29 



On Osculating Focal Systems. 



11. The equations which we have thus obtained, as transforma- 

 tions of the formulae (B'), are not only remarkable in an analytic 

 view, but contain an interesting geometrical property of the caustic 

 surfaces. To explain this property, it is necessary to introduce the 

 consideration of osculating systems of rays. Let us therefore con- 

 ceive a system, placed in the same medium, and composed of the 

 same kind of light, as that given system of rays which has W for its 

 characteristic function, but converging to or diverging from some one 

 point X, Y, Z; and let us denote by W', the corresponding charac- 

 teristic function of this new system, which becomes equal to the W' 

 of the preceding number, when the point X, Y, Z, coincides with the 

 point .t", ?/", z" ; then the general expression for this function W is 



(_!j ( d» 6/3 Oy 



C being an arbitrary constant: and the system which thus has W' for 

 its characteristic function, we shall call ajocal system. The four arbi- 

 trary quantities, X, Y, Z, C, which enter into the general expression 

 (<S0 for JV', may be determined by the condition that for some given 

 ray of the given system, that is, for some given values of a, fB, y, 

 certain of the first terms of the development of TV', according to the 

 positive powers of the variations of a, (3, y, may be equal to the cor- 

 responding terms in the development of the given function W ; and 

 when the form of W' has been particularized by this condition, we 

 shall call the corresponding system of rays, an osculating focal system. 

 Now, if we suppose a, /3, y, to be changed into a + ^a, /3 + ^/3, 

 y + iy, we may express the altered values of W and IV by means of 

 the following developments : 



