Hence BP is the

BC meeting in C

PC, is the point of contact B on CB. chord of contact of the tangents PC and at right angles. Therefore C is a point on the directrix (Drew, ch. i., Prop. VI.). Join BP and bisect the chord in k, determining the direction Ck of the principal axis. From C let fall the perpendicular CS, meeting BP in the focus S, (Drew, ch. i., Prop. IV.). Wherefore the axis coincides with the line SA drawn through S parallel to Ck.

To deduce the projective properties of the curve, choose any two poles O and O', lying on the same tangent line. (24-24). From one of these poles O project two points (9 and 13) of line CP, and from the other O' the correlative points (9 and 13) of line CB. Let the two rays (9) meet in b, and the two rays (13) in a; then ab is the perspective line. From the pole O draw a line parallel to PC intersecting the line ab in ∞, the projection of the point at infinity on CP. The correlative point on CB produced is found by joining O'∞ and producing it to meet CB. But O'∞o is parallel to CB; therefore the points at infinity on CP and CB are correlative, which is a distinctive property of a parabola. The focus and direction of the axis may now be graphically determined as explained in the analytical treatment of the problem.

15. POLE AND POLAR.-If pABCDq (Fig. 37) be a complete quadrangle inscribed in a conic, then it follows from §§ 9, 10 that the points of intersection of pairs of opposite sides as well as the points of intersection of the pairs of tangents at opposite points all lie on one and the same line pq.

The point where the diagonals AC and BD intersect is termed the pole of the line pq, which is called the polar of z.

Upon projecting the point-range pqwv from the centres B and C respectively, it follows from the harmonic properties of the complete quadrangle proved in Art. 2 that the

point v is harmonically separated from z by A and C, and that z and w are harmonically separated by B and D. general terms the polar pq is the locus of

1o. The points of intersection of the opposite pairs of sides of every inscribed quadrangle whose diagonals pass through the pole.

2o. The points of intersection of pairs of tangents whose points of contact lie in a straight line with the pole.

3°. The points harmonically separated from the pole by two points on the curve.

Similarly the pole z is the point traversed

1o. By every ray harmonically separated from the polar pq by two tangents issuing from a point on the polar. Thus, for instance, since the points CzAv are projected from r in harmonic ratio the pencil r)CzAv is harmonic.

2o. By the diagonals of every quadrilateral formed by the four tangents issuing in pairs from any two points of the polar. Thus, for instance, in the quadrilateral mont formed by the tangents issuing from r and s (beyond the Fig.) on pq, the diagonals mn and of meet in the same point z as the lines BD and AC joining the opposite points of contact (Art. 13, Cor. I.).

The harmonic ratio tzoq is projected from r in perspective with the harmonic ratio CzAv and the points tzoq lie in one and the same straight line. Similarly the ratio nzmp is projected from s in perspective with the ratio BzDw and the points nzmp lie in one and the same straight line.

3°. By every line joining the points of contact of pairs of tangents issuing from points on the polar pq.

To find the polar of a point p on the polar of z, draw any pair of transversals pCB and pDA determining the inscribed quadrangle ACBD, whose pairs of opposite sides meet in z and q respectively; then, according to what has already been shewn, the line qz produced will be the polar

of p. Hence the rule that the polars of all points on any line pq traverse the pole z of pq; and, conversely, the poles of all rays traversing the pole z of pq lie upon the polar pq. Thus, for instance, r is the pole of the line of contact CA, and s the pole of BD.

Since the line pz traverses z, its pole must lie on the polar of z or on pq. For a similar reason it must lie on the polar of p or on zq; therefore it coincides with q, the point of intersection of zq and pq. A triangle, such as pqz, of which each vertex is the pole of the opposite side is termed a polar triangle. But p, q, z are the three points in which the three pairs of opposite sides of the complete quadrangle ABCD intersect each other; therefore the pairs of opposite sides of every quadrangle inscribed in a conic intersect in three vertices of a polar triangle. Similarly the three pairs of opposite points or apices mn, ot, and rs of a circumscribed quadrilateral lie upon the three sides of a polar triangle pzq.

16. If a point p (Fig. 37) describe a line pq, its polar qz with respect to a conic will simultaneously describe a pencil of rays projective of the points so described by the point p. In order that two punctuated lines may be projective, one of the other, it is necessary and sufficient that any four points of the one should bear a determinate ratio to the correlative four points of the other. But, as the most general of all ratios is the anharmonic, it follows that two punctuated lines will be projective when four segments of the one are in anharmonic proportion with the correlative four segments of the other.

Now corresponding to any polar triangle pqz (Fig. 37) there is an infinite number of inscribed quadrangles ABCD whose pairs of opposite sides intersect in pq; so that if A and C be chosen as fixed centres of projection from which to project the variable points B on the conic, it follows from Art. 7 that the pencils A and C will be projective,

and therefore any four positions of the point q, determined by means of the pencil A, will be projective of, or in anharmonic ratio with, the corresponding four positions of p, determined by means of pencil C. Hence, if the positions

be projected from any point z, the pencil z will be projective of pencil C, where qz is the polar of p in this case.

DEF. Two points in the same plane are termed reciprocal or conjugate when one of them is situate on the polar of the other with respect to a conic.

Similarly two lines are termed reciprocal or conjugate when one passes through the pole of the other.

M

P

B

Fig. 38.

If A

17. If a plane triangle AMB' (Fig. 38) be inscribed in a conic, every line which is reciprocal of one of the sides AB' cuts the other two sides in points P and P' which are reciprocal of each other. and B' be chosen as centres of projection, and M'B', MA as punctuated lines to describe the conic, the point M or M' will be a double point. Again, the point A' correlative of the point of

contact A will lie on the tangent AA', which is also the polar of the point of contact A, so that A and A' are reciprocal points. Similarly B and B' are reciprocal points. Hence, since three points of the system M, A, B are correlative and reciprocal of the corresponding points M', A', B' of the second system, any fourth ray PP' traversing the centre of perspective z will intersect MA and MB' in points P and P', which are reciprocal and correlative.

For similar reasons, if a triangle ABC (Fig. 39) be cir

cumscribed to a conic, every point reciprocal of one of the vertices B is projected from the other vertices by two reciprocal rays. Thus

of itself. Therefore (§ 7, II.) all the homologous rays of

pencils b and c are reciprocal. 18. If a conic be cut by two

Now project the same range from C instead of p and it will be found that the four rays, or harmonic pencil

F