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and E, F the intersections of tangents at opposite vertices. The curve of the second class having the same point of contact. In four points P, Q, E, F lic therefore in a line. The quadrilateral other words, the curve of second order is a curve of second class, ACBD gives us in the same way the four points Q, R, G, H in a line, and vice versa. Hence the important theoremsa and the quadrilateral ABDG a line containing the four points R, P, Every curve of second order is Every curve of second class is a I, K. These three lines form a triangle PQR. She a curve of second class. Susis

curve of second order. The relation between the points and lines in this figure may be The curves of second order and of second class, having thus been expressed more clearly if we consider ABCD as a four-point inscribed proved to be identical, shall henceforth be called by the common in a conic, and the tangents at these points as a four-side circumscribed name of Conics.iggers about it,

- viz. it will be seen that P, Q, R are the diagonal points For these curves hold, therefore, all properties which have been of the four-point ABCD, whilst the sides of the triangle PQR are proved for curves of second order or of second class. We may the diagonals of the circumscribing four-side. Hence the theorem- therefore now state Pascal's and Brianchon's theorem thusod Any four point on a curse of ihe second order and the four-side Pascal's Theorem. If a hexagon be inscribed in a conic, then formed by the tangents at these points stand in this relation that the the intersections of opposite sides lie in a line. diagonal points of the four-point lie in the diagonals of the four-side. Brianchon's Theorem.-If a hexagon be circumscribed about a And conversely,

conic, then the diagonals forming opposite centres meet in a point. If a four-point and a circumscribed four-side stand in the above $57. If we suppose in fig. 21 that the point D together with the relation, then a curve of the second order may be described which passes tangent d moves along the curve, whilst A, B, C and their tangents through the four points and touches there the four sides of these figures. a, b, cremain fixed, then the ray DA will describe a pencil about

That the last part of the theorem is true follows from the fact A, the point Q a projective row on the fixed line BC, the point F that the four points A, B, C, D and the linee, as tangent at A, deter- 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 6. At the same time the line BD describes a pencil about B projective to that described by AD ($ 53). 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 ike curve is equal lo the cross-ratio of the points in which the tangents al these points cut the tangent of A.

58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to $43. We 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. Tina

A cone of second order is uniquely determined by five of its edges amoodle adott

or by five of its tangent planes, or by four edges and the tangent plane Flats!

at one of them, &c. &c. 09 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 fnode 10

are three lines in a plane. A

10. Song

Brianchon's Theorem.-- If a solid angle of six edges be circumமாயா|Paa

Podle scribed about a cone of the second order, then the planes through

opposite edges meet in a line. 10 B

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

$ 59. Projeclive 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 17

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). II, lastly, the line at infinity cuts the
curve in two points, it Tot
consists of two separate to Indt 23
parts which each extend out the line

in two branches to the DO TRIGA K

points at infinity where
FIG. 21.

they meet. The curve is A
in this case called an

Hyperbola (see fig. 20). mine a curve of the second order, and the tangents to this curve at the tangents at the

Nude the other points B, C, D are given by the construction which leads two points at infinity to fig. 21.

finite because the The theorem reciprocal to the last is

line at infinity is not

al Any four-side circumscribed about a curve of second class and the a

tangent. They are four-point formed by the points of contact stand in this relation that the called Å sym plotes. The diagonals of the four-side pass through the diagonal points of the branches of the hyper. p.

If a four-side and an inscribed four-point stand in the above relation, indefinitely as a point on then a curve of the second class may be described which touches the sides the curves moves to inof the four side at the points of the four-point.

finity. $ 56. The four-point and the four-side in the two reciprocal $ 60. That the circle

bos theorems are alike. Hence if we have a four-point ABCD and a bclongs to the curves of

PX four-side abcd related in the manner described, then not only may the second order is scen

fra FIG. 22. a curve of the second order be drawn, but also a curve of the second at once if we state in class, which both touch the lines a, b, c, d at the points A, B, C, D. a slightly different form the theorem that in a circle all angles at The curve of second order is already more than determined by the the circumference standing

upon the same arc are equal. If two points A, B, C and the tangents a, b, c at A, B and C. The point D points S, S, on a circle be joined to any other two points A and B may therefore be any point on this curve, and d any tangent to the on the circle, then the angle included by the rays SIA and SB is curve. On the other hand the curve of the second class is more equal to that between the rays SA and S,B, so that as A moves than determined by the three tangents a, b, c and their points of along the circumference the rays S.A and SA describe equal and contact A, B, C. so that d is any tangent to this curve. It follows therefore projective

pencils. The circle can

thus be generated by that every tangent to the curve of second order is a tangent of a two projective pencils, and is a curve of the second order.

