Chapter XXXI: Book XIII (3)

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 S 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.

S 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.

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 four-point 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._

This is a very important theorem. It may also be stated thus--

_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.

S 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.

S 68. If two conics intersect in four points (they cannot have more
points in common, S 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

S 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_ (S 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.

S 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--

_Every parallelogram inscribed in a conic has its sides parallel to
two conjugate 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 polar-triangle. 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 conjugate diameters._

S 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 at 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 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 S 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 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.

S 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 (S 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 or of an hyperbola_.

_A parabola has one axis_, if we define an axis as a diameter
perpendicular to the chords 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.

S 73. The first part of the right-hand theorem in S 64 may be stated
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 (S 23)--

_The axes of an hyperbola bisect the angles between the asymptotes._

Let O be the centre of the hyperbola (fig. 26), t 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 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 ED = CF; 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.

S 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._

S 75. Two asymptotes and any two tangents to an hyperbola may be
considered as a quadrilateral circumscribed about the hyperbola. But
in such a quadrilateral the intersections of the diagonals and the
points of contact of opposite sides lie in a line (S 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 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 axes of co-ordinates, the lines OQ
and QP will be the co-ordinates of P, and will satisfy the equation xy
= const. = a^2.

_For the asymptotes as axes of co-ordinates the equation of the
hyperbola is xy = const._

INVOLUTION

S 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 each 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, 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. 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-ratio equals (BA, DD'), which,
according to S 15, equals (AB, D'D); so that the equation becomes

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

This requires that C and D' coincide.

S 77. 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.

2. If we call a point which coincides with its mate a "focus" or
"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 (S 34).

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

For if A, A' be two conjugate points, F1, F2 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, each focus corresponding to itself. Hence (F1F2, AA')
= (F1F2, 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 (S 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, the centre,
I the point at infinity, then

(AB, OI) = (A'B', IO),

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

OF^2 = c; OF = [+-][root]c.

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

If c = 0, OF = 0, and the two foci both coincide with the centre. If c
is negative, [root]c becomes imaginary, and there are no foci. Hence
we may write--

In an hyperbolic involution, OA.OA' = k^2,
In a parabolic involution, OA.OA' = 0,
In an elliptic involution, OA.OA' = -k^2.

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' = k^2 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--

([alpha]) By two pairs of conjugate points. Hence also
([beta]) By one pair of conjugate points and the centre;
([gamma]) By the two foci;
([delta]) By one focus and one pair of conjugate points;
([epsilon]) 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--

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

or (AB, CA') = (A'B', C'A),

or (AB, C'A') = (A'B', CA),

for each expresses that in the two projective rows in which A, B, C
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).

S 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 S 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--

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.

S 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 (S 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.

S 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:

_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.

S 81. _Involution Range on a Conic._--By the aid of S 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 S 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._

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

S 82. The polars, with regard to a conic, of points in a row p form a
pencil P projective to the row (S 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._

If, on the other hand, we take a point P in the plane of a conic, we
get to each line a through P one conjugate line which joins P to the
pole of a. These pairs of conjugate lines through P form an involution
in the pencil at P. The focal rays of this involution are the tangents
drawn from P to the conic. This gives the theorem reciprocal to the
last, viz:--

_A conic determines in every pencil in its plane an involution,
corresponding lines being conjugate lines with regard to the conic._

_If the point is without the conic the involution is hyperbolic, the
tangents from the points being the focal rays._

_If the point lies on the conic the involution is parabolic, the
tangent at the point counting for coincident focal rays._

_If the point is within the conic the involution is elliptic, having
no focal rays._

It will further be seen that the involution determined by a conic on
any line p is a section of the involution, which is determined by the
conic at the pole P of p.

S 83. _Foci._--The centre of a pencil in which the conic determines a
circular involution is called a "focus" of the conic.

In other words, a focus is such a point that every line through it is
perpendicular to its conjugate line. The polar to a focus is called a
_directrix_ of the conic.

