**CURVE** (Lat. *curvus*, bent), a word commonly meaning a
shape represented by a line bending continuously out of the
straight without making an angle, but only properly to be defined
in its geometrical sense in the terms set out below. This subject
is treated here from an historical point of view, for the purpose
of showing how the different leading ideas were successively
arrived at and developed.

1. A curve is a line, or continuous singly infinite system of points. We consider in the first instance, and chiefly, a plane curve described according to a law. Such a curve may be regarded geometrically as actually described, or kinematically as in the course of description by the motion of a point; in the former point of view, it is the locus of all the points which satisfy a given condition; in the latter, it is the locus of a point moving subject to a given condition. Thus the most simple and earliest known curve, the circle, is the locus of all the points at a given distance from a fixed centre, or else the locus of a point moving so as to be always at a given distance from a fixed centre. (The straight line and the point are not for the moment regarded as curves.)

Next to the circle we have the conic sections, the invention
of them attributed to Plato (who lived 430–347 B.C.); the
original definition of them as the sections of a cone was by the
Greek geometers who studied them soon replaced by a proper
definition *in plano* like that for the circle, viz. a conic section
(or as we now say a “conic”) is the locus of a point such that its
distance from a given point, the focus, is in a given ratio to its
(perpendicular) distance from a given line, the directrix; or it is
the locus of a point which moves so as always to satisfy the
foregoing condition. Similarly any other property might be
used as a definition; an ellipse is the locus of a point such that
the sum of its distances from two fixed points (the foci) is constant,
&c., &c.

The Greek geometers invented other curves; in particular,
the conchoid (*q.v.*), which is the locus of a point such that its
distance from a given line, measured along the line drawn through
it to a fixed point, is constant; and the cissoid (*q.v.*), which is
the locus of a point such that its distance from a fixed point is
always equal to the intercept (on the line through the fixed
point) between a circle passing through the fixed point and the
tangent to the circle at the point opposite to the fixed point.
Obviously the number of such geometrical or kinematical
definitions is infinite. In a machine of any kind, each point
describes a curve; a simple but important instance is the
“three-bar curve,” or locus of a point in or rigidly connected
with a bar pivoted on to two other bars which rotate about
fixed centres respectively. Every curve thus arbitrarily defined
has its own properties; and there was not any principle of
classification.

2. *Cartesian Co-ordinates.*—The principle of classification first
presented itself in the *Géometrie* of Descartes (1637). The idea
was to represent any curve whatever by means of a relation
between the co-ordinates (*x*, *y*) of a point of the curve, or say to
represent the curve by means of its equation. (See Geometry:
*Analytical*.)

Any relation whatever between (*x*, *y*) determines a curve,
and conversely every curve whatever is determined by a relation
between (*x*, *y*).

Observe that the distinctive feature is in the *exclusive* use of
such determination of a curve by means of its equation. The
Greek geometers were perfectly familiar with the property of an
ellipse which in the Cartesian notation is *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1, the
equation of the curve; but it was as one of a number of properties,
and in no wise selected out of the others for the characteristic
property of the curve.

3. *Order of a Curve.*—We obtain from the equation the notion
of an algebraical as opposed to a transcendental curve, viz.
an algebraical curve is a curve having an equation F(*x*, *y*) = 0
where F(*x*, *y*) is a rational and integral function of the co-ordinates
(*x*, *y*); and in what follows we attend throughout
(unless the contrary is stated) only to such curves. The
equation is sometimes given, and may conveniently be used,
in an irrational form, but we always imagine it reduced to the
foregoing rational and integral form, and regard this as the
equation of the curve. And we have hence the notion of a curve
of a *given order*, viz. the order of the curve is equal to that of
the term or terms of highest order in the co-ordinates (*x*, *y*)
conjointly in the equation of the curve; for instance, *xy* − 1 = 0
is a curve of the second order.

It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.

We hence divide curves according to their order, viz. a curve
is of the first order, second order, third order, &c., according as
it is represented by an equation of the first order, *ax* + *by* + *c* = 0,
or say (*≬ *x*, *y*, 1) = 0; or by an equation of the second order,
*ax*^{2} + 2*hxy* + *by*^{2} + 2*fy* + 2*gx* + *c* = 0, say (*≬ *x*, *y*, 1)^{2} = 0; or by an
equation of the third order, &c.; or what is the same thing,
according as the equation is linear, quadric, cubic, &c.

A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.

A curve of the order *m* has for its equation (≬ *x*, *y*, 1)^{m} = 0;
and when the coefficients of the function are arbitrary, the curve
is said to be the general curve of the order m. The number of
coefficients is ½(*m* + 1)(*m* + 2); but there is no loss of generality
if the equation be divided by one coefficient so as to reduce the
coefficient of the corresponding term to unity, hence the number
of coefficients may be reckoned as ½(*m* + 1)(*m* + 2) − 1, that is,
½*m*(*m* + 3); and a curve of the order *m* may be made to satisfy
this number of conditions; for example, to pass through ½*m*(*m* + 3)
points.

It is to be remarked that an equation may *break up*; thus a
quadric equation may be (*ax* + *by* + *c*)(*a*′*x* + *b*′*y* + *c*′) = 0, breaking
up into the two equations *ax* + *by* + *c* = 0, *a*′*x* + *b*′*y* + *c*′ = 0, viz.
the original equation is satisfied if either of these is satisfied.
Each of these last equations represents a curve of the first order,
or right line; and the original equation represents this pair of
lines, viz. the pair of lines is considered as a quadric curve.
But it is an *improper* quadric curve; and in speaking of curves
of the second or any other given order, we frequently imply that
the curve is a proper curve represented by an equation which
does not break up.

4. *Intersections of Curves.*—The intersections of two curves
are obtained by combining their equations; viz. the elimination
from the two equations of *y* (or *x*) gives for *x* (or *y*) an equation
of a certain order, say the resultant equation; and then to each
value of *x* (or *y*) satisfying this equation there corresponds in
general a single value of *y* (or *x*), and consequently a single point
of intersection; the number of intersections is thus equal to the
order of the resultant equation in *x* (or *y*).

Supposing that the two curves are of the orders *m*, *n*, respectively,
then the order of the resultant equation is in general and
at most = *mn*; in particular, if the curve of the order *n* is an
arbitrary line (*n* = 1), then the order of the resultant equation
is = *m*; and the curve of the order *m* meets therefore the line in
m points. But the resultant equation may have all or any of its
roots imaginary, and it is thus not always that there are *m* real
intersections.

The notion of imaginary intersections, thus presenting itself,
through algebra, in geometry, must be accepted in geometry—and
it in fact plays an all-important part in modern geometry.
As in algebra we say that an equation of the *m*th order has
*m* roots, viz. we state this generally without in the first instance,
or it may be without ever, distinguishing whether these are real
or imaginary; so in geometry we say that a curve of the *m*th
order is met by an arbitrary line in *m* points, or rather we thus,
through algebra, obtain the proper geometrical definition of a
curve of the *m*th order, as a curve which is met by an arbitrary
line in *m* points (that is, of course, in *m*, and not more than *m*,
points).

The theorem of the *m* intersections has been stated in regard
to an *arbitrary* line; in fact, for particular lines the resultant
equation may be or appear to be of an order less than *m*; for
instance, taking *m* = 2, if the hyperbola *xy* − 1 = 0 be cut by the
line *y* = β, the resultant equation in *x* is β*x* − 1 = 0, and there is
apparently only the intersection (*x* = 1/β, *y* = β); but the theorem
is, in fact, true for every line whatever: a curve of the order *m*
meets every line whatever in precisely *m* points. We have, in the
case just referred to, to take account of a point at infinity on the
line *y* = β; the two intersections are the point (*x* = 1/β, *y* = β),
and the point at infinity on the line *y* = β.

It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is necessary to take account of these various circumstances.

5.*Line at Infinity.*—The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line,

*i.e.*this projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection is

*always*a line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane.

^{[1]}

Reverting to the purely plane theory, infinity is a line, related
like any other right line to the curve, and thus intersecting it
in *m* points, real or imaginary, distinct or coincident.

Descartes in the *Géométrie* defined and considered the remarkable
curves called after him the ovals of Descartes, or simply
Cartesians, which will be again referred to. The next important
work, founded on the *Géométrie*, was Sir Isaac Newton’s *Enumeratio*
*linearum tertii ordinis* (1706), establishing a classification of
cubic curves founded chiefly on the nature of their infinite
branches, which was in some details completed by James Stirling
(1692–1770), Patrick Murdoch (d. 1774) and Gabriel Cramer;
the work also contains the remarkable theorem (to be again referred
to), that there are five kinds of cubic curves giving by their
projections every cubic curve whatever. Various properties of
curves in general, and of cubic curves, are established in Colin
Maclaurin’s memoir, “De linearum geometricarum proprietatibus
generalibus Tractatus” (posthumous, say 1746, published in
the 6th edition of his *Algebra*). We have in it a particular kind
of *correspondence* of two points on a cubic curve, viz. two points
correspond to each other when the tangents at the two points
again meet the cubic in the same point.

6. *Reciprocal Polars. Intersections of Circles. Duality.*
*Trilinear and Tangential Co-ordinates.—The Géométrie descriptive*,
by Gaspard Monge, was written in the year 1794 or 1795 (7th
edition, Paris, 1847), and in it we have stated, *in plano* with
regard to the circle, and in three dimensions with regard to
a surface of the second order, the fundamental theorem of
reciprocal polars, viz. “Given a surface of the second order
and a circumscribed conic surface which touches it ... then
if the conic surface moves so that its summit is always in the same
plane, the plane of the curve of contact passes always through
the same point.” The theorem is here referred to partly on
account of its bearing on the theory of imaginaries in geometry.
It is in Charles Julian Brianchon’s memoir “Sur les surfaces du
second degré” (*Jour. Polyt.* t. vi. 1806) shown how for any given
position of the summit the plane of contact is determined,
or reciprocally; say the plane XY is determined when the point
P is given, or reciprocally; and it is noticed that when P is
situate in the interior of the surface the plane XY does not cut
the surface; that is, we have a real plane XY intersecting the
surface in the imaginary curve of contact of the imaginary
circumscribed cone having for its summit a given real point P
inside the surface.

