PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
561 
Scrolls with two directrix lines, Article Nos. 6 to 11. 
6. Consider now a scroll having two directrix lines : it may be assumed that these do 
not intersect ; for if they did, then any generating line, qua line meeting the two direc- 
trix lines, would either lie in the plane of the two lines, or else would pass through 
their point of intersection ; that is, the scroll would break up into the plane of the two 
lines, considered as the locus of the tangents of a plane curve, and into a cone having 
for its vertex the point of intersection of the two lines. Each generating line meets 
any plane section of the scroll in the point where such generating line meets the plane 
of the section ; the plane section constitutes a third directrix ; or the scrolls in question 
are all included in the form S(l, 1, m), where m is a plane curve. The order of the scroll 
S(l, 1, m) is in general =2m; hut if the one line meets the curve a times, that is, in 
an os-tuple point of the curve, and the other line meets the curve (3 times, that is, in a 
/3-tuple point of the curve, then by the general formula (ante, No. 5) the order of the 
scroll is =2 m—a—(3; and in particular if a+/3=m, then the order is =m. 
7. We may without loss of generality attend only to the last-mentioned case. To 
show how this is, suppose for a moment that the two lines do not either of them meet 
the curve ; the scroll is then of the order 2m. Call the point' in which each line meets 
the plane of the curve the foot of this line, then the line joining the two feet meets the 
curve in m points ; and it is in respect of each of these points a generating line of the 
scroll ; that is, it is an m-tuple generating line : the section of the scroll by the plane of 
the curve m is in fact this line counting m times, and the curve m; m+m=2m, the 
order of the scroll. And in like manner the section by any plane through the m-tuple 
line is this line counting m times, and a curve of the order m not meeting either of the 
directrix lines. But the section by any other plane is a curve of the order 2m meeting 
each of the directrix lines in a point which is an m-tuple point of the section (each direc- 
trix line is in fact an m-tuple line of the scroll) ; and by considering, in place of the par- 
ticular section m, this general section, we have the scroll of the order 2m in the form 
S(l, 1, 2m), where the two directrix lines each meet the section m times; so that the 
order is 4m— m— m=2m. 
8. And so in general, m being a plane curve, when the scroll S(l, 1, m) is of an order 
superior to m, say =m-\-k, this only means that the section chosen for the directrix 
curve m is not the complete section by the plane of such curve, but that the line join- 
ing the feet of the two directrix lines is a &-tuple generating line of the scroll, and that 
the complete section is made up of this line counting Jc times and of the curve m. So 
that taking, not the section through the multiple generating line, but the general sec- 
tion, for the plane directrix curve, the only case to be considered is that in which the 
section is a proper curve of an order equal to that of the scroll ; or, what is the same 
thing, we have only to consider the scrolls S(l, 1, m) for which the order is depressed 
from 2m to m in consequence of the directrix lines meeting the plane section os times 
and /3 times, that is, in an os-tuple point and a /3-tuple point respectively, where 
a-b/3 =m. 
