468 
CUKVE. 
[785 
the point (a, b, c). Thus (£, y, £) are the line-coordinates of any line whatever; but 
when these, instead of being absolutely arbitrary, are subject to the restriction 
+ by + = 0, this obliges the line to pass through a point (a, b, c); and the last- 
mentioned equation ai; + by + c£= 0 is considered as the line-equation of this point. 
A line has only a point-equation, and a point has only a line-equation; but any other 
curve has a point-equation and also a line-equation; the point-equation (*$#, y, z) m = 0 
is the relation which is satisfied by the point-coordinates (x, y, z) of each point of 
the curve; and similarly the line-equation (*}££, y, £) n = 0 is the relation which is 
satisfied by the line-coordinates (£, y, £) of each line (tangent) of the curve. 
There is in analytical geometry little occasion for any explicit use of line-coordinates; 
but the theory is very important; it serves to show that, in demonstrating by point- 
coordinates any purely descriptive theorem whatever, we demonstrate the correlative 
theorem; that is, we do not demonstrate the one theorem, and then (as by the method 
of reciprocal polars) deduce from it the other, but we do at one and the same time 
demonstrate the two theorems; our {x, y, z) instead of meaning point-coordinates may 
mean line-coordinates, and the demonstration is then in every step of it a demonstration 
of the correlative theorem. 
The above dual generation explains the nature of the singularities of a plane 
curve. The ordinary singularities, arranged according to a cross division, are 
Proper. Improper. 
The stationary point, 2. The double point, or node; 
cusp, or spinode; 
The stationary tangent, 4. The double tangent:— 
or inflexion ; 
1. The cusp: the point as it travels along the line may come to rest, and then 
reverse the direction of its motion. 
3. The stationary tangent: the line may in the course of its rotation come to 
rest, and then reverse the direction of its rotation. 
2. The node: the point may in the course of its motion come to coincide with 
a former position of the point, the two positions of the line not in general coinciding. 
4. The double tangent: the line may in the course of its motion come to coin 
cide with a former position of the line, the two positions of the point not in general 
coinciding. 
It may be remarked that we cannot with a real point and line obtain the node 
with two imaginary tangents (conjugate or isolated point, or acnode), nor again the real 
double tangent with two imaginary points of contact; but this is of little consequence, 
since in the general theory the distinction between real and imaginary is not 
attended to. 
The singularities (1) and (3) have been termed proper singularities, and (2) and 
(4) improper; in each of the first-mentioned cases there is a real singularity, or 
Point-singularities— 
Line-singularities— 
arising as follows :—