Stating the theorem in regard to a conic, we have a real point
P (called the pole) and a real line XY (called the polar), the line
joining the two (real or imaginary) points of contact of the (real
or imaginary) tangents drawn from the point to the conic; and
the theorem is that when the point describes a line the line
passes through a point, this line and point being polar and pole
to each other. The term “pole” was first used by François
Joseph Servois, and “polar” by Joseph Diez Gergonne (*Gerg.*
t. i. and iii., 1810–1813); and from the theorem we have the
method of reciprocal polars for the transformation of geometrical
theorems, used already by Brianchon (in the memoir above
referred to) for the demonstration of the theorem called by his
name, and in a similar manner by various writers in the earlier
volumes of Gergonne. We are here concerned with the method
less in itself than as leading to the general notion of duality.

Bearing in a somewhat similar manner also on the theory of
imaginaries in geometry (but the notion presents itself in a more
explicit form), there is the memoir by L. Gaultier, on the graphical
construction of circles and spheres (*Jour. Polyt.* t. ix., 1813).
The well-known theorem as to radical axes may be stated as
follows. Consider two circles partially drawn so that it does not
appear whether the circles, if completed, would or would not
intersect in real points, say two arcs of circles; then we can,
by means of a third circle drawn so as to intersect in two real
points each of the two arcs, determine a right line, which, if
the complete circles intersect in two real points, passes through
the points, and which is on this account regarded as a line
passing through two (real or imaginary) points of intersection
of the two circles. The construction in fact is, join the two
points in which the third circle meets the first arc, and join also
the two points in which the third circle meets the second arc,
and from the point of intersection of the two joining lines, let
fall a perpendicular on the line joining the centre of the two
circles; this perpendicular (considered as an indefinite line) is
what Gaultier terms the “radical axis of the two circles”;
it is a line determined by a real construction and itself always
real; and by what precedes it is the line joining two (real or
imaginary, as the case may be) intersections of the given circles.

The intersections which lie on the radical axis are two out of the
four intersections of the two circles. The question as to the
remaining two intersections did not present itself to Gaultier, but
it is answered in Jean Victor Poncelet’s *Traité des propriétés*
*projectives* (1822), where we find (p. 49) the statement, “deux
circles placés arbitrairement sur un plan ... ont idéalement
deux points imaginaires communs à l’infini”; that is, a circle
*qua* curve of the second order is met by the line infinity in two
points; but, more than this, they are the same two points for
any circle whatever. The points in question have since been
called (it is believed first by Dr George Salmon) the circular points
at infinity, or they may be called the circular points; these are
also frequently spoken of as the points I, J; and we have thus
the circle characterized as a conic which passes through the two
circular points at infinity; the number of conditions thus imposed
upon the conic is = 2, and there remain three arbitrary
constants, which is the right number for the circle. Poncelet
throughout his work makes continual use of the foregoing theories
of imaginaries and infinity, and also of the before-mentioned
theory of reciprocal polars.

Poncelet’s two memoirs *Sur les centres des moyennes harmoniques*
and *Sur la théorie générale des polaires réciproques*, although
presented to the Paris Academy in 1824, were only published
(*Crelle*, t. iii. and iv., 1828, 1829) subsequent to the memoir by
Gergonne, *Considérations philosophiques sur les élémens de la*
*science de l’étendue* (*Gerg.* t. xvi., 1825–1826). In this memoir
by Gergonne, the theory of duality is very clearly and explicitly
stated; for instance, we find “dans la géométrie plane, à chaque
théorème il en répond nécessairement un autre qui s’en déduit en
échangeant simplement entre eux les deux mots *points* et *droites*;
tandis que dans la géométrie de l’espace ce sont les mots *points*
et *plans* qu’il faut échanger entre eux pour passer d’un théorème
à son corrélatif”; and the plan is introduced of printing correlative
theorems, opposite to each other, in two columns. There
was a reclamation as to priority by Poncelet in the *Bulletin*
*universel* reprinted with remarks by Gergonne (*Gerg.* t. xix.,
1827), and followed by a short paper by Gergonne, *Rectifications*
*de quelques théorèmes, &c.*, which is important as first introducing
the word *class*. We find in it explicitly the two correlative
definitions: “a plane curve is said to be of the *m*th degree
(order) when it has with a line *m* real or ideal intersections,” and
“a plane curve is said to be of the *m*th class when from any point
of its plane there can be drawn to it *m* real or ideal tangents.”

It may be remarked that in Poncelet’s memoir on reciprocal
polars, above referred to, we have the theorem that the number
of tangents from a point to a curve of the order *m*, or say the class
of the curve, is in general and at most = *m*(*m* − 1), and that he
mentions that this number is subject to reduction when the curve
has double points or cusps.

The theorem of duality as regards plane figures may be
thus stated: two figures may correspond to each other in such
manner that to each point and line in either figure there correspond
in the other figure a line and point respectively. It is
to be understood that the theorem extends to all points or lines,
drawn or not drawn; thus if in the first figure there are any
number of points on a line drawn or not drawn, the corresponding
lines in the second figure, produced if necessary, must meet in
a point. And we thus see how the theorem extends to curves,
their points and tangents; if there is in the first figure a curve
of the order *m*, any line meets it in *m* points; and hence from the
corresponding point in the second figure there must be to the
corresponding curve *m* tangents; that is, the corresponding
curve must be of the class *m*.

Trilinear co-ordinates (see Geometry: *Analytical*) were first
used by E. E. Bobillier in the memoir *Essai sur un nouveau mode*
*de recherche des propriétés de l’étendue* (*Gerg.* t. xviii., 1827–1828).
It is convenient to use these rather than Cartesian co-ordinates.
We represent a curve of the order *m* by an equation (*≬ *x*, *y*, *z*)^{m} = 0,
the function on the left hand being a homogeneous rational and
integral function of the order *m* of the three co-ordinates (*x*, *y*, *z*);
clearly the number of constants is the same as for the equation
(*≬ *x*, *y*, 1)^{m} = 0 in Cartesian co-ordinates.

The theorem of duality is considered and developed, but chiefly
in regard to its metrical applications, by Michel Chasles in the
*Mémoire de géométrie sur deux principes généraux de la science,*
*la dualité et l’homographie*, which forms a sequel to the *Aperçu*
*historique sur l’origine et le développement des méthodes en géométrie*
(*Mém. de Brux.* t. xi., 1837).

We now come to Julius Plücker; his “six equations” were
given in a short memoir in *Crelle* (1842) preceding his great work,
the *Theorie der algebraischen Curven* (1844). Plücker first gave
a scientific dual definition of a curve, viz.; “A curve is a
locus generated by a point, and enveloped by a line—the point
moving continuously along the line, while the line rotates
continuously about the point”; the point is a point (ineunt.)
of the curve, the line is a tangent of the curve. And, assuming
the above theory of geometrical imaginaries, a curve such that
*m* of its points are situate in an arbitrary line is said to be of the
order *m*; a curve such that *n* of its tangents pass through an
arbitrary point is said to be of the class *n*; as already appearing,
this notion of the order and class of a curve is, however, due to
Gergonne. Thus the line is a curve of the order 1 and class 0;
and corresponding dually thereto, we have the point as a curve
of the order 0 and class 1.

Plücker, moreover, imagined a system of line-co-ordinates
(tangential co-ordinates). (See Geometry: *Analytical*.) The
Cartesian co-ordinates (*x*, *y*) and trilinear co-ordinates (*x*, *y*, *z*)
are point-co-ordinates for determining the position of a
point; the new co-ordinates, say (ξ, η, ζ) are line-co-ordinates
for determining the position of a line. It is possible, and
(not so much for any application thereof as in order to
more fully establish the analogy between the two kinds of
co-ordinates) important, to give independent quantitative
definitions of the two kinds of co-ordinates; but we may also
derive the notion of line-co-ordinates from that of point-co-ordinates;
viz. taking ξx + ηy + ζz = 0 to be the equation of
a line, we say that (ξ, η, ζ) are the line-co-ordinates of this line.
A linear relation aξ + bη + cζ = 0 between these co-ordinate
determines a point, viz. the point whose point-co-ordinates are
(a, b, c); in fact, the equation in question aξ + bη + cζ = 0 expresses
that the equation ξx + ηy + ζz = 0, where (*x*, *y*, *z*) are
current point-co-ordinates, is satisfied on writing therein
*x*, *y*, *z* = a, b, c; or that the line in question passes through the
point (a, b, c). Thus (ξ, η, ζ) are the line-co-ordinates of any line
whatever; but when these, instead of being absolutely arbitrary,
are subject to the restriction aξ + bη + cζ = 0, this obliges the line
to pass through a point (a, b, c); and the last-mentioned equation
aξ + bη + cζ = 0 is considered as the line-equation of this point.

A line has only a point-equation, and a point has only a line-equation;
but any other curve has a point-equation and also a
line-equation; the point-equation (*≬ *x*, *y*, *z*)^{m} = 0 is the relation
which is satisfied by the point-co-ordinates (*x*, *y*, *z*) of each point
of the curve; and similarly the line-equation (*≬ ξ, η, ζ)^{n} = 0 is
the relation which is satisfied by the line-co-ordinates (ξ, η, ζ)
of each line (tangent) of the curve.

There is in analytical geometry little occasion for any explicit
use of line-co-ordinates; but the theory is very important; it
serves to show that in demonstrating by point-co-ordinates any
purely descriptive theorem whatever, we demonstrate the correlative
theorem; that is, we do not demonstrate the one theorem,
and then (as by the method of reciprocal polars) deduce from it
the other, but we do at one and the same time demonstrate the
two theorems; our (*x*, *y*, *z*.) instead of meaning point-co-ordinates
may mean line-co-ordinates, and the demonstration is then in
every step of it a demonstration of the correlative theorem.

7. *Singularities of a Curve. Plücker’s Equations.*—The above
dual generation explains the nature of the singularities of a plane
curve. The ordinary singularities, arranged according to a cross
division, are

Proper. | Improper. | |

Point-singularities— | 1. The stationary point, cusp or spinode; | 2. The double point or node; |

Line-singularities— | 3. The stationary tangent or inflection; | 4. The double tangent; |

arising as follows:—

1. The cusp: the point as it travels along the line may come to rest, and then reverse the direction of its motion.

3. The stationary tangent: the line may in the course of its rotation come to rest, and then reverse the direction of its rotation.

2. The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.

4. The double tangent: the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding.

It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.