From the definition it follows that _every focus lies on an axis_, for
the line joining a focus to the centre of the conic is a diameter to
which the conjugate lines are perpendicular; and _every line joining
two foci is an axis_, for the perpendiculars to this line through the
foci are conjugate to it. These conjugate lines pass through the pole
of the line, the pole lies therefore at infinity, and the line is a
diameter, hence by the last property an axis.

It follows that all _foci lie on one axis_, for no line joining a
point in one axis to a point in the other can be an axis.

As the conic determines in the pencil which has its centre at a focus
a circular involution, no tangents can be drawn from the focus to the
conic. Hence _each focus lies within a conic_; and _a directrix does
not cut the conic_.

Further properties are found by the following considerations:

S 84. Through a point P one line p can be drawn, which is with regard
to a given conic conjugate to a given line q, viz. that line which
joins the point P to the pole of the line q. If the line q is made to
describe a pencil about a point Q, then the line p will describe a
pencil about P. These two pencils will be projective, for the line p
passes through the pole of q, and whilst q describes the pencil Q, its
pole describes a projective row, and this row is perspective to the
pencil P.

We now take the point P on an axis of the conic, draw any line p
through it, and from the pole of p draw a perpendicular q to p. Let q
cut the axis in Q. Then, in the pencils of conjugate lines, which have
their centres at P and Q, the lines p and q are conjugate lines at
right angles to one another. Besides, to the axis as a ray in either
pencil will correspond in the other the perpendicular to the axis (S
72). The conic generated by the intersection of corresponding lines in
the two pencils is therefore the circle on PQ as diameter, _so that
every line in P is perpendicular to its corresponding line in Q_.

To every point P on an axis of a conic corresponds thus a point Q,
such that conjugate lines through P and Q are perpendicular.

We shall show that these _point-pairs_ P, Q _form an involution_. To
do this let us move P along the axis, and with it the line p, keeping
the latter parallel to itself. Then P describes a row, p a perspective
pencil (of parallels), and the pole of p a projective row. At the same
time the line q describes a pencil of parallels perpendicular to p,
and perspective to the row formed by the pole of p. The point Q,
therefore, where q cuts the axis, describes a row projective to the
row of points P. The two points P and Q describe thus two projective
rows on the axis; and not only does P as a point in the first row
correspond to Q, but also Q as a point in the first corresponds to P.
The two rows therefore form an involution. _The centre of this
involution, it is easily seen, is the centre of the conic._

_A focus of this involution has the property that any two conjugate
lines through it are perpendicular; hence, it is a focus to the
conic._

Such involution exists on each axis. But only one of these can have
foci, because all foci lie on the same axis. The involution on one of
the axes is elliptic, and appears (S 80) therefore as the section of
two circular involutions in two pencils whose centres lie in the other
axis. These centres are foci, hence the one axis contains two foci,
the other axis none; _or every central conic has two foci which lie on
one axis equidistant from the centre_.

The axis which contains the foci is called the _principal axis_; in
case of an hyperbola it is the axis which cuts the curve, because the
foci lie within the conic.

In case of the parabola there is but one axis. The involution on this
axis has its centre at infinity. One focus is therefore at infinity,
the one focus only is finite. _A parabola has only one focus._

S 85. If through any point P (fig. 34) on a conic the tangent PT and
the normal PN (i.e. the perpendicular to the tangent through the point
of contact) be drawn, these will be conjugate lines with regard to the
conic, and at right angles to each other. They will therefore cut the
principal axis in two points, which are conjugate in the involution
considered in S 84; hence they are harmonic conjugates with regard to
the foci. If therefore the two foci F1 and F2 be joined to P, these
lines will be harmonic with regard to the tangent and normal. As the
latter are perpendicular, they will bisect the angles between the
other pair. Hence--

_The lines joining any point on a conic to the two foci are equally
inclined to the tangent and normal at that point._

In case of the parabola this becomes--

_The line joining any point on a parabola to the focus and the
diameter through the point, are equally inclined to the tangent and
normal at that point._

From the definition of a focus it follows that--

_The segment of a tangent between the directrix and the point of
contact is seen from the focus belonging to the directrix under a
right angle_, because the lines joining the focus to the ends of this
segment are conjugate with regard to the conic, and therefore
perpendicular.

