Page images
PDF
EPUB
[graphic]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

curve of the second class having the same point of contact. In other words, the curve of second order is a curve of second class, and vice versa. Hence the important theoremsEvery curve of second order is Every curve of second class is a a curve of second class. curve of second order. The curves of second order and of second class, having thus been proved to be identical, shall henceforth be called by the common name of Conics.

For these curves hold, therefore, all properties which have been proved for curves of second order or of second class. We may therefore now state Pascal's and Brianchon's theorem thusPascal's Theorem.-If a hexagon be inscribed in a conic, then the intersections of opposite sides lie in a line.

Brianchon's Theorem.-If a hexagon be circumscribed about a conic, then the diagonals forming opposite centres meet in a point. § 57. If we suppose in fig. 21 that the point D together with the tangent d moves along the curve, whilst A, B, C and their tangents a, b, cremain fixed, then the ray DA will describe a pencil about A, the point Q a projective row on the fixed line BC, the point F the row b, and the ray EF a pencil about E. But EF passes always through Q. Hence the pencil described by AD is projective to the pencil described by EF, and therefore to the row described by F on b. At the same time the line BD describes a pencil about B projective to that described by AD (853). Therefore the pencil BD and the row F on b are projective. Hence

If on a conic a point A be taken and the tangent a at this point, then the cross-ratio of the four rays which join A to any four points on the curve is equal to the cross-ratio of the points in which the tangents at these points cut the tangent at A.

$58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to § 43We mention only a few of the more important ones.

The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone of the second order.

The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone of the second class.

Cones of second order and cones of second class are identical.
Every plane cuts a cone of the second order in a conic.

A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c.

Pascal's Theorem.-If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane.

Brianchon's Theorem.-If a solid angle of six edges be circumscribed about a cone of the second order, then the planes through opposite edges meet in a line.

Each of the other theorems about conics may be stated for cones of the second order.

859. Projective Definitions of the Conics.-We now consider the shape of the conics. We know that any line in the plane of the conic, and hence that the line at infinity, either has no point in common with the curve, or one (counting for two coincident points) or two distinct points. If the line at infinity has no point on the curve the latter is altogether finite, and is called an Ellipse (fig. 21). If the line at infinity has only one point in common with the conic, the latter extends to infinity, and has the line at infinity a tangent. It is called a Parabola (fig. 22). If, lastly, the line at infinity cuts the curve in two points, it

consists of two separate parts which each extend in two branches to the points at infinity where they meet. The curve is in this case called an Hyperbola (see fig. 20). The tangents at the two points at infinity are finite because the line at infinity is not a tangent. They are called Asymptotes. The branches of the hyper- p bola approach these lines indefinitely as a point on the curves moves to infinity.

§ 60. That the circle belongs to the curves of the second order is seen at once if we state in

P

FIG. 22.

a slightly different form the theorem that in a circle all angles at the circumference standing upon the same arc are equal. If two points S, S on a circle be joined to any other two points A and B on the circle, then the angle included by the rays SA and S,B is equal to that between the rays SA and SB, so that as A moves along the circumference the rays SA and SA describe equal and therefore projective pencils. The circle can thus be generated by two projective pencils, and is a curve of the second order.

If we join a point in space to all points on a circle, we get a (circular) cone of the second order (§ 43). Every plane section of this cone is a conic. This conic will be an ellipse, a parabola, or an hyperbola, according as the line at infinity in the plane has no, one or two points in common with the conic in which the plane at infinity cuts the cone. It follows that our curves of second order may be obtained as sections of a circular cone, and that they are identical with the "Conic Sections" of the Greek mathematicians.

$61. Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar. Hence the theorem

The tangents to a parabola cut each other proportionally.

POLE AND POLAR

§ 62. We return once again to fig. 21, which we obtained in § 55. If a four-side be circumscribed about and a four-point inscribed in a conic, so that the vertices of the second are the points of contact of the sides of the first, then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other.

