394 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
Parabolic hyperbolas. Osculating asymptote and six-pointic asymptotic parabola. 
The satellite line is here at infinity, and there is no new distinction of groups. The 
groups therefore are 
Asymptote cuts parabola, 
XLV. 
Asymptote does not cut parabola, 
XLVI. 
Asymptote touches parabola, 
XLVII. 
As to the Groups of the Central and Parabolic Hyperbolisms. Article No. 116. 
116. For the Hyperbolisms, Central and Parabolic, since these have a node or a 
cusp at infinity, they cannot acquire a new node, and the theory of critic centres does 
not arise. There is, however, as regards the Hyperbolisms of the Hyperbola a dis 
tinction in the position of the satellite line, viz. this may lie outside, or between, the 
parallel asymptotes. The groups are 
Hyperbolisms of the Hyperbola. Ordinary asymptote. The satellite line is not at 
infinity, and it may lie in either of the positions just mentioned. We have therefore 
XLVIII. Satellite line lies between the parallel asymptotes. 
XLIX. „ „ outside „ „ 
Osculating asymptote ; the satellite line is at infinity. We have 
L. 
Hyperbolisms of the Ellipse. Ordinary asymptote. The satellite line is not at 
infinity, and we have 
LI. 
Osculating asymptote. Satellite line is at infinity, 
LII. 
Hyperbolisms of the Parabola. Ordinary asymptote. Satellite line is not at infinity 
we have 
LIII. 
Osculating asymptote: satellite line is at infinity: we have 
LIV.