350] 
ON THE CLASSIFICATION OF CUBIC CUBYES. 
391 
Hence the critic centres are real if X 2 > 3g 2 , that is, if the satellite line is 
inclined to the asymptote at an angle > 60°; imaginary if X 2 < 3/r, that is, if the 
' satellite line is inclined to the asymptote at an angle < 60°: and there is a twofold 
centre if X 2 = 3g 2 , that is, if the inclination is = 60°. This assumes, however, that v 
is not = 0, that is that the satellite line does not pass through the asymptote point; 
when it does the distinction of the cases disappears. Hence the groups are 
110. Hyperbolas © Defective. No osculating asymptote. The Satellite line is 
not parallel to the asymptote, and the groups are, 
Satellite line not passing through asymptote point. 
XXIV. Satellite line inclined to asymptote at angle > 60°. 
XXV. „ „ „ „ = 60°. 
XXVI. „ „ „ „ < 60°. 
Satellite line passes through asymptote point, the single group 
XXVII. 
Hyperbolas © Defective. Real osculating asymptote. The satellite line is 
parallel to the asymptote, and we have the single group 
XXXIV. 
Hyperbolas © Defective. Three osculating asymptotes. Satellite line is at 
infinity and we have the single group 
XXXVI. 
The foregoing theory of the hyperbolas A and © completes the enumeration of 
the groups I. to XXXVI. 
As to the groups of the parabolic hyperbolas. Article Nos. Ill to 115. 
111. I consider the equation in the form 
\x(by- + cz 9 - + 2 gzx) + kz 2 (gy + vz) = 0, 
viz. the cubic x (by 2 + cz- + 2gzx) = 0 is made up of a conic by 1 + cz- + 2gzx = 0, and a 
line x = 0; the other cubic & (gy + vz) — 0 is made up of a tangent of the conic, 
regarded as a twofold line, z 9 - — 0, and of a line gy + vz = 0 through the point of 
contact of such tangent. 
112. To determine the critic centres we have 
x. gz + } (by 2 + cz- 4- 2gzx) = 0, 
x .by + kz . gz =0, 
x (cz + gz) + kz (2gy + 3vz) = 0 ;