With equal ease the following theorem is proved:

_The two lines which join the points of contact of two tangents each
to one focus, but not both to the same, are seen from the intersection
of the tangents under equal angles._

S 86. Other focal properties of a conic are obtained by the following
considerations:

Let F (fig. 35) be a focus to a conic, f the corresponding directrix,
A and B the points of contact of two tangents meeting at T, and P the
point where the line AB cuts the directrix. Then TF will be the polar
of P (because polars of F and T meet at P). Hence TF and PF are
conjugate lines through a focus, and therefore perpendicular. They are
further harmonic conjugates with regard to FA and FB (SS 64 and 13),
so that they bisect the angles formed by these lines. This by the way
proves--

_The segments between the point of intersection of two tangents to a
conic and their points of contact are seen from a focus under equal
angles._

If we next draw through A and B lines parallel to TF, then the points
A1, B1 where these cut the directrix will be harmonic conjugates with
regard to P and the point where FT cuts the directrix. The lines FT
and FP bisect therefore also the angles between FA1 and FB1. From this
it follows easily that the triangles FAA1 and FBB1 are equiangular,
and therefore similar, so that FA : AA1 = FB : BB1.

The triangles AA1A2 and BB1B2 formed by drawing perpendiculars from A
and B to the directrix are also similar, so that AA1 : AA2 = = BB1 :
BB2. This, combined with the above proportion, gives FA : AA2 = FB :
BB2. Hence the theorem:

_The ratio of the distances of any point on a conic from a focus and
the corresponding directrix is constant._

To determine this ratio we consider its value for a vertex on the
principal axis. In an ellipse the focus lies between the two vertices
on this axis, hence the focus is nearer to a vertex than to the
corresponding directrix. Similarly, in an hyperbola a vertex is nearer
to the directrix than to the focus. In a parabola the vertex lies
halfway between directrix and focus.

It follows in an ellipse the ratio between the distance of a point
from the focus to that from the directrix is less than unity, in the
parabola it equals unity, and in the hyperbola it is greater than
unity.

It is here the same which focus we take, because the two foci lie
symmetrical to the axis of the conic. If now P is any point on the
conic having the distances r1 and r2 from the foci and the distances
d1 and d2 from the corresponding directrices, then r1/d1 = r2/d2 =
e, where e is constant. Hence also r1 [+-] r2 / d1 [+-] d2 = e.

In the ellipse, which lies between the directrices, d1 + d2 is
constant, therefore also r1 +r2. In the hyperbola on the other hand d1
- d2 is constant, equal to the distance between the directrices,
therefore in this case r1 - r2 is constant.

If we call the distances of a point on a conic from the focus its
focal distances we have the theorem:

_In an ellipse the sum of the focal distances is constant; and in an
hyperbola the difference of the focal distances is constant._

_This constant sum or difference equals in both cases the length of
the principal axis._

PENCIL OF CONICS

S 87. Through four points A, B, C, D in a plane, of which no three lie
in a line, an infinite number of conics may be drawn, viz. through
these four points and any fifth one single conic. This system of
conics is called a pencil of conics. Similarly, all conics touching
four fixed lines form a system such that any fifth tangent determines
one and only one conic. We have here the theorems:

The pairs of points in which The pairs of tangents which
any line is cut by a system of can be drawn from a point to
conics through four fixed points a system of conics touching four
are in involution. fixed lines are in involution.

We prove the first theorem only. Let ABCD (fig. 36) be the four-point,
then any line t will cut two opposite sides AC, BD in the points E,
E', the pair AD, BC in points F, F', and any conic of the system in M,
N, and we have A(CD, MN) = B(CD, MN).

If we cut these pencils by t we get

(EF, MN) = (F'E', MN)

or (EF, MN) = (E'F', NM).

But this is, according to S 77 (7), the condition that M, N are
corresponding points in the involution determined by the point pairs
E, E', F, F' in which the line t cuts pairs of opposite sides of the
four-point ABCD. This involution is independent of the particular
conic chosen.

