390 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
by the two real lines 
106. The critic centres are given by the equations 
2 (Aa? + /xy) — 3v = 0, 
A« 2 — 2 (A + /x) xy + /,t/y 2 = 0; 
or, what is the same thing, the} r are the intersections of the line 
2 (Xx + /xy) — 3r = 0 
Xx — [(A -(- fx) + VA 2 -f- A/X -(- yUr] y. 
107. The groups are 
Hyperbolas © Redundant. No osculating asymptote. 
The satellite line not parallel to any asymptote, that is A = 0, /x = 0, A + /x = 0, We 
have the two groups 
VII. Satellite line does not pass through asymptote point (v not = 0). 
VIII. Satellite line passes through asymptote point (v = 0). 
Hyperbolas © Redundant. One osculating asymptote. Satellite line is parallel to 
an asymptote, suppose to the asymptote x = 0; that is, /x = 0, or the satellite line is 
Air + v = 0. We have only the group 
XV. 
Hyperbolas © Redundant. Three osculating asymptotes. Satellite line lies at 
infinity, that is, A = 0, ¡x = 0. We have only the group 
XVII. 
The Hyperbolas © Defective. Article Nos. 108 to 110. 
108. The equation may be taken to be 
^ x {x" + y 2 ) + kz 2 (Xx + ¡xy + vz) = 0, 
or writing z= 1, then it is 
^ x (x 2 + y 2 ) + k (Xx + /xy + v) = 0, 
and if to fix the ideas we take the case where the two imaginary, asymptotes are 
the asymptotes of a circle, then x, y will be ordinary rectangular coordinates. 
109. The critic centres are given by the equations 
2 (\x + /xy) — Sv= 0, 
3 /xx 2 — 2A xy + /xy 2 = 0, 
that is they are the intersections of the line 2 (\x + /xy) — Sv = 0 (which is a line 
parallel to the satellite line, on the other side of the asymptote point and at a 
distance from it = f distance of satellite line) by the pair of lines 
S/xx = (A + Va 2 — 3/x 2 ) y.