16^2 



(«) /5i y) being the cosines of the angles which the reflected ray makes with the axes. To de- 

 termine the remaining constant ({), by the condition that the paraboloid shall osculate to the 

 mirror in a given direction, we are to employ the formula 



(/_r) dx'' + 2(j'— «) ds.djf + (V—t) dy* — 0, 



r, s, t, being the given partial differential coefficients, second order, of the mirror, and r', s, t', 

 the corresponding coefficients of the paraboloid, which involve the unknown distance (j), being 

 determined by the equations, 



e-(v+vO-^ = 1 +i»'- {«'+y'i') * 

 ?.(y + v')-*' = jJ? — (*' + Vp ) ( /3' *- Vq) 

 ?-(y+-/K=l+5''-{/3'+y'9)'. ^ 



To simplify our calculations, let us, as in [30.] , take the normal to the mirror for the axis of 

 (2), and the tangents to the lines of curvature for the axes of (x) and (_y) ; we shall then have 



p = 0, q = 0, s=0, « + «' = 0, /3 + /3' = 0, y = y, 



275.?^=^'+ y% 2y{. «'= — «^, 2y^.i' = »*+v; 



and the condition of osculation becomes 



2yf.(r + <t') = /3^ + v" — 2«/3r + («' + y^) r\ (Y) 



if we put dj/ = T.dx. This formula (Y) determines the osculating paraboloid for any given value 

 of (t), that is, for any given direction of osculation ; differentiating it with respect to (t), in 

 order to find the greatest and least osculating paraboloids, we get 



2yft.r = — a/3 + («^ + y') t, 

 2yjj- = ^" + y* — «/3t, 



equations which give, by elimination, 



I 2yef - («» + y') } { 2yr - (H' + y^ } - »'/3' =0, 



«^(<.T» — r)+ |(«' + y»)r_(^^ + y«y I r=0: 



and since these coincide with the formulx (S) (V) of the two preceding sections, it follows, that 

 when parallel rays are incident upon a mirror, the two foci of any given reflected ray, that is, 

 the two points in which it touches the caustic surfaces, are the foci of the greatest and least pa- 

 raboloids, which having their axis parallel to the incident rays, osculate to the mirror at the 

 given point of incidence ; and that the directions of the two lines of reflexion passing through 

 that point, are the directions of osculation corresponding. 



[38.] In general when the incident system is rectangular, which is always the case in nature 

 it follows from the principles already established that we can find an infinite number of focal 



