'204 A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. | 340 
8. And so in general, m being a plane curve, when the scroll S (1, 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 joining 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 k times and 
of the curve m. So that taking, not the section through the multiple generating line, 
but the general section, 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 (1, 1, m) 
for which the order is depressed from 2m to m in consequence of the directrix lines 
meeting the plane section a times and /3 times, that is, in an a-tuple point and a 
/3-tuple point respectively, where a + /3 = m. 
9. It is clear that in the case in question the directrix lines are an a-tuple line 
and a /3-tuple line respectively. The generation is as follows: Scroll S (1, 1, m) of the 
order m; the curve m being a plane curve of the order m having an a-tuple point 
and a /3-tuple point, where a + /3 = m: the directrix lines, say 1 and 1', pass through 
these points respectively, and they do not intersect each other. The generating lines 
pass through the directrix lines 1 and 1' and the curve m, and we have thence the 
scroll S (1, 1, m). Taking at pleasure any point on the curve m, we can through this 
point draw a single line meeting each of the directrix lines 1, T ; that is, the curve 
m is a simple curve on the scroll. Taking at pleasure a point on the directrix line 1, 
and making this the vertex of a cone standing on the curve m, this cone has an 
a-tuple line (the line 1) and a /3-tuple line (the line joining the vertex with the foot 
of the line 1/); the line T meets this cone in the foot of the line 1', counting ¡3 
times, and besides in m — ¡3, = a points; the lines joining the vertex with the last- 
mentioned points respectively (or, what is the same thing, the lines, other than the 
/3-tuple line, in which the plane through the vertex and the line T meets the cone) 
are the a generating lines through the assumed point on the line 1; and the line 
1 is thus an a-tuple line of the scroll. And in like manner, through an assumed 
point of the directrix line 1', we construct /3 generating lines of the scroll; and the 
line T is a /3-tuple line of the scroll. 
10. The scroll >3(1, 1, m) now in question has not in general any multiple gene 
rating line ; in fact a multiple generating line would imply a corresponding multiple 
point on the section m; and this section, assumed to be a curve having an a-tuple 
point and a /3-tuple point, has not in general any other multiple point. But it may 
have other multiple points; and if there is, for example, a 7-tuple point, then the 
line from this point which meets the two directrix lines counts 7 times, or it is a 
7-tuple generating line; and so for all the multiple points of m other than the 
a-tuple point and the /3-tuple point which correspond to the directrix lines respectively. 
It is to be noticed that the multiplicity 7 of any such multiple generating line is at 
most equal to the smallest of the two numbers a and /3; for suppose 7 > a, then, 
since a + /3 = m, we should have 7 + /3 > m, and the line joining the 7-tuple point and 
the /3-tuple point would meet the curve m in 7 + /3 points, which is absurd. In the