22 



ex cS oy 



in which the symbols 



ex Ssc 



have the same meanings as before. Knowing, in this manner, the 

 differential coefficients of V, before the first reflexion or refraction, 

 we can, by the method of the preceding number, calculate the cor- 

 responding coefficients of V, and thence of TV, immediately after that 

 change ; the coefficients of W, thus deduced, will remain the same, 

 in passing from the point of first reflexion or refraction to the second 

 point at which the direction of the ray is altered, and, by the me- 

 thods of the fifth number, we can deduce from these coefficients of 

 7Fthe corresponding coefficients of V, immediately before that second 

 change ; and so proceeding, we can calculate the alterations in the 

 partial differentials of the two characteristic functions, produced by 

 any finite number of successive reflexions or refractions, 



Conneximi of the two Characteristic Functions with the Developable 

 Pencils and the Caustic Curves and Surfaces. 



8. Let us now suppose these partial differentials known, and let 

 us examine their connexion with the geometrical properties of the 

 system. One of the most remarkable of these geometrical properties 

 is, that the rays are in general tangents to two series of caustic curves, 

 which are contained upon two caustic surfaces, and form the aretes de 

 rebroussejnent of two series of developable pencils; that is, two series 

 of developable surfaces, composed by rays of the system : a property 

 which was first discovered by Malcs, and to which I also had 



