114 
PEOFESSOE CATLET ON SKEW SUEFACES, OTHEEWISE SCEOLLS. 
will be 
(H, F, C, B, A-F, -GJp, q, r)*= 0, 
viz. this is a quadric equation in (p, q, r), with arbitrary coefficients. 
61. Comparing with the result, Second Memoir, Nos. 47 to 50, we see that in the 
particular case where the coefficients (A, B, C, F, G, H) satisfy the relation 
AF+BG+CH=0, we have the eighth species , S(l, 3 2 ), with a directrix line and a 
directrix skew cubic met twice by each generating line. We exclude this particular case, 
and in the tenth species consider the relation AF+BG+CH=0 as not satisfied, and 
therefore the scroll as not having a directrix line. 
62. I consider how the scroll may be obtained as a scroll S(m 2 , ri) generated by a line 
meeting a curve of the order m twice and a curve of the order n once. The first curve 
will be the skew cubic, that ism=3; the second curve may be any plane section of the 
scroll ; such a section will be a quartic curve having three nodes, one at each intersection 
of its plane with the skew cubic. Conversely, if we have a skew cubic, and a plane 
quartic meeting the skew cubic in three points, each of them a node on the quartic, then 
the scroll generated by the lines which meet the skew cubic twice and the quartic once 
will be a quartic scroll. In fact (see First Memoir, No. 10, and Second Memoir, No. 5) 
the order of the scroll is given by the formula S(m 2 , n) = n{ [m] 2 + M ) — reduction, 
= 16— reduction. And in the present case the reduction arises (Second Memoir, No. 4) 
from the cones having their vertices at the intersections of the skew cubic and the 
quartic, and passing through the skew cubic. Each cone is of the order 2, and each 
intersection qua, double point on the quartic gives a reduction 2 X order of cone, =4; 
that is, the reduction arising from the three intersections is = 12 ; or the order of the 
scroll is 16-12, =4. 
63. We may, instead of the section by a plane in general, consider the section by a 
plane through a generating line ; the section is here made up of the generating line and 
of a plane cubic passing through each of the two points of intersection of the generating 
line with the skew cubic, and having a node at the remaining intersection of its plane 
with the skew cubic. Or we may consider the section by a plane through the two 
generating lines at any point of the skew cubic ; the section is here made up of the two 
generating lines and of a conic passing through the second intersections of the two gene- 
rating lines with the skew cubic ; that is, meeting the skew cubic twice. 
64. Conversely, consider a skew cubic, and a conic meeting it twice ; the lines which 
meet the skew cubic twice, and also the conic, generate a quartic scroll ; this appears 
by the before-mentioned formula S(m 2 , n)=n{[mf-\-M.)— reduction; viz. we have m= 3, 
n=2, and the order is =8 — reduction; the reduction arises from the cones having their 
vertices at the intersections of the skew cubic and the conic. Each cone is of the order 2, 
and ( qua simple point on the conic) each intersection gives a reduction = order of the cone ; 
that is, the total reduction is =4, and the order of the scroll is 8 — 4, =4 as above. 
65. But a more elegant mode of generation of the scroll may be obtained by means 
of the skew cubic alone ; viz. considering the system of lines which are in involution 
