Page:Optics.djvu/52

 Let $Q$, (Fig. 17.) be the radiant point, $MqN$ the caustic, $Rq$ a reflected ray touching it in $q$. All we know about the reflecting curve $ARB$ is, that every pair of lines such as $QR$, $Rq$, make equal angles with it. Now it will easily be seen, that there is no limit to the number of curves that will answer this condition, any more than to the number of ellipses that may have the same two foci. If $QRq$ were part of a string fastened at $Q$ and $N$, so as to lap round the part $Nq$ of the caustic, and extended by a pin, this pin would describe the reflecting curve $ARB$, and of course it is only necessary to change the length of the string to get any number of different curves.

The algebraical explanation of this seems to be, that the equation $du+dv=0$, requires for its integration a constant, which is just as arbitrary here as in the case of the ellipse in which the line represented by $v$ is always measured from the same point.

36. a sufficient quantity of light falls on an object which is neither transparent nor specular, (that is, polished sufficiently to reflect properly,) it is dispersed from every point in all directions, and makes the object visible to a spectator placed on any side of it. If this light meet with a specular surface, either plane or curved, it will be reflected regularly according to the laws we have investigated in the preceding chapters, and since for every pencil of rays falling any how on a plane mirror, or nearly perpendicularly on a curved one, there is a focus of reflected rays, it follows, that the principal part of the reflected light will as it were proceed from the various foci of reflected rays, and the effect will be the same as if the same quantity of light came from an actual substance, of which each point should correspond or coincide with one of those foci; this is technically expressed by saying, that they