The singularities (1) and (3) have been termed proper singularities,
and (2) and (4) improper; in each of the first-mentioned
cases there is a real singularity, or peculiarity in the motion;
in the other two cases there is not; in (2) there is not when the
point is first at the node, or when it is secondly at the node, any
peculiarity in the motion; the singularity consists in the point
coming twice into the same position; and so in (4) the singularity
is in the line coming twice into the same position. Moreover
(1) and (2) are, the former a proper singularity, and the latter
an improper singularity, *as regards the motion of the point*; and
similarly (3) and (4) are, the former a proper singularity, and the
latter an improper singularity, *as regards the motion of the line*.

But as regards the representation of a curve by an equation, the case is very different.

First, if the equation be in point-co-ordinates, (3) and (4) are
in a sense not singularities at all. The curve (*≬ *x*, *y*, *z*)^{m} = 0,
or general curve of the order m, has double tangents and inflections;
(2) presents itself as a singularity, for the equations
d_{x}(*≬ *x*, *y*, *z*)^{m} = 0, d_{y}(*≬ *x*, *y*, *z*)^{m} = 0, dz(*≬ *x*, *y*, *z*)^{m} = 0, implying
(*≬ *x*, *y*, *z*)^{m} = 0, are not in general satisfied by any values (a, b, c)
whatever of (*x*, *y*, *z*), but if such values exist, then the point
(a, b, c) is a node or double point; and (1) presents itself as a
further singularity or sub-case of (2), a cusp being a double point
for which the two tangents becomes coincident.

In line-co-ordinates all is reversed:—(1) and (2) are not singularities; (3) presents itself as a sub-case of (4).

The theory of compound singularities will be referred to farther on.

In regard to the ordinary singularities, we have

m, | the | order, |

n | ” | class, |

δ | ” | number of double points, |

κ | ” | ” cusps, |

τ | ” | ” double tangents, |

ι | ” | ” inflections; |

and this being so, Plücker’s “six equations” are

(1) | n = m (m − 1) − 2 δ − 3κ, |

(2) | ι = 3m (m − 2) − 6δ − 8κ, |

(3) | τ = ½m (m − 2) (m^{2} − 9) − (m^{2} − m − 6) (2δ + 3κ) + 2δ (δ − 1)
+ 6δκ + 92κ (κ − 1), |

(4) | m = n (n − 1) − 2τ − 3ι, |

(5) | κ = 3n (n − 2) − 6τ − 8ι, |

(6) | δ = ½n (n − 2) (n^{2} − 9) − (n^{2} − n − 6) (2τ + 3ι) + 2τ (τ − 1) + 6τι
+ 92ι (ι − i). |

It is easy to derive the further forms—

(7) | ι − κ | = 3 (n − m), |

(8) | 2 (τ − δ) | = (n − m) (n + m − 9), |

(9) | ½m (m + 3) − δ − 2κ | = ½n (n + 3) − τ − 2ι, |

(10) | ½ (m − 1) (m − 2) − δ − κ | = ½ (n − 1) (n − 2) − τ − ι, |

(11, 12) | m^{2} − 2δ − 3κ | = n^{2} − 2τ − 3^{ι}, = m + n,— |

the whole system being equivalent to three equations only; and
it may be added that using a to denote the equal quantities
3*m* + ι and 3*n* + κ everything may be expressed in terms of *m*, *n*, *a*.
We have

κ = a − 3n,ι = a − 3m,2δ = m^{2} − m + 8n − 3a.2τ = n^{2} − n + 8m − 3a. |

It is implied in Plücker’s theorem that, *m*, *n*, δ, κ, τ, ι signifying
as above in regard to any curve, then in regard to the reciprocal
curve, *n*, *m*, τ, ι, δ, κ will have the same significations, viz. for the
reciprocal curve these letters denote respectively the order, class,
number of nodes, cusps, double tangent and inflections.

The expression ½*m*(*m* + 3) − δ − 2κ is that of the number of the disposable constants in a curve of the order *m* with δ nodes and κ
cusps (in fact that there shall be a node is 1 condition, a cusp 2
conditions) and the equation (9) thus expresses that the curve and
its reciprocal contain each of them the same number of disposable
constants.

For a curve of the order m, the expression ½*m*(*m* − 1) − δ − κ is
termed the “deficiency” (as to this more hereafter); the equation
(10) expresses therefore that the curve and its reciprocal have each
of them the same deficiency.

The relations *m*^{2} − 2δ − 3κ = *n*^{2} − 2τ − 3ι = *m* + *n*, present themselves
in the theory of envelopes, as will appear farther on.

With regard to the demonstration of Plücker’s equations it
is to be remarked that we are not able to write down the equation
in point-co-ordinates of a curve of the order m, having the given
numbers δ and κ of nodes and cusps. We can only use the general
equation (*≬ *x*, *y*, *z*)^{m} = 0, say for shortness *u* = 0, of a curve of the
*m*th order, which equation, so long as the coefficients remain
arbitrary, represents a curve without nodes or cusps. Seeking
then, for this curve, the values, *n*, ι, τ of the class, number of
inflections, and number of double tangents,—first, as regards
the class, this is equal to the number of tangents which can be
drawn to the curve from an arbitrary point, or what is the same
thing, it is equal to the number of the points of contact of these
tangents. The points of contact are found as the intersections
of the curve *u* = 0 by a curve depending on the position of the
arbitrary point, and called the “first polar” of this point;
the order of the first polar is = *m* − 1, and the number of intersections
is thus = *m*(*m* − 1). But it can be shown, analytically
or geometrically, that if the given curve has a node, the first
polar passes through this node, which therefore counts as two
intersections, and that if the curve has a cusp, the first polar
passes through the cusp, touching the curve there, and hence
the cusp counts as three intersections. But, as is evident, the
node or cusp is not a point of contact of a proper tangent from the
arbitrary point; we have, therefore, for a node a diminution 2,
and for a cusp a diminution 3, in the number of the intersections;
and thus, for a curve with δ nodes and κ cusps, there is a diminution
2δ + 3κ, and the value of *n* is *n* = *m*(*m* − 1) − 2δ −
3κ.

Secondly, as to the inflections, the process is a similar one; it
can be shown that the inflections are the intersections of the
curve by a derivative curve called (after Ludwig Otto Hesse
who first considered it) the Hessian, defined geometrically as
the locus of a point such that its conic polar (§8 below) in regard
to the curve breaks up into a pair of lines, and which has an
equation H = 0, where H is the determinant formed with the
second differential coefficients of *u* in regard to the variables
(*x*, *y*, *z*); H = 0 is thus a curve of the order 3(*m* − 2), and the
number of inflections is = 3*m*(*m* − 2). But if the given curve
has a node, then not only the Hessian passes through the node,
but it has there a node the two branches at which touch respectively
the two branches of the curve; and the node thus
counts as six intersections; so if the curve has a cusp, then the
Hessian not only passes through the cusp, but it has there a cusp
through which it again passes, that is, there is a cuspidal branch
touching the cuspidal branch of the curve, and besides a simple
branch passing through the cusp, and hence the cusp counts as
eight intersections. The node or cusp is not an inflection, and we
have thus for a node a diminution 6, and for a cusp a diminution 8,
in the number of the intersections; hence for a curve with δ nodes
and κ cusps, the diminution is = 6δ + 8κ, and the number of
inflections is ι = 3*m*(*m* − 2) − 6δ − 8κ.

Thirdly, for the double tangents; the points of contact of
these are obtained as the intersections of the curve by a curve
Π = 0, which has not as yet been geometrically defined, but which
is found analytically to be of the order (*m* − 2)(*m*^{2} − 9); the
number of intersections is thus = *m*(*m* − 2)(*m*^{2} − 9); but if the
given curve has a node then there is a diminution = 4(*m*^{2} − *m* − 6),
and if it has a cusp then there is a diminution = 6(*m*^{2} − *m* − 6),
where, however, it is to be noticed that the factor (*m*^{2} − *m* − 6)
is in the case of a curve having only a node or only a cusp the
number of the tangents which can be drawn from the node or cusp
to the curve, and is used as denoting the number of these tangents,
and ceases to be the correct expression if the number of nodes
and cusps is greater than unity. Hence, in the case of a curve
which has δ nodes and κ cusps, the apparent diminution
2(*m*^{2} − *m* − 6)(2δ + 3κ) is too great, and it has in fact to be
diminished by 2{2δ(δ − 1) + 6δκ +
92κ(κ − 1)}, or the half thereof is
4 for each pair of nodes, 6 for each combination of a node and
cusp, and 9 for each pair of cusps. We have thus finally an expression
for 2τ, = *m*(*m* − 2)(*m*^{2} − 9) − &c.; or dividing the whole
by 2, we have the expression for τ given by the third of Plücker’s
equations.

It is obvious that we cannot by consideration of the equation
*u* = 0 in point-co-ordinates obtain the remaining three of Plücker’s
equations; they might be obtained in a precisely analogous
manner by means of the equation *v* = 0 in line-co-ordinates, but they
follow at once from the principle of duality, viz. they are obtained
by the mere interchange of *m*, δ, κ, with *n*, τ, ι respectively.

To complete Plücker’s theory it is necessary to take account
of compound singularities; it might be possible, but it is at any
rate difficult, to effect this by considering the curve as in course of
description by the point moving along the rotating line; and it
seems easier to consider the compound singularity as arising
from the variation of an actually described curve with ordinary
singularities. The most simple case is when three double points
come into coincidence, thereby giving rise to a triple point;
and a somewhat more complicated one is when we have a cusp
of the second kind, or node-cusp arising from the coincidence
of a node, a cusp, an inflection, and a double tangent, as shown
in the annexed figure, which represents the singularities as on the
point of coalescing. The general conclusion (see Cayley, *Quart.*
*Math. Jour.* t. vii., 1866, “On the higher singularities of plane
curves”; *Collected Works*, v. 520) is that every singularity
whatever may be considered as compounded of ordinary singularities,
say we have a singularity = δ′ nodes, κ′ cusps, τ′ double
tangents and ι′ inflections. So that, in fact, Plücker’s equations
properly understood apply to a curve with any singularities
whatever.

