316 A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [410 
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 inter 
section 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 generating 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 1 2 , n) = n ([m] 2 -f 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 with five given lines, or say simply the lines which belong to an involution^), 
I say that the locus of a line belonging to the involution, and meeting the skew cubic 
twice is the quartic scroll, tenth species, S (3 2 ). In the particular case where the line 
(instead of belonging to a proper involution) meets a given line, the locus is a quartic 
scroll, eighth species, >8(1, 3 2 ). 
66. The analysis is almost identical with that given (Second Memoir, Nos. 47 to 50) 
in regard to the scroll >8(1, 3 2 ). Considering a line defined by its “six coordinates” 
(a, b, c, f, g, h), the condition which expresses that the line shall belong to an 
involution is 
(A, B, G, F, G, H\a, b, c, f, g, h) = 0, 
where (M, B, G, F, G, H) are arbitrary coefficients; if they are the coordinates of a line, 
that is, if AF + BG + CH = 0, then the condition expresses that the line (a, b, c, f g, h), 
instead of belonging to a proper involution, meets the line (F, G, H, A, B, G). 
1 The theory is explained in my memoir “On the Six Coordinates of a Line,” Camb. Phil. Trans, vol. xi. 
1868, [348]. In explanation of the subsequent analytical investigations of the present memoir, it is convenient to 
remark that if on a given line we have the two points (a, /3, y, 5) and (a', /3', y', 5'), and through the given 
line two planes Ax + By + Cz + Dw = 0 and A'x + B'y + C'z + D'w = 0 ; then we have 
jSy' — f3'y : ya' — y'a : a/3' - a'/3 : a5' — a'5 : ¡35' - ¡3'8 : yd’ - y'5 
= AD’-A'D : BD'-B'D : CD'-C'D : BC'-B'C : CA'-C'A : AB'-A'B; 
and denoting either of these sets of equal ratios by 
a : b : c : f : g : h, 
then (a, b, c, f, g, h) satisfy identically the relation af+ bg + ch = 0, and are said to be the six coordinates of 
the line.