340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 205 
case of several multiple lines, there are other conditions of inequality preventing self 
contradictory results ( x ). 
11. The general section is a curve of the order m, having an a-tuple point and 
a /3-tuple point corresponding to the directrix lines respectively, and a y-tuple point, 
&c. ... corresponding to the other multiple points (if any). A section through the 
directrix line 1 is in general made up of this line, counting a times, and of /3 gene 
rating lines passing through one and the same point of the directrix line 1'; if the 
section pass also through a 7-tuple generating line, then, of the /3 generating lines in 
question, 7 (which, as has been seen, is $> ¡3) unite together in the 7-tuple generating 
line; and so for the sections through the directrix line T. The general section through 
a 7-tuple generating line is this line counting 7 times, and a curve of the order 
m — 7, which has an (a—7) tuple point at its intersection with the directrix line 1, 
and a (/3 — 7) tuple point at its intersection with the directrix line 1'; it has a S-tuple 
point, &c... at its intersections with the other multiple generating lines, if any. 
Scrolls with a twofold directrix line, Article Nos. 12 to 16. 
12. But there is a case included indeed as a limiting one in the foregoing general 
case, but which must be specially considered; viz. the two directrix lines 1 and 1' may 
coincide, giving rise to a twofold directrix line. To show how this is, I return for the 
moment to the case of the scroll S (1, 1, m) with two distinct directrix lines 1 and T, 
and, to fix the ideas, I suppose that the directrix lines do not either of them meet 
the curve m, so that the order of the scroll is = 2m. Through the line 1 imagine the 
series of planes A, B, G,... meeting the line T in the points a', b', c'..; the generating 
lines through the point a are the lines in the plane A to the points in which this 
plane meets the curve m; the generating lines through the point h' are the lines in 
the plane B to the points where this plane meets the curve m; and so for the 
generating lines through the points d, d!... ; and it is clear that the points a', V, c',.. 
correspond homographically with the planes A, B, G,.. . This gives immediately the 
construction for the case where the two directrix lines come to coincide. In fact, on 
the twofold directrix line 1 = 1' take the series of points a, b, c.., and through the 
same line, corresponding homographically to these points, the series of planes A, B, C,..; 
the generating lines through the point a are the lines through this point, in the 
plane A, to the points in which this plane meets the curve m; and so for the entire 
series of points b, c,.. of the line 1 = 1'; the resulting scroll, which I will designate 
as the scroll 3(1, 1, m), remains of the order = 2m. If there is given a point of the 
curve m, then the plane through this point and the directrix line is the plane A ; 
and the point a is then also given by the homographic correspondence of the series 
of planes and points, and the generating line through the given point on the curve 
m is the line joining this point with the point a. 
1 Suppose, for example (see next paragraph of the text), that there were a 7-tuple generating line and a 
5-tuple generating line lying in piano with the line 1; these lines counting as (7+ 5) lines, must he included 
among the /3 generating lines through the plane in question ; this implies that 7 + 5 :|> /3, a conclusion which 
must be obtainable from consideration of the curve m irrespectively of the scroll.