313
457] ON THE QUARTIC SURFACES V, W) 2 = 0.
which may also be written
c (x + y + z + w) {lixy (z +w) — nzw (x + y)} 4- (hxy + nzw) 2 = 0, (MIZ)
the equation of a quartic surface with the four nodes A, B, C, D; it is to be observed
that the lines AC, AD, BC, BD are, the lines AB and CD are not, lines on the
surface.
A more simple form may be given to the equation as follows; using the second
of the above forms, multiplying the equation by 4, and writing therein
p = V(c) (x +y + z + w),
qr — c(z + wf + 4 nzw,
st — c(x + y) 2 — 4>hxw,
q, r, and s, t being the linear factors of the two quadric functions respectively, we have
and thence
qr — st = c (x + y + z + w)(—x — y + z + iu) + Hixy + 4nzw,
p 2 + qr — st= 2c (z + w) {pc 4- y + z + w) + 4hxy + 4nzw,
wherefore the equation is
or, what is the same thing,
(p 2 + qr — st) 2 = 4p 2 qr,
P + \Z{qr) + \/(st) = o,
(AH)
(AH)
where p, q, r, s, t are any linear functions of the coordinates; this is the equation of
a quartic surface having the nodal conic p = 0, qr — st— 0; and the four nodes
= 0, r = 0, p 2 — st = 0) and (s = 0,t=0,p 2 — qr = 0). It includes the Cyclide, the equation
of which may be written
b 2 = \/{(ax — eh) 2 + b 2 y 2 } + V {(ex — ah) 2 — b 2 z 2 ).
I remark that Prof. Kummer in his most valuable Memoir, “ Ueber die Flächen
vierten Grades auf welchen Schaaren von Kegelschnitten liegen,” Grelle, t. lxvi. (1864),
pp # 60—76, has considered several of the cases of a quartic surface with a nodal conic,
viz. no node, (AC)\ a single node, (AD)\ two nodes (the case AF); and four nodes,
(AH)-, but he has not considered two nodes, the case (AE); nor three nodes, (AG).
In reference to the general case of a quartic surface with a nodal conic, some
most interesting properties have recently been obtained by Prof. Clebsch, see Berl.
Monatsb., April 30, 1868, where it is shown that there are on the surface 16 right
lines forming 20 systems of double-fours, analogous in some respect to the 27 lines
and 36 systems of double-sixes of a cubic surface.
