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PROFESSOR CAYLEY ON SKEW SERFAGES, OTHERWISE SCROLLS. 
the consecutive point to a along the line of approach. The expression “ the line of 
approach,” used absolutely, has always the signification just explained, viz. it is the inter- 
section of the plane of the curve m by the plane corresponding to the foot of the direc- 
trix line. 
14. Suppose now that the line 1 meets the curve m, or, more generally, meets it a 
times, that is, in an a-tuple point ; it might at first sight appear that the coincident 
line 1' should also be considered as meeting the curve a times, and that the resulting 
scroll should be of the order 2 m — a — a= 2m— 2a. But this is not the case; so long 
as the direction of the line of approach is arbitrary, the line 1' must be considered as a 
line indefinitely near to the line 1, but nevertheless as a line not meeting the curve at 
all; and the order of the scroll is thus =2 m — a. If, however, the line of approach is 
the tangent to a branch through the a-tuple point — that is, if the plane corresponding 
to the a-tuple point meet the plane of the curve in such tangent, then the coincident 
line 1' is to be considered as meeting the curve m in a consecutive point on such branch, 
and the order of the scroll is = 2m — a — 1. And so if at the multiple point there are 
/ 3 branches having a common tangent, then the coincident line 1' is to be considered 
as meeting the curve m in a consecutive point along each of such branches, or say 
in a consecutive /3-tuple point along the branch, and the order of the scroll sinks to 
2m— a — /3. The point spoken of as the a-tuple point is, it should be observed, more 
than an a-tuple point with a /3-fold tangent ; it is really a point of union of an a-tuple 
point and a /3-tuple point, or say a united a(+/3)tuple point, equivalent to 
ia(«-l) + i/3(/3— 1) 
double points or nodes; and the case is precisely analogous to that of the scroll 
S(l, 1, m), where the two directrix lines pass through an a-tuple point and a /3-tuple 
point of the curve m respectively. It may be added that if at the multiple point in 
question, besides the /3 branches having a common tangent, there are y branches having 
a common tangent, then the point is, so to .speak, a united a(+(3, -|-y)tuple point 
equivalent to |a(a — 1) + ^/3(/3 — l)-j-^y(y— 1) double points or nodes; but the order of 
the scroll is still =2m — a — (3. 
15. In the same way as the scrolls S(l, 1, m) are all included in the case where the 
order of the scroll, instead of being =2m, is =m, so the scrolls S(l, 1, m) are all 
included in the case where the order of the scroll, instead of being =2 m, is =m. That 
is, we may suppose that the curve m has a united a( + /3)tuple point (a+/3=m), and may 
take the directrix line to pass through this point, and the line of approach to be the 
common tangent of the (3 branches ; and this being so, the order of the scroll will be 
2m — a— /3, =m. It may be added that if the curve m has, besides the a(-{-/3)tuple 
point, a y-tuple point, then the scroll will have a y-tuple generating line, and so for 
the other multiple points of the curve m. 
16. We may, in the same way as for the scroll S(l, 1, m), consider the different 
sections of the scroll S(l, 1, m) of the order m. The general section is a curve of the 
order m, having an a(+/3)tuple point at the intersection with the directrix line, and a 
