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 ;