76 



Remarks on other questions of the same kind, which conduct to equations in ordinary differ- 

 entials ..••••.. 98 



The partial differential equation which expresses the condition for the two caustic surfaces 

 coinciding, resolves itself into two other equations which are peculiar to the focal reflectors or 

 refractors : however, these two equations determine, on some particular reflecting and refracting 

 surfaces, a line qfjjcal curvature, analogous to the line of spheric curvature, which exists upon 

 some curved surfaces ..... 99 



XX. On the caustics of a given rejlecting or refracting curve. 



General theorem respecting the arc of a caustic curve ; means of finding by this theorem, 

 the curves corresponding to a given caustic . . . . 100. 



As an infinite number of curves correspond to the same given caustic, so also an infinite num- 

 ber of caustics correspond to the same given curve; these caustics have for their locus a surface 

 of circular -profile, and are the shortest lines between two points upon it ; this surface envelopes 

 all the caustic surfaces, corresponding to the reflectors or refractors, upon which the given curve 

 is a line of reflection or refraction . . . . 101,102. 



XXI. On the conditions of Achromatism, 



The coordinates of the image of a luminous point, formed by any given combination of lenses 

 and mirrors, are functions of the colour of the rays ; hence may be deduced series for the chro- 

 matic aberrations, and conditions for achromatism, perfect or approximate 103, lOt. 



XXII. On systems of atmospheric rays. 



• Equations of an atmospheric ray .... 105. 



These rays are perpendicular to the surfaces of constant action ; and the properties of the 



system may all be deduced from the form of one characteristic function . 106, 



PART THIRD. ON EXTRAORDINARY SYSTEMS, AND SYSTEMS OF RAYS 



IN GENERAL. 



XXIII. On plane systems of rays. 



n.i..; "h) iiLti* ji 'jo -Kir.- 



Object of this third part i« iic4fc 5i's::3!:i jji-od^i" mvi,; u. . 107. 



Equation of a ray in a plane system , formulie for 'the focal coordinates, and for the arc and 



curvature of the caustic curve, . . . . 108. 



