785] 
CURVE. 
471 
the number of intersections is thus = m (to — 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 3 nodes and k cusps, there is a diminution 28 + 3«, and 
the value of n is n = to (to — 1) — 23 — 3k. 
Secondly, as to the inflexions, the process is a similar one; it can be shown that 
the inflexions are the intersections of the curve by a derivative curve called (after 
Hesse, who first considered it) the Hessian, defined geometrically as the locus of a 
point such that its conic polar 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 (to — 2), and the number of inflexions is = 3to (to — 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 inflexion, 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 8 nodes and k cusps, the diminution is 
= 63 + 8k, and the number of inflexions is i = 3to (to — 2) — 63 — 8k. 
Thirdly, for the double tangents; the points of contact of these are obtained as the 
intersections of the curve by a curve n = 0, which has not as yet been geometrically 
defined, but which is found analytically to be of the order (to — 2) (to 2 — 9); the 
number of intersections is thus = to (to — 2) (to 2 — 9); but if the given curve has a node 
then there is a diminution = 4 (to 2 — to — 6), and if it has a cusp then there is a 
diminution = 6 (to 2 — to — 6), where, however, it is to be noticed that the factor 
(to 2 —to—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 3 nodes and k cusps, the apparent diminution 2 (to 2 — to — 6) (23 + 3k) is 
too great, and it has in fact to be diminished by 2 {23 (3 — 1) + 63« + §k {k — 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 2t, =to(to—2)(to 2 —9)—&c.; 
or dividing the whole by 2, we have the expression for r given by the third of 
Pliicker’s equations. 
It is obvious that we cannot by consideration of the equation u = 0 in point- 
coordinates obtain the remaining three of Plticker’s equations; they might be obtained