Such a triangle will be called a polar-triangle of the conic, so that PQR in fig. 21 is a polar-triangle. It has the property that on the side opposite P meet the tangents at A and B, and also those at C and D. From the harmonic properties of four-points and four-sides it follows further that the points L, M, where it cuts the lines AB and CD, are harmonic conjugates with regard to AB and CD respectively.

If the point P is given, and we draw a line through it, cutting the conic in A and B, then the point Q harmonic conjugate to P with regard to AB, and the point H where the tangents at A and B meet, are determined. But they lie both on p, and therefore this line is determined. If we now draw a second line through P, cutting the conic in C and D, then the point M harmonic conjugate to P with regard to CD, and the point G where the tangents at C and D meet, must also lie on p. As the first line through P already determines p, the second may be any line through P. Now every two lines through P determine a four-point ABCD on the conic, and therefore a polar-triangle which has one vertex at P and its opposite side at p. This result, together with its reciprocal, gives the theoremsAll polar-triangles which have one vertex in common have also the opposite side in common.

All polar-triangles which have one side in common have also the opposite vertex in common.

63. To any point P in the plane of, but not on, a conic corresponds thus one line p as the side opposite to P in all polar-triangles which have one vertex at P, and reciprocally to every line p corresponds one point P as the vertex opposite to p in all triangles which have p as one side.

We call the line p the polar of P, and the point P the pole of the line p with regard to the conic.

If a point lies on the conic, we call the tangent at that point its polar; and reciprocally we call the point of contact the pole of tangent.

§64. From these definitions and former results followThe polar of any point P not on the conic is a line p, which has the following properties:

1. On every line through P which cuts the conic, the polar of P contains the harmonic conjugate of P with regard to those points on the conic.

2. If tangents can be drawn from P, their points of contact lie on p.

3. Tangents drawn at the points where any line through P cuts the conic meet on p; and conversely,

4. If from any point on p. tangents be drawn, their points of contact will lie in a line with P. 5. Any four-point on the conic which has one diagonal point at P has the other two lying on p.

The pole of any line p not a tangent to the conic is a point P, which has the following properties:

1. Of all lines through a point on from which two tangents may be drawn to the conic, the pole P contains the line which is harmonic conjugate to p, with regard to the two tangents.

2. If p cuts the conic, the tangents at the intersections meet at P.

3. The point of contact of tangents drawn from any point on p to the conic lie in a line with P; and conversely,

4. Tangents drawn at points where any line through P cuts the conic meet on p.

5. Any four-side circumscribed about a conic which has one diagonal on p has the other two meeting at P.

The truth of 2 follows from 1. If T be a point where p cuts the conic, then one of the points where PT cuts the conic, and which are harmonic conjugates with regard to PT, coincides with T; hence the other does-that is, PT touches the curve at T.

That 4 is true follows thus: If we draw from a point H on the polar one tangent a to the conic, join its point of contact A to the pole P, determine the second point of intersection B of this line with the conic, and draw the tangent at B, it will pass through H, and will therefore be the second tangent which may be drawn from H to the curve.

65. The second property of the polar or pole gives rise to the

theorem

From a point in the plane of a conic, two, one or no tangents may be drawn to the conic, according as its polar has two, one, or no points in common with

the curve.

A line in the plane of a conic has two, one or no points in common with the conic, according as two, one or no tangents can be drawn from its pole to the conic.

Of any point in the plane of a conic we say that it was without, on or within the curve according as two, one or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.

The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straight-edge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic (fig. 22).

These constructions also solve the problem

From a point without a conic, to draw the two tangents to the conic by aid of the straight-edge only.

For we need only draw the polar of the point in order to find the points of contact.

$66. The property of a polar-triangle may now be stated thus→→→ In a polar-triangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.