By means of Plücker’s equations we may form a table—

m | n | δ | κ | τ | ι |

0 | 1 | − | − | 0 | 0 |

1 | 0 | 0 | 0 | − | − |

2 | 2 | 0 | 0 | 0 | 0 |

3 | 6 | 0 | 0 | 0 | 9 |

” | 4 | 1 | 0 | 0 | 3 |

” | 3 | 0 | 1 | 0 | 1 |

4 | 12 | 0 | 0 | 28 | 24 |

” | 10 | 1 | 0 | 16 | 18 |

” | 9 | 0 | 1 | 10 | 16 |

” | 8 | 2 | 0 | 8 | 12 |

” | 7 | 1 | 1 | 4 | 10 |

” | 6 | 0 | 2 | 1 | 8 |

” | 6 | 3 | 0 | 4 | 6 |

” | 5 | 2 | 1 | 2 | 4 |

” | 4 | 1 | 2 | 1 | 2 |

” | 3 | 0 | 3 | 1 | 0 |

The table is arranged according to the value of m; and we have
m = 0, *n* = 1, the point; *m* = 1, *n* = 0, the line; *m* = 2, *n* = 2, the
conic; of *m* = 3, the cubic, there are three cases, the class being
6, 4 or 3, according as the curve is without singularities, or as it
has 1 node or 1 cusp; and so of *m* = 4, the quartic, there are ten
cases, where observe that in two of them the class is = 6,—the
reduction of class arising from two cusps or else from three nodes.
The ten cases may be also grouped together into four, according
as the number of nodes and cusps (δ + κ) is = 0, 1, 2 or 3.

The cases may be divided into sub-cases, by the consideration
of compound singularities; thus when *m* = 4, *n* = 6, δ = 3, the
three nodes may be all distinct, which is the general case, or two
of them may unite together into the singularity called a tacnode,
or all three may unite together into a triple point or else into an
oscnode.

We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

The expression for the number of inflections 3*m*(*m* − 2) for a
curve of the order *m* was obtained analytically by Plücker,
but the theory was first given in a complete form by Hesse in
the two papers “Über die Elimination, u.s.w.,” and “Über
die Wendepuncte der Curven dritter Ordnung” (*Crelle*, t. xxviii.,
1844); in the latter of these the points of inflection are obtained
as the intersections of the curve *u* = 0 with the Hessian, or curve
Δ = 0, where Δ is the determinant formed with the second derived
functions of *u*. We have in the Hessian the first instance of a
covariant of a ternary form. The whole theory of the inflections
of a cubic curve is discussed in a very interesting manner by
means of the canonical form of the equation *x*^{3} + *y*^{3} + *z*^{3} + 6*lxyz* = 0;
and in particular a proof is given of Plücker’s theorem that the
nine points of inflection of a cubic curve lie by threes in twelve
lines.

It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin. For an acnodal cubic the six imaginary inflections disappear, and there remain three real inflections lying in a line. For a crunodal cubic the six inflections which disappear are two of them real, the other four imaginary, and there remain two imaginary inflections and one real inflection. For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.

A quartic curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H. G. Zeuthen that at most eight of these are real.

The expression ½*m*(*m* − 2)(*m*^{2} − 9) for the number of double
tangents of a curve of the order *m* was obtained by Plücker only
as a consequence of his first, second, fourth and fifth equations.
An investigation by means of the curve Π = 0, which by its intersections
with the given curve determines the points of contact of
the double tangents, is indicated by Cayley, “Recherches sur
l’élimination et la théorie des courbes” (*Crelle*, t. xxxiv., 1847;
*Collected Works*, vol. i. p. 337), and in part carried out by Hesse
in the memoir “Über Curven dritter Ordnung” (*Crelle*, t.
xxxvi., 1848). A better process was indicated by Salmon in
the “Note on the Double Tangents to Plane Curves,” *Phil. Mag.*,
1858; considering the *m* − 2 points in which any tangent to
the curve again meets the curve, he showed how to form the
equation of a curve of the order (*m* − 2), giving by its intersection
with the tangent the points in question; making the
tangent touch this curve of the order (*m* − 2), it will be a double
tangent of the original curve. See Cayley, “On the Double
Tangents of a Plane Curve” (*Phil. Trans.* t. cxlviii., 1859;
*Collected Works*, iv. 186), and O. Dersch (*Math. Ann.* t. vii.,
1874). The solution is still in so far incomplete that we have no
properties of the curve Π = 0, to distinguish one such curve from
the several other curves which pass through the points of contact
of the double tangents.

A quartic curve has 28 double tangents, their points of contact
determined as the intersections of the curve by a curve Π = 0
of the order 14, the equation of which in a very elegant form was
first obtained by Hesse (1849). Investigations in regard to them
are given by Plücker in the *Theorie der algebraischen Curven*,
and in two memoirs by Hesse and Jacob Steiner (*Crelle*, t. xlv.,
1855), in respect to the triads of double tangents which have their
points of contact on a conic and other like relations. It was
assumed by Plücker that the number of real double tangents
might be 28, 16, 8, 4 or 0, but Zeuthen has found that the last
case does not exist.

8. *Invariants and Covariants.* *Polar Curves.*—The Hessian Δ
has just been spoken of as a covariant of the form u; the notion
of invariants and covariants belongs rather to the form *u* than
to the curve *u* = 0 represented by means of this form; and the
theory may be very briefly referred to. A curve *u* = 0 may have
some invariantive property, viz. a property independent of
the particular axes of co-ordinates used in the representation
of the curve by its equation; for instance, the curve may have
a node, and in order to this, a relation, say A = 0, must exist
between the coefficients of the equation; supposing the axes
of co-ordinates altered, so that the equation becomes u′ = 0, and
writing A′ = 0 for the relation between the new coefficients, then
the relations A = 0, A′ = 0, as two different expressions of the
same geometrical property, must each of them imply the other;
this can only be the case when A, A′ are functions differing
only by a constant factor, or say, when A is an invariant of *u*.
If, however, the geometrical property requires two or more relations
between the coefficients, say A = 0, B = 0, &c., then we must
have between the new coefficients the like relations, A′ = 0, B′ = 0,
&c., and the two systems of equations must each of them imply
the other; when this is so, the system of equations, A = 0, B = 0,
&c., is said to be invariantive, but it does not follow that A, B,
&c., are of necessity invariants of *u*. Similarly, if we have a
curve U = 0 derived from the curve *u* = 0 in a manner independent
of the particular axes of co-ordinates, then from the transformed
equation u’ = 0 deriving in like manner the curve U′ = 0, the
two equations U = 0, U′ = 0 must each of them imply the other;
and when this is so, U will be a covariant of *u*. The case is less
frequent, but it may arise, that there are covariant systems
U = 0, V = 0, &c., and U′ = 0, V′ = 0, &c., each implying the other,
but where the functions U, V, &c., are not of necessity covariants
of *u*.

If we take a fixed point (*x*′, *y*′, *z*′) and a curve *u* = 0 of order
m, and suppose the axes of reference altered, so that *x*′, *y*′, *z*′
are linearly transformed in the same way as the current *x*, *y*, *z*,
the curves [*x*′(∂/∂x) + *y*′(∂/∂y) + *z*′(∂/∂z)]^{r}*u* = 0, (*r* = 1, 2, ... *m* − 1) have the
covariant property. They are the polar curves of the point with
regard to *u* = 0.

The theory of the invariants and covariants of a ternary cubic
function *u* has been studied in detail, and brought into connexion
with the cubic curve *u* = 0; but the theory of the invariants and
covariants for the next succeeding case, the ternary quartic
function, is still very incomplete.

9. *Envelope of a Curve.*—In further illustration of the Plückerian
dual generation of a curve, we may consider the question of the
*envelope* of a variable curve. The notion is very probably older,
but it is at any rate to be found in Lagrange’s *Théorie des fonctions*
*analytiques* (1798); it is there remarked that the equation obtained
by the elimination of the parameter a from an equation ƒ(*x*, *y*, *a*) = 0
and the derived equation in respect to a is a curve, the envelope
of the series of curves represented by the equation ƒ(*x*, *y*, *a*) = 0
in question. To develop the theory, consider the curve corresponding
to any particular value of the parameter; this has
with the consecutive curve (or curve belonging to the consecutive
value of the parameter) a certain number of intersections and
of common tangents, which may be considered as the tangents
at the intersections; and the so-called envelope is the curve
which is at the same time generated by the points of intersection
and enveloped by the common tangents; we have thus a dual
generation. But the question needs to be further examined.
Suppose that in general the variable curve is of the order *m* with
δ nodes and κ cusps, and therefore of the class *n* with τ double
tangents and ι inflections, *m*, *n*, δ, κ, τ, ι being connected by the
Plückerian equations,—the number of nodes or cusps may be
greater for particular values of the parameter, but this is a
speciality which may be here disregarded. Considering the variable
curve corresponding to a given value of the parameter,
or say simply the variable curve, the consecutive curve has then
also δ and κ nodes and cusps, consecutive to those of the variable
curve; and it is easy to see that among the intersections of the
two curves we have the nodes each counting twice, and the cusps
each counting three times; the number of the remaining intersections
is = *m*^{2} − 2δ − 3κ. Similarly among the common tangents
of the two curves we have the double tangents each counting
twice, and the stationary tangents each counting three times, and
the number of the remaining common tangents is = *n*^{2} − 2τ − 3ι
(= *m*^{2} − 2δ − 3κ, inasmuch as each of these numbers is as was
seen = *m* + *n*). At any one of the *m*^{2} − 2δ − 3κ points the variable
curve and the consecutive curve have tangents distinct from yet
infinitesimally near to each other, and each of these two tangents
is also infinitesimally near to one of the *n*^{2} − 2τ − 3ι common
tangents of the two curves; whence, attending only to the
variable curve, and considering the consecutive curve as coming
into actual coincidence with it, the *n*^{2} − 2τ − 3ι common tangents
are the tangents to the variable curve at the *m*^{2} − 2δ − 3κ points
respectively, and the envelope is at the same time generated
by the *m*^{2} − 2δ − 3κ points, and enveloped by the *n*^{2} − 2τ − 3ι
tangents; we have thus a dual generation of the envelope,
which only differs from Plücker’s dual generation, in that in place
of a single point and tangent we have the group of *m*^{2} − 2δ − 3κ
points and *n*^{2} − 2τ − 3ι tangents.