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If we join a point in space to all points on a circle, we get a (circular) $65. The second property of the polar or pole gives rise to the cone of the second order (6 43).

Every plane section of this cone is a theoremconic. This conic will be an ellipse, a parabola, or an hyperbola, From a point in the plane of a A line in the plane of a conic according as the line at infinity in the plane has no, one or two points conic, two, one or no tangents has two, one or no points in in common with the conic in which the plane at infinity cuts the may be drawn to the conic, common with the conic, accordcone. It follows that our curves of second order may be obtained according as its polar has two, ing as two, one or no tangents as sections of a circular cone, and that they are identical with the one, or no points in common with can be drawn from its pole to the Conic Sections" of the Greek mathematicians.

the curve.

conic. $61. Any two tangents to a parabola are cut by all others in Of any point in the plane of a conic we say that it was without, projective rows; but the line at infinity being one of the tangents, on or within the curve according as two, one or no tangents to the the points at infinity on the rows are corresponding points, and the curve pass through it. The points on the conic separate those within rows therefore similar. Hence the theorem

the conic from those without. That this is true for a circle is known The langents to a parabola cut each other proportionally.

from elementary geometry.

That it also holds other conics POLE AND POLAR

follows from the fact that every conic may be considered as the

projection of a circle, which will be proved later on. $ 62. We return once again to fig. 21, which we obtained in $55. The fifth property of pole and polar stated in 64 shows how

if a four-side be circumscribed about and a four-point inscribed to find the polar of any point and the pole of any line by aid of the in a conic, so that the vertices of the second are the points of contact straight-edge only. Practically it is often convenient to draw three of the sides of the first, then the triangle formed by the diagonals secants through the pole, and to determine only one of the diagona! of the first is the same as that formed by the diagonal points of the points for two of the four-points formed by pairs of these lines and other.

the conic (fig. 22). Such a triangle will be called a polar-triangle of the conic, so that These constructions also solve the problemPQR in fig. 21 is a polar-triangle. It has the property that on the From a point without a conic, to draw the two tangents to the side Ropposite P meet the tangents at A and B, and also those at C conic by aid of the straight-edge only: and b. From the harmonic properties of four-points and four-sides For we need only draw the polar of the point in order to find the it follows further that the points L, M, where it cuts the lines AB points of contact. and CD, are harmonic conjugates with regard to AB and CD $ 66. The property of a polar-triangle may now be stated thusrespectively

In a polar-triangle each side is the polar

of the opposite vertex, If the point P is given, and we draw a line through it, cutting and cach vertex is the pole of the opposite side. the conic in A and B, then the point Q harmonic conjugate to P 11 P is one vertex of a polar-triangle, then the other vertices, Q 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

and R, lie on the polar p of P. One of these vertices we may choose

For if from line is determined. If we now draw a second line through P, cutting any point Q on the polar IB the conic in C and D, then the point M harmonic conjugate to P a secant be drawn cutting with regard to CD, and the point G where the tangents at C and Dthe conic in A and D (fig. meet, must also lie on p. As the first line through P already deter-23), and if the lines joining mines p, the second may be any line through P. Now every two these points to P cut the lines through P determine a four-point ABCD on the conic, and conic again at B and C, therefore a polar-triangle which has one vertex at P and its opposite then the line BC will pass side at p.' This result, together with its reciprocal, gives the through Q. Hence P and theorems

Q are two of the vertices All polar-triangles which have one vertex in common have also the on the polar-triangle which opposile side in common.

is determined by the fourAll polar-triangles which have one side in common have also the point ABCD. The third opposite vertex in common.

vertex R lies also on the 63. To any point P in the plane of, but not on, a conic corresponds line p. It follows, therefore, thus one line p as the side opposite to P in all polar-triangles which also have one vertex at P, and reciprocally to every line p corresponds IQ is a point on the polar one point P as the vertex opposite to pin all triangles which have pof P. Then P is a point on the as one side.

