Page:Optics.djvu/105

From Wikisource
Jump to navigation Jump to search
This page has been validated.

81

LetAQ =∆,
AQR =θ, AqR=θ′,
Am =x,
mP =y.

Then tan θ′=NR/NP=ARMP/AM=∆tanθy/x,

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+. 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−∆/cosθ2=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′.