\B

If P is one vertex of a polar-triangle, then the other vertices, Q and R, lie on the polar p of P. One of these vertices we may choose arbitrarily. For if from any point Q on the polar a secant be drawn cutting the conic in A and D (fig. 23), and if the lines joining these points to P cut the conic again at B and C, then the line BC will pass through Q. Hence P and Q are two of the vertices on the polar-triangle which is determined by the fourpoint ABCD. The third vertex R lies also on the line p. It follows, therefore, also

If Q is a point on the polar of P, then P is a point on the polar of Q; and reciprocally,

If q is a line through the pole of p, then p is a line through the pole of q.

thus

FIG. 23.

This is a very important theorem. It may also be stated If a point moves along a line describing a row, its polar turns about the pole of the line describing a pencil. This pencil is projective to the row, so that the cross-ratio of four poles in a row equals the cross-ratio of its four polars, which pass through the pole of the row.

To prove the last part, let us suppose that P, A and B in-fig. 23 remain fixed, whilst Q moves along the polar p of P.This will make CD turn about P and move R along p, whilst QD and RD describe projective pencils about A and B. Hence Q and R describe projective rows, and hence PR, which is the polar of Q, describes a pencil projective to either.

$67. Two points, of which one, and therefore each, lies on the polar of the other, are said to be conjugate with regard to the conic: and two lines, of which one, and therefore each, passes through the pole of the other, are said to be conjugate with regard to the conic. Hence all points conjugate to a point P lie on the polar of P; all lines conjugate to a line p pass through the pole of p.

If the line joining two conjugate poles cuts the conic, then the poles are harmonic conjugates with regard to the points of intersection; hence one lies within the other without the conic, and all points conjugate to a point within a conic lie without it.

Of a polar-triangle any two vertices are conjugate poles, any two sides conjugate lines. If, therefore, one side cuts a conic, then one of the two vertices which lie on this side is within and the other without the conic. The vertex opposite this side lies also without, for it is the pole of a line which cuts the curve. In this case therefore one vertex lies within, the other two without. If, on the other hand, we begin with a side which does not cut the conic, then its pole lies within and the other vertices without. Hence→→ Every polar-triangle has one and only one vertex within the conic. We add, without a proof, the theorem

The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.

§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one

four-point which is inscribed in both, and therefore one polar-triangle common to both.

Theorem.-Two conics which intersect in four points have always one and only one common polar-triangle; and reciprocally, Two conics which have four common tangents have always one and only one common polar-triangle.

DIAMETERS AND AXES OF CONICS

§ 69. Diameters.-The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity, it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems.

The middle points of parallel chords of a conic lie in a line-viz. on the polar to the point at infinity on the parallel chords. This line is called a diameter.

The polar of every point at infinity is a diameter.

The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.

All diameters pass through a common point, the pole of the line at infinity.

All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity. In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.

A centre of a conic bisects every chord through it.

The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.

The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also

From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).

To construct a diameter of a conic, draw two parallel chords and join their middle points.

To find the centre of a conic, draw two diameters; their intersection will be the centre.

§70. Conjugate Diameters.-A polar-triangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.

Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.

Further

[ocr errors]

Every parallelogram inscribed in a conic has its sides parallel to two conjugale diameters; and

Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.

This will be seen by considering the parallelogram in the first case as an inscribed four-point, in the other as a circumscribed four-side, and determining in each case the corresponding polartriangle. The first may also be enunciated thus

The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugale diameters.

§ 71. If every diameter is perpendicular to its conjugate the conic is a circle.

For the lines which join the ends of a diameter to any point on the curve include a right angle.

A conic which has more than one pair of conjugate diameters al right angles to each other is a circle.

Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right angles to each other, CC' and DD' a second pair. If we draw

D'

B

A

through the end point A of one diameter a chord AP parallel to DD', and join P to A', then PA and PA' are, according to § 70, parallel to two conjugate diameters. But PA is parallel to DD', hence PA' is parallel to CC', and therefore PA and PA' are perpendicular. If we further draw the tangents to the conic, at A and A', these will be perpendicular to AA'. they being parallel to the conjugate diameter BB'. We know thus five points on the conic, viz. the points A and A' with their tangents, and the point P. Through these a FIG. 24. circle may be drawn having AA' as diameter; and as through five points one conic only can be drawn, this circle must coincide with the given conic.