polar of Q; and reciprocally. We call the line p the polar of P, and the point P the pole of the If q is a line through the line p with regard to the conic.

pole of p, then p is a line it'a point lies on the conic, we call the tangent at that point its through the pole of a polar; and reciprocally we call the point of contact the pole of This is a very important theorem. It may also be stated tangent. 64. From these definitions and former results follow

Ithus

If a point moves along a line describing a row, ils polar turns about The polar of any point P not The pole of any line p not a The pole of the line describing a pencil. on the conic is a line P, which has tangent to the conic is a point This pencil is projective to the row, so that the cross-ratio of four the following properties : P, which has the following pro- poles in a row equals the cross-ratio of its four polars, which pass perties:

through the pole of the row. 1. On every line through P 1. Of all lines through a point To prove the last part, let us suppose that P, A and B in-fig. 23 which

cuts the conic, the polar on P from which two tangents remain fixed, whilst moves along the polar p of P. This will of P contains the harmonic con- may be drawn to the conic, the make CD turn about P and move R along, pwhilst QD and RD jugate of P with regard to those pole P contains the line which is describe projective pencils about A and B. Hence Q and R describe points on the conic.

harmonic conjugate to ø, with projective rows, and hence PR, which is the polar of Q, describes a

regard to the two tangents. pencil projective to either. 2. !f tangents can be drawn 2. II p cuts the conic, the $ 67. Two points, of which one, and therefore each, lies on the from P, their points of contact lie tangents at the intersections polar of the other, are said to be conjugate with regard to the conic; on p. mect at P.

and two lines, of which one, and therefore each, passes through the 3. Tangents drawn at the 3. The point of contact of pole of the other, are said to be conjugale with regard to the conic. points where any line through P tangents drawn from any point Hence all points conjugate to a point P lie on the polar of P; all lines cuts the conic meet on pi and on p to the conic lic in a line with conjugate to a line o pass through the pole of p. conversely, P; and conversely,

If the line joining two conjugate poles cuts the conic, then the 4. If from any point on p. 4. Tangents drawn at points poles are harmonic conjugates with regard to the points of intertangents be drawn, their points where any line through P cuts the section; hence one lies within the other without the conic, and all of contact will lie in a line with P. conic meet on po

points conjugate to a point within a conic lie without it. 5. Any four-point on the conic S. Any four-side circumscribed Of a polar-triangle any two vertices are conjugate poles, any two which has one diagonal point at about a conic which has one sides conjugate lines. Hi therefore, one side cuts a conic, then P has the other two lying on p. diagonal on p has the other two one of the two vertices which lie on this side is within and the other meeting at P.

without the conic. The vertex opposite this side lies also without, The truth of 2 follows from I, it T be a point where p cuts the for it is the pole of a line which cuts the curve. In this case thereconic, then one of the points where PT cuts the conic, and which fore one vertex lies within, the other two without. If, on the are harmonic conjugates with regard to PT, coincides with T; hence othero hand, we begin with a side which does not cut the conic, the other does-that is, PT touches the curve at T.

then its pole lies within and the other vertices without. HenceThat 4 is true follows thus: If we draw from a point H on the Every polar-triangle has one and only one vertex within the conic. polar one tangent a to the conic, join its point of contact A to the We add, without a proof, the theorem pole P, determine the second point of intersection B of this line with The four points in which a conic is cut by two conjugate polars the conic, and draw the tangent at B, it will pass through H. and are four harmonic points in the conic will therefore be the second tangent which may be drawn from H to $68. If two conics intersect in four points (they cannot have the curve.

more points in common, $ 52), there exists one and only one

FIG. 33

axes.

four-point which is inscribed in both, and therefore one polar-triangle If we describe on a diameter AB of an ellipse

, or hyperbola a circle common to both.

concentric to the conic, it will cut the latter in A and B (fig. 25). Theorem.-Two conics which intersect in four points have always Each of the semicircles in which it is divided by AB will be partly one and only one common polar-triangle; and reciprocally, within, partly without the curve, and must cut the latter therefore

Two conics which have four common tangents have always one again in a point. The circle and the conic have thus four points and only one common polar-triangle.

