LiS. [340 
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
211 
orders m — y, 
?he scroll has 
3; in fact for 
least, and of 
ht line. And 
s a scroll; in 
line counting 
nsequently on 
re to examine 
[uation of the 
Suppose first 
uation of the 
rolls the epithets 
so that, besides the nodal directrix line (x = 0, y = 0), the scroll has the simple directrix 
line (z = 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 (z — 0, w = 0); that is, it is a cubic scroll of the 
kind $ (1, 1, 3); and since for m = 3 the only partition m — a + /3 is m — 2 + 1, there 
is only one kind of cubic scroll S’ (1, 1, 3), and we may say simpliciter 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 + ¡¿y), then 
zU+wV will contain this same factor, and the remaining factor will be of the form 
z (ax + /3y) + w (yx + 8y), = y (y3z + 8w) + x (az + 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 
(Xx + yy) (yw — xz) 4- (* $#, y) 3 = 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 $(1, 1, 3), viz. it is the scroll of the kind where the section is a cubic 
curve with a 2 (+1) tuple point (ordinary double point, or node), the line of approach 
being one of the two tangents at the node; and since for m = 3 the only partition 
m = a + /3 is m — 2 + 1, there is only one kind of cubic scroll S ( 1, 1, 3), and we may 
say simpliciter that the scroll in question is the cubic scroll S (1, 1, 3). The conclusion 
therefore is that for cubic scrolls we have only the two kinds, $(], 1, 3) and 8 (1, 1, 3). 
The foregoing equations of these scrolls admit however of simplification; and I will 
further consider the two kinds respectively. 
The Cubic Scroll S (1, 1, 3). 
28. Starting from the equation 
(*$«, yf (z, w) = 0, 
or, writing it at full length, 
z(a, b, c§x, y) 2 + w(a', b', c^jx, y)- = 0, 
we may find 6 1} 6 2 so that 
(a, b, c$x, yf + 6 1 (a, b', c'\x, yj = (p 1 x + q 1 yf, 
(a, b, c\x, y) 2 + 6 2 (a, V, c'^x, yf = (p. 2 x + q. 2 yf, 
6 2 and 6. 2 being unequal, since by hypothesis (a, b, cffx, yf and (a', b', c'\x, yf have no 
common factor. This gives 
(a, b, c\x, yf = a(p x x + q 2 yf + /3 (p 2 x + q. 2 yf, 
(a', b', c'][x, yf = 7 (p Y x + q x yf + 8 (p,x + q,yf ; 
or the equation becomes 
(az + 7iv) (p x x + q 2 yf + (J3z + 8w) (p,x + q 2 yf = 0 ; 
27—2