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 analogous to normals drawn at different points of a curve, and that as these by their mutual intersections produce a broken line which becomes a regular curve, when their number is increased without limit, so the reflected rays should give rise to a similar broken line and curve, which is, in fact, the case, the curve being what is technically termed a caustic.

22. . Given a point from which a thin pencil of rays proceeding, fall on a spherical reflector, to determine their intersections after the reflexion.

Let $QR$, $QR′$ (Fig. 16.) be two incident rays, $Rq$, $R′q$ the reflected rays meeting in $q$, $RE$, $R′E$ the normals at $R$, $R′$ meeting in $E$, which if we suppose the distance $RR′$ to be, according to the phrase, infinitely small, will be the centre of the osculating circle. Then since a small variation in the place of $QE$ causes an infinitely less variation in that of $=q$, we may establish the following equations by differentiating those above,

Moreover, since $Eq$, $=t$, $q^{2}=u^{2}+r^{2}−2rucosφ$, $QR$ make respectively equal angles with the curve as in an ellipse, of which $=u$ and $t^{2}=v^{2}+r^{2}−3rvcosφ$ would be the foci,

so that our equations now stand