Page:Optics.djvu/103

79

y2	=(e2−1) {(x−ae)2−a2}
	=(e2−1) {x2−2aex+(e2−1)a2},

which agrees with that above, if

m=e,

mn/m2−1=	−ae; ∴ a=−n/m2−1,
n2/m2−1=	(e2−1)a2, or a2=n2/(m2−1)2, as before.

If m be less than unity, the integrated equation becomes

y²=n²+2mnx−(1−m²)x²,

which agrees with the equation to the ellipse,

y2	=(1−e2) {a2−(x−ae)2},
or y2	=(1−e2) {(1−e2)a2+2aex−x2}.

The general equation integrated gives

u+mv=n,

that is, √x²+y²+m√(c−x)²+y²=n.

The curve to which this belongs is a certain oval, which Descartes has described.

105. Newton's second proposition is:

To find the form of a convex lens, that shall refract light accurately from one point to another.

He supposes the first surface given, and determines the second thus, (Fig. 102.)

Let A be the focus of incident, B of refracted rays; ADFB the course of a ray; CP, ER, circular arcs with centers A, B; CQ, ES, orthogonal trajectories to DF.

Let AB, AD, DF, be produced so that BG=(m−1)CE, AH=AG; DK=1/mDH. Join KB, and let a circle with centre D, and radius DH cut it in L. Draw BF parallel to DL. In the first place PD=m·DQ; FR=Fn·FS. For let AD, AE