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PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
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(l, 1, to) of the order 
to ; the curve to being a plane curve of the order to having an a-tuple point and a 
/3-tuple point, where a-j-/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 to, and we have thence the scroll 
S(l, 1, to). Taking at pleasure any point on the curve to, we can through this point 
draw a single line meeting each of the directrix lines 1, 1' ; that is, the curve to 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 to, 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 1' meets this cone in the foot of the line 1', counting /3 times, and besides in 
to— / 3, = a points; the lines joining the vertex with the last-mentioned points respect 
ively (or, what is the same thing, the lines, other than the /3-tuple line, in which the 
plane through the vertex and the line 1' 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 con- 
struct /3 generating lines of the scroll ; and the line 1' is a /3-tuple line of the scroll. 
10. The scroll S(l, 1, to) now in question has not in general any multiple generating 
line ; in fact a multiple generating line would imply a corresponding multiple point on 
the section to ; 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 y-tuple point, then the line from this 
point which meets the two directrix lines counts y times, or it is a y-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 y of any such multiple generating line is at most equal to the 
smallest of the two numbers a and /3; for suppose y> a, then, since a+/3=m, we 
should have y+/3>m, and the line joining the y-tuple point and the /3-tuple point, 
would meet the curve m in y+/3 points, which is absurd. In the case of several 
multiple lines, there are other conditions of inequality preventing self-contradictory 
results *. 
11. The general section is a curve of the order to, having an a-tuple point and a 
/3-tuple point corresponding to the directrix lines respectively, and a y-tuple point, 
&c. . . . corresponding to the other multiple points (if any). A section through the 
directrix line 1 is in general made up of this line, counting a times, and of /3 generating 
lines passing through one and the same point of the directrix line 1' ; if the section pass 
* Suppose, for example (see next paragraph of the text), that there were a y-tuple generating line and a 
S - tuple generating line lying in piano with the line 1; these lines counting as (y+$) lines, must he included 
among the /3 generating lines through the plane in question; this implies that y+£;J>/ 3, a conclusion which 
must he obtainable from consideration of the curve m irrespectively of the scroll. 
