14 REPORT — 1860. 



Note on the Caustics produced by Reflexion. 

 By L. L. Lindelof, Professor at Helsingfors. 



There are, no doubt, few branches of mathematical physics that have been more 

 often discussed than the reflexion and refraction of light, and the theory of these 

 phenomena has consequently been gradually reduced to the greatest simplicity. The 

 whole doctrine of catoptrics and dioptrics may indeed be said to be implicitly con- 

 tained in the elegant principle successively developed by Dupin, Quetelet, and Gor- 

 gonne, namely, that a system of rays that can be cut orthogonally by a particular sur- 

 face, preserves this property after any number of reflexions and refractions. Never- 

 theless, it appears to me that the theory of caustics has been somewhat neglected. 

 Not but what there are many interesting researches on this subject that have been 

 conducted with abundance of care, but because these, for the most part, refer to cer- 

 tain very restricted cases, as for example, to reflecting surfaces of a particular kind. 

 In examining from a somewhat more general point of view the theory of caustics 

 produced by reflexion, I have arrived at certain results, which appear to me to be 

 sufficiently curious to deserve a short notice. 



I suppose the reflecting surface to be of any kind whatever, and that it is illumi- 

 nated by a bundle of parallel rays. Suppose, now, that two of these rays impinge on 

 the surface at two points A and A' infinitely near each other. Unless certain par- 

 ticular conditions are fulfilled, the corresponding reflected rays will not be in the same 

 plane. In order, therefore, that the two rays may meet after reflexion so as to form 

 a point in a caustic, the points A and A' must be related in a certain manner. Now 

 it will be found that, starting from any point A, there will always be two different 

 directions in which the consecutive reflected rays intersect, and by following these 

 directions from point to point, certain curves will be traced on the surface, which play 

 an important part in the theory of caustics, and which may be called catoptrical lines. 

 These lines bear some analogy to the lines of greatest and least curvature, with which 

 they sometimes coincide. Their form and situation depend not only on the nature of 

 the surface, but also on the direction of the incident rays. Each point of the surface is 

 the intersection of two catoptrical lines, which possess the remarkable property that 

 their projections on the plane perpendicular to the incident rays, cut each other at 

 right angles. To each catoptrical line there is a corresponding caustic formed by the 

 rays reflected from the catoptric, and these caustic lines themselves form a caustic 

 surface, which in general consists of two sheets, corresponding to the two systems 

 of catoptrical lines. 



Let x, y, and z be the coordinates of any point in the reflecting surface, and let the 

 axis of z be parallel to the incident rays. Calling, as usual, the partial differential 

 coefficients of z with respect to x and y of the first orders and q, those of the second 

 r, s, and t, we have for the catoptrical lines the simple equation 



dp . dy=dq . dr, 

 which may be put in the form 



21 +*=-'£-'-. 



since dp—rdx-\-sdy, dq = sdx-\-tdy. 



The quantities/?, q, r, s, and t being all expressible in terms of # and y by means of 



the equation to the reflecting surface, the two values of -i. derived from the above 



dx 

 equation can also be expressed in terms of x and y. If this differential equation can 

 be integrated, the resulting relation between x and y, together with the equation to 

 the surface, determine the catoptrical lines. 



The point £, t], and £ of the caustic corresponding to x, y, z of the reflecting sur- 

 face, is determined by the following equations : — 



t-s^l 

 |-' r -9-y_ t *- __ dx 



■2p 2q p 2 +f-l 2^-rt) 



Eliminating x, y, z by means of these three equations and that of the given surfaces, 

 we obviously get the equation to the caustic surface ; and eliminating the same quan- 



