568 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
Cubic Scrolls , Article Nos. 25 to 35. 
25. In the case of a cubic scroll there is necessarily a nodal* line; in fact for the 
TO-thic scroll there is a nodal curve which is of the order to— 2 at least, and of the order 
\{jfi — 1)(to — 2) at most, and which for to= 3 is therefore a right line. And moreover 
we see at once that every cubic surface having a nodal line is a scroll ; in fact any plane 
whatever through the nodal line meets the surface in this line counting as 2 lines, and 
in a curve of the order 1, that is, a line; there are consequently on the surface an 
infinity of lines, or the surface is a scroll. We have therefore to examine the cubic 
surfaces which have a nodal line. 
26. Let the equations of the nodal line be #=0, y= 0; then the equation of thfe 
surface is 
U2+Yw+Q=0, 
where U, V, Q are functions of (x, y) of the orders 2, 2, 3 respectively. Suppose first 
that U, Y have no common factor, then we may write 
Q=(ax -}-(3y)TJ + (y#+ ty)Y ; 
and substituting this value, and changing the values of z and w, the equation of the 
surface is of the form 
Vz+Yw=0, 
or, what is the same thing, 
y)\z, w)=0; 
so that, besides the nodal directrix line (#=0, y — 0), the scroll has the simple directrix 
line (2=0, w=0): it is clear that the section by any plane whatever is a cubic curve 
having a node at the foot of the nodal directrix line (x=0, y=0), and passing through 
the foot of the simple directrix line (2=0, w= 0 ) ; that is, it is a cubic scroll of the kind 
S(l, 1, 3); and since for to= 3 the only partition TO=a-f- j3 is to= 2 + 1, there is only 
one kind of cubic scroll S(l, 1, 3), and we may say sim/pliciter that the scroll in question 
is the cubic scroll S(l, 1, 3). 
27. If however the functions U, V have a common factor, say (Xx-^-py), then 
2U +wY will contain this same factor, and the remaining factor will be of the form 
z{ccx + (fyy + w(yx + ty), =y(Pz+$w)+ x(uz + yw\ 
or, changing the values of z and w, the remaining factor will be of the form yw—xz , 
and the equation of the scroll thus is 
{7,x + py) (yw — xz ) +(*)>, yf= 0, 
where it is clear that the section by any plane whatever is a cubic curve having a node 
at the foot of the directrix line x=0, y= 0. The scroll is thus a cubic scroll of the 
form S(l, 1, 3), viz. it is the scroll of the kind where the section is a cubic curve with 
a 2(+l)tuple point (ordinary double point, or node), the line of approach being one 
of the two tangents at the node; and since for to— 3 the only partition TO=a+/3 is 
* The nodal line of a cubic scroll is of course a double line, and in regard to these scrolls the epithets ‘ nodal’ 
and 4 double’ may be used indifferently. 
