PEOEESSOE CAYLEY ON SKEW SUEEACES, OTHEEWISE SCEOLLS. 
571 
34. The Hessian of the form is # 4 , and it thus appears that the plane x=0 is a deter- 
minate plane through the double line. But y=0 is not a determinate plane; in fact, 
if for y we write y-\-Xx, the equation is 
— x 2 z-\- xw(y-\-Xx) + (y- i rXx) 3 = 0, 
that is, 
— x\z — 7M) — 3 X 2 y — X 3 x) + xy{ w -f- 3Xx) -\-y 3 = 0 , 
which, changing z and w, is still of the form x(yw-xz)-\-y 3 = 0. 
The planes z— 0, w = 0 will alter with the plane y — 0, but they are not determined 
even when the plane y=0 is determined; in fact we may, without altering the equa- 
tion, change w, z into w-\-dy, z-\-dx respectively. 
35. In the equation x{yw-xz)-\-y 3 = 0, writing y=Xx, we find for the equations 
of a generating line, y = Xx, z = Xw-\-X 3 x. Considering the section by the plane 
ux-\-fiy-{-yz-\-lw=§, we have 
X : y : z : w= — yX — § : — yX 2 — tA : — cA 3 +(3?t 2 -j-a\ : yX 3 -f-(3X+a 
for the coordinates of the point where the generating line meets the section. 
The generating line meets the nodal line at the intersection of the nodal line by the 
plane z=Xw; that is, the points z=Xw on the nodal line correspond to the planes y—Xx 
through the nodal line. In particular the point w=0 on the nodal line corresponds 
to the plane x=0 through the nodal line: the point yz -(- biv = 0 on the nodal line 
(that is, the point where this line is met by the plane ux-\-fiy-{-yz-\-'bw = ti) corre- 
sponds to the plane yx-\-'6y=Q through the nodal line; the intersections of the plane 
a#-f-j 3 ^+y 2 +($w =0 by this plane y#+ciy=0, and by the plane x—0, are the tangents 
of the section at the node. 
Quartic Scrolls , Article Nos. 36 to 50. 
36. We may consider, first, the quartic scrolls S(l, 1, 4). The section is a quartic 
curve having an a-tuple point and a /3-tuple point, where a-f-/3=4; that is, we have 
a=2, /3=2, a quartic with two nodes (double points), or else a=3, (3=1, a quartic 
I with a triple point. But the case a=2, /3=2 gives rise to two species: viz., in general 
the quartic has only the two double points, and we have then a scroll with two nodal 
(2-tuple) directrix lines, and without any nodal generator ; the section may however 
have a third double point, and the scroll has then a nodal (double) generator. For the 
case K= 3, (3=1, the section admits of no further singularity, and we have a quartic 
scroll with a triple directrix line and a single directrix line. 
37. Next for the quartic scrolls S(l, 1, 4). The section is here a quartic curve with 
an a(+|3)tuple point, where a+/3=4; that is, a=2, (3=2, or else a= 3, (3=1. In 
the former case the section has a 2(+2)tuple point, that is, a double point where the 
two branches have a common tangent — otherwise, two coincident double points: say 
the curve has a tacnode; the line of approach is the tangent at the tacnode. We have 
here a scroll with a twofold double line; there are however two cases: viz., in general 
