97 



and the formula (R) ftw the two foci, which may be thus written 



becomes 



4rt 



.«'— -^ • {(«' + y') ^+(/3'+y'K } + 1 =0. (S) 



If the incident rays be perpendicular to the mirror, at the given point of incidence, then 



y=l, « = 0, /3=0, 



and the two roots of (S) are 



11 11 



that is the two focal distances are the halves of the two radii of curvature of the mirror. 



If without being perpendicular to the mirror, the incident ray is contained in the plane of (xz), 

 that is in the plane of the greatest or the least, osculating circle to the mirror, we shall have 

 /3=:0, «* + y*c:l, and the two roots of ( S) will be 



11 111 



the first root is quarter of the chord of curvature, that is, 'quarter of the portion of the 

 reflected ray intercepted within the osculating circle before mentioned ; and the other root is 

 equal to the distance of the point, where the reflected ray meets a parallel to the incident rays, 

 passing through the centre of the other osculating circle. In general, it will appear, when we 

 come to treat of osculating focal mirrors, that the two foci determined by the formula (S), are 

 the foci of the greatest and least paraboloids of revolution, which having their axis parallel to the 

 incident rays, osculate to the mirror at the point of incidence. 



[31.] I shall conclude this section by remarking, that the equation of the caustic surfaces is 

 a singular primitive of the partial differential equation (O), which we found in the preceding sec- 

 tion to represent all the pencils of the system, and that the equations, 



dx _ dy _ dz 



cc fi y 



of which the complete integral represents all the rays, are also satisfied, as a singular solution, by 

 the equations of the caustic curves : from which it may be proved, that the portion of any ray, 

 or the arc of any caustic curve, intercepted between any two given points, is equal to the incre- 

 ment that the characteristic function (V) receives in passing from the one point to the other. 



f2 



