Chapter X: Part 10

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.

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 " " 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.)

FOOTNOTE:

[1] 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.

CURVILINEAR, in architecture, that which is formed by curved or flowing lines; the roofs over the domes and vaults of the Byzantine churches were generally curvilinear. The term is also given to the flowing tracery of the Decorated and the Flamboyant styles.

CURWEN, HUGH (d. 1568), English ecclesiastic and statesman, was a native of Westmorland, and was educated at Cambridge, afterwards taking orders in the church. In May 1533 he expressed approval of Henry VIII.'s marriage with Anne Boleyn in a sermon preached before the king. In 1541 he became dean of Hereford, and in 1555 Queen Mary nominated him to the archbishopric of Dublin, and in the same year he was appointed lord chancellor of Ireland. He acted as one of the lords justices during the absence from Ireland of the lord deputy, the earl of Sussex, in 1557. On the accession of Elizabeth, Curwen at once accommodated himself to the new conditions by declaring himself a Protestant, and was continued in the office of lord chancellor. He was accused by the archbishop of Armagh of serious moral delinquency, and his recall was demanded both by the primate and the bishop of Meath. In 1567 Curwen resigned the see of Dublin and the office of lord chancellor, and was appointed bishop of Oxford. He died on the 1st of November 1568.

See John Strype, _Life and Acts of Archbishop Parker_ (3 vols.,
Oxford, 1824), and _Memorials of Thomas Cranmer_ (2 vols., Oxford,
1840); John D'Alton. _Memoirs of the Archbishops of Dublin_ (Dublin,
1838).