+-------+-------+-------+-------+-------+-------+
| m | n |[delta]|[kappa]| [tau] | [iota]|
+-------+-------+-------+-------+-------+-------+
| 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
([delta] + [kappa]) 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, [delta] = 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 3m(m - 2) for a curve of
the order m was obtained analytically by Plucker, but the theory was
first given in a complete form by Hesse in the two papers "Uber die
Elimination, u.s.w.," and "Uber 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 [Delta] = 0, where [Delta] 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 +
6lxyz = 0; and in particular a proof is given of Plucker'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 1/2m(m - 2)(m^2 - 9) for the number of double tangents of
a curve of the order m was obtained by Plucker only as a consequence of
his first, second, fourth and fifth equations. An investigation by means
of the curve [Pi] = 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'elimination et la theorie des courbes"
(_Crelle_, t. xxxiv., 1847; _Collected Works_, vol. i. p. 337), and in
part carried out by Hesse in the memoir "Uber 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 [Pi] = 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 [Pi] = 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
Plucker 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 Plucker 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 [Delta] 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'([dP]/[dP]x) + y'([dP]/[dP]y) + z'([dP]/[dP]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 Pluckerian
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 _Theorie des fonctions
analytiques_ (1798); it is there remarked that the equation obtained by
the elimination of the parameter a from an equation f(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 f(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 [delta] nodes and [kappa]
cusps, and therefore of the class n with [tau] double tangents and
[iota] inflections, m, n, [delta], [kappa], [tau], [iota] being
connected by the Pluckerian 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 [delta] and [kappa]
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 - 2[delta] - 3[kappa].
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
-2[tau] - 3[iota] (= m^2 - 2[delta] - 3[kappa], inasmuch as each of
these numbers is as was seen = m + n). At any one of the m^2 - 2[delta]
-3[kappa] 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
-2[tau] - 3[iota] 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 - 2[tau] - 3[iota]
common tangents are the tangents to the variable curve at the m^2 -
2[delta] -3[kappa] points respectively, and the envelope is at the same
time generated by the m^2 - 2[delta] - 3[kappa] points, and enveloped by
the n^2 - 2[tau] - 3[iota] tangents; we have thus a dual generation of
the envelope, which only differs from Plucker's dual generation, in that
in place of a single point and tangent we have the group of m^2 -
2[delta] -3[kappa] points and n^2 - 2[tau] - 3[iota] 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 limacon
(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 [Omega] 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 [Omega] 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 [Omega] touch at b the arc aa' so that the two portions ab, ba' lie
on the same side of the line [Omega], then projecting the figure as
before, the tangent at b, that is, the line [Omega] 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 [Omega] is a tangent at an inflection, passes
through a crunode, or touches one of the branches of a crunode, &c.;
thus, if the line [Omega] 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.