81
LetAQ | =∆, |
∠ AQR | =θ, ∠ AqR=θ′, |
Am | =x, |
mP | =y. |
Then tan θ′=NRNP=AR−MPAM=∆tanθ−yx,
xtanθ′−∆tanθ+y=0.
This is the equation to any point P on the refracted ray. If this point be on the caustic, it must be common to two successive refracted rays infinitely near each other, that is, x and y must be the same for the refracted rays answering to θ and θ+dθ. We may therefore equate to nothing the differential of our equation with respect to θ and θ′, considering x and y as invariable. This gives us
xdθ′cosθ′2−∆dθcosθ2=0.
We have, moreover, between θ and θ′ the equation
sinθ=m·sinθ′.
These three equations must, by the elimination of the functions of θ and θ′, give the one containing only x, y, and ∆, which will be the equation to the caustic.
107. Prop. Required the focus of a thin pencil of rays after being refracted obliquely at a curved surface.
Let QR, QR′, (Fig. 105.) represent two rays inclined to each other at an infinitely small angle, incident obliquely on a curved surface at R, R′; Rq, R′q the refracted rays; RE, R′E, normals.
LetEQ | =q, |
Eq | =t, |
ER | =r, |
QR | =u, |
Rq | =v, |
∠ QRZ | =φ, |
∠ ERq | =φ′. |