472 
CURVE. 
[785 
in a precisely analogous manner by means of the equation v = 0 in line-coordinates, 
but they follow at once from the principle of duality, viz. they are obtained by the 
mere interchange of m, S, k with n, r, l respectively. 
To complete Pliicker’s theory it is necessary to take account of compound singu 
larities ; 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 inflexion, 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, 
[374], “ On the higher singularities of a plane curve ”) is that every singularity whatever 
may be considered as compounded of ordinary singularities, say we have a singularity = S' 
nodes, k cusps, r double tangents, and i inflexions. So that, in fact, Plticker’s equations 
properly understood apply to a curve with any singularities whatever. 
By means of Plucker’s equations we may form a table— 
m 
n 
8 
K 
T 
i 
0 
1 
0 
0 
1 
0 
0 
0 
— 
— 
2 
2 
0 
0 
0 
0 
3 
6 
0 
0 
0 
9 
4 
1 
0 
0 
3 
3 
0 
1 
0 
1 
4 
12 
0 
0 
28 
24 
10 
1 
0 
16 
18 
9 
0 
1 
10 
16 
8 
2 
0 
8 
12 
7 
1 
1 
4 
10 
6 
0 
2 
1 
8 
6 
3 
0 
4 
6 
?5 
5 
2 
1 
2 
4 
5 J 
4 
1 
2 
1 
2 
J? 
3 
0 
3 
1 
0