362 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
a non-singular asymptote, an asymptote is in general an ordinary tangent or asymptote 
of two-pointic intersection; if, however, the point of contact is an inflexion, then the 
asymptote is an asymptote of three-pointic intersection, or osculating asymptote. In 
particular for the Hyperbolas, the asymptotes may be all ordinary, or they may be two 
ordinary and one osculating, or all three osculating; but they cannot be only two of 
them osculating; for the line through two inflexions meets a cubic curve in a third 
point which is also an inflexion; that is, if two asymptotes are osculating, the third 
is also an osculating asymptote. The foregoing remarks apply as well to the defective 
as the redundant Hyperbolas; it is to be noticed, however, as regards the defective 
Hyperbolas that the osculating asymptote, when there is only one, is necessarily the 
real asymptote, and consequently that the cases are—asymptotes ordinary; the real 
asymptote alone osculating; three osculating asymptotes. For the Parabolic Hyperbolas 
the asymptote, and for the Central Hyperbolisms and the Parabolic Hyperbolisms the 
onefold asymptote, may be ordinary or osculating. 
37. The distinction of ordinary and osculating asymptotes has reference to the 
position of the satellite line; viz. for the Hyperbolas, when there is a single osculating 
asymptote, the satellite line passes through the point at infinity of the osculating 
asymptote, or what is the same thing, the satellite line is parallel to the osculating 
asymptote: and when there are three osculating asymptotes, the satellite line coincides 
with the line infinity. And, conversely, when the satellite line is parallel to an 
asymptote such asymptote is an osculating one, and when the satellite line is at infinity 
the three asymptotes are osculating. For the Parabolic Hyperbolas the asymptote, and 
for the Hyperbolisms the onefold asymptote, is an osculating asymptote when the 
satellite line is at infinity; and conversely. 
38. There is in regard to the Divergent Parabolas a distinction which may be 
mentioned here; viz. the satellite line may disappear altogether (/u, = 0), and the curve 
thus coincide with the asymptotic semicubical parabola. Or, what is the general case, 
the satellite line may be distinct from the line infinity,—and it may cut in two real 
points, touch, or cut in two imaginary points the asymptotic semicubical parabola: or 
the satellite line may coincide with the line infinity, the asymptotic semicubical parabola 
being in this case of nine-pointic intersection. 
39. The term “diameter” is used by Newton in the Enumeratio in two different 
senses; viz. for any given direction of the ordinates there exists a right line or 
“ diameter,” such that measuring the ordinates from this line the sum y + y' + y" of 
the three ordinates is =0. Such diameter is in fact the second or line polar in regard 
to the cubic of an arbitrary point on the line infinity. But the term diameter is 
afterwards and will be here used to denote a diameter absolute dictum, viz. for a 
direction of the ordinates parallel to a non-singular asymptote there may exist a right 
line or “diameter” such that the ordinates measured from this point are equal and 
opposite to each other, or what is the same thing, such that the sum y + y' of the 
two ordinates is = 0; this implies that the asymptote is an osculating asymptote. In 
fact, the first or conic polar of any inflexion of the cubic breaks up into a pair of 
lines, one of which is the tangent at the inflexion, the other of them, the ‘ polar ’ of