C, pass through the four points of the curve; hence the four points A, B, C, D are projected from any point in the curve by a harmonic pencil.

Again, the tangent at C intersects itself in C, the tangent at D in d, that at B in b, and that at A in q; and, as already mentioned, the pencil projecting these points from p is harmonic; hence the tangents at ABCD are cut by the tangent at C, or any other tangent to the curve, in four harmonic points.

19. INVOLUTION. We have already shewn (Art. 16) that, if the point p (Fig. 37) describe pq or the polar of the point, the polar of p will determine a series of points q on the line pq, which are projective of the points p. Thus, if p' be any second position of p on the line pq, the corresponding point q' may be found by projection in the same way that the point q was derived from the position of p; that is, by producing the line p'C to meet the curve in B', then producing the line B'z to meet the conic in D', and projecting D' from C upon pq in q', or, more simply, by producing B'A to meet pq in q'. Now let us reverse the operation, assuming q' to be a second position of p, or to belong to the system p instead of the system q; and then proceed as before to find the correlative of q' in the system p by first joining q' to C to meet the conic in D', then drawing D'z to intersect the curve in B', and finally producing B'C to meet the curve in p', the required point of the system q which is correlative of q' in the system p. Thus it will be seen that the points p' and q' are mutually projective and interchangeable. Any series of pairs of points pq which are mutually projective are said to be in involution.

For the sake of clearness we shall represent any pair of points in involution by the same letters, the points of the second system being distinguished by a dash. Thus for Pq, p'q', p"q" we shall substitute aa', bb', cc' respectively,

which being mutually projective constitute the forms aa'be' and a'ab'c projective, and therefore in anharmonic propor

There is a simple mechanical way of evolving this equation. Thus, take any pair of points, such as ab', consisting of one point from each system; next, the pair bc', consisting of the conjugate b of the second point of the first pair, and any point c' of the second system; and lastly, take the pair ca', consisting of the conjugate of c of the second point of the second pair and the conjugate a' of the first point of the first pair.

To form the right-hand member, interchange the points composing the pairs of the left; thus

b'a. c'b. a'c,

and then permute the indices, thus

ba'. cb'. ac'.

Since the points c and c' are mutually projective, we may permute them in equation (1), thus obtaining the equation

As a second example take ab as the first pair, and write down the terms in succession as follows:

ab. b'c. c'a'b'a'. c'b. ac.

∞

If the third point p" or c of the first system pass to infinity on the line pq (Fig. 37), the line drawn from p to C will be parallel to pq; and the correlative of p or c, found in the usual way, will lie at c' or q%, in which case we have

Hence, substituting in equations (1) and (2), we obtain

so that, representing the point in the system q corresponding to infinity in the system p by O, we have

Oa. Oa' Ob. Ob', a constant.

When Od Od', we have

=

Oa. Oa' = Od2.

Evidently the line Oz coincides with the diameter conjugate to that parallel to pq.

The point c' or 。, which we have agreed to term O, is sometimes called the centre of involution of the superposed systems p and q, the points of which are mutually projective.

second line; then, since the two systems aa'bb' and a'ab'b are in anharmonic ratio, the point b taken as an element of the second system is the correlative of b' considered as an element of the first.

21. If any two points a and a' of a double system situated on a conic (Figs. 41 and 42) be mutually projective, all the points of the two systems are mutually projective. Let z be the pole

of the line xy (Figs. 41 and 42), b and b', a and a' correlative points on the curve in perspective from z.

Assume b

and b'as centres of projection from which to project the system of points a and a' respectively. If a and a' be double points, i.e. mutually projective, the pencils b'(aa'b) and b(a'ab') are perspective of the polar xy (§§ 15, 16). Apart from the proof deduced from §§ 15, 16, it will be seen that the rays b'b and bb' of these two pencils are coincident; and, therefore, the pencils are perspective of the same line (Lemma II., § 7), which in this case coincides with the polar of the point z. The pole z is sometimes called the centre and the polar xy the axis of involution. The tangents at any two correlative points, such as bb', meet upon the polar and are mutually projective, i.e. any tangent of the group is cut by the correlative tangents in a double system of points in involution. Given two pairs of correlative points on a conic, such as aa' and bb' (Figs. 41 and 42), we can find the pole z, and thereby the sixth point c' corresponding to any fifth point c.

If it be required to couple a given system of lines in involution, it is necessary to cut the system by a conic or circle and proceed as just explained. If, moreover, the given double system of lines form two pencils of rays in involution issuing from the same point, it is necessary that the given circle should pass through that point (Fig. 43).

DEF. When the line zbb' becomes a tangent at M or N the points band b' coalesce and form what is termed a double point (Fig. 41).

b

2

C

22. The double points of a system in involution are harmonically separated by two conjugate points. Let M and N be the double points (Fig. 41), a and a' the given conjugate points, b and b' a second pair of conjugate points. The opposite sides of the quadrangle ab'a'b intersect in two points x and x' which are harmonically separated by M and N. Hence the pencil b(aMa'N) is harmonic (§§ 15, 16).

Fig. 43.

x

Fig. 44.

y

23. A punctuated line xy (Fig. 44) and a pencil of rays z, such that xy does not traverse z and is projective of the pencil, are in involution when either of two points p and p' on xy is situate on the ray of the pencil which is correlative of the second point. The points p and p' in this case are mutually projective ; and, therefore, the section of the pencil by the line is mutually projective of, or in involution with, the line (§ 20).

COR. I. The polars of all the points of a punctuated line xy with respect to a curve of the second order, which does not meet the line, traverse the pole z of xy and form a pencil of rays in involution with the punctuated line (SS 15, 16).