45 



in this kind of osculation, therefore, as in the former, the distances ot 

 the variable focus or centre from the points where the ray touches 

 the two caustic surfaces, are proportional to the squares of the sines 

 of the angles which the plane of osculation makes with the tangent 

 planes to the developable pencils. 



On Osculating Focal Reflectors or Refractors. 



16. Besides the two preceding kinds of osculation, it is interest- 

 ing to consider a third kind, which exists between the last reflecting 

 or refracting surface, and certain other surfaces, which would have 

 reflected or refracted to or from one focus the rays of the last incident 

 system, and which we shall therefore call focal reflectors or refrac- 

 tors. Let Fi , Vi, denote, as in the sixth number, any two succes- 

 sive forms of the characteristic function V, of which we shall suppose 

 that Vi belongs to the system in its given state, and Vi to the same 

 system before its last reflexion or refraction ; then, by the number 

 cited, the equation Fj — Fg z= 0, will be a form for the equation of 

 the reflector or refractor, at which the state of the system was last 

 changed, and which we shall consider as known. Let V'„ be tire 

 form which V^ would have, if the rays of the tinal system all con- 

 verged to or diverged from one focus, this form being such as was 

 assigned in the fifteenth number, and depending only on the nature 

 of the light and of the final medium, but involving four arbitrary con- 

 stants, of which three are the coordinates of the focus ; then it is easy 

 to prove that the equation with four arbitrary constants, of the focal 

 surface, which would have reflected or refracted to or from one focus 

 the rays of the last incident system, is 



F, — F', = . (E'") 



