano concave lens P' Q' q p, having one hyperbolic surface 1"Kg, and a plane surface A q, upon which the incident rays fall perpendicularly.
'Po determine thc figure of a refracting surface which shall refract accurately to one point all the incident rays.
Let A, Fig. 15. be the point from which the rays AK, AI) diverge, B the focus in which it is required to unite them after refraction, and let D be taken at pleasure as the summit of the surface which is to refract them. Flom A and B, as centres, and with the radii All, BK, describe the circles DX, KY, then they will all meet in one point B, provided the curve KGD is such that lines AK, KB, drawn from any point K, have such a ratio that the excess of AK above AD or KX, shall be to that of BD above BK, or DY, as the sine of incidence is to the sine of refraction. Take the point G infinitely near K, so that K G may be considered as a tangent to the curve at the point K, and draw ZKH perpendicular to it, so that AKZ will be the angle of incidence, and IIK the angle of refraction. Now, since in every point
of the curve the difference between AK and AD is to the difference between BD and BK, in the sante ratio, the difference of lines infinitely near to KA, viz. AG and BG will be in the same ratio; that is, GQ : GL : Sin. Incidence. Sin. Refraction ; but •LGQ, and GKQ, are respectively equal to AKZ and BKII, the angles of incidence and refraction ; consequently KL and KQ, which are the sines of the angles LGQ, GKQ, being to one another as the sine of incidence to the sine of refraction, it follows that the sines of the angles AKZ and BKH are in the same ratio, and therefore that KB is the refracted ray when AK is the incident ray.
In order to describe this curve, viz. KD, Fig. 16. take in the line AB any point C, and let CD be divid ed at the point E, in such a manner that DE : DC, as the sine of refraction is to the sine of incidence. From the point A, and with AC as radius, and from the point B with the radius BE, describe circles C c, E e, and their intersection, or the point K, will be a point in the curve required ; for it is obvious that AK will decrease, and BK increase, in such a manner that their respective increments and decrements will be in the same ratio as DC and DE, which is the property found in the proposition. See Newton's Lectiones Opticce, part i sect iv. Prop. xxxiv. p. 141.; Alontucla's toire des Illathematiques, Scc. tom. p. 260 and 266 ; and Huygens's Traite de la Lumiere,p. 111.
shall refract incident rays accurately to one focus, when the other side or the lens is given, and is such that tangents can be drawn to it.
Let thc figure of the given surface be such as is ge nerated by the revolution of the curve KAK, Fig. 17. round the axis AN', and let this surface AK receive the raj's diverging front the point L. Let AB, the thick ness or the lens, he also given, and let F be the focus to which all the rays arc to be accurately converged, whatever may have been the refraction of the first sur face AK.