366
ON THE CLASSIFICATION OF CUBIC CURVES.
[350
6. Defective Hyperbolas with a diameter.
Sp. 39, 40, 41, 42, 43, 44, 45.
7. Parabolic Hyperbolas without a diameter.
Sp. 46, 47, 48, 49, 50, 51, 52.
8. Parabolic Hyperbolas with a diameter.
Sp. 53, 54, 55, 56, 56', 56".
9. Hyperbolisms of the hyperbola.
Without a diameter, Sp. 57, 58, 59. With a diameter, Sp. 60. Sp. 59 has a centre.
10. Hyperbolisms of the ellipse.
Without a diameter, Sp. 61, 62. With a diameter, Sp. 63. Sp. 62 has a centre.
11. Hyperbolisms of the parabola.
Without a diameter, Sp. 64. With a diameter, Sp. 65.
12. Trident Curve, Sp. 66.
13. Divergent Parabolas. Sp. 67, 68, 69, 70, 71.
14. Cubical Parabola. Sp. 72.
PlücJcers Classification. Article Nos. 47 to 49.
47. Pliicker in the first instance obtains by analytical considerations his six head
divisions, corresponding to the seven divisions in the present Memoir, the Central
Hyperbolisms and the Parabolic Hyperbolisms forming with him a single division. The
equations are obtained in the form already mentioned, the only difference being that he
writes z = l; his six head divisions with their equations thus are
Hyperbolas
Parabolic Hyperbolas
Hyperbolisms
Divergent Parabolas
Trident Curve
Cubical Parabola
pqr + ps = 0,
(p + *) (q 2 + \p) + p (q + <r) = 0,
p(q 2 + K) + p(q +a) = 0,
p 3 + \q 2 + p(p + er) = 0,
p(p 2 + \q) + p = 0,
p 3 + ps = 0.
48. He then divides the Hyperbolas into the redundant and defective. The
redundant hyperbolas are then divided as they have no osculating asymptote, one
osculating asymptote, or three osculating asymptotes ; and each of these according as
the asymptotes form a triangle or meet in a point. As regards the defective hyperbolas
he attends to the imaginary asymptotes, represented by means of their real point of
intersection, the “ asymptote-point,” and the division is thus similar to that of the
redundant hyperbolas, viz. the defective hyperbolas are distinguished according as they
have no osculating asymptote, a real osculating asymptote, or three osculating asymp-
