without, or on the curve; we shall now extend this conclusion to all the conic sections. It is evident that, the nearer any line or point is to the origin, the farther the corresponding point or line will be; that if any line passes through the origin, the corresponding point must be at an infinite distance; and that the line corresponding to the origin itself must be altogether at an infinite distance. To two tangents, therefore, through the origin on one figure, will correspond two points at an infinite distance on the other; hence, if two real tangents can be drawn from the origin, the reciprocal curve will have two real points at infinity, that is, it will be a hyperbola; if the tangents drawn from the origin be imaginary, the reciprocal curve will be an ellipse; if the origin be on the curve, the tangents from it coincide, therefore the points at infinity on the reciprocal curve coincide, that is, the reciprocal curve will be a parabola. Since the line at infinity corresponds to the origin, we see that, if the origin be a point on one curve, the line at infinity will be a tangent to the reciprocal curve; and we are again led to the theorem (Art. 254) that every parabola has one tangent situated at an infinite distance.

315. To the points of contact of two tangents through the origin must correspond the tangents at the two points at infinity on the reciprocal curve, that is to say, the asymptotes of the reciprocal curve. The eccentricity of the reciprocal hyperbola depending solely on the angle between its asymptotes, depends therefore on the angle between the tangents drawn from the origin to the original curve.

Again, the intersection of the asymptotes of the reciprocal curve (i.e. its centre) corresponds to the chord of contact of tangents from the origin to the original curve. We met with a particular case of this theorem when we proved that to the centre of a circle corresponds the directrix of the reciprocal conic, for the directrix is the polar of the origin which is the focus of that conic.

Ex. 1. The reciprocal of a parabola with regard to a point on the directrix is an equilateral hyperbola. (See Art. 221).

Ex. 2. Prove that the following theorems are reciprocal:
The intersection of perpendiculars of

a triangle circumscribing a parabola is a
point on the directrix,

The intersection of perpendiculars of

a triangle inscribed in an equilateral hyperbola lies on the curve.

Ex. 3. Derive the last from Pascal's theorem. (See Ex. 3, p. 247).

Ex. 4. The axes of the reciprocal curve are parallel to the tangent and normal of a conic drawn through the origin confocal with the given one. For the axes of the reciprocal curve must be parallel to the internal and external bisectors of the angle between the tangents drawn from the origin to the given curve. The theorem stated follows by Art. 189.

316. Given two circles, we can find an origin such that the reciprocals of both shall be confocal conics. For, since the reciprocals of all circles must have one focus (the origin) common; in order that the other focus should be common, it is only necessary that the two reciprocal curves should have the same centre, that is, that the polar of the origin with regard to both circles should be the same, or that the origin should be one of the two points determined in Art. 111. Hence, given a system of circles, as in Art. 109, their reciprocals with regard to one of these limiting points will be a system of confocal conics.

The reciprocals of any two conics will, in like manner, be concentric if taken with regard to any of the three points (Art. 282) whose polars with regard to the curves are the same. Confocal conics cut at right angles (Art. 188).

The tangents from any point to two confocal conics are equally inclined to each other. (Art. 189).

The locus of the pole of a fixed line with regard to a series of confocal conics is a line perpendicular to the fixed line. (Art. 226, Ex. 3).

The common tangent to two circles subtends a right angle at either limiting point.

If any line intersect two circles, its two intercepts between the circles subtend equal angles at either limiting point.

The polar of a fixed point, with regard to a series of circles having the same radical axis, passes through a fixed point; and the two points subtend a right angle at either limiting point.

317. We may mention here that the method of reciprocal polars affords a simple solution of the problem, "to describe a circle touching three given circles." The locus of the centre of a circle touching two of the given circles (1), (2), is evidently a hyperbola, of which the centres of the given circles are the foci, since the problem is at once reduced to-" Given base and difference of sides of a triangle." Hence (Art. 308) the polar of the centre with regard to either of the given circles (1) will always touch a circle which can be easily constructed. In like manner, the polar of the centre of any circle touching (1) and (3) must also touch a given circle. Therefore, if we draw a common tangent to the two circles thus determined, and take the pole

of this line with respect to (1), we have the centre of the circle. touching the three given circles.

