LS. 
[340 
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 203 
Tiction is the 
oils S', S",..., 
however that 
, to make at 
is a torse, or 
rate of proper 
Dropriety throw 
iting into the 
a line which 
sume that the 
■ve to meeting 
3 in fi points, 
e of 2 is afiy 
be a proper 
two of them 
r appear. And 
r 8 as degene- 
lay degenerate; 
then the lines 
r to which will 
intersect in a 
7 points, then 
%, and 7 cones 
le cone-factors; 
i + fin + 7p. It 
f the curves is 
the number of 
multiplicity in 
curves n and p 
this cone; this 
have S = G 6 S', 
= 6 x order of 
complete inter- 
[y (this implies 
rss through the 
-tuple point on 
simultaneously; 
reduction is in 
all cases the sum of the separate reductions. But the aggregate reduction once ascer 
tained, then writing S (to, n, p) for the order of the reduced scroll, we shall have 
S (to, 11, p) = 2mnp — Reduction. 
In particular, in the case above referred to, where the curves n and p, p and to, 
to and n meet in a, fi, 7 points respectively, but there is no other cause of reduction, 
S (to, 11, p) = 2m np — a to — fin — 7p, 
which is a formula which will be made use of. 
The foregoing investigations apply, mutatis mutandis, to the scrolls 8 (to 2 , n), 8 (in 3 ); 
but I do not at present enter into the development of them in regard to these 
scrolls. 
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 
directrix lines, would either lie in the plane of the two lines, or else would pass 
through them 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 gene 
rating 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 8 (1, 1, to), where to is a plane 
curve. The order of the scroll S (1, 1, to) is in general = 2to ; but if the one line 
meets the curve a times, that is, in an a-tuple point of the curve, and the other 
line meets the curve fi 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 —2m — a — fi; and in particular 
if a. + fi = to, then the order is = to. 
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 2to. 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 to points; and it is in respect of each of these points 
a generating line of the scroll; that is, it is an TO-tuple generating line : the section 
of the scroll by the plane of the curve in is in fact this line counting to times, and 
the curve to ; to + to = 2to, the order of the scroll. And in like manner the section 
by any plane through the m-tuple line is this line counting to 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 2to meeting each of the directrix lines in a point which 
is an TO-tuple point of the section (each directrix line is in fact an TO-tuple line of 
the scroll) ; and by considering, in place of the particular section to, this general section, 
we have the scroll of the order 2to in the form S(l, 1, 2to), where the two directrix 
lines each meet the section to times ; so that the order is 4to — to — to = 2to. 
26—2