188 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
point, counting for the three curves respectively as mn — 1, n — 1, and m — 1 points 
respectively. Moreover the construction (ante, Art. 43) for the Nodal Residue 
NR(1, to, n) shows that a generating line meets this curve in (to — l)(?i — 1) points; 
and since the curve is merely a double curve, these count each as a single point; 
and the generating line does not meet the Nodal Generator NG (1, m, n). The 
number of intersections therefore is 
which is 
mn — 1 + (m — 1) 4- (n — 1) + (m — 1) (n — 1), 
= 2 mn — 2, = S — 2. 
60. Similarly for the scroll $(1, m 2 ); the directrix curves are multiple curves, 
viz. the line 1 is a (±\m\ 2 + M)tuple curve, and the curve m a (to —l)tuple curve; 
the generating line meets the former in a single point, counting as \ [to] 2 + M — 1 
points, and the latter in two points, each counting as (m — 2) points. The construction 
(ante, Art. 45) for the Nodal Residue NR(1, to 2 ) shows that the generating line meets 
this curve in \ [to — 2] 2 points; and since the curve is merely a double curve, these 
count each as a single point. Finally, the generating line does not meet the Nodal 
Generator NG( 1, m 2 ). The number of intersections thus is 
l [m] 2 — l + M + 2(m — 2) + %[m — 2] 2 , 
which is 
= [to] 2 — 2 + M, = S — 2. 
In the remaining cases we may use the theorem to find the number of points in 
which the generating line meets the Nodal Residue. Using II as the symbol for the 
points in question (il (m, n, p) for the scroll S (m, n, p), &c.), we find 
61. For the scroll S(m, n, p), 
(mn — 1) + (np — 1) + (mp — 1) +11 (m, n, p) = S — 2 = 2mnp — 2, 
which gives 
II (m, n, p) = 2mnp — mn — mp — np + 1. 
This includes the before-mentioned case 
U(l, m, n) = (ni — 1) (n — 1), 
and the more particular one 
II (1, 1, m) = 0. 
62. For the scroll S(m 2 , n), 
\ [m] 2 — 1 + M + 2 ((m — 1) n — l) + II (ra 2 , n) = S — 2 = n ([m] 2 + M) — 2, 
which gives 
II (m 2 , n) =n ([to] 2 — 2to + 2 + M) — i [to] 2 +1 — M. 
This includes the before-mentioned particular case 
II (1, to 2 ) = i — 2] 2 .