93 



* dz fi dz . ,_.. 



which expresses that the tangent plane to the pencil contftins the ray passing through the point 

 of contact. 



VI. On the developable pencils, the two foci of a fay, and the caustic curves and surfaces. 



[26.] Among all the pencils of a given rectangular system, there is only a certain series deve- 

 lopable ; namely, those which pass through the lines of curvature on the surfaces that cut the 

 rays perpendicularly. It follows from the known properties of normals to surikces, that each ray 

 has two of these developable pencils passing through it, and is therefore a common tangent to 

 two caustic curves, the aretes de rebroussement of those pencils ; the points in which it touches 

 those two caustic curves may be called the two foci of the ray ; and the locus of these foci forms 

 two caustic surfaces, touched by all the rays. 



[27.] To determine analytically these several properties of the system, let us represent by 

 (a, b, c) the coordinates of the point in which a ray crosses a given perpendicular surface ; these 

 coordinates will be determined, if the ray be given, so that they may be considered as functions 

 of («, fi) ; we may therefore put their differentials under the form 



, da , , da , 



da = —r • det-U —r . ds. 



da ^ dfi 



„ db , db ^ 



do= -T—. d» + -r- . ds. 



d» ^ dfi 



J dc , , do 

 dc^-. d.+ -. da. 



we have also itda •{■ fidb -|- ydc = 0, which gives 



dc / » da li db\ 



d» \y d» y d») 



dc i» da fi db\ 



~dfi (V ^ y ' "dfi)' 



_. . da da db db 



with respect to the coefficients — , — , — - , — , these are to be determined by differen* 



tiating the two following equations, 



dV _ dV _ 



dT-"' lb~^' 

 which give 



