218 
[747 
747. 
NOTE ON THE DEGENERATE FORMS OF CURVES. 
[From Salmons Higher Plane Curves, (3rd ed., 1879), pp. 383—385.] 
Some remarks may be added as to the analytical theory of the degenerate forms 
of curves. As regards conics, a line-pair can be represented in point-coordinates by an 
equation of the form xy = 0 ; and reciprocally a point-pair can be represented in line- 
coordinates by an equation £77 = 0, but we have to consider how the point-pair can be 
represented in point-coordinates: an equation x 2 = 0 is no adequate representation of 
the point-pair, but merely represents (as a two-fold or twice repeated line) the line 
joining the two points of the point-pair, all traces of the points themselves being lost 
in this representation : and it is to be noticed, that the conic, or two-fold line x' 2 = 0, 
or say (ax + fiy + yz)* 2 = 0 is a conic which, analytically, and (in an improper sense) 
geometrically, satisfies the condition of touching any line whatever; whereas the only 
proper tangents of a point-pair are the lines which pass through one or other of the 
two points of the point-pair. 
The solution arises out of the notion of a point-pair, considered as the limit of 
a conic, or say as an indefinitely flat conic; we have to consider conics certain of the 
coefficients whereof are infinitesimals, and which, when the infinitesimal coefficients 
actually vanish, reduce themselves to two-fold lines; and it is, moreover, necessary to 
consider the evanescent coefficients as infinitesimals of different orders. Thus consider 
the conics which pass through two given points, and touch two given lines (four con 
ditions); take y — 0, z — 0 for the given lines, x—0 for the line joining the given 
points, and (x = 0, y — az = 0), (x = 0, y — fiz — 0) for the given points ; the equation of 
a conic satisfying the required conditions and containing one arbitrary parameter 6, is 
x 2 + 2Oxy + 26 *d(a(3) xz + 6- (y — az) (y — $z) = 0 ;