origin to two points at infinity; their points of contact X and Y reciprocate into two real tangents to the conic, meeting in C the correspondent of XY, whose points of contact are at infinity.

These lines are termed the Asymptotes of the hyperbola. They are imaginary for the ellipse, though they intersect in a real point, and coincident with the line at infinity for the parabola.

The tangents A' and B' at the extremities of the diameter OS correspond to points A and B called the Vertices of the conic; also since the distances of S from A', XY, B', are in H.P., SA, SC, SB their reciprocals are in A.P.; hence C is the middle point of the segment AB, and it is obviously the point at which the asymptotes intersect. *

* When the origin is outside the circle its polar divides the circumference into two parts which are respectively concave and convex to it. These portions reciprocate into two distinct curves convex and concave to the origin as shown in the figure, and both branches reach to

Also since SA', SO and SB' are in A.P., their reciprocals SA, SL, SB respectively are in H.P.

The tangents from any point K, on XY, to the circle, with XY and KS form an harmonic pencil (Art. 78, Ex. 5); hence by reciprocation any line through C meets the conic in an harmonic row of points, one of which, corresponding to the ray KS, is at infinity. Thus every chord of the conic through C is bisected. On account of this property C is termed the Centre of the curve.

Again, the tangents to the circle from any point on the perpendicular through S to RS and the lines joining that point to R and S form an harmonic pencil; hence by reciprocation any line parallel to OS meets the conic in an harmonic row of points, one of which, corresponding to the ray through S, is at infinity; another, that corresponding to the ray through R, is on M the perpendicular through C to OS. It is therefore manifest that the conic is symmetrically situated with respect to this line. It is moreover symmetrical with respect to ON. These rectangular lines OM, ON through the centre Care termed the Axes of the curve.

infinity. If, however, we assume in general that consecutive tangents to the circle reciprocate into consecutive points on the conic, by taking two tangents indefinitely near, one on the convex and the other on the concave part of the circle, we are led to the conclusions that the points at infinity on the opposite branches of the curves are indefinitely near, that the asymptotes are tangents at the points of coincidence, and that the hyperbola is a continuous curve.

EXAMPLES.

1. A circle, any point and its polar with respect to the circle, e.g.

Circle, centre and line at infinity.

Circle, origin and polar of origin.

Circle and inscribed polygon. Circle (or conic) and self conjugate triangle.*

2. The opposite sides of a cyclic hexagon meet in three collinear points. (Pascal.)

A conic, a line and its pole with respect to the conic.

Conic, directrix and focus.

Conic, line at infinity and centre of conic.

Conic and escribed polygon. Conic and self conjugate triangle.

The opposite vertices of an escribed polygon connect by three concurrent lines. (Brianchon.)

This result follows when the circle described about the hexagon is taken as the circle of reciprocation.

In general, from any origin, the theorem of Pascal with respect to a circle reciprocates into Brianchon's property for a conic.

3. Four points on a circle subtend at a variable point on it equianharmonic pencils.

Four fixed tangents to a circle meet a variable tangent to it in equianharmonic rows;

hence, generally from any origin, the property of Euc. III. 21 becomes :-A variable tangent to a conic meets four fixed tangents in rows of points which are equianharmonic; and reciprocally four fixed points on a conic subtend equianharmonic rows at a variable fifth point on it.

And again it follows conversely that, if two points connect equianharmonically with four others, all six lie on a conic; hence :-Any two of the hexad of points connect equianharmonically with the remaining four. This system is sometimes called an Equianharmonic Hexagon. (Townsend, Mod. Geom. vol. II. p. 168.) 4. Concentric Circles. Conics having same focus (origin) and directrix.

* If the origin is taken at one of the vertices of the triangle the corresponding side of the reciprocal triangle is therefore at infinity, and its other two sides are diameters (conjugate) of the conic. See Exs. 8, 9.

5. Circles having a common

Conics having a common focus

pair of inverse points (from and centre. either point as origin).

From the synimetry of the conic we infer that such a system has a second common focus; hence :-Coaxal Circles reciprocate from either of their common pair of inverse points into a system of Confocal Conics.

6. Euc. III. 35, 36.

The rectangle under the distances of either focus from a pair of parallel tangents is constant ;

hence from symmetry we infer that the rectangle under the distances of the foci from any tangent is constant; and conversely, the envelope of a variable line, the product of whose distances from two fixed points is constant, is a conic having the fixed points for foci.

7. A chord of a circle which subtends a right angle at the origin envelopes a conic.

8. A variable chord of a circle passing through a fixed origin is divided harmonically by the point and its polar.

The locus of the intersection of rectangular tangents to a conic is a circle.

(Director Circle.)

The variable chord of contact of two parallel tangents passes through and is bisected at the

centre of the conic.

Def. The diameter of a conic parallel to a tangent is said to be Conjugate to that which passes through its point of contact.

19. Conjugate points with respect to a circle (from the pole of line joining them as origin).

10. If a variable point P moves on a line through the origin, Sits polar passes through

the pole of the line with respect to the circle; and the tangents from P and the lines PQ and PS form an harmonic pencil.

Conjugate diameters of a

conic.

If a variable chord of a conic moves parallel to a fixed direction, the harmonic conjugates of the points on it at infinity (i.e. the middle points) are collinear;

hence the locus of the middle points of any system of parallel chords is a line.

or the circles on the diagonals are concurrent.