210 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[340 
where U, V, W, ... are functions of x, y of the forms 
(*$>, y) m , (*$>, y) m ~\ (*$>, y) m ~\ ...; 
assuming that these contain respectively the factors 
(y ~ fcx)y, (y - Kx)y~\ (y - K xy~ 1 2 
where 7 ;j> ^m, then the equation takes the form 
(U', V', W'...\y — kx, w (y— kx) + x (iciv — z))y = 0, 
where the coefficients U', V', W', ... are functions of x, y, z, w of the orders m — 7, 
m — 7 — 1, m — 7— 2,...; or, what is the same thing, the equation is 
(U", V", W",.Jy- K x, KW-z)y = 0, 
where U", V", W", ... are functions of x, y, z, w of the order m — 7. The scroll has 
thus the 7-tuple generating line 
y — kx — 0, kw — z = 0. 
Cubic Scrolls, Article Nos. 25 to 35. 
25. In the case of a cubic scroll there is necessarily a nodal( J ) line; in fact for 
the m-thic scroll there is a nodal curve which is of the order m — 2 at least, and of 
the order \{m — l)(m — 2) at most, and which for m = 3 is therefore a right line. And 
moreover we see at once that every cubic surface having a nodal line is a scroll; in 
fact any plane whatever through the nodal line meets the surface in this line counting 
as 2 lines, and in a curve of the order 1, that is, a line; there are consequently on 
the surface an infinity of lines, or the surface is a scroll. We have therefore to examine 
the cubic surfaces which have a nodal line. 
26. Let the equations of the nodal line be x = 0, y = 0; then the equation of the 
surface is 
Uz + Vw +()=0, 
where U, V, Q are functions of (x, y) of the orders 2, 2, 3 respectively, Suppose first 
that U, V have no common factor, then we may write 
Q = (ax + Sy) U + (yx + Sty) V; 
and substituting this value, and changing the values of z and w, the equation of the 
surface is of the form 
Uz+ Vw = 0, 
or, what is the same thing, 
(*$>, yf (z, w) = 0 ; 
1 The nodal line of a cubic scroll is of course a double line, and in regard to these scrolls the epithets 
‘ nodal ’ and ‘ double ’ may be used indifferently.