318. To find the equation of the reciprocal of a conic with regard to its centre.

We found, in Art. 178, that the perpendicular on the tangent could be expressed in terms of the angles it makes with the axes, p2= a* cos10+ b2 sin*0.

IIence the polar equation of the reciprocal curve is

a concentric conic, whose axes are the reciprocals of the axes of the given conic.

319. To find the equation of the reciprocal of a conic with regard to any point (x'y').

The length of the perpendicular from any point is (Art. 178)

Ρ √(a* cos30 + b2 sin30) - x' cos 0y' sin 0;

=

k2 ρ

- =

therefore the equation of the reciprocal curve is

(xx' + yy' + k3)* = a*x2 + b2y3.

320. Given the reciprocal of a curve with regard to the origin of coordinates, to find the equation of its reciprocal with regard to any point (x'y').

If the perpendicular from the origin on the tangent be P, the perpendicular from any other point is (Art. 34)

P-x cos 0 y' sin 0,

we must therefore substitute, in the equation of the given

P

The effect of this substitution may be very simply written as follows: Let the equation of the reciprocal with regard to the origin be

where u denotes the terms of the nth degree, &c., then the reciprocal with regard to any point is

a curve of the same degree as the given reciprocal.

321. To find the reciprocal with respect to x2+y-k" of the conic given by the general equation.

We find the locus of a point whose polar xx+yy' — k3 shall touch the given conic by writing x', y', - k" for λ, μ, v in the tangential equation (Art. 151). The reciprocal is therefore

C or ab - h2=0, and the
We can, in like manner,

Ax* +2Hxy + By3 – 2 Gk3x – 2Fk2y + Ck* = 0. Thus, if the curve be a parabola, reciprocal passes through the origin. verify by this equation other properties proved already geometrically. If we had, for symmetry, written k2 = - z3, and looked for the reciprocal with regard to the curve x2 + y2 + z2 = 0, the polar would have been xx' + yy′+zz', and the equation of the reciprocal would have been got by writing x, y, z for λ, μ, v in the tangential equation. In like manner, the condition that λx + μy + vz may touch any curve, may be considered as the equation of its reciprocal with regard to x2 + y2 + z2.

A tangential equation of the nth degree always represents a curve of the nth class; since if we suppose λx + μy + vz to pass through a fixed point, and therefore have λx' +μy' + vz′ = 0; eliminating between this equation and the given tangential equation, we have an equation of the nth degree to determine λ: μ; and therefore n tangents can be drawn through the given point.

322. Before quitting the subject of reciprocal polars, we wish to mention a class of theorems, for the transformation of which M. Chasles has proposed to take as the auxiliary conic a parabola instead of a circle. We proved (Art. 211) that the intercept made on the axis of the parabola between any two

lines is equal to the intercept between perpendiculars let fall on the axis from the poles of these lines. This principle then enables us readily to transform theorems which relate to the magnitude of lines measured parallel to a fixed line. We shall give one or two specimens of the use of this method, premising that to two tangents parallel to the axis of the auxiliary parabola correspond the two points at infinity on the reciprocal curve, and that consequently the curve will be a hyperbola or ellipse, according as these tangents are real or imaginary. The reciprocal will be a parabola if the axis pass through a point at infinity on the original curve.

"Any variable tangent to a conic intercepts on two parallel tangents, portions whose rectangle is constant."

To the two points of contact of parallel tangents answer the asymptotes of the reciprocal hyperbola, and to the intersections of those parallel tangents with any other tangent answer parallels to the asymptotes through any point; and we obtain, in the first instance, that the asymptotes and parallels to them through any point on the curve intercept on any fixed line portions whose rectangle is constant. But this is plainly equivalent to the theorem: "The rectangle under parallels drawn to the asymptotes from any point on the curve is constant."

Chords drawn from two fixed points of a hyperbola to a variable third point intercept a constant length on the asymptote. (Art. 199, Ex. 1).

If any tangent to a parabola meet two fixed tangents, perpendiculars from its extremities on the tangent at the vertex will intercept a constant length on that line.

This method of parabolic polars is plainly very limited in its application.