§72. Axes.-Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.

In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection.

If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig. 25). Each of the semicircles in which it is divided by AB will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the conic have thus four points A, B, C, D, and therefore one polar-triangle, in common (§ 68). Of this the centre is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The

other two sides are conjugate diameters of both, hence perpendicular to each other. This gives

An ellipse as well as an hyperbola has one pair of

axes.

This reasoning shows at the same time how to construct the axis of an ellipse an hyperbola.

or

FIG. 25.

of if we define an axis as a diameter perpendicular to the chords A parabola has one axis, which it bisects. It is easily constructed. The line which bisects. any two parallel chords is a diameter. Chords perpendicular to it will be bisected by a parallel diameter, and this is the axis. stated the first part of the right-hand theorem in §64 may be thus: any two conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic. If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence

Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.

As the axes are conjugate diameters at right angles to one another, it follows (23)

The axes of an hyperbola bisect the angles between the asymptotes. Let O be the centre of the hyperbola (fig. 26), any secant which cuts the hyperbola in C,D and the asymptotes in E.F, then the line OM which bisects the chord CD is a diameter conjugate to the

FIG. 26.

D

diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point M therefore bisects EF. But by construction, M bisects CD. It follows that DF = EC, and EDCF; or

On any secant of an hyperbola the segments between the curve and the asymptotes are equal.

If the chord is changed into a tangent, this gives

The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.

The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.

§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. Hence

The segment PM which joins the middle point M of a chord of a parabola to the pole P of the chord is bisected by the parabola at A.

$75. Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the

1

hyperbola. But in such a quadrilateral the intersections of the | which, according to § 15, equals (AB, D'D); so that the equation diagonals and the points of contact of opposite sides lie in a line becomes

M

(54). If therefore DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE will meet on the line which joins the points of contact of the asymptotes, that is, on the line at infinity; hence they are parallel. From this the following theorem is a simple deduction:

All triangles formed by a tangent and the asymptotes of an hyperbola are equal in

area.

If we draw at a point P (fig. 28) on an hyperbola a tangent, the part HK between the asymptotes FIG. 27. is bisected at P. The parallelogram PQOQ' formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique

FIG. 28.

axes of co-ordinates, the lines OQ and QP will be the co-ordinates of P, and will satisfy the equation xy-const. a'. For the asymptotes as axes of co-ordinates the equation of the hyperbola is xy-const.

A

B

INVOLUTION

$76. If we have two projective rows, ABC on u and A'B'C' on u', and place their bases on the same line, then cach point in this line counts twice, once as a point in the row u and once as a point in the row u'. In fig. 29 we denote the points as points in the one row by letters above the line A, B, C and as points in the second row by A', B', 'C'... below the line. Let now A and B' be the same point, then to A will correspond a point A' in the second, FIG. 29. and to B' a point B in the first row. In general these points A' and B will be different. It may, however, happen that they coincide. Then the correspondence is a peculiar one, as the following theorem

shows:

If two projective rows lie on the same base, and if it happens that to one point in the base the same point corresponds, whether we consider the point as belonging to the first or to the second row, then the same will happen for every point in the base-that is to say, to every point in the line corresponds the same point in the first as in the second row. In order to determine the correspondence, we may assume three pairs of corresponding points in two projective rows.

A

D

B

+

[blocks in formation]

C + D'

FIG. 30.

Let then

A', B', C', in fig. 30, correspond to A, B, C, so that A and B', and also B and A', denote the same point. Let us further denote the point C' when considered as a point in the first row by D; then it is to be proved that the point D', which corresponds to D, is the same point as C. We know that the cross-ratio of four points is equal to that of the corresponding row. Hence

(AB, CD) = (A'B', C'D')

but replacing the dashed letters by those undashed ones which denote the same points, the second cross-ratic equais (BA DD

(AB, CD) = (AB, D'D).

This requires that C and D' coincide.