The parameter which determines the variable curve may be given as a point upon a given curve, or say as a parametric point; that is, to the different positions of the parametric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the given curve; we have thus the envelope as a derivative curve of the given curve. Many well-known derivative curves present themselves in this manner; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve; the intersection of the consecutive normals is the centre of curvature; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal. It may be added that the given curve is one of a series of curves, each cutting the several normals at right angles. Any one of these is a “parallel” of the given curve; and it can be obtained as the envelope of a circle of constant radius having its centre on the given curve. We have in like manner, as derivatives of a given curve, the caustic, catacaustic or diacaustic as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays.

10. *Forms of Real Curves.*—We have in much that precedes
disregarded, or at least been indifferent to, reality; it is only thus
that the conception of a curve of the *m*-th order, as one which
is met by every right line in *m* points, is arrived at; and the curve
itself, and the line which cuts it, although both are tacitly
assumed to be real, may perfectly well be imaginary. For
real figures we have the general theorem that imaginary intersections,
&c., present themselves in conjugate pairs; hence, in
particular, that a curve of an even order is met by a line in an
even number (which may be = 0) of points; a curve of an odd
order in an odd number of points, hence in one point at least;
it will be seen further on that the theorem may be generalized in
a remarkable manner. Again, when there is in question only
one pair of points or lines, these, if coincident, must be real;
thus, a line meets a cubic curve in three points, one of them
real, and other two real or imaginary; but if two of the intersections
coincide they must be real, and we have a line cutting
a cubic in one real point and touching it in another real point.
It may be remarked that this is a limit separating the two cases
where the intersections are all real, and where they are one real,
two imaginary.

Considering always real curves, we obtain the notion of a
branch; any portion capable of description by the continuous
motion of a point is a branch; and a curve consists of one or
more branches. Thus the curve of the first order or right line
consists of one branch; but in curves of the second order, or
conics, the ellipse and the parabola consist each of one branch, the
hyperbola of two branches. A branch is either re-entrant, or
it extends both ways to infinity, and in this case, we may regard
it as consisting of two legs (*crura*, Newton), each extending one
way to infinity, but without any definite separation. The branch,
whether re-entrant or infinite, may have a cusp or cusps, or it may
cut itself or another branch, thus having or giving rise to crunodes
or double points with distinct real tangents; an acnode, or
double point with imaginary tangents, is a branch by itself,—it
may be considered as an indefinitely small re-entrant branch.
a branch may have inflections and double tangents, or there
may be double tangents which touch two distinct branches;
there are also double tangents with imaginary points of contact,
which are thus lines having no visible connexion with the curve.
A re-entrant branch not cutting itself may be everywhere
convex, and it is then properly said to be an oval; but the term
oval may be used more generally for any re-entrant branch not
cutting itself; and we may thus speak of a once indented, twice
indented oval, &c., or even of a cuspidate oval. Other descriptive
names for ovals and re-entrant branches cutting themselves
may be used when required; thus, in the last-mentioned case
a simple form is that of a figure of eight; such a form may break
up into two ovals or into a doubly indented oval or hour-glass.
A form which presents itself is when two ovals, one inside the
other, unite, so as to give rise to a crunode—in default of a better
name this may be called, after the curve of that name, a limaçon
(*q.v.*). Names may also be used for the different forms of infinite
branches, but we have first to consider the distinction of hyperbolic
and parabolic. The leg of an infinite branch may have at
the extremity a tangent; this is an asymptote of the curve,
and the leg is then hyperbolic; or the leg may tend to a fixed
direction, but so that the tangent goes further and further off
to infinity, and the leg is then parabolic; a branch may thus
be hyperbolic or parabolic as to its two legs; or it may be hyperbolic
as to one leg and parabolic as to the other. The epithets
hyperbolic and parabolic are of course derived from the conic
hyperbola and parabola respectively. The nature of the two
kinds of branches is best understood by considering them as
projections, in the same way as we in effect consider the hyperbola
and the parabola as projections of the ellipse. If a line Ω cut
an arc *aa*′ at *b*, so that the two segments *ab*, *ba*′ lie on opposite
sides of the line, then projecting the figure so that the line Ω goes
off to infinity, the tangent at *b* is projected into the asymptote,
and the arc *ab* is projected into a hyperbolic leg touching the
asymptote at one extremity; the arc *ba*′ will at the same time
be projected into a hyperbolic leg touching the same asymptote
at the other extremity (and on the opposite side), but so that the
two hyperbolic legs may or may not belong to one and the same
branch. And we thus see that the two hyperbolic legs belong
to a simple intersection of the curve by the line infinity. Next,
if the line Ω touch at *b* the arc *aa*′ so that the two portions
*ab*, *ba*′ lie on the same side of the line Ω, then projecting the
figure as before, the tangent at b, that is, the line Ω itself, is
projected to infinity; the arc *ab* is projected into a parabolic
leg, and at the same time the arc *ba*′ is projected into a parabolic
leg, having at infinity the same direction as the other leg, but so
that the two legs may or may not belong to the same branch.
And we thus see that the two parabolic legs represent a contact
of the line infinity with the curve,—the point of contact being
of course the point at infinity determined by the common direction
of the two legs. It will readily be understood how the like
considerations apply to other cases,—for instance, if the line Ω
is a tangent at an inflection, passes through a crunode, or touches
one of the branches of a crunode, &c.; thus, if the line Ω passes
through a crunode we have pairs of hyperbolic legs belonging
to two parallel asymptotes. The foregoing considerations also
show (what is very important) how different branches are connected
together at infinity, and lead to the notion of a complete
branch or circuit.

The two legs of a hyperbolic branch may belong to different
asymptotes, and in this case we have the forms which Newton
calls inscribed, circumscribed, ambigene, &c.; or they may
belong to the same asymptote, and in this case we have the
serpentine form, where the branch cuts the asymptote, so as
to touch it at its two extremities on opposite sides, or the
conchoidal form, where it touches the asymptote on the same
side. The two legs of a parabolic branch may converge to
ultimate parallelism, as in the conic parabola, or diverge to
ultimate parallelism, as in the semi-cubical parabola *y*^{2} = *x*^{3}, and
the branch is said to be convergent, or divergent, accordingly;
or they may tend to parallelism in opposite senses, as in the
cubical parabola *y* = *x*^{3}. As mentioned with regard to a branch
generally, an infinite branch of any kind may have cusps, or,
by cutting itself or another branch, may have or give rise to a
crunode, &c.

11. *Classification of Cubic Curves.*—We may now consider
the various forms of cubic curves as appearing by Newton’s
*Enumeratio*, and by the figures belonging thereto. The species
are reckoned as 72, which are numbered accordingly 1 to 72;
but to these should be added 10^{a}, 13^{a}, 22^{a} and 22^{b}. It is not
intended here to consider the division into species, nor even
completely that into genera, but only to explain the principle of
classification. It may be remarked generally that there are at
most three infinite branches, and that there may besides be a
re-entrant branch or oval.

The genera may be arranged as follows:—

1,2,3,4 | redundant hyperbolas |

5,6 | defective hyperbolas |

7,8 | parabolic hyperbolas |

9 | hyperbolisms of hyperbola |

10 | hyperbolisms of ellipse |

11 | hyperbolisms of parabola |

12 | trident curve |

13 | divergent parabolas |

14 | cubic parabola; |

and thus arranged they correspond to the different relations of the line infinity to the curve. First, if the three intersections by the line infinity are all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the defective hyperbolas, with one hyperbolic branch. Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point,—viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola. As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity. Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection, a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch, and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.

It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves; Newton gives (in two short paragraphs without any development) the remarkable theorem that the five divergent parabolas by their shadows generate and exhibit all the cubic curves.

The five divergent parabolas are curves each of them symmetrical
with regard to an axis. There are two non-singular
kinds, the one with, the other without, an oval, but each of them
has an infinite (as Newton describes it) *campaniform* branch;
this cuts the axis at right angles, being at first concave, but
ultimately convex, towards the axis, the two legs continually
tending to become at right angles to the axis. The oval may
unite itself with the infinite branch, or it may dwindle into a
point, and we have the crunodal and the acnodal forms respectively;
or if simultaneously the oval dwindles into a point and
unites itself to the infinite branch, we have the cuspidal form.
(See Parabola.) Drawing a line to cut any one of these curves
and projecting the line to infinity, it would not be difficult to
show how the line should be drawn in order to obtain a curve
of any given species. We have herein a better principle of classification;
considering cubic curves, in the first instance, according
to singularities, the curves are non-singular, nodal (viz. crunodal
or acnodal), or cuspidal; and we see further that there are two
kinds of non-singular curves, the complex and the simplex.
There is thus a complete division into the five kinds, the complex,
simplex, crunodal, acnodal and cuspidal. Each singular kind
presents itself as a limit separating two kinds of inferior singularity;
the cuspidal separates the crunodal and the acnodal, and
these last separate from each other the complex and the simplex.

The whole question is discussed very fully and ably by A. F.
Möbius in the memoir “Ueber die Grundformen der Linien
dritter Ordnung” (*Abh. der K. Sachs. Ges. zu Leipzig*, t. i., 1852).
The author considers not only plane curves, but also cones, or,
what is almost the same thing, the spherical curves which are
their sections by a concentric sphere. Stated in regard to the
cone, we have there the fundamental theorem that there are two
different kinds of sheets; viz., the single sheet, not separated
into two parts by the vertex (an instance is afforded by the plane
considered as a cone of the first order generated by the motion
of a line about a point), and the double or twin-pair sheet,
separated into two parts by the vertex (as in the cone of the
second order). And it then appears that there are two kinds
of non-singular cubic cones, viz. the simplex, consisting of a
single sheet, and the complex, consisting of a single sheet and a
twin-pair sheet; and we thence obtain (as for cubic curves)
the crunodal, the acnodal and the cuspidal kinds of cubic cones.
It may be mentioned that the single sheet is a sort of wavy form,
having upon it three lines of inflection, and which is met by any
plane through the vertex in one or in three lines; the twin-pair
sheet has no lines of inflection, and resembles in its form a cone
on an oval base.

