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ON THE CLASSIFICATION OF CUBIC CURVES. 
367 
totes ; and each of these according as the asymptotes form a triangle or meet in a 
point ; that is, according as the asymptote-point does not, or does, lie on the real 
asymptote. 
The Parabolic Hyperbolas are distinguished according as the asymptote is ordinary 
and the asymptotic parabola one of five-pointic intersection ; or, as the asymptote is 
osculating and the parabola one of six-pointic intersection ; and each of these according 
as the asymptote cuts, touches, or does not cut, the parabola. The Hyperbolisms are 
distinguished into those of the hyperbola, ellipse, and parabola, and each of these 
according as the asymptote is ordinary or osculating. The Divergent Parabolas are 
distinguished in the manner already mentioned; viz. according as the curve is the 
semicubical parabola ; or, as there is a satellite line not at infinity, and an asymptotic 
semicubical parabola of seven-pointic intersection, and which is cut, touched, or not 
cut, by the satellite line ; or, as the satellite line is at infinity, and the semicubical 
parabola is of nine-pointic intersection. The Trident Curve and the Cubical Parabola are 
not divided. 
49. I annex the following Table showing the Groups included in each division : 
for shortness I use the before-mentioned symbols A, © to denote that the asymptotes 
form a triangle or meet in a point respectively. 
Table of the Plückerian Divisions. 
Hyperbolas : 
Redundant ; 
No osculating asymptote, 
A i, ii, hi, iv, v, vi 
© VII, VIII 
One osculating asymptote, 
A ix, x, xi, xii, xiii, xiv 
© xv 
Three osculating asymptotes, 
A xvi 
© XVII 
Defective ; 
No osculating asymptote, 
A XVIII, XIX, XX, XXI, XXII, XXIII 
© XXIV, XXV, XXVI, XXVII 
Real osculating asymptote, 
A XXVIII, XXIX, XXX, XXXI, XXXII, XXXIII 
© XXXIV