314 
A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[410 
three generating lines, but through each point of either of the plane quartics only a 
single generating line; that is, that the line is a triple directrix line, but each of 
the plane quartics a simple directrix curve. 
58. We may instead of the section by any plane, consider the section by a plane 
through a generating line, or by a plane through two of the three generating lines 
which meet at any point of the directrix line; if (to consider only the most simple 
case) each of the planes be thus a plane through two generating lines, the section 
by either of these planes is made up of the two generating lines, and of a conic 
passing through the directrix line; the directrices are thus the line and two conics 
each of them meeting the line; we have therefore in the foregoing formula 
m =1, n — 2, p = 2, « = 0, /3 = 1, 7 = 1, 
and the order of the scroll is 8 — 2 — 2, =4 as before. 
Quartic Scroll, Tenth Species, (3 1 2 ), with a directrix skew cubic met twice by each 
generating line( l ). 
59. Consider a line, the intersection of two planes; and let the equation of each 
plane contain in the order 2 a variable parameter 0; the equations of the two planes 
may be taken to be 
(p, q, r\6, 1) 2 = 0, (p\ q', r'\6, 1) 2 = 0, 
where (p, q, r, p', q, r) are linear functions of the coordinates (x, y, z, w); hence 
•eliminating 0, we have as the equation of the scroll generated by the line in question, 
□ = 0, where □ is the resultant of the two quadric functions. The equation may be 
written 
4 ( pq' — p'q) (rq — r'q) — (pr' — p'rY = 0 ; 
and the scroll has thus as a nodal (double) line the skew cubic determined by the 
equations 
P’ 
q, r 
p', 
t / 
q, r 
It is easy to see (and indeed it will be shown presently) that this curve is met twice 
by each generating line of the scroll, and that the scroll is consequently a quartic 
scroll as described above. 
1 I have worded this heading in accordance with that of the eighth species, Second Memoir, No. 47, but 
the two headings might be expressed more completely thus : 
Eighth Species, /S(l, 3 2 2 ), with a directrix line and a double directrix skew cubic met twice by each 
generating line; 
Tenth Species, S (3 2 2 ), icith a double directrix skew cubic met twice by each generating line; 
viz. the subscript 2 would indicate that the skew cubic is a nodal (double) line on the scroll, the exponent 
2 indicating that it is met twice by each generating line.