In general a cone consists of one or more single or twin-pair
sheets, and if we consider the section of the cone by a plane,
the curve consists of one or more complete branches, or say
circuits, each of them the section of one sheet of the cone; thus,
a cone of the second order is one twin-pair sheet, and any section
of it is one circuit composed, it may be, of two branches. But
although we thus arrive by projection at the notion of a circuit,
it is not necessary to go out of the plane, and we may (with
Zeuthen, using the shorter term *circuit* for his *complete branch*)
define a circuit as any portion (of a curve) capable of description
by the continuous motion of a point, it being understood that
a passage through infinity is permitted. And we then say that
a curve consists of one or more circuits; thus the right line, or
curve of the first order, consists of one circuit; a curve of the
second order consists of one circuit; a cubic curve consists of one
circuit or else of two circuits.

A circuit is met by any right line always in an even number, or always in an odd number, of points, and it is said to be an even circuit or an odd circuit accordingly; the right line is an odd circuit, the conic an even circuit. And we have then the theorem, two odd circuits intersect in an odd number of points; an odd and an even circuit, or two even circuits, in an even number of points. An even circuit not cutting itself divides the plane into two parts, the one called the internal part, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.

We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none,—four if the point is on the odd circuit, none if it is on the even circuit. It at once appears from inspection of the figure of a non-singular cubic curve, which is the odd and which the even circuit. The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode must be regarded as an even circuit.

12. *Quartic Curves.*—The analogous question of the classification
of quartics (in particular non-singular quartics and nodal
quartics) is considered in Zeuthen’s memoir “Sur les différentes
formes des courbes planes du quatrième ordre” (*Math. Ann.*
t. vii., 1874). A non-singular quartic has only even circuits;
it has at most four circuits external to each other, or two circuits
one internal to the other, and in this last case the internal circuit
has no double tangents or inflections. A very remarkable
theorem is established as to the double tangents of such a quartic:
distinguishing as a double tangent of the first kind a real double
tangent which either twice touches the same circuit, or else
touches the curve in two imaginary points, the number of the
double tangents of the first kind of a non-singular quartic is
= 4; it follows that the quartic has at most 8 real inflections.
The forms of the non-singular quartics are very numerous, but
it is not necessary to go further into the question.

We may consider in relation to a curve, not only the line
infinity, but also the circular points at infinity; assuming the
curve to be real, these present themselves always conjointly;
thus a circle is a conic passing through the two circular points,
and is thereby distinguished from other conics. Similarly a
cubic through the two circular points is termed a circular cubic;
a quartic through the two points is termed a circular quartic,
and if it passes twice through each of them, that is, has each of
them for a node, it is termed a bicircular quartic. Such a quartic
is of course binodal (*m* = 4, δ = 2, κ = 0); it has not in general,
but it may have, a third node or a cusp. Or again, we may have
a quartic curve having a cusp at each of the circular points:
such a curve is a “Cartesian,” it being a complete definition of
the Cartesian to say that it is a bicuspidal quartic curve (*m* = 4,
δ = 0, κ = 2), having a cusp at each of the circular points. The
circular cubic and the bicircular quartic, together with the
Cartesian (being in one point of view a particular case thereof),
are interesting curves which have been much studied, generally,
and in reference to their *focal* properties.

13. *Foci.*—The points called *foci* presented themselves in the
theory of the conic, and were well known to the Greek geometers,
but the general notion of a focus was first established by Plücker
(in the memoir “Über solche Puncte die bei Curven einer
höheren Ordnung den Brennpuncten der Kegelschnitte entsprechen”
(*Crelle*, t. x., 1833). We may from each of the circular
points draw tangents to a given curve; the intersection of two
such tangents (belonging of course to the two circular points
respectively) is a focus. There will be from each circular point
λ tangents (λ, a number depending on the class of the curve and
its relation to the line infinity and the circular points, = 2 for
the general conic, 1 for the parabola, 2 for a circular cubic, or
bicircular quartic, &c.); the λ tangents from the one circular
point and those from the other circular point intersect in λ real
foci (viz. each of these is the only real point on each of the tangents
through it), and in λ^{2} − λ imaginary foci; each pair of real foci
determines a pair of imaginary foci (the so-called antipoints
of the two real foci), and the ½λ(λ − 1) pairs of real foci thus
determine the λ^{2} − λ imaginary foci. There are in some cases
points termed centres, or singular or multiple foci (the nomenclature
is unsettled), which are the intersections of improper
tangents from the two circular points respectively; thus, in the
circular cubic, the tangents to the curve at the two circular
points respectively (or two imaginary asymptotes of the curve)
meet in a centre.

14. *Distance and Angle.* *Curves described mechanically.*—The
notions of *distance* and of lines *at right angles* are connected with
the circular points; and almost every construction of a curve
by means of lines of a determinate length, or at right angles
to each other, and (as such) mechanical constructions by means
of linkwork, give rise to curves passing the same definite number
of times through the two circular points respectively, or say to
circular curves, and in which the fixed centres of the construction
present themselves as ordinary, or as singular, foci. Thus the
general curve of three bar-motion (or locus of the vertex of a
triangle, the other two vertices whereof move on fixed circles)
is a tricircular sextic, having besides three nodes (*m* = 6,
δ = 3 + 3 + 3 = 9), and having the centres of the fixed circles each
for a singular focus; there is a third singular focus, and we have
thus the remarkable theorem (due to S. Roberts) of the triple
generation of the curve by means of the three several pairs of
singular foci.

Again, the normal, *qua* line at right angles to the tangent,
is connected with the circular points, and these accordingly
present themselves in the before-mentioned theories of evolutes
and parallel curves.

15. *Theories of Correspondence.*—We have several recent
theories which depend on the notion of *correspondence*: two
points whether in the same plane or in different planes, or on
the same curve or in different curves, may determine each other
in such wise that to any given position of the first point there
correspond α′ positions of the second point, and to any given
position of the second point a positions of the first point; the
two points have then an (α, α) correspondence; and if α, α are
each = 1, then the two points have α (1, 1) or rational correspondence.
Connecting with each theory the author’s name, the
theories in question are G. F. B. Riemann, the rational transformation
of a plane curve; Luigi Cremona, the rational transformation
of a plane; and Chasles, correspondence of points on
the same curve, and united points. The theory first referred to,
with the resulting notion of “Geschlecht,” or *deficiency*, is more
than the other two an essential part of the theory of curves, but
they will all be considered.

Riemann’s results are contained in the memoirs on “Abelian
Integrals,” &c. (*Crelle*, t. liv., 1857), and we have next R. F. A.
Clebsch, “Über die Singularitäten algebraischer Curven”
(*Crelle*, t. lxv., 1865), and Cayley, “On the Transformation of
Plane Curves” (*Proc. Lond. Math. Soc.* t. i., 1865; *Collected*
*Works*, vol. vi. p. 1). The fundamental notion of the rational
transformation is as follows:—

Taking *u*, X, Y, Z to be rational and integral functions (X, Y, Z
all of the same order) of the co-ordinates (*x*, *y*, *z*), and *u*′, X′, Y′, Z′
rational and integral functions (X′, Y′, Z′, all of the same order) of
the co-ordinates ( *x*′, *y*′, *z*′), we transform a given curve *u* = 0, by the
equations of *x*′ : *y*′ : *z*′ = X : Y : Z, thereby obtaining a transformed
curve *u*′ = 0, and a converse set of equations *x* : *y* : *z* = X′ : Y′ : Z′;
viz. assuming that this is so, the point (*x*, *y*, *z*) on the curve *u* = 0
and the point ( *x*′, *y*′, *z*′) on the curve *u*′ = 0 will be points having a
(1, 1) correspondence. To show how this is, observe that to a given
point (*x*, *y*, *z*) on the curve *u* = 0 there corresponds a single point
( *x*′, *y*′, *z*′) determined by the equations *x*′ : *y*′ : *z*′ = X : Y : Z; from
these equations and the equation *u* = 0 eliminating *x*, *y*, *z*, we obtain
the equation *u*′ = 0 of the transformed curve. To a given point
( *x*′, *y*′, *z*′) not on the curve *u*’ = 0 there corresponds, not a single point,
but the system of points (*x*, *y*, *z*) given by the equations *x*′ : *y*′ : *z*′ =
X : Y : Z, viz., regarding *x*′, *y*′, *z*′ as constants (and to fix the ideas,
assuming that the curves X = 0, Y = 0, Z = 0, have no common intersections),
these are the points of intersection of the curves X : Y : Z,
= *x*′ : *y*′ : *z*′, but no one of these points is situate on the curve *u* = 0.
If, however, the point ( *x*′, *y*′, *z*′) is situate on the curve *u*′ = 0, then
one point of the system of points in question is situate on the curve
u = 0, that is, to a given point of the curve *u*′ = 0 there corresponds
a single point of the curve *u* = 0; and hence also this point must
be given by a system of equations such as *x* : *y* : *z* = X′ : Y′ : Z′.

It is an old and easily proved theorem that, for a curve of
the order *m*, the number δ + κ of nodes and cusps is at most
= ½(*m* − 1)(*m* − 2); for a given curve the deficiency of the actual
number of nodes and cusps below this maximum number, viz.
½(*m* − 1)(*m* − 2) − δ − κ, is the “Geschlecht” or “deficiency,”
of the curve, say this is = D. When D = 0, the curve is said to be
unicursal, when = 1, bicursal, and so on.

The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this rational or (1, 1) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]

A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D; and a curve of any order deficiency = or > 3 can be rationally transformed into a curve of the order D + 3, deficiency D.

Taking *x*′, *y*′, *z*′ as co-ordinates of a point of the transformed curve, and in its equation writing *x*′ : *y*′ : *z*′ = 1 : θ : φ we have φ a certain
irrational function of θ, and the theorem is that the co-ordinates *x*, *y*, *z*
of any point of the given curve can be expressed as proportional to
rational and integral functions of θ, φ, that is, of θ and a certain
irrational function of θ.

In particular if D = 0, that is, if the given curve be unicursal, the
transformed curve is a line, φ is a mere linear function of θ, and
the theorem is that the co-ordinates *x*, *y*, *z* of a point of the unicursal
curve can be expressed as proportional to rational and integral
functions of θ; it is easy to see that for a given curve of the order
*m*, these functions of θ must be of the same order *m*.

If D = 1, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, φ
can be expressed as the square root of a quartic function of θ; and
the theorem is that the co-ordinates *x*, *y*, *z* of a point of the bicursal
curve can be expressed as proportional to rational and integral
functions of θ, and of the square root of a quartic function of θ.