S 88. There follow several important theorems:

_Through four points two, one, or no conics may be drawn which touch
any given line, according as the involution determined by the given
four-point on the line has real, coincident or imaginary foci._

_Two, one, or no conics may be drawn which touch four given lines and
pass through a given point, according as the involution determined by
the given four-side at the point has real, coincident or imaginary
focal rays._

For the conic through four points which touches a given line has its
point of contact at a focus of the involution determined by the
four-point on the line.

As a special case we get, by taking the line at infinity:

_Through four points of which none is at infinity either two or no
parabolas may be drawn._

The problem of drawing a conic through four points and touching a
given line is solved by determining the points of contact on the line,
that is, by determining the foci of the involution in which the line
cuts the sides of the four-point. The corresponding remark holds for
the problem of drawing the conics which touch four lines and pass
through a given point.

RULED QUADRIC SURFACES

S 89. We have considered hitherto projective rows which lie in the
same plane, in which case lines joining corresponding points envelop a
conic. We shall now consider projective rows whose bases do not meet.
In this case, corresponding points will be joined by lines which do
not lie in a plane, but on some surface, which like every surface
generated by lines is called a _ruled_ surface. This surface clearly
contains the bases of the two rows.

If the points in either row be joined to the base of the other, we
obtain two axial pencils which are also projective, those planes being
corresponding which pass through corresponding points in the given
rows. If A', A be two corresponding points, [alpha], [alpha]' the
planes in the axial pencils passing through them, then AA' will be the
line of intersection of the corresponding planes [alpha], [alpha]' and
also the line joining corresponding points in the rows.

If we cut the whole figure by a plane this will cut the axial pencils
in two projective flat pencils, and the curve of the second order
generated by these will be the curve in which the plane cuts the
surface. Hence

_The locus of lines joining corresponding points in two projective
rows which do not lie in the same plane is a surface which contains
the bases of the rows, and which can also be generated by the lines of
intersection of corresponding planes in two projective axial pencils.
This surface is cut by every plane in a curve of the second order,
hence either in a conic or in a line-pair. No line which does not lie
altogether on the surface can have more than two points in common with
the surface, which is therefore said to be of the second order or is
called a ruled quadric surface._

That no line which does not lie on the surface can cut the surface in
more than two points is seen at once if a plane be drawn through the
line, for this will cut the surface in a conic. It follows also that a
line which contains more than two points of the surface lies
altogether on the surface.

S 90. Through any point in space one line can always be drawn cutting
two given lines which do not themselves meet.

If therefore three lines in space be given of which no two meet, then
through every point in either one line may be drawn cutting the other
two.

_If a line moves so that it always cuts three given lines of which no
two meet, then it generates a ruled quadric surface._

Let a, b, c be the given lines, and p, q, r ... lines cutting them in
the points A, A', A" ...; B, B', B" ...; C, C', C" ... respectively;
then the planes through a containing p, q, r, and the planes through b
containing the same lines, may be taken as corresponding planes in two
axial pencils which are projective, because both pencils cut the line
c in the same row, C, C', C" ...; the surface can therefore be
generated by projective axial pencils.

Of the lines p, q, r ... no two can meet, for otherwise the lines a,
b, c which cut them would also lie in their plane. There is a single
infinite number of them, for one passes through each point of a. These
lines are said to form a set of lines on the surface.

If now three of the lines p, q, r be taken, then every line d cutting
them will have three points in common with the surface, and will
therefore lie altogether on it. This gives rise to a second set of
lines on the surface. From what has been said the theorem follows:

_A ruled quadric surface contains two sets of straight lines. Every
line of one set cuts every line of the other, but no two lines of the
same set meet._

_Any two lines of the same set may be taken as bases of two projective
rows, or of two projective pencils which generate the surface. They
are cut by the lines of the other set in two projective rows._

The plane at infinity like every other plane cuts the surface either
in a conic proper or in a line-pair. In the first case the surface is
called an _Hyperboloid of one sheet_, in the second an _Hyperbolic
Paraboloid_.

