1,2,3,4 redundant hyperbolas
5,6 defective hyperbolas
7,8 parabolic hyperbolas
9 hyperbolisms of hyperbola
10 " " ellipse
11 " " 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. Mobius 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 differentes formes des courbes
planes du quatrieme 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, [delta] = 2, [kappa] = 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, [delta] =
0, [kappa] = 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 Plucker (in the
memoir "Uber solche Puncte die bei Curven einer hoheren 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 [lambda] tangents ([lambda], 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 [lambda] tangents from
the one circular point and those from the other circular point intersect
in [lambda] real foci (viz. each of these is the only real point on each
of the tangents through it), and in [lambda]^2 - [lambda] imaginary
foci; each pair of real foci determines a pair of imaginary foci (the
so-called antipoints of the two real foci), and the 1/2[lambda]([lambda]
-1) pairs of real foci thus determine the [lambda]^2 - [lambda]
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,
[delta] = 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 [alpha]' 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 ([alpha], [alpha])
correspondence; and if [alpha], [alpha] are each = 1, then the two
points have a (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, "Uber
die Singularitaten 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 [delta] + [kappa] of nodes and cusps is at most = 1/2(m -
1)(m - 2); for a given curve the deficiency of the actual number of
nodes and cusps below this maximum number, viz. 1/2(m - 1)(m - 2) -
[delta] - [kappa], 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 : [theta] : [phi] we have
[phi] a certain irrational function of [theta], 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
[theta], [phi], that is, of [theta] and a certain irrational function
of [theta].
In particular if D = 0, that is, if the given curve be unicursal, the
transformed curve is a line, [phi] is a mere linear function of
[theta], 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 [theta]; it is easy to see that for a given
curve of the order m, these functions of [theta] 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, [phi] can
be expressed as the square root of a quartic function of [theta]; 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 [theta], and of the square root of a quartic
function of [theta].
And so if D = 2, then the transformed curve is a nodal quartic; [phi]
can be expressed as the square root of a sextic function of [theta]
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 [theta], and of the square root of a sextic
function of [theta]. But D = 3, we have no longer the like law, viz.
[phi] is not expressible as the square root of an octic function of
[theta].
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 snu, cnu, dnu; in fact,
taking the radical to be [root] [1 - [theta]^2.1 - k^2[theta]^2], and
writing [theta] = snu, the radical becomes = cnu, dnu; 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 [alpha]1 + [alpha]2 +
[alpha]3 ... + [alpha]_(n-1) common intersections, where the [alpha]1
points are ordinary points, the [alpha]2 points are double points, the
[alpha]3 points are triple points, &c., on each curve, we have the
condition
[alpha]1 + 4[alpha]2 + 9[alpha]3 + ... (n - 1)^2[alpha]_(n-1)
= n^2 - 1;
but to this must be joined the condition
[alpha]1 + 3[alpha]2 + 6[alpha]3 ... + 1/2n(n - 1)[alpha]_(n-1)
= 1/2n(n + 3) - 2
(without which the transformation would be illusory); and the
conclusion is that [alpha]1, [alpha]2, ... [alpha]_(n-1) may be any
numbers satisfying these two equations. It may be added that the two
equations together give
[alpha]2 + 3[alpha]3 ... + 1/2(n - 1)(n - 2)[alpha]_(n-1)
= 1/2(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 a double courbure
dont les points peuvent se determiner individuellement--application du
principe de correspondance dans la theorie de ces courbes."
The theorem is as follows: if in a unicursal curve two points have an
([alpha], [beta]) correspondence, then the number of united points (or
points each corresponding to itself) is = [alpha] + [beta]. 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 ([alpha], [beta]) correspondence between the two points
is given by an equation of the form (*() [theta], 1)^[alpha]([Phi],
1)^[beta] = 0 between their parameters [theta] and [phi]; at a united
point [phi] = [theta], and the value of [theta] is given by an
equation of the order [alpha] + [beta]. 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 ([alpha], [alpha]') 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 [Theta] 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 = [alpha] + [alpha]' + 2kD; and more
generally, if the curve [Theta] 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 ([alpha], [alpha]') correspondence with a
number = a of united points, (P, Q') points having a ([beta], [beta]')
correspondence with a number = b of united points, and so on, the
theorem is that we have
p(a - [alpha] - [alpha]') + q(b - [beta] - [beta]') + ... = 2kD.
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,
Plucker'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 = 2m - 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 +
[kappa] = 2m - 2 + 2D; or, writing D = 1/2(m - 1)(m - 2) - [delta]
-[kappa], this is n = m(m - 1) - 2[delta] - 3[kappa], 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 ([alpha], [alpha]')
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 Jonquieres, "Theoremes generaux concernant les courbes
geometriques 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 1/2n(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 ([mu],
[nu]) of such a system of conics, viz. [mu] is the number of the conics
which pass through a given arbitrary point, and [nu] 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 ([mu], [nu]) contains 2[nu] - [mu] line-pairs (that is,
conics, each of them a pair of lines), and 2[mu] - [nu] 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 (4p) of the conics which pass through 4p points are (5p),
(4p, 1l), 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 (9p), (8p, 1l), &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 (8p), &c., which satisfy 8 instead of 9 conditions.