A, B, C, D, and therefore
DIAMETERS AND AXES OF CONICS

one polar-triangle, in com$ 69. Diameters.—The theorems about the harmonic properties mon (9.68). Of this the of poles and polars contain, as special cases, a number of important centre is one vertex, for metrical properties of conics. These are obtained if either the pole the line at infinity is the or the polar is moved to infinity;-it being remembered that the polar to this point, harmonic conjugate to a point at infinity, with regard to two points with regard to the circle A, B, is the middle point of the segment AB. The most important and the other conic. The properties are stated in the following theorems.

other two sides are conThe middle points of parallel chords of a conic lie in a line-vie. on jugate diameters of both, the polar to the point al infinity on the parallel chords.

hence perpendicular to This line is called a diameler.

each other. This gives The polar of every point at infinity is a diameler.

An ellipse as well as an The tangenis al {he end points of a diameter are parallel, and are hyperbola has one pair of parallel to the chords bisected by the diameler.

All diameters pass through a common point, the pole of the line ol This reasoning shows at infinity

the same time how to conAu diameters of a parabola are parallel, the pole to the line at

struct the axis of an ellipse

FIG. 25. infinity being the point where the curve touches the line at infinity. or of an hyperbola. In case of the ellipse and hyperbola, the pole to the line at infinity Wir we define an axis as a diameter perpendicular to the chords

A parabola has one axis, is a finite point called the centre of the curve. A centre of a conic bisects every chord through il.

which it bisects. It is casily constructed. The line which bisects The centre of an ellipse is wilhin the curve, for the line at infinity any two parallel chords is a diameter. Chords perpendicular to it does not cut the ellipse.

will be bisected by a parallel diameter, and this is the axis. The centre of an hyperbola is wilhout the curve, because the line at

$ 73. The first part of the right-hand theorem in $64 may be infinity cuts the curve. Hence also

stated thus: any two conjugate lines through a point P'without a From the centre of an hyperbola two langenls can be drawn to the conic are harmonic conjugates with regard to the two tangents curve which have their point of contact al infinity. These are called that may be drawn from P to the conic. Asym plotes ($ 59).

11 we take instead of P the centre C of an hyperbola, then the To construct a diameler of a conic, draw two parallel chords and conjugate lines become conjugate diameters, and the tangents join their middle points.

asymptotes. Hence To find the centre of a conic, draw two diameters; their inter

Any two conjugale diamelers of an hyperbola are harmonic conjugales section will be the centre.

with regard to the asymploles. $ 70. Conjugale Diamelers.--A polar-triangle with one vertex at As the axes are conjugate diameters at right angles to one another, the centre will have the opposite side at infinity. The other two

it follows (823)sides pass through the centre, and are called conjugale diamelers,

The axes of an hyperbola bisecl the angles between the asymploles. each being the polar of the point at infinity on the other.

Let O be the centre of the hyperbola (fig. 26), 1 any

secant which of two conjugale diameters each bisects the chords parallel to the cuts the hyperbola “in C,D and the asymptotes in E.F, then the olher, and if one culs the curve, the langenis al ils ends are parallel 10 line OM which bisects the chord CD is a diameter conjugate to the the other diameter.

Further

Every parallelogram inscribed in a conic has its sides parallel lo two conjugale diamelers; and

Every parallelogran circumscribed about a conic has as diagonals two conjugale diamelers.

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 10 the ends of a diameter are parallel to two.conjugale diameters.

$71. If wery diameter is perpendicular lo ils conjugate the conic is e circle.

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

A conic which kas more than one pair of conjugale diamelers al right angles dans each other in a circle AA' and BB' (fig. 24) be one pair of conjugate diameters at