The latter may be generated by a line cutting three lines of which one
lies at infinity, that is, cutting two lines and remaining parallel to
a given plane.

QUADRIC SURFACES

S 91. The conics, the cones of the second order, and the ruled quadric
surfaces complete the figures which can be generated by projective
rows or flat and axial pencils, that is, by those aggregates of
elements which are of one dimension (SS 5, 6). We shall now consider
the simpler figures which are generated by aggregates of two
dimensions. The space at our disposal will not, however, allow us to
do more than indicate a few of the results.

S 92. We establish a correspondence between the lines and planes in
pencils in space, or reciprocally between the points and lines in two
or more planes, but consider principally pencils.

In two pencils we may either make planes correspond to planes and
lines to lines, or else planes to lines and lines to planes. If hereby
the condition be satisfied that to a flat, or axial, pencil
corresponds in the first case a projective flat, or axial, pencil, and
in the second a projective axial, or flat, pencil, the pencils are
said to be _projective_ in the first case and _reciprocal_ in the
second.

For instance, two pencils which join two points S1 and S2 to the
different points and lines in a given plane [pi] are projective (and
in perspective position), if those lines and planes be taken as
corresponding which meet the plane [pi] in the same point or in the
same line. In this case every plane through both centres S1 and S2 of
the two pencils will correspond to itself. If these pencils are
brought into any other position they will be projective (but not
perspective).

_The correspondence between two projective pencils is uniquely
determined, if to four rays (or planes) in the one the corresponding
rays (or planes) in the other are given, provided that no three rays
of either set lie in a plane._

Let a, b, c, d be four rays in the one, a', b', c', d' the
corresponding rays in the other pencil. We shall show that we can find
for every ray e in the first a single corresponding ray e' in the
second. To the axial pencil a (b, c, d ...) formed by the planes which
join a to b, c, d ..., respectively corresponds the axial pencil a'
(b', c', d' ... ), and this correspondence is determined. Hence, the
plane a'e' which corresponds to the plane ae is determined. Similarly
the plane b'e' may be found and both together determine the ray e'.

Similarly the correspondence between two reciprocal pencils is
determined if for four rays in the one the corresponding planes in the
other are given.

S 93. We may now combine--

1. Two reciprocal pencils.

Each ray cuts its corresponding plane in a point, the locus of these
points is a quadric surface.

2. Two projective pencils.

Each plane cuts its corresponding plane in a line, but a ray as a
rule does not cut its corresponding ray. The locus of points where a
ray cuts its corresponding ray is a twisted cubic. The lines where a
plane cuts its corresponding plane are secants.

3. Three projective pencils.

The locus of intersection of corresponding planes is a cubic
surface.

Of these we consider only the first two cases.

S 94. If two pencils are reciprocal, then to a plane in either
corresponds a line in the other, to a flat pencil an axial pencil, and
so on. Every line cuts its corresponding plane in a point. If S1 and
S2 be the centres of the two pencils, and P be a point where a line a1
in the first cuts its corresponding plane [alpha]2, _then the line b2
in the pencil S2 which passes through P will meet its corresponding
plane [beta]1 in P_. For b2 is a line in the plane [alpha]2. The
corresponding plane [beta]1 must therefore pass through the line a1,
hence through P.

The points in which the lines in S1 cut the planes corresponding to
them in S2 are therefore the same as the points in which the lines in
S2 cut the planes corresponding to them in S1.

_The locus of these points is a surface which is cut by a plane in a
conic or in a line-pair and by a line in not more than two points
unless it lies altogether on the surface. The surface itself is
therefore called a quadric surface, or a surface of the second order._

To prove this we consider any line p in space.

The flat pencil in S1 which lies in the plane drawn through p and the
corresponding axial pencil in S2 determine on p two projective rows,
and those points in these which coincide with their corresponding
points lie on the surface. But there exist only two, or one, or no
such points, unless every point coincides with its corresponding
point. In the latter case the line lies altogether on the surface.