(7) [iota] - [kappa] = 3(n - m),
(8) 2([tau] - [delta]) = (n - m)(n + m - 9),
(9) 1/2m(m + 3) - [delta]-2[kappa] = 1/2n(n + 3) - [tau] - 2[iota],
(10) 1/2(m - 1)(m - 2) - [delta] - [kappa]
= 1/2(n - 1)(n - 2) - [tau] - [iota],
(11, 12) m^2 - 2[delta] - 3[kappa] = n^2 - 2[tau] - 3^[iota],
= m + n,--
the whole system being equivalent to three equations only; and it may be
added that using a to denote the equal quantities 3m + [iota] and 3n +
[kappa] everything may be expressed in terms of m, n, a. We have
[kappa] = a - 3n,
[iota] = a - 3m,
2[delta] = m^2 - m + 8n - 3a.
2[tau] = n^2 - n + 8m - 3a.
It is implied in Plucker's theorem that, m, n, [delta], [kappa],
[tau], [iota] signifying as above in regard to any curve, then in
regard to the reciprocal curve, n, m, [tau], [iota], [delta], [kappa]
will have the same significations, viz. for the reciprocal curve these
letters denote respectively the order, class, number of nodes, cusps,
double tangent and inflections.
The expression 1/2m(m + 3) - [delta] - 2[kappa] is that of the number
of the disposable constants in a curve of the order m with [delta]
nodes and [kappa] cusps (in fact that there shall be a node is 1
condition, a cusp 2 conditions) and the equation (9) thus expresses
that the curve and its reciprocal contain each of them the same number
of disposable constants.
For a curve of the order m, the expression 1/2m(m - 1) - [delta] -
[kappa] is termed the "deficiency" (as to this more hereafter); the
equation (10) expresses therefore that the curve and its reciprocal
have each of them the same deficiency.
The relations m^2 - 2[delta] - 3[kappa] = n^2 - 2[tau] - 3[iota] = m +
n, present themselves in the theory of envelopes, as will appear
farther on.
With regard to the demonstration of Plucker's equations it is to be
remarked that we are not able to write down the equation in
point-co-ordinates of a curve of the order m, having the given numbers
[delta] and [kappa] of nodes and cusps. We can only use the general
equation (*() x, y, z)^m = 0, say for shortness u = 0, of a curve of the
_m_th order, which equation, so long as the coefficients remain
arbitrary, represents a curve without nodes or cusps. Seeking then, for
this curve, the values, n, [iota], [tau] of the class, number of
inflections, and number of double tangents,--first, as regards the
class, this is equal to the number of tangents which can be drawn to the
curve from an arbitrary point, or what is the same thing, it is equal to
the number of the points of contact of these tangents. The points of
contact are found as the intersections of the curve u = 0 by a curve
depending on the position of the arbitrary point, and called the "first
polar" of this point; the order of the first polar is = m - 1, and the
number of intersections is thus = m(m - 1). But it can be shown,
analytically or geometrically, that if the given curve has a node, the
first polar passes through this node, which therefore counts as two
intersections, and that if the curve has a cusp, the first polar passes
through the cusp, touching the curve there, and hence the cusp counts as
three intersections. But, as is evident, the node or cusp is not a point
of contact of a proper tangent from the arbitrary point; we have,
therefore, for a node a diminution 2, and for a cusp a diminution 3, in
the number of the intersections; and thus, for a curve with [delta]
nodes and [kappa] cusps, there is a diminution 2[delta] + 3[kappa], and
the value of n is n = m(m - 1) - 2[delta] -3[kappa].
Secondly, as to the inflections, the process is a similar one; it can be
shown that the inflections are the intersections of the curve by a
derivative curve called (after Ludwig Otto Hesse who first considered
it) the Hessian, defined geometrically as the locus of a point such that
its conic polar (S8 below) in regard to the curve breaks up into a pair
of lines, and which has an equation H = 0, where H is the determinant
formed with the second differential coefficients of u in regard to the
variables (x, y, z); H = 0 is thus a curve of the order 3(m - 2), and
the number of inflections is = 3m(m - 2). But if the given curve has a
node, then not only the Hessian passes through the node, but it has
there a node the two branches at which touch respectively the two
branches of the curve; and the node thus counts as six intersections; so
if the curve has a cusp, then the Hessian not only passes through the
cusp, but it has there a cusp through which it again passes, that is,
there is a cuspidal branch touching the cuspidal branch of the curve,
and besides a simple branch passing through the cusp, and hence the cusp
counts as eight intersections. The node or cusp is not an inflection,
and we have thus for a node a diminution 6, and for a cusp a diminution
8, in the number of the intersections; hence for a curve with [delta]
nodes and [kappa] cusps, the diminution is = 6[delta] + 8[kappa], and
the number of inflections is [iota] = 3m(m - 2) - 6[delta] - 8[kappa].
Thirdly, for the double tangents; the points of contact of these are
obtained as the intersections of the curve by a curve [Pi] = 0, which
has not as yet been geometrically defined, but which is found
analytically to be of the order (m - 2)(m^2 - 9); the number of
intersections is thus = m(m - 2)(m^2 - 9); but if the given curve has a
node then there is a diminution = 4(m^2 - m - 6), and if it has a cusp
then there is a diminution = 6(m^2 - m - 6), where, however, it is to be
noticed that the factor (m^2 - m - 6) is in the case of a curve having
only a node or only a cusp the number of the tangents which can be drawn
from the node or cusp to the curve, and is used as denoting the number
of these tangents, and ceases to be the correct expression if the number
of nodes and cusps is greater than unity. Hence, in the case of a curve
which has [delta] nodes and [kappa] cusps, the apparent diminution 2(m^2
- m - 6)(2[delta] + 3[kappa]) is too great, and it has in fact to be
diminished by 2{2[delta]([delta] - 1) + 6[delta][kappa] +
(9/2)[kappa]([kappa] - 1)}, or the half thereof is 4 for each pair of
nodes, 6 for each combination of a node and cusp, and 9 for each pair of
cusps. We have thus finally an expression for 2[tau], = m(m - 2)(m^2 - 9)
- &c.; or dividing the whole by 2, we have the expression for [tau]
given by the third of Plucker's equations.
It is obvious that we cannot by consideration of the equation u = 0 in
point-co-ordinates obtain the remaining three of Plucker's equations;
they might be obtained in a precisely analogous manner by means of the
equation v = 0 in line-co-ordinates, but they follow at once from the
principle of duality, viz. they are obtained by the mere interchange of
m, [delta], [kappa], with n, [tau], [iota] respectively.
[Illustration]
To complete Plucker's theory it is necessary to take account of compound
singularities; it might be possible, but it is at any rate difficult, to
effect this by considering the curve as in course of description by the
point moving along the rotating line; and it seems easier to consider
the compound singularity as arising from the variation of an actually
described curve with ordinary singularities. The most simple case is
when three double points come into coincidence, thereby giving rise to a
triple point; and a somewhat more complicated one is when we have a cusp
of the second kind, or node-cusp arising from the coincidence of a node,
a cusp, an inflection, and a double tangent, as shown in the annexed
figure, which represents the singularities as on the point of
coalescing. The general conclusion (see Cayley, _Quart. Math. Jour._ t.
vii., 1866, "On the higher singularities of plane curves"; _Collected
Works_, v. 520) is that every singularity whatever may be considered as
compounded of ordinary singularities, say we have a singularity =
[delta]' nodes, [kappa]' cusps, [tau]' double tangents and [iota]'
inflections. So that, in fact, Plucker's equations properly understood
apply to a curve with any singularities whatever.