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given involving six unknown quantities; of these if three are given the other three can be found; so that we have altogether 60 different formulæ.

For other properties of reciprocal curves I must refer the reader to the original memoir by M. Poncelet in Crelle's Journal, Vol. IV, 1839; to several memoirs by Steiner and others in the same Journal; to Analytisch-Geometrische Entwickelungen of Plücker, Zweiter Band, p. 259; and to the Annales des Mathématiques of Terquem and Gerono, Vol. XII.

SECTION 4.-On caustics.

322.] A particular class of envelopes formed by straight lines exists in optics which is of too great importance to be passed over in silence in the present Chapter; we will therefore first give some general notions of the formation of such envelopes which are called Caustics, and then consider some particular examples and general properties of them.

In fig. 118 suppose s to be a source of light from which rays proceed and fall on a highly polished surface, which is perpendicular to the plane of the paper, and of which AP is a section made by the paper; and let SP be a type-ray of such a system incident on the surface at P. Now it is a physical property of a ray, that it is reflected or turned back in the direction PR; so that, if po is the normal to the curve at P, the angle of incidence, as it is called, SPO is equal to the angle of reflexion RPO. The envelope of the lines of which PR is the type, is called the caustic by reflexion of the surface.

And again in fig. 122, suppose s to be the source of a system of rays, of which let SP be the type-ray; and suppose the rays to fall on a medium different to that in which s is, of which the bounding surface is perpendicular to the plane of the paper: and of which let AP be the section made by the paper; then by a physical law of optics, called the law of refraction, the ray sp does not proceed in the same straight line, but at p is bent or refracted into the direction Pr, which is so related to sp that, if nPN is the normal to the surface at P, sin SPN μ sin ren, where is constant for a given medium, but varies for different media; that is, the sine of the angle of incidence bears a constant ratio to the sine of the angle of refraction. The envelope

μ

of all the refracted rays is called the caustic by refraction of the given surface *.

323.] To determine the caustic by reflexion of a system of parallel rays falling on a plane curve.

Suppose the source of light to be at an infinite distance, such as the sun is, and therefore all the incident rays to be parallel; and first let us suppose them to be parallel to the axis of x. In fig. 114 let QP be the incident type-ray, and PR the reflected type-ray Ps being the reflecting curve, and PG its normal at the point P.

η

Let y = f(x) be the equation to the reflecting curve, and 7 and the current coordinates of the reflected ray; then the equation to PR is

n-y= =tan PRG (-x);

of which straight line we have to find the envelope.

Since the angles of incidence and reflexion are equal, QPG RPG, and therefore their complements are equal, viz.

-1

dy dx

QPL = RPT; but QPL PTG = tan ;

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*In the figure the line AP is straight, but the matter of the text is ex

pressed as though it were a curve.

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By means of which equations, and that to the reflecting curve, we may eliminate x and y, and thereby obtain a relation between έ and ŋ, which will be the equation to the caustic.

Ex. 1. Let the reflecting curve be the parabola.

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All the reflected rays therefore pass through the focus, which is their envelope; the caustic therefore is a point.

Ex. 2. Let the reflecting curve be a circle.

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the equation to an epicycloid, the radius

is, and of whose generating circle is

of whose fixed circle

a

4'

as will

appear on eliminating between the two equations (37), Art. 204, having

a

first replaced b by and a by. See fig. 119.

4

The above examples are sufficient for illustration, but the difficulties of elimination are in most cases insurmountable; the semi-cubical parabola is another curve admitting of solution.

324.] Again, suppose the incident rays to be parallel to the axis of y, see fig. 115, of which let MP be the incident type-ray, and PR the reflected type-ray; let y = f(x) be the equation to the reflecting curve; and let έ, n be the current coordinates of the reflected ray; then the equation to PR is

η

n-y=

1 dy 2dx

dy} (&−x);

(81)

of which line we have to determine the envelope.

Differentiating, and proceeding as in the last Article, it will

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By means of which, and the equation to the curve, we may eliminate x and y, and determine a relation between έ and ŋ, which will be the equation to the caustic.

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which represents a curve symmetrical with respect to the axis of x, passing through the origin where it touches the axis of y; with a double point on the axis of x at a distance 9 a from the origin, and a loop between the origin and that point; and approaching to the semi-cubical parabola as an asymptotic curve.

Ex. 2. Find the equation to the caustic of the cycloid, the incident rays being perpendicular to the base.

Take the starting point o for the origin, as in fig. 117; then the equation to the curve is

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a

{− (an)* + (an−n2)* ;

-

.'. §—(an) + (an-n2) = a versin-1

} + (an-n2) = versin-1 27;

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a

& = versin- -1 - (an-n2);

2

a

(83)

which equation is that to a cycloid, of which the starting point is the origin, and the radius of whose generating circle is onehalf of that of the generating circle of the original cycloid.

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