410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 315 
60. The coordinates (x, y, z, w) may be fixed in such manner that the equations 
of the skew cubic shall be 
X, 
y> 2 
y> 
z, w 
or, what is the same thing, 
yw — z- = 0, zy — xw = 0, xz — y 2 = 0 ; 
each of the equations pq — p'q = 0, rq —r'q = 0, pr'—p'r — 0 is then the equation of a 
quadric surface passing through the skew cubic, or, what is the same thing, each of 
the functions pq'—p'q, rq —rq, pr — p’r is a linear function of yw — z 2 , zy — xw, xz — y 2 \ 
and the equation of the scroll is given as a quadric equation in the last-mentioned 
quantities. It will be convenient to represent the equation in the form 
(H, F, C, B, A — F, — G\yw — z 2 , zy — xw, xz — y 2 ) 2 = 0, 
or, writing for shortness 
yw — z 2 , zy — xw, xz — y 2 =p, q, r, 
which letters (p, q, r) are used henceforward in this signification only, the equation 
will be 
(H, F, G, B, A-F, -G\p, q, r) 2 = 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 + GH = 0, we have the eighth species, S (1, 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 -f GH= 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 , n) 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 is m= 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, [339], and Second Memoir, No. 5) the order of the scroll is given by the formula 
$ (m 2 , w) == ?i([m]' 2 + iff) — 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. 
40—2