206 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[340 
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 V 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 m), and the plane through a be 
the plane of the curve in, 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 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 intersection of the plane of the curve m by the plane 
corresponding to the foot of the directrix line. 
14. Suppose now that the line 1 meets the curve in, 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 2m — 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 = 2m — 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 A 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 2in —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(a—l) + |/3(/3—1) 
double points or nodes; and the case is precisely analogous to that of the scroll 
$(1, 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 A branches having a common tangent, there are 7 branches having 
a common tangent, then the point is, so to speak, a united a (+ A, + 7) tuple point 
equivalent to | a (a — 1) + \ /3 (A — 1) + ^7 (7 — 1) double points or nodes; but the order 
of the scroll is still = 2m — a — /3. 
15. In the same way as the scrolls 8(1, 1, m) are all included in the case where 
the order of the scroll, instead of being = 2m, is = m, so that the scrolls S(l, 1, m) are 
all included in the case where the order of the scroll, instead of being = 2m, is =m.