PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
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also through a y-tuple generating line, then, of the (3 generating lines in question, y 
(which, as has been seen, is ^>/3) unite together in the y-tuple generating line ; and so 
for the sections through the directrix line 1\ The general section through a y-tuple 
generating line is this line counting y times, and a curve of the order to — y, which has 
an (a — y) tuple point at its intersection with the directrix line 1, and a (/3—y) tuple point 
at its intersection with the directrix line 1' ; it has a ^-tuple point, &c. . . at its inter- 
sections 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(l, 1, to) with two distinct directrix lines 1 and 1', 
and, to fix the ideas, I suppose that the directrix lines do not either of them meet the 
curve to, so that the order of the scroll is =2 to. Through the line 1 imagine the series of 
planes A, B, C, . . . meeting the line 1' in the points a!, b\ d . . ; the generating lines through 
the point a! are the lines in the plane A to the points in which this plane meets the 
curve to ; the generating lines through the point V are the lines in the plane B to the 
points where this plane meets the curve to ; and so for the generating lines through the 
points c', d' . . . . And it is clear that the points a', b', c\ . . correspond homographi- 
cally with the planes A, B, C, . . . 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 to; 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 S (1, 1, to), 
remains of the order =2 m. If there is given a point of the curve to, 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 to is the line joining this point 
with the point a. 
13. We may say that, in regard to any point a of the line 1, the corresponding plane 
A is the plane of approach of the coincident line 1' ; and that in regard to the same 
point a and to any plane through it, the trace on that plane of the plane of approach is 
the line of approach of 1' ; that is, we may consider that the coincident directrix line 1' 
meets the plane through a in a consecutive point on the line of approach. In particular 
if the point a be the foot of the directrix line 1 (that is, the point where this line meets 
the plane of the curve to), and the plane through a be the plane of the curve to, then 
the intersection of the last-mentioned plane by the plane A which corresponds to the 
point a is the line of approach, and the foot of the coincident directrix line 1' is 