KS right angles to each other, CC' and DD' a second pair. ll we draw

through the end point A of one
diameter a chord AP parallel to

21
DD', and join P to A', then PA and DU):14

FIG. 26.
PA' are, according to $ 70, parallel to
two conjugate diameters. But PA is diameter OK which is parallel to the secant i, so that OK and OM
parallel to DD', hence PA' is parallel are harmonic with regard to the asymptotes. The point M there-
to CC', and therefore PA and PA

fore bisects EF. But by construction M bisects CD. It follows
are perpendicular, If we further that DF = EC, and ED=CF; or
draw the tangents to the conic at A On any secant of an hyperbola lhe segments between the curve and the
and A', these will be perpendicular asymptoies are equal.
to AA'. they being parallel to the In the chord is changed into a tangent, this gives--
conjugate diameter BB'. We know The segment between

the asym plotes on any tangent to an kyperbola
thus five points on the conic, viz. the is bisected by the point of contact.
points A and A' with their tangents, The first part allows a simple solution of the problem to find any
and the point P. Through these a number of points on an hyperbola, of which the asymptotes and onc
circle may be drawn having AA' as point are given. This is equivalent to three points and the tangents

diameter; and as through five points at two of them. This construction requires measurement. one conic only can be drawn, this circle must coincide with the $ 74. For the parabola, too, follow some metrical properties. A given canic.

diameter PM (fig. 27) bisects every chord conjugate to it, and the $ 72. Axes.--Conjugate diameters perpendicular to cach other pole P of such a chord BC lies on the diameter. But a diameter cuts are called axes, and the points where they cut the curve verlites the parabola once at infinity. Henceof the conic. In a circle every diameter is an axis, every point on it is a vertex;

The segment PM which

joins the middle point M of a chord of a para

bola to the pole P of the chord is bisected by the parabola at A. and any two lines at right angles to each other may be taken as a $75. Two asymptotes and any two tangents to an hyperbola Days of axes of any circle which has its centre at their intersection. may be considered as a quadrilateral circumscribed about the

.

BI

FIG. 24.

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arco.

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 54). If therefore DEFG

(AB, CD) - (AB, D'D). (hg. 28) is such a quadri This requires that C and D' coincide. lateral, then the diagonals $77. Two projective rows on the same base, which have the above DF and GE will meet on property, that to every point, whether it be considered as a point in the line which joins the the one or in the other row, corresponds the same point, are said points of contact of the

to be in involution, or to form an involution of points on the line. asymptotes, that is, on the

We mention, but without proving it, that any two projective line at inhnity; hence they rows may be placed so as to form an involution. are parallel. From this

An involution may be said to consist of a row of pairs of points, the following theorem is to every point A corresponding a point A', and to A again the a simple deduction:

point A. These points are said to be conjugate, or, better, one point All triangles formed by a is termed the “ mate" of the other. tangent and the asymptotes From the definition, according to which an involution may be of an hyperbola are equal in considered as made up of two projective rows, follow at once the

following important properties: If we draw at a point P !. The cross-ratio of lour points equals that of the four conjugate (fig. 28) on an hyperbola points. a tangent, the part HK 2. If we call a point which coincides with its mate a "focus"

between the asymptotes or “ double point of the involution, we may say: An involution FIG. 27.

is bisected at P. The has either two foci, or one, or none, and is called respectively a formed by the asymptotes and lines parallel to them through

parallelogram poog hyperbolic, parabolic or elliptic involution ( 34).

3. In a hyperbolic involution any two conjugate points are P will be hall the triangle OHK, and will therefore be con harmonic conjugates with regard to the two foci. stant. If we now take the asymptotes OX and OY as oblique For if A, A' be two conjugate points, F1, F, the two foci, then to the

points F, F3, A, A' in the one row correspond the points F, F2, A', A in the other, cach focus corresponding to itself. Hence (F.F;, AA') = (F;F>,A'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'=(.

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

(AB, OI) = (A'B', 10),
or

OA.OA'=OB. OB'.
In order to determine the distances of the foci from the centre,

we write F for A and A' and get
FIG. 28.

OF? =c; OF=#Vc.

Hence if c is positive OF is real, and has two values, equal and axes of co-ordinates, the lines OQ and QP will be the co-ordinates of opposite. The involution is hyperbolic. P, and will satisfy the equation xy=const. = q*.