877. Two projective rows on the same base, which have the above property, that to every point, whether it be considered as a point in the one or in the other row, corresponds the same point, are said to be in involution, or to form an involution of points on the line.

We mention, but without proving it, that any two projective rows may be placed so as to form an involution.

An involution may be said to consist of a row of pairs of points, to every point A corresponding a point A', and to A' again the point A. These points are said to be conjugate, or, better, one point is termed the "mate" of the other.

From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties:

1. The cross-ratio of four points equals that of the four conjugate points.

or

2. If we call a point which coincides with its mate a "focus double point" of the involution, we may say: An involution has either two foci, or one, or none, and is called respectively a hyperbolic, parabolic or elliptic involution (§ 34)..

3. In a hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.

For if A, A' be two conjugate points, F1, F, the two foci, then to the points F1, F2, A, A' in the one row correspond the points F1, F2, A', A in the other, cach focus corresponding to itself. Hence (FF, AA') = (FIFA'A)—that is, we may interchange the two points AA' without altering the value of the cross-ratio, which is the characteristic property of harmonic conjugates (§ 18).

"

4. The point conjugate to the point at infinity is called the centre of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity.

In an hyperbolic involution the centre is the middle point between the foci.

5. The product of the distances of two conjugate points A, A' from the centre O is constant: OA. OA'=c.

For let A, A and B, B' be two pairs of conjugate points, O the centre, I the point at infinity, then

[blocks in formation]

Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic.

If c=o, OF=o, and the two foci both coincide with the centre. If c is negative, c becomes imaginary, and there are no foci. Hence we may write

In an hyperbolic involution, OA . OA'=k2, In a parabolic involution, OA. OA'=0, In an elliptic involution, OA. OA'=-k2. From these expressions it follows that conjugate points A, A' in an hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides with the centre.

In the first case, for instance, OA. OA' is positive; hence OA and OA' have the same sign.

It also follows that two segments, AA' and BB', between pairs of conjugate points have the following positions: in an hyperbolic' involution they lie either one altogether within or altogether without each other; in a parabolic involution they have one point in common; and in an elliptic involution they overlap, each being partly within and partly without the other.

Proof. We have OA. OA'=OB. OB' = k2 in case of an hyperbolic involution. Let A and B be the points in each pair which are nearer to the centre O. If now A, A' and B, B' lie on the same side of O, and if B is nearer to O than A, so that OB<OA, then OB'>OA', hence B' lies farther away from O than A', or the segment AA' lies within BB'. And so on for the other cases. 6. An involution is determined

(a) By two pairs of conjugate points. Hence also
(8) By one pair of conjugate points and the centre;
(7) By the two foci;

(8) By one focus and one pair of conjugate points;
(e) By one focus and the centre.

7. The condition that A, B, C and A', B', C' may form an involution may be written in one of the forms

[blocks in formation]

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' A'B. B'C.C'A ̄A'B. B'C'.CA ̄A'B'.BC.C'AA'B'. BC'. CA-1. 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 A,B,C,D, (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, CD, cuts the line A,B in C, and if we project the row A,B,C,C top once from D, 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.

Or again

[blocks in formation]

the property that there are two Every elliptical involution has which any two conjugate points definite points in the plane from are seen under a right angle. At the same time the follow

ing problem has been solved: To determine the centre and also the point corresponding

FIG. 32.

B

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

The projections from any point on to any line of the six vertices from its definition, and from § 53) that the lines which join conjugate

[blocks in formation]

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

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.

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 A A' and BB meet in P. If we P 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'B B') and A'(AB'B) are projective, and in perspective posi tion, because AA' corresponds to

A'A.

B

FIG. 33.

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."

form

INVOLUTION DETERMINED BY A CONIC ON A LINE.-FOCI §82. The polars, with regard to a conic, of points in a row p pencil P projective to the row (§ 66). This pencil cuts the base of the row p in a projective row.

a

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.

[blocks in formation]
« ՆախորդըՇարունակել »