105 



— = 0, tan.V = — — • 



? Si 



I shall however continue to employ it, both on account of the analytic theorem which it ex- 

 presses, and also on account of its analogy to the received phrase of greatest osculating sphere, 

 to which the same objection may be made, when the two concavities of the surface are turned in 

 opposite directions. 



[40.] I shall conclude this section, by pointing out another remarkable property of the osculat- 

 ing focal mirrors ; wliich is, that if upon the plane, passing through a given direction of osculation, 

 we project the ray reflected from the consecutive point on that direction, the projection will cross 

 the given ray in the osculating focus corresponding. To prove this theorem, I observe, that 

 when the given ray is taken for axis of (2), the point where it meets the mirror for origin, and 

 the tangent planes of the developable pencils for the planes of (x, z), [y, 2), the partial differ- 

 entials second order of the characteristic function (V) become, at the origin, 



d^V 1_ <i'F _ d^V _ 1 d^V _ d'V _ i^V ^ 



dx^ ~ s, dx.dy— ' ~d^~~~^ 'dldz~'^' d^z~^' 1^ ~ ^' 



and therefore the cosines of the angles which an infinitely near ray makes with the axes of (x) 

 and (y), are 



■, _ «?i; di/ . 



~ it' e^ 



Hence it follows that the equations of this infinitely near ray are of the form 



«i ^ + (z' — ei)rfx = o, g,y + (2' — 5,)flr^ = = 0; 



and if we project this ray on the plane 



x' dy —y' dx :z: 0, 



which passes through the given ray and through the consecutive point on the mirror, the pro- 

 jecting plane will have for equation 



i,.i^+(^—i,).dx u-y' + (^ — e)'dy _o 



ik-i,).dx (h-^^).dy 



(k) being the height of the point where the projection crosses the given ray, which is to be de- 

 termined by the condition that the latter plane shall be perpendicular to the former, that is, by 

 the equation, 



(h-^^)dx^ (h-^,) dy -"' 

 which, when we put dy = dx, tan. ^, becomes 



h= irh (Jj/) 



s2 



