72
ON THE CURVES SITUATE ON A SURFACE OF THE SECOND ORDER. [314
For n = 4, I take first the species (2, 2) [the quadriquadric curve] which is repre
sented by an equation of the form
{a, b, c^jX, /j,) 2 v~ + 2 (o', b', c'^X, pif vp + (a", b", c"^X, g)‘ 2 p 2 = 0,
which in fact belongs to a quartic curve, the intersection of two quadric surfaces.
For, reverting to the original coordinates, the equation becomes
(a, b, c§x, yf sc 2 + 2 (a', b', c'^cc, y) 2 xz + (a", b", c"\x, y'fz- — 0,
which by means of the equation xw--yz = 0 of the quadric surface is at once reduced to
(a, b, c][x, yf + 2axz + 4b'yz + 2c'yw + a'z 2 + 2b"zw + c"iv 2 =■ 0,
which is the equation of a quadric surface intersecting the given quadric surface
xw — yz=0 in the curve in question.
Consider next the species (3, 1) [the excubo-quartic curve] represented by an
equation of the form
(a, b, c, d][X, yf v + (a', b', c', d' $\, pf p = 0,
which is the other species of quartic curve situate on only a single quadric surface.
Reverting to the original coordinates, the equation becomes
(a, b, c, d\x, y) 3 x + (a', b', c', d'][x, y) 3 z = 0;
and by means of the equation xw — yz = 0 of the quadric surface this is reduced to
(a, b, c, d\x, yf + a'x-z + Sb'xyz + ocy 2 z + d'yhu — 0,
which is the equation of a cubic surface containing the line (x = 0, y = 0) twice, and
therefore along this line touching the quadric surface xw — yz = 0; and consequently
intersecting it besides in a quartic curve. And in like manner for the curves of the
fifth and higher orders which lie upon a quadric surface.
The combination of the equations
(* PF O, p) q = 0,
(*'$>, p.y (y, pY = 0,
show's at once that two curves on the same quadric surface of the species (p, q)
and (pq') respectively intersect in a number (pq' +p'q) °f points. Thus if the curves
are (1, 0) and (1, 0), or (0, 1) and (0, 1), i.e. generating lines of the same kind, the
number of intersections is 1.0 + 0.1=0; but if the curves are (1, 0) and (0, 1),
i.e. generating lines of different kinds, the number of intersections is 1.1+0.0 = 1.
The notion of the employment of hyperboloidal coordinates presented itself several
years ago to Prof. Plticker (see his paper “ Die analytische Geometrie der Curven auf
den Flächen zweiter Ordnung und Classe,” Grelle, vol. xxxiv. pp. 341—359, 1847);
but the systems made use of, e.g. £ = — — -, y — — - -, with z {z + d) + pxy = 0 for
A L V P x
the equation of the surface of the second order, is less simple; and the question of
the classification of the curves on the surface is not entered on.
2, Stone Buildings, W.G., May 24, 1861.
