*134.* If a conic of the system should go through the fixed point, it is clear that the two tangents would coincide and indicate a double ray of the involution. The theorem, therefore, follows:

_Two conics or none may be drawn through a fixed point to be tangent to four fixed lines._

*135. Double correspondence.* It further appears that two projective pencils of rays which have the same center are in involution if two pairs of rays correspond to each other doubly. From this it is clear that we might have deemed six rays in involution as six rays which pa.s.s through a point and also through six points in involution. While this would have been entirely in accord with the treatment which was given the corresponding problem in the theory of harmonic points and lines, it is more satisfactory, from an aesthetic point of view, to build the theory of lines in involution on its own base. The student can show, by methods entirely a.n.a.logous to those used in the second chapter, that involution is a projective property; that is, six rays in involution are cut by any transversal in six points in involution.

*136. Pencils of rays of the second order in involution.* We may also extend the notion of involution to pencils of rays of the second order.

Thus, _the tangents to a conic are in involution when they are corresponding rays of two protective pencils of the second order superposed upon the same conic, and when they correspond to each other doubly._ We have then the theorem:

*137.* _The intersections of corresponding rays of a pencil of the second order in involution are all on a straight line __u__, and the intersection of any two tangents __ab__, when joined to the intersection of the corresponding tangents __a'b'__, gives a line which pa.s.ses through a fixed point __U__, the pole of the line __u__ with respect to the conic._

*138. Involution of rays determined by a conic.* We have seen in the theory of poles and polars (-- 103) that if a point _P_ moves along a line _m_, then the polar of _P_ revolves about a point. This pencil cuts out on _m_ another point-row _P'_, projective also to _P_. Since the polar of _P_ pa.s.ses through _P'_, the polar of _P'_ also pa.s.ses through _P_, so that the correspondence between _P_ and _P'_ is double. The two point-rows are therefore in involution, and the double points, if any exist, are the points where the line _m_ meets the conic. A similar involution of rays may be found at any point in the plane, corresponding rays pa.s.sing each through the pole of the other. We have called such points and rays _conjugate_ with respect to the conic (-- 100). We may then state the following important theorem:

*139.* _A conic determines on every line in its plane an involution of points, corresponding points in the involution __ being conjugate with respect to the conic. The double points, if any exist, are the points where the line meets the conic._

*140.* The dual theorem reads: _A conic determines at every point in the plane an involution of rays, corresponding rays being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic._

PROBLEMS

1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pa.s.s through a fixed point.

2. Two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point. Show that the lines joining the two points where the lines meet the conic again all pa.s.s through a fixed point.

3. Four lines divide the plane into a certain number of regions.

Determine for each region whether two conics or none may be drawn to pa.s.s through points of it and also to be tangent to the four lines.

4. If a variable quadrangle move in such a way as always to remain inscribed in a fixed conic, while three of its sides turn each around one of three fixed collinear points, then the fourth will also turn around a fourth fixed point collinear with the other three.

5. State and prove the dual of problem 4.

6. Extend problem 4 as follows: If a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic, while all its sides but one pa.s.s through as many fixed collinear points, then the last side will also pa.s.s through a fixed point collinear with the others.

7. If a triangle _QRS_ be inscribed in a conic, and if a transversal _s_ meet two of its sides in _A_ and _A'_, the third side and the tangent at the opposite vertex in _B_ and _B'_, and the conic itself in _C_ and _C'_, then _AA'_, _BB'_, _CC'_ are three pairs of points in an involution.

8. Use the last exercise to solve the problem: Given five points, _Q_, _R_, _S_, _C_, _C'_, on a conic, to draw the tangent at any one of them.

9. State and prove the dual of problem 7 and use it to prove the dual of problem 8.

10. If a transversal cut two tangents to a conic in _B_ and _B'_, their chord of contact in _A_, and the conic itself in _P_ and _P'_, then the point _A_ is a double point of the involution determined by _BB'_ and _PP'_.

11. State and prove the dual of problem 10.

12. If a variable conic pa.s.s through two given points, _P_ and _P'_, and if it be tangent to two given lines, the chord of contact of these two tangents will always pa.s.s through a fixed point on _PP'_.

13. Use the last theorem to solve the problem: Given four points, _P_, _P'_, _Q_, _S_, on a conic, and the tangent at one of them, _Q_, to draw the tangent at any one of the other points, _S_.

14. Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve and by the asymptotes respectively have the same middle point.

15. In a triangle circ.u.mscribed about a conic, any side is divided harmonically by its point of contact and the point where it meets the chord joining the points of contact of the other two sides.

CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS

[Figure 39]

FIG. 39

*141. Introduction of infinite point; center of involution.* We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called the _center_ of the involution. Since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points, the center is midway between the double points when they exist. To construct the center (Fig. 39) we draw as usual through _A_ and _A'_ any two rays and cut them by a line parallel to _AA'_ in the points _K_ and _M_. Join these points to _B_ and _B'_, thus determining on _AK_ and _AN_ the points _L_ and _N_. _LN_ meets _AA'_ in the center _O_ of the involution.

*142. Fundamental metrical theorem.* From the figure we see that the triangles _OLB'_ and _PLM_ are similar, _P_ being the intersection of KM and LN. Also the triangles _KPN_ and _BON_ are similar. We thus have

_OB : PK = ON : PN_

and

_OB' : PM = OL : PL;_