Ill 



with the given mirror, complete contact of the second order. A point of contact of this kind, 

 that is, a point where the given mirror is met by an axis of the reflected system, I shall call a 

 vertex of the mirror. 



[48.] Another remarkable property of the principal foci, is that they are the centres of 

 spheres, which have complete contact of the second order with the surfaces that cut the rays 

 perpendicularly; which may be proved by means of the following formulae, deduced from (S') 

 and (T'), 



rf^F _ »' — 1 dW _ /3'— 1 d''y,_ y-—l 



^ >. ^ (U') 



d'V _ x^ d'V _ jcy_ £V _ y3y ' ^ ' 



dx.dy { ' dx,dz { ' dy.dz g 



And if we substitute these expressions (U') in the formulae of [27.], vve find the following equa- 

 tions, 



which determine the vertices, the axes, and the principal foci, when we know the equation of 

 the mirror, and the characteristic function of the incident system. These formulae (V) may 

 also be deduced from the equations (Z) of Section VIII. by means of the theorem that we have 

 already established, respecting the complete contact of the second order, which exists, at a 

 vertex, between the given mirror and tiie osculating focal surface corresponding : and they may 

 be reduced to the two following ; that is, to the equations (T) of Section VII. . 



^.^y-{-^/).dq-\.d^'-\-q.d-/^—dy->fq.dz — {^-\-yq){xdx-\-^y)rydz), , , 



by observing, that these equations, which in general determine the lines of reflexion on the 

 mirror, are, at a vertex, satisfied independently of the ratio between the differentials {dx, dy), 

 provided that we assign to (j) its proper value, namely the distance of the principal focus. 



[49.] As an application of the preceding theory, let us suppose that the incident rays di- 

 verge from a luminous point {X', Y', Z'), and let us seek the vertices, the axes, and the princi- 

 pal foci of the reflected system. In this question, the equations (T) become, by [35-] 



VOL. XV. s 



