174 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
the lines G (in, n, p, q). The like remarks apply to the lines G (to 2 , n, p), G (m 2 , n-), 
G (to 3 , n), and G (to 4 ); but I will develope them somewhat more particularly as regards 
the lines G (to 4 ). 
16. Given a curve to, then (as in fact mentioned in the investigation for JV7) (to 3 )) 
through any point whatever of the curve there can be drawn 
(h — to -f- 2 = ) [to] 2 + m —2 + M 
lines meeting the curve in two other points, or say [to] 2 + to — 2 + M lines through three 
points. But in general no one of these lines meets the curve in a fourth point; that 
is, we cannot through every point of the curve to draw a line through four points; 
there are, however, on the curve in a certain number (= 4G {in*)) of points through 
which can be drawn a line through four points, or line G (to 4 ). 
17. But the curve to may be such that through every point of the curve there 
passes a line through four points. In fact, assume any skew surface or scroll whatever, 
and upon this surface a curve meeting each generating line in four points (e.g. the 
intersection of the scroll by a quartic surface). Taking the curve in question for the 
curve in, then it is clear that through every point of this curve there passes a line 
(the generating line of the assumed scroll) which is a line through four points, or 
line G (to 4 ). 
18. It is to be noticed, moreover, that if we take on the curve in any point 
whatever, then of the [to] 2 + m — 2 + M lines through three points which can be drawn 
through this point, three will unite together in the generating line of the assumed 
scroll (for if 0 be the point on the curve to, and 1, 2, 3 the other points in which 
the generating line of the assumed scroll meets the curve in, then such generating 
line unites the three lines 012, 013, 023, each of them a line through three points); 
and there will be besides \ [to] 2 + to — 5 + M mere lines through three points. The 
line through four points generates the assumed scroll taken (| [3] 2 =) 3 times, or 
considered as three coincident scrolls; the remaining lines generate a scroll S' (to 3 ), 
which is such that the curve in is on this scroll a (^ [to] 2 + to — 5 + J/)tuple line; the 
assumed scroll three times and the scroll S' (to 3 ) make up the entire scroll S(m 3 ) 
derived from the curve to, or say S (to 3 ) = 3 (assumed scroll) + S' (to 3 ). 
19. The case just considered is that of a curve in such that through every point 
of it there passes a line through four points counting as (£ [3] 2 =) 3 lines through 
three points, and that all the other lines through three points are mere lines through 
three points. But it is clear that we may in like manner have a line through p 
points counting as — l] 2 lines through three points; and more generally if p, q, ... 
are numbers all different and not < 3, and if 
i [to] 2 - to + 2 + M= a. \ [p - l] 2 + /3.1 [<? - l] 2 + ..., 
then we may have a curve to such that through every point of it there pass a. lines 
each through p points and counting as | [p — l] 2 lines, /3 lines each through q points 
and counting as $[q — l] 2 lines, &c....: the case p = 3 gives of course a lines each