CURVE. 
469 
peculiarity in the motion; in the other two cases there is not; in (2) there is not 
when the point is first at the node, or when it is secondly at the node, any peculiarity 
in the motion; the singularity consists in the point coming twice into the same 
position; and so in (4) the singularity is in the line coming twice into the same 
position. Moreover (1) and (2) are, the former a proper singularity, and the latter an 
improper singularity, as regards the motion of the point; and similarly (3) and (4) are, 
the former a proper singularity, and the latter an improper singularity, as regards the 
motion of the line. 
But as regards the representation of a curve by an equation, the case is very 
different. 
First, if the equation be in point-coordinates, (3) and (4) are in a sense not 
singularities at all. The curve (*fx, y, z) m = 0, or general curve of the order to, has 
double tangents and inflexions; (2) presents itself as a singularity, for the equations 
d x y, z) m = 0, d y (*\x, y, z) m = 0, d z (*$x, y, z) m = 0, implying (*$>, y, z) m = 0, are 
not in general satisfied by any values (a, h, c) whatever of (x, y, z), but if such 
values exist, then the point (a, h, c) is a node or double point; and (1) presents 
itself as a further singularity or sub-case of (2), a cusp being a double point for which 
the two tangents become coincident. 
In line-coordinates all is reversed:—(1) and (2) are not singularities; (3) pre 
sents itself as a sub-case of (4). 
The theory of compound singularities will be referred to further on. 
In regard to the ordinary singularities, we have 
to, the order, 
n „ class, 
8 „ number of double points, 
i „ „ cusps, 
r „ „ double tangents, 
k „ „ inflexions; 
and this being so, Pliicker’s “six equations” are 
(1) 
n 
— TO (to 
-l)-28- 
- 3/e, 
(2) 
i 
= 3to (to ■ 
- 2) - 68 - 
- 8/e, 
(3) 
T 
= \m (to 
— 2) (to 2 — 
9) — (to 2 — 
TO - 6) (28 + 3/e) + 28 (8 - 1) + 68/e + |/e (k 
(4) 
TO 
= ft (ft — 
1) - 2t - 
3 l, 
(5) 
K 
= 3 ft (ft — 
2) - 6t - 
Si, 
(6) 
8 
=\n(n — 
2) (ft 2 — 9) 
— (ft 2 — ft - 
■ 6) (2t 4- 3e) + 2t (t — 1) + 6tì + ft (i — 1).