CONTENTS OF THE SUPPLEMENT. 



FuNDAJMENTAi, formula of optical systems ordinary and extraordinary, or equa- 

 tion of the characteristic function ; 



V =fvds; JF = 4^ J* + J^ Ji/ -)- ^ 3« ; ... numbers 1, 2, 3. 

 Other characteristic function, for systems of straight rays ; 



Connexions between the partial differential coefficients of the two characteristic 

 functions, and changes produced by reflexion or refraction, ordinary or extra- 

 ordinary ; 



JaF = a3m; aW'= a ( X s^ + y e- + z j-j; •.. ••• 6,6,7. 



-Connexions of the characteristic functions with the developable pencils, and with 

 the caustic curves and surfaces j 



'i7=^'ir''-w = '-W''-h^^i^' - ''''^'- 



On osculating focal systems ; the foci of the extreme systems are contained upon 

 the caustic surfaces, and their planes of osculation touch the developable pen- 

 cils; the foci of the other osculating systems are inside or outside the interval 

 of the extreme foci, according to the sign of a certain function v", which does 

 not change by transformation of coordinates ; the distances of any one of these 

 other foci from the two caustic surfaces are proportional to the squares of the 

 sines of the angles which the corresponding plane of osculation makes with 

 the tangent planes to the two developable pencils ; 



„^rw;'^P^^,{^iS^=^)', .4>o;... M, ...... 



Principal foci, and principal rays : the principal foci belong to those osculating 

 systems, for which the osculation is most complete ; they are the only points in 



*b2 



