101 



VIII, On osculating focal mirrors. 



[36.] It has long been known that a paraboloid of revolution possesses the property of re- 

 flecting to its focus, rays which are incident parallel to its axis ; and that an ellipsoid in like 

 manner will reflect to one of its two foci, rays that diverge from the other : but I do not know 

 that any one has hitherto applied these properties of accurately reflecting mirrors, to the investi- 

 gation of the caustic surfaces, and lines of reflexion of mirrors in general. There exists how- 

 ever a remarkable connexion between them, analogous to the connexion between the properties 

 of spheres and of normals; and it is this connexion, not only for paraboloids and ellipsoids, but 

 also for that general class of focal mirrors, pointed out in Section II. of this Essay, that we are 

 now going to consider. 



[37.] To begin with the simplest case, I observe that the general equation of a paraboloid 

 of revolution may be put under the form 



J = P + <.'.(x-X) + /S'.G'- Y) + v'.(z_Z), 



(P) being the semiparameter, (j) the distance from the focus (X, F, Z), and «', ^', y, the co- 

 sines of the angles which the axis of the paraboloid, measured from the vertex, makes with the 

 axes of coordinates : and that the partial differential coefficients of (z), of the first and second 

 orders, which we shall denote by (p', q', r', «', f), are determined by the following equations, 



x-x + p'.(z-Z) = e.(«'-t-y/) 

 1 + y^ + /.{z — Z) = gy/ + («' + y'p'y 



j/q'+ s'\z—Z) = iy'f + («' + 7'p')(fi'+Vq')- 



This being laid down, if we suppose the three constants (»', /3', y') determined by the con- 

 dition that the axis of the paraboloid shall be parallel to a given system of incident rays, we may 

 propose to determine the other four constants (X, Y, Z, P) by the condition of osculating to a 

 given mirror, at a given point, in a given direction. The condition of passing through the given 

 point, will serve to determine, or rather eliminate (P), and the condition of contact produces 

 the two equations 



p'=p, q' = q, 



which express that the focus of the paraboloid is somewhere on the reflected ray, and which are 

 therefore equivalent to the three following. 



X — a = «{, y — y^=^^i, Z — z=- 



V4» 