If c=0, OF=0, and the two foci both coincide with the contre. For the asymptotes as axes of co-ordinates the equation of the hyperbolo Hence we may write

If c is negative, ve becomes imaginary, and there are no loci. is xy=const. INVOLUTION

In an hyperbolic involution, OA. OA' =k?,

In a parabolic involution, OA.OA' =0, $76. If we have two projective rows, ABC on u and A'B'C' on

OA.OA' =-k. ', and place their bases on the same line, then cach point in this

In an elliptic involution, line counts twice, once as a point in the row u and once as a point From these expressions it follows that conjugate points A, A' in an in the row u'. In fig. 29. we denote the points as points in the one hyperbolic involution lie on the same side of the centre, and in an row by letters above the line A, B, C ..., and as points in the second eliptic involution on opposite sides of the centre, and that in a

row by A', B', C'... below the parabolic involution one coincides with the centre.
B

line. Let now A and B' be the In the first case, for instance, OA.OA' is positive; hence OA
same point, then to A will" corre- and OA' have the same sign.
spond a point A' in the second, It also follows that two segments, AA' and BB', between pairs of
and to B' a point B in the first conjugate points have the following positions: in an hyperbolic

row. In general these points A involution they lie either one altogether within or altogether without and B will be different. It may, however, happen that they coincide each other; in a parabolic involution they have one point in common; Then the correspondence is a peculiar one, as the following theorem and in an elliptic involution they overlap, each being partly within shows:

and partly without the other. If two projective rows lie on the same base, and if it happens that to one Proof.-We have OA. OA'=OB.OB'= k· in case of an hyperbolic point in the base the same point corresponds, whether we consider the involution. Let A and B be the points in each pair which are point as belonging to the first or to the second row, then the same will nearer to the centre 0. If now A, A'and B, B' lie on tlte same side of kappen for every point in the base-thal is to say, to every point in the O, and if B is nearer to Othan A, so that OB<OA, then OB'>OA'; line corresponds the same point in the first as in the second row. hence B' lies farther away from than A', or the segment AA' lies

In order to determine the correspondence, we may assume three within BB'. And so on (or the other cases. pairs of corresponding points in two projective rows. Let then 6. An involution is determined

A', B', C', in fig. 30, correspond to (a) By two pairs of conjugate points. Hence also
A, B, C, so that A and B', and also
B and A, denote the same point.

(B) By one pair of conjugate points and the centre; B'

D'
Let us further denote the point

(7) By the two foci;.
C' when considered as a point in

(0) By one focus and one pair of conjugate points; the first row by D; then it is to

(e) By one focus and the centre. be proved that the point D', which corresponds to D, is the same 7. The condition that A, B, C and A’, B', C' may form an inpoint as C. We know that the cross-ratio of four points is equal volution may be written in one of the forms-to that of the corresponding row. Hence

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

(AB, CA) = (A'B', C'A), but replacing the dashed letters by those undashed ones whịch

(AB, C'A') =(A'B', CA), denote the same points, the second cross-ratic equais (BA DD2 is not expresses that in the two projective rows in which A, B, C:

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B'

FIG. 29.

A

D

B

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C'

A

FIG. 30.

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FIG. 32.

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and A', B', C' are conjugate points two conjugate elements may be base in the required point C' for OC. OC'EQA.OA': But EC and interchanged.

EC' are at right angles. Hence the involution which is obtained 8. Any three pairs, A, A', B, Bʻ, Ç, C', of conjugate points are by joining E or E' to the points connected by the relations:

in the given involution is cirAB'. BC'.CA' AB'. BC.C'A' AB. B'C'.CA'AB. B'C.C'A'

cular. This may also be exA'B. B'C.C'AA'B. B'C'.CAA'B'.BC.C'AA'B'.BC'. CA=-1. pressed thus:

Every elliplical involution has These relations readily follow by working out the relations in (7) the property that there are two (above). $78. Involution of a quadrangle. - The sides of any four-point are which any two conjugale points

definite points in the plane from cut by any line in six points in involution, opposite sides being cut in are seen under a right angle. conjugale points

At the same time the followLet A,B,C,D, (fig. 31) be the four-point. If its sides be cut by ing problem has been solved: the line p in the points A, A', B, B'; C, C', if further, C.D. cuts the To determine the centre and line A,Bi in Cs, and if we project the row A,B,C;C to p once from also the point corresponding 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 conjugate points are given.

to any given point in an elliptical involution of which two pairs of (A'B', C'C) = (ÅB, CC'). Hence by $ 77 (7) the points are in in $ 8i. Involution Range on a Conic.-By the aid of $ 53, the points volution.

