11
the one most likely to occur, is that in which the radiant point is at the opposite point of the sphere from the centre of the mirror.
Making ∆=2r, we find here ∆′=∆r2∆−r=23r.
It will occur to every one, that of the two foci Q and q, that which lies between E and A moves much more slowly than the other, when their places are changed; in fact, we have seen that by merely bringing up Q from E to F, q was sent from E to an infinite distance, and that when Q moved on from F towards A, q came back from an infinite distance on the reverse side of the reflector to meet Q at A.
13. We have hitherto considered only one species of spherical reflector, the concave; let us now take the convex, (Fig. 11.) where as before, E is the centre, Q the radiant point, QR, RS an incident and a reflected ray, making equal angles with ER the radius or normal. Let SR cut AE in q.
Then we have, keeping the same notation as before,
EREQ=sinEQRsinERQ;
that is, as before, rq=sin(π−φ)−θsinφ=sin(θ+φ)sinφ,
EREq=sinEqRsinERq;
that is, rq′=sin(π−φ)+θsinφ=sin(θ−φ)sinφ;
∴ rq′−rq=sin(θ−φ)−sin(θ+φ)sinφ=−2cosθ,
and finally, 1q′=1q−2cosθr,
which is the same result as before, except the sign of the 2d term; it will however immediately occur that they may be reconciled completely by supposing the radius r to be positive in the one case, and negative in the other, which is exactly true in algebraical
