and A′, B′, C′ are conjugate points two conjugate elements may be interchanged.
8. Any three pairs. A, A′, B, B′, C, C′, of conjugate points are connected by the relations:
| AB′ · BC′ · CA′ | = | AB′ · BC · C′A′ | = | AB · B′C′ · CA′ | = | AB · B′C · C′A′ | = −1. |
| A′B · B′C · C′A | A′B · B′C′ · CA | A′B′ · BC · C′A | A′B′ · BC′ · CA |
These relations readily follow by working out the relations in (7) (above).
§ 78. Involution of a quadrangle.—The sides of any four-point are cut by any line in six points in involution, opposite sides being cut in conjugate points.
Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the line p in the points A, A′, B, B′, C, C′, if further, C1D1 cuts the line A1B1 in C2, and if we project the row A1B1C2C to p once from D1 and once from C1, we get (A′B′, C′C) = (BA, C′C).
Interchanging in the last cross-ratio the letters in each pair we get (A′B′, C′C) = (AB, CC′). Hence by § 77 (7) the points are in involution.
The theorem may also be stated thus:
The three points in which any line cuts the sides of a triangle and the projections, from any point in the plane, of the vertices of the triangle on to the same line are six points in involution.
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| Fig. 31. |
Or again—
The projections from any point on to any line of the six vertices of a four-side are six points in involution, the projections of opposite vertices being conjugate points.
This property gives a simple means to construct, by aid of the straight edge only, in an involution of which two pairs of conjugate points are given, to any point its conjugate.
§ 79. Pencils in Involution.—The theory of involution may at once be extended from the row to the flat and the axial pencil—viz. we say that there is an involution in a flat or in an axial pencil if any line cuts the pencil in an involution of points. An involution in a pencil consists of pairs of conjugate rays or planes; it has two, one or no focal rays (double lines) or planes, but nothing corresponding to a centre.
An involution in a flat pencil contains always one, and in general only one, pair of conjugate rays which are perpendicular to one another. For in two projective flat pencils exist always two corresponding right angles (§ 40).
Each involution in an axial pencil contains in the same manner one pair of conjugate planes at right angles to one another.
As a rule, there exists but one pair of conjugate lines or planes at right angles to each other. But it is possible that there are more, and then there is an infinite number of such pairs. An involution in a flat pencil, in which every ray is perpendicular to its conjugate ray, is said to be circular. That such involution is possible is easily seen thus: if in two concentric flat pencils each ray on one is made to correspond to that ray on the other which is perpendicular to it, then the two pencils are projective, for if we turn the one pencil through a right angle each ray in one coincides with its corresponding ray in the other. But these two projective pencils are in involution.
A circular involution has no focal rays, because no ray in a pencil coincides with the ray perpendicular to it.
§ 80. Every elliptical involution in a row may be considered as a section of a circular involution.
In an elliptical involution any two segments AA′ and BB′ lie partly within and partly without each other (fig. 32). Hence two circles described on AA′ and BB′ as diameters will intersect in two points E and E′. The line EE′ cuts the base of the involution at a point O, which has the property that OA . OA′ = OB · OB′, for each is equal to OE . OE′. The point O is therefore the centre of the involution. If we wish to construct to any point C the conjugate point C′, we may draw the circle through CEE′. This will cut the base in the required point C′ for OC · OC′ = OA · OA′. But EC and EC′ are at right angles. Hence the involution which is obtained by joining E or E′ to the points in the given involution is circular. This may also be expressed thus:
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| Fig. 32. |
Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle.
At the same time the following problem has been solved:
To determine the centre and also the point corresponding to any given point in an elliptical involution of which two pairs of conjugate points are given.
§ 81. Involution Range on a Conic.—By the aid of § 53, the points on a conic may be made to correspond to those on a line, so that the row of points on the conic is projective to a row of points on a line. We may also have two projective rows on the same conic, and these will be in involution as soon as one point on the conic has the same point corresponding to it all the same to whatever row it belongs. An involution of points on a conic will have the property (as follows from its definition, and from § 53) that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil, and that a fixed tangent is cut by the tangents at conjugate points on the conic in points which are again conjugate points of an involution on the fixed tangent. For such involution on a conic the following theorem holds:
The lines which join corresponding points in an involution on a conic all pass through a fixed point; and reciprocally, the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line.
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| Fig. 33 |
We prove the first part only. The involution is determined by two pairs of conjugate points, say by A, A′ and B, B′ (fig. 33). Let AA′ and BB′ meet in P. If we join the points in involution to any point on the conic, and the conjugate points to another point on the conic, we obtain two projective pencils. We take A and A′ as centres of these pencils, so that the pencils A(A′BB′) and A′(AB′B) are projective, and in perspective position, because AA′ corresponds to A′A. Hence corresponding rays meet in a line, of which two points are found by joining AB′ to A′B and AB to A′B′. It follows that the axis of perspective is the polar of the point P, where AA′ and BB′ meet. If we now wish to construct to any other point C on the conic the corresponding point C′, we join C to A′ and the point where this line cuts p to A. The latter line cuts the conic again in C′. But we know from the theory of pole and polar that the line CC′ passes through P. The point of concurrence is called the “pole of the involution,” and the line of collinearity of the meets is called the “axis of the involution.”
Involution Determined by a Conic on a Line.—Foci
§ 82. The polars, with regard to a conic, of points in a row p form a pencil P projective to the row (§ 66). This pencil cuts the base of the row p in a projective row.
If A is a point in the given row, A′ the point where the polar of A cuts p, then A and A′ will be corresponding points. If we take A′ a point in the first row, then the polar of A′ will pass through A, so that A corresponds to A′—in other words, the rows are in involution. The conjugate points in this involution are conjugate points with regard to the conic. Conjugate points coincide only if the polar of a point A passes through A—that is, if A lies on the conic. Hence—
A conic determines on every line in its plane an involution, in which those points are conjugate which are also conjugate with regard to the conic.
If the line cuts the conic the involution is hyperbolic, the points of intersection being the foci.
If the line touches the conic the involution is parabolic, the two foci coinciding at the point of contact.
If the line does not cut the conic the involution is elliptic, having no foci.