on a conic may be made to correspond to those on a line, so that the The theorem may also be stated thus: The three points in which any line cuts the sides of a triangle and the We may also have two projective rows on the same conic, and these

row of points on the conic is projective to a row of points on a line. projections, from any point in the plane, of the vertices of the triangle will be in involution as soon as one point on the conic has the same on to the same line are six points in involution.

point corresponding to it all the same to whatever row it belongs. Or againThe projections from any point on to any line of the six vertices from its definition, and from $ 53) that the lines which join conjugate

An involution of points on a conic will have the property (as follows
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 invol on 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 interşeclion of conjugale lines in an involution anong langents 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 Α Α ́ and B'B' 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,

obtain two projective pencils. We take A and

A' as centres of of a four-side are six points in involution, the projections of opposite these pencils, so vertices being conjugate points.

that the pencils This property gives a simple means to construct, by aid of the A (AB B'). straight edge only, in an involution of which two pairs of conjugate A'(AB'B) are propoints are given, to any point its conjugate.

jective, and in $ 79. Pencils in Involution. --The theory of involution may at once perspective posi; be extended from the row to the flat and the axial pencil-viz. we say tion, because AA that there is an involution in a flat or in an axial pencil if any line corresponds to cuts the pencil in an involution of points. An involution in a pencil

A'A.

Hence corconsists of pairs of conjugate rays or planes; it has two, one or no

responding rays focal rays (double lines) or planes, but nothing corresponding to a

meet in a line, of which two points are found by joining AB' to centre.

A'B and AB to A'B'. It follows that the axis of perspective is the An involution in a flat pencil contains always one, and in general polar of the point P, where AA' and BB' meet. If we now wish only one, pair of conjugate rays which are perpendicular to one to construct to any other point C on the conic the corresponding another. For in two projective flat pencils exist always two corre

point C', we join C to A' and the point where this line cuts p to A. sponding right angles ( 40).

The latter line cuts the conic again in C'. But we know from the Each involution in an axial pencil contains in the same manner theory of pole and polar that the line CC' passes through P. The one pair of conjugate planes at right angles to one another.

point of concurrence is called the “pole of the involution," and As a rule, there exists but one pair of conjugate lines or planes the line of collinearity of the meets' is called the “ axis of the at right angles to each other. But it is possible that there are

involution." more, and then there is an infinite number of such pairs. An in INVOLUTION DETERMINED BY A CONIC ON A LINE.-FOCI volution in a flat pencil, in which every ray is perpendicular to its $ 82. The polars, with regard to a conic, of points in a row p form conjugate ray, is said to be circular. That such involution is a pencil P projective to the row (866). This pencil cuts the base of possible is easily seen thus: if in two concentric flat pencils each the row p in a projective row. ray on one is made to correspond to that ray on the other which II A is a point in the given row, A'the point where the polar of is perpendicular to it, then the two pencils are projective, for if A cuts p. then A and A’ will be corresponding points. If we take we turn the one pencil through a right angle each ray in one coincides A' a point in the first row, then the polar of A will pass through with its corresponding ray in the other. But these two projective A, so that A corresponds to A-in other words, the rows are in pencils are in involution.

involution. The conjugate points in this involution are conjugate A circular involution has no focal rays, because no ray in a pencil points with regard to the conic. Conjugate points coincide only if coincides with the ray perpendicular to it.

the polar of a point A passes through å-that is, il A lies on the $ 80. Every elliptical involution in a row may be considered as a conic. Hencesection of a circular involution.

A conic determines on every line in its plane an involution, in which In an elliptical involution any two segments AA' and BB' lie those points are conjugate which are also conjugate with regard to the partly within and partly without each other (fig. 32). Hence two conic. circles described on AA and BB' as diameters will intersect in two Is the line culs the conic the involution is hyperbolic, the points of points E and E'. The line EE' cuts the base of the involution at a intersection being the foci. point 0. which has the property that OA.OA'=OB. OB', for If the line touches the conic the involution is parabolic, the two foci each is equal to OE.OE'. 'The point O is therefore the centre of coinciding at the point of contact. the involution. If we wish to construct to any point C the conjugate If the line does not cul the conic the involution is elliptic, having no point C', we may draw the circle through CEE'. This will cut the foci.

B

C'

A

B

с

we

FIG. 31.

and

FIG. 33

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