And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and
the theorem is, that the co-ordinates *x*, *y*, *z* of a point of the tricursal
curve can be expressed as proportional to rational and integral
functions of θ, and of the square root of a sextic function of θ. But
D = 3, we have no longer the like law, viz. φ is not expressible as
the square root of an octic function of θ.

Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.

It is a form of the theorem for the case D = 1, that the co-ordinates
*x*, *y*, *z* of a point of the bicursal curve, or in particular
the co-ordinates of a point of the cubic, can be expressed as
proportional to rational and integral functions of the elliptic
functions sn*u*, cn*u*, dn*u*; in fact, taking the radical to be
√1 − θ²·1 − *k*²θ², and writing θ = sn*u*, the radical becomes
= cn*u*, dn*u*; and we have expressions of the form in question.

It will be observed that the equations *x*′ : *y*′ : *z*′ = X : Y : Z
before mentioned do not of themselves lead to the other system
of equations *x* : *y* : *z* = X′ : Y′ : Z′, and thus that the theory does
not in anywise establish a (1, 1) correspondence between the
points (*x*, *y*, *z*) and (*x*′, *y*′, *z*′) of two planes or of the same
plane; this is the correspondence of Cremona’s theory.

In this theory, given in the memoirs “Sulle trasformazioni geometriche
delle figure piani,” *Mem. di Bologna*, t. ii. (1863) and t. v.
(1865), we have a system of equations *x*′ : *y*′ : *z*′ = X : Y : Z which
*does* lead to a system *x* : *y* : *z* = X′ : Y′ : Z′, where, as before, X, Y, Z
denote rational and integral functions, all of the same order, of the
co-ordinates *x*, *y*, *z*, and X′, Y′, Z′ rational and integral functions, all
of the same order, of the co-ordinates *x*′, *y*′, *z*′, and there is thus a
(1, 1) correspondence given by these equations between the two
points (*x*, y, z) and (*x*′, *y*′, *z*′). To explain this, observe that starting
from the equations of *x*′ : *y*′ : *z*′ = X : Y : Z, to a given point (*x*, *y*, *z*)
there corresponds one point ( *x*′, *y*′, *z*′), but that if n be the order of
the functions X, Y, Z, then to a given point *x*′, *y*′, *z*′ there would, if
the curves X = 0, Y = 0, Z = 0 had no common intersections, correspond
*n*^{2} points (*x*, *y*, *z*). If, however, the functions are such that
the curves X = 0, Y = 0, Z = 0 have *k* common intersections, then
among the *n*^{2} points are included these *k* points, which are fixed
points independent of the point ( *x*′, *y*′, *z*′); so that, disregarding
these fixed points, the number of points (*x*, *y*, *z*) corresponding to
the given point ( *x*′, *y*′, *z*′) is = *n*^{2} − *k*; and in particular if *k* = *n*^{2} − 1,
then we have one corresponding point; and hence the original
system of equations *x*′ : *y*′ : *z*′ = X : Y : Z must lead to the equivalent
system *x* : *y* : *z* = X′ : Y′ : Z′; and in this system by the like reasoning
the functions must be such that the curves X′ = 0, Y′ = 0, Z′ = 0
have n′^{2} − 1 common intersections. The most simple example is
in the two systems of equations *x*′ : *y*′ : *z*′ = *yz* : *zx* : *xy* and *x* : *y* : *z* =
*y*′*z*′ : *z*′ *x*′ : *x*′*y*′; where yz = 0, zx = 0, xy = 0 are conics (pairs of lines)
having three common intersections, and where obviously either
system of equations leads to the other system. In the case where
X, Y, Z are of an order exceeding 2 the required number *n*^{2} − 1 of
common intersections can only occur by reason of common multiple
points on the three curves; and assuming that the curves X = 0,
Y = 0, Z = 0 have α_{1} + α_{2} + α_{3} ... +
α_{n−1} common intersections, where
the α_{1} points are ordinary points, the α_{2} points are double points, the
α_{3} points are triple points, &c., on each curve, we have the condition

_{1}+ 4α

_{2}+ 9α

_{3}+ ... (

*n*− 1)

^{2}α

_{n−1}=

*n*

^{2}− 1;

but to this must be joined the condition

_{1}+ 3α

_{2}+ 6α

_{3}... + ½

*n*(

*n*− 1) α

_{n−1}= ½

*n*(

*n*+ 3) − 2

(without which the transformation would be illusory); and the
conclusion is that α_{1}, α_{2}, ... α_{n−1} may be any numbers satisfying
these two equations. It may be added that the two equations
together give

_{2}+ 3α

_{3}... + ½ (

*n*− 1) (

*n*− 2) α

_{n−1}= ½ (

*n*− 1) (

*n*− 2),

which expresses that the curves X = 0, Y = 0, Z = 0 are unicursal.
The transformation may be applied to any curve *u* = 0, which is
thus rationally transformed into a curve *u*′ = 0, by a rational transformation
such as is considered in Riemann’s theory: hence the two
curves have the same deficiency.

Coming next to Chasles, the principle of correspondence is
established and used by him in a series of memoirs relating to the
conics which satisfy given conditions, and to other geometrical
questions, contained in the *Comptes rendus*, t. lviii. (1864) et seq.
The theorem of united points in regard to points in a right line
was given in a paper, June–July 1864, and it was extended to
unicursal curves in a paper of the same series (March 1866), “Sur
les courbes planes ou à double courbure dont les points peuvent
se déterminer individuellement—application du principe de correspondance
dans la théorie de ces courbes.”

The theorem is as follows: if in a unicursal curve two points
have an (α, β) correspondence, then the number of united points
(or points each corresponding to itself) is = α + β. In fact in a
unicursal curve the co-ordinates of a point are given as proportional
to rational and integral functions of a parameter, so that any point
of the curve is determined uniquely by means of this parameter;
that is, to each point of the curve corresponds one value of the
parameter, and to each value of the parameter one point on the
curve; and the (α, β) correspondence between the two points is given
by an equation of the form (*≬ θ, 1)^{α}(φ, 1)^{β} = 0 between their parameters
θ and φ; at a united point φ = θ, and the value of θ is given by
an equation of the order α + β. The extension to curves of any given
deficiency D was made in the memoir of Cayley, “On the correspondence
of two points on a curve,”—*Proc. Lond. Math. Soc.* t. i.
(1866; *Collected Works*, vol. vi. p. 9),—viz. taking P, P′ as the corresponding
points in an (α, α′) correspondence on a curve of deficiency
D, and supposing that when P is given the corresponding points P′
are found as the intersections of the curve by a curve Θ containing
the co-ordinates of P as parameters, and having with the given curve
k intersections at the point P, then the number of united points is
a = α + α′ + 2*k*D; and more generally, if the curve Θ intersect the
given curve in a set of points P′ each p times, a set of points Q′ each
g times, &c., in such manner that the points (P, P′) the points (P, Q′)
&c., are pairs of points corresponding to each other according to
distinct laws; then if (P, P′) are points having an (α, α′) correspondence
with a number = a of united points, (P, Q′) points having a (β, β′)
correspondence with a number = *b* of united points, and so on, the
theorem is that we have

*p*(

*a*− α − α′) +

*q*(

*b*− β − β′) + ... = 2

*k*D.

The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every geometrical problem. We can by means of it investigate the class of a curve, number of inflections, &c.—in fact, Plücker’s equations; but it is necessary to take account of special solutions: thus, in one of the most simple instances, in finding the class of a curve, the cusps present themselves as special solutions.

Imagine a curve of order *m*, deficiency D, and let the corresponding points P, P′ be such that the line joining them passes through a given
point O; this is an (*m* − 1, *m* − 1) correspondence, and the value of *k* is = 1, hence the number of united points is = 2*m* − 2 + 2D; the united points are the points of contact of the tangents from O and (as special solutions) the cusps, and we have thus the relation *n* + κ = 2*m* − 2 + 2D; or, writing D = ½(*m* − 1)(*m* − 2) − δ − κ, this is *n* = *m*(*m* − 1) − 2δ − 3κ, which is right.

The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other geometrical problems.

There is one application of the theory of the (α, α′) correspondence between two planes which it is proper to notice.

Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates *u*, *u*′; then, writing *u* = *x* + iy, *u*′ = *x*′ + *iy*′, the equation determines real values of (*x*, *y*), and of (*x*′, *y*′), corresponding to any given real values of (*x*′, *y*′) and (*x*, *y*) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (*x*, *y*) and (*x*′, *y*′); for example in the imaginary circle *u*^{2} + *u*′^{2} = (*a* + *bi*)^{2}, the correspondence is given by the two equations *x*^{2} − *y*^{2} + *x*′^{2} − *y*′^{2} = *a*^{2} − *b*^{2}, *xy* + *x*′*y*′ = *ab*. We have thus a means of geometrical representation for the portions, as well imaginary as real, of any real or imaginary curve. Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.

16. *Systems of Curves satisfying Conditions.*—The researches of Chasles (*Comptes Rendus*, t. lviii., 1864, et seq.) refer to the conics which satisfy given conditions. There is an earlier paper by J. P. E. Fauque de Jonquières, “Théorèmes généraux concernant les courbes géométriques planes d’un ordre quelconque,” *Liouv.* t. vi. (1861), which establishes the notion of a system of curves (of any order) of the index N, viz. considering the curves of the order *n* which satisfy ½*n*(*n* + 3) − 1 conditions, then the index N is the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers (February 1864), considering the conics which satisfy four conditions, establishes the notion of the two characteristics (μ, ν) of such a system of conics, viz. μ is the number of the conics which pass through a given arbitrary point, and ν is the number of the conics which touch a given arbitrary line. And he gives the theorem, a system of conics satisfying four conditions, and having the characteristics (μ, ν) contains 2ν − μ line-pairs (that is, conics, each of them a pair of lines), and 2μ − ν point-pairs (that is, conics, each of them a pair of points,—coniques infiniment aplaties), which is a fundamental one in the theory. The characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate conics in the system, and how often each is to be counted. It was thus that Zeuthen (in the paper *Nyt Bydrag*, “Contribution to the Theory of Systems of Conics which satisfy four Conditions” (Copenhagen, 1865), translated with an addition in the *Nouvelles* *Annales*) solved the question of finding the characteristics of the systems of conics which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conics which satisfy five conditions of contact with a given curve or curves (Cayley, *Comptes Rendus*, t. lxiii., 1866; *Collected Works*, vol. v. p. 542), and “On the Curves which satisfy given Conditions” (*Phil.* *Trans.* t. clviii., 1868; *Collected Works*, vol. vi. p. 191).

