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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.[1]
22. Prop. 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.
| Let QE | =q | |
| Eq | =t | q2=u2+r2−2rucosφ |
| QR | =u | t2=v2+r2−3rvcosφ |
| Rq | =v | |
| ∠QRE, or ERq=φ | ||
| ER | =r. | |
Then since a small variation in the place of R′ causes an infinitely less variation in that of q, we may establish the following equations by differentiating those above,
0=udu−rducosφ+rusinφdφ
0=vdv−rdvcosφ+rvsinφdφ.
Moreover, since QR, Rq, QR′, R′q make respectively equal angles with the curve as in an ellipse, of which Q and q would be the foci,
QR+Rq=QR′+R′q or d(u+v)=0,
so that our equations now stand
0=−udu−rducosφ+rusinφdφ,
0=−vdu+rducosφ+rvsinφdφ,
- ↑ Strictly speaking this curve is but a section of the caustic, which is a surface like the reflector which produces it.