The elementary problems in regard to cubics are solved very completely
by S. Maillard in his _These_, _Recherche des caracteristiques des
systemes elementaires des courbes planes du troisieme 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 ([mu], [nu]) of the
corresponding systems of cubics (4p), (3p, 1l), &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
([mu], [nu]) 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 [mu],
[nu], 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 P1, P2, ... be rational and integral functions of the
co-ordinates (x, y, z) of the orders m1, m2 ... respectively, we have
the curve P1^([alpha]1)P2^([alpha]2) ... = 0, of the order m, =
[alpha]1m1 + [alpha]2m2 + ..., composed of the curve P1 = 0 taken
[alpha]1 times, the curve P2 = 0 taken [alpha]2 times, &c.
Instead of the equation P1^([alpha]1)P2^([alpha]2) ... = 0, we may
start with an equation u = 0, where u is a function of the order m
containing a parameter [theta], and for a particular value say [theta]
= 0, of the parameter reducing itself to
P1^([alpha]1)P2^([alpha]2).... Supposing [theta] indefinitely small,
we have what may be called the penultimate curve, and when [theta] = 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 P1 = 0, P2 = 0 respectively; a summit [Sigma] is a
point such that, drawing from an arbitrary point O the tangents to the
penultimate curve, we have O[Sigma] 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 [theta], [theta]^2, ...;
they are thus in some measure arbitrary points as regards the ultimate
curve P1^([alpha]1)P2^([alpha]2) ... = 0.
It may be added that we have summits only on the component curves P1 =
0, of a multiplicity [alpha]1 > 1; the number of summits on such a
curve is in general = ([alpha]1^2 - [alpha]1)m1^2. Thus assuming that
the penultimate curve is without nodes or cusps, the number of the
tangents to it is = m^2 - m, = ([alpha]1m1 + [alpha]2m2 + ...)^2 -
([alpha]1m1 + [alpha]2m2 + ...). Taking P1 = 0 to have [delta]1 nodes
and [kappa]1 cusps, and therefore its class n1 to be = m1^2 - m1 -
2[delta]1 - 3[kappa]1, &c., the expression for the number of tangents
to the penultimate curve is
= ([alpha]1^2 - [alpha]1)m1^2 + ([alpha]2^2 - [alpha]2)m2^2 +
+ ... + 2[alpha]1[alpha]2m1m2 + [alpha]1(n1 + 2[delta]1 +
+ 3[kappa]1) + [alpha]2(n2 + 2[delta]2 + 3[kappa]2) + ...
where a term 2[alpha]1[alpha]2m1m2 indicates tangents which are in the
limit the lines drawn to the intersections of the curves P1 = 0, P2 =
0 each line 2[alpha]1[alpha]2 times; a term [alpha]1(n1 + 2[delta]1 +
3[kappa]1) tangents which are in the limit the proper tangents to P1
= 0 each [alpha]1 times, the lines to its nodes each 2[alpha]1 times,
and the lines to its cusps each 3[alpha]1, times; the remaining terms
([alpha]1^2 - [alpha]1)m1^2 + ([alpha]2^2 - [alpha]2)m2^2 + ...
indicate tangents which are in the limit the lines drawn to the
several summits, that is, we have ([alpha]1^2 - [alpha]1)m1^2 summits
on the curve P1 = 0, &c.
There is, of course, a precisely similar theory as regards
line-co-ordinates; taking [Pi]1, [Pi]2, &c., to be rational and
integral functions of the co-ordinates ([xi], [eta], [zeta]) we
connect with the ultimate curve [Pi]1^([alpha]1)[Pi]2^([alpha]2) ... =
0, and consider as belonging to it, certain lines, which for the
moment may be called "axes" tangents to the component curves [Pi]1 =
01, [Pi]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 L1^([gamma]1) ... P1^([alpha]1) ... = 0, where L1 = 0 is a right
line, P1 = 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[Lambda]^([delta]1) ...
[Pi]1^([alpha]1) = 0, where [Lambda]1 = 0,... are the equations of the
summits in question, [Pi]1 = 0, &c., are the line-equations
corresponding to the several point-equations P1 = 0, &c.; and this
curve will contain as part of itself axes not exhibited by this
equation, but which are the lines L1 = 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 a 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 Plucker'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 a l'analyse des lignes
courbes algebriques_ (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 Flachen," Bd. iii. 3 (1902); F. R. v.
Lilienthal, "Die auf einer Flache 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 hoheren ebenen algebraischen
Kurven," Bd. iii. 2 (1906). Also A. Brill and M. Noether, "Die
Entwicklung der Theorie der algebraischen Funktionen in alterer und
neuerer Zeit" (_Jahresb. der deutschen math. ver._, 1894); E. Kotter,
"Die Entwickelung der synthetischen Geometrie" (_Jahresb. der
deutschen math. ver._, 1898-1901); E. Pascal, _Repertorio di
matematiche superiori_, ii. "Geometria" (Milan, 1900); H. Wieleitner,
_Bibliographie der hoheren algebraischen Kurven fur 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 hoheren ebenen Kurven_ (Leipzig,
2te Aufl., 1882); L. Cremona, _Introduzione ad una teoria geometrica
delle curve piane_ (Bologna, 1861); J. H. K. Durege, _Die ebenen
Kurven dritter Ordnung_ (Leipzig, 1871); R. F. A. Clebsch and C. L. F.
Lindemann, _Vorlesungen uber Geometrie_, Band i. and i2 (Leipzig,
1875-1876); H. Schroeter, _Die Theorie der ebenen Kurven dritter
Ordnung_ (Leipzig, 1888); H. Andoyer, _Lecons sur la theorie des
formes et la geometrie analytique superieure_ (Paris, 1900);
Wieleitner, _Theorie der ebenen algebraischen Kurven hoherer Ordnung_
(Leipzig, 1905). (A. Ca.; E. B. El.)