It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of conics satisfying 4 conditions, yet what is really determined is in every case the number of the conics which satisfy 5 conditions; the characteristics of the system (4*p*) of the conics which pass through 4*p* points are (5*p*), (4*p*, 1*l*), the number of the conics which pass through 5 points, and which pass through 4 points and touch 1 line: and so in other cases. Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the number of the cubics (9*p*), (8*p*, 1*l*), &c., which pass through 9 points, pass through 8 points and touch 1 line, &c.; but it is in the investigation convenient to seek for the characteristics of the systems of cubics (8*p*), &c., which satisfy 8 instead of 9 conditions.

The elementary problems in regard to cubics are solved very completely by S. Maillard in his *Thèse*, *Recherche des caractéristiques* *des systèmes élémentaires des courbes planes du troisième ordre* (Paris, 1871). Thus, considering the several cases of a cubic

No. of consts. | |

1. With a given cusp | 5 |

2. ” cusp on a given line | 6 |

3. ” cusp | 7 |

4. ” a given node | 6 |

5. ” node on given line | 7 |

6. ” node | 8 |

7. non-singular | 9 |

he determines in every case the characteristics (μ, ν) of the corresponding systems of cubics (4*p*), (3*p*, 1*l*), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir “Almindelige Egenskaber, &c.,” *Danish Academy*, t. x. (1873), considers the problem in reference to curves of any order, and applies his results to cubic and quartic curves.

The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics (μ, ν) of a system of curves by consideration of the special or degenerate forms of the curves included in the system. The quantities which have to be considered are very numerous. Zeuthen in the case of curves of any given order establishes between the characteristics μ, ν, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23 independent equations), involving (besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves. It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.

17. *Degeneration of Curves.*—The remarks which follow have reference to the analytical theory of the degenerate curves which present themselves in the foregoing problem of the curves which satisfy given conditions.

A curve represented by an equation in point-co-ordinates may break up: thus if P_{1}, P_{2}, ... be rational and integral functions of the co-ordinates (*x*, *y*, *z*) of the orders *m*_{1}, *m*_{2} ... respectively, we have the curve P_{1}α1P_{2}α2 ... = 0, of the order *m*, = α_{1}*m*_{1} + α_{2}*m*_{2} + ..., composed of the curve P_{1} = 0 taken α_{1} times, the curve P_{2} = 0 taken α_{2} times, &c.

Instead of the equation P_{1}α1P_{2}α2 ... = 0, we may start with an equation *u* = 0, where *u* is a function of the order *m* containing a parameter θ, and for a particular value say θ = 0, of the parameter reducing itself to P_{1}α1P_{2}α2.... Supposing θ indefinitely small, we have what may be called the penultimate curve, and when θ = 0 the ultimate curve. Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called “summits” on the component curves P_{1} = 0, P_{2} = 0 respectively; a summit Σ is a point such that, drawing from an arbitrary point O the tangents to the penultimate curve, we have OΣ as the limit of one of these tangents. The ultimate curve together with its summits may be regarded as a degenerate form of the curve *u* = 0. Observe that the positions of the summits depend on the penultimate curve *u* = 0, viz. on the values of the coefficients in the terms multiplied by θ, θ^{2}, ...; they are thus in some measure arbitrary points as regards the ultimate curve P_{1}α1P_{2}α2 ... = 0.

It may be added that we have summits only on the component curves P_{1} = 0, of a multiplicity α_{1} > 1; the number of summits on such a curve is in general = (α_{1}^{2} − α_{1})*m*_{1}^{2}. Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = *m*^{2} − *m*, = (α_{1}*m*_{1} + α_{2}*m*_{2} + ...)^{2} − (α_{1}*m*_{1} + α_{2}*m*_{2} + ...). Taking P_{1} = 0 to have δ_{1} nodes and κ_{1} cusps, and therefore its class *n*_{1} to be = *m*_{1}^{2} − *m*_{1} − 2δ_{1} − 3κ_{1}, &c., the expression for the number of tangents to the penultimate curve is

_{1}

^{2}− α

_{1})

*m*

_{1}

^{2}+ (α

_{2}

^{2}− α

_{2})

*m*

_{2}

^{2}+ ... + 2α

_{1}α

_{2}

*m*

_{1}

*m*

_{2}+

+ α

_{1}(

*n*

_{1}+ 2δ

_{1}+ 3κ

_{1}) + α

_{2}(

*n*

_{2}+ 2δ

_{2}+ 3κ

_{2}) + ...

where a term 2α_{1}α_{2}*m*_{1}*m*_{2} indicates tangents which are in the limit the lines drawn to the intersections of the curves P_{1} = 0, P_{2} = 0 each line 2α_{1}α_{2} times; a term α_{1}(*n*_{1} + 2δ_{1} + 3κ_{1}) tangents which are in the limit the proper tangents to P_{1} = 0 each α_{1} times, the lines to its
nodes each 2α_{1} times, and the lines to its cusps each 3α_{1}, times;
the remaining terms (α_{1}^{2} − α_{1})*m*_{1}^{2} + (α_{2}^{2} − α_{2})*m*_{2}^{2} + ... indicate tangents
which are in the limit the lines drawn to the several summits, that is,
we have (α_{1}^{2} − α_{1})*m*_{1}^{2} summits on the curve P_{1} = 0, &c.

There is, of course, a precisely similar theory as regards line-co-ordinates; taking Π_{1}, Π_{2}, &c., to be rational and integral functions of the co-ordinates (ξ, η, ζ) we connect with the ultimate curve Π_{1}^{α1}Π_{2}^{α2} ... = 0, and consider as belonging to it, certain lines, which for the moment may be called “axes” tangents to the component curves Π_{1} = 0_{1}, Π_{2} = 0 respectively. Considering an equation in point-co-ordinates, we may have among the component curves right lines, and if in order to put these in evidence we take the equation to be L_{1}^{γ1} .. P_{1}^{α1} ... = 0, where L_{1} = 0 is a right line, P_{1} = 0 a curve of the second or any higher order, then the curve will contain as part of itself summits not exhibited in this equation, but the corresponding line-equation will be _{1}Λ^{δ1} ... Π_{1}^{α1} = 0, where Λ_{1} = 0,... are the equations of the summits in question, Π_{1} = 0, &c., are the line-equations corresponding to the several point-equations P_{1} = 0, &c.; and this curve will contain as part of itself axes not exhibited by this equation, but which are the lines L_{1} = 0,... of the equation in point-co-ordinates.

18. *Twisted Curves.*—In conclusion a little may be said as to curves of double curvature, otherwise twisted curves or curves in space. The analytical theory by Cartesian co-ordinates was first considered by Alexis Claude Clairaut, *Recherches sur les courbes* *à double courbure* (Paris, 1731). Such a curve may be considered as described by a point, moving in a line which at the same time rotates about the point in a plane which at the same time rotates about the line; the point is a point, the line a tangent, and the plane an osculating plane, of the curve; moreover the line is a generating line, and the plane a tangent plane, of a developable surface or torse, having the curve for its edge of regression. Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one), viz. if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of the system, then m, r, n are the order, rank and class respectively. The system has singularities, and there exist between m, r, n and the numbers of the several singularities equations analogous to Plücker’s equations for a plane curve.

It is a leading point in the theory that a curve in space cannot in general be represented by means of two equations U = 0, V = 0; the two equations represent surfaces, intersecting in a curve; but there are curves which are not the complete intersection of any two surfaces; thus we have the cubic in space, or skew cubic, which is the residual intersection of two quadric surfaces which have a line in common; the equations U = 0, V = 0 of the two quadric surfaces represent the cubic curve, not by itself, but together with the line.

Authorities.—In addition to the copious authorities mentioned in the text above, see Gabriel Cramer, *Introduction à l’analyse des*
*lignes courbes algébriques* (Geneva, 1750). Bibliographical articles are given in the *Ency. der math. Wiss.* Bd. iii. 2, 3 (Leipzig, 1902–1906); H. C. F. von Mangoldt, “Anwendung der Differential- und Integralrechnung auf Kurven und Flächen,” Bd. iii. 3 (1902); F. R. v. Lilienthal, “Die auf einer Fläche gezogenen Kurven,” Bd. iii. 3 (1902); G. W. Scheffers, “Besondere transcendente Kurven,” Bd. iii. 3 (1903); H. G. Zeuthen, “Abzahlende Methoden,” Bd. iii. 2 (1906); L. Berzolari, “Allgemeine Theorie der höheren ebenen algebraischen Kurven,” Bd. iii. 2 (1906). Also A. Brill and M. Noether, “Die Entwicklung der Theorie der algebraischen Funktionen in älterer und neuerer Zeit” (*Jahresb. der deutschen math. ver.*, 1894); E. Kötter, “Die Entwickelung der synthetischen Geometrie” (*Jahresb.* *der deutschen math. ver.*, 1898–1901); E. Pascal, *Repertorio di* *matematiche superiori*, ii. “Geometrìa” (Milan, 1900); H. Wieleitner, *Bibliographie der höheren algebraischen Kurven für den Zeitabschnitt* *von 1890–1894* (Leipzig, 1905).

*Text-books*:—G. Salmon, *A Treatise on the Higher Plane Curves* (Dublin, 1852, 3rd ed., 1879); translated into German by O. W. Fiedler, *Analytische Geometrie der höheren ebenen Kurven* (Leipzig, 2te Aufl., 1882); L. Cremona, *Introduzione ad una teoria geometrica* *delle curve piane* (Bologna, 1861); J. H. K. Durège, *Die ebenen* *Kurven dritter Ordnung* (Leipzig, 1871); R. F. A. Clebsch and C. L. F. Lindemann, *Vorlesungen über Geometrie*, Band i. and i_{2} (Leipzig, 1875–1876); H. Schroeter, *Die Theorie der ebenen Kurven* *dritter Ordnung* (Leipzig, 1888); H. Andoyer, *Leçons sur la théorie* *des formes et la géométrie analytique supérieure* (Paris, 1900); Wieleitner, *Theorie der ebenen algebraischen Kurven höherer Ordnung* (Leipzig, 1905). (A. Ca.; E. B. El.)

- ↑ In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane.