182
ON SKEW SURFACES, OTHERWISE SCROLLS.
[339
scroll /8(1, тег, те), and these lines intersect in a point which belongs to the Nodal
Residue NR; and in like manner the lines m 1 щ and т. 2 щ are generating lines of the
scroll, and they intersect on a point of NR; we have thus
(2 . % [тег] 2 . ¡2 [те] 2 =) i [mf [те] 2
points on NR, that is, the arbitrary plane through the line 1 cuts NR in [тег] 2 [те] 2
points. But the plane also cuts NR in certain points lying on the line 1, and if
the number of these be (a), then
NR( 1, m, n) = ^ [тег] 2 [те] 2 + a.
44. The points (a) are included among the cuspidal points on the line 1.
Taking for a moment x = 0, y = 0 for the equations of the line 1 (which, as we have
seen, is a тегте-tuple line on the scroll), the equation of the scroll is of the form
(A, ...\x, y) mn — 0, where A, ... are functions of the coordinates of the degree mn.
The entire number of cuspidal points on the line 1 is thus = 2 [mn] 2 ; but these
include different kinds of cuspidal points, viz. we have
2 [?тете] 2 = 2a + 2« + 2a' + R,
if (a) be the number of points in which the line 1 meets NR,
» ® r> » » » /8 (ni 2 , ii),
» „ » ,, „ S(m, n 2 ),
„ R » „ „ „ Torse (w, n),
Avhere by Torse (m, n) I denote the developable surface or “ Torse ” generated by a
line which meets each of the curves m and n. The order of the Torse in question is
R = (n {[m] 2 — 2h) + m ([n] 2 — 2k) =) — 2 (nM + mN),
see post, Annex No. 4. And then observing that we have
a = S (m 2 , n) = n ([m] 2 + M),
a.' = S (m, n 2 ) = m ([те] 2 + N),
these values give
2a + 2a! + R = 2n [тег.] 2 + 2тег. [те] 2 ,
and we have
a = \ (2 [mnf — 2a — 2a' — R),
= [тегте] 2 — те [тег.] 2 — тег [те] 2 ,
= № №,
and thence
NR (1, тег, n) = f [ггг] 2 [те] 2 .
45. NR(1, тег 2 ).—Through the line 1 take any arbitrary plane meeting the curve
тег in тег points; if m 1} тег 2 , m 3 , тег 4 be any four of these, then the lines m 1 m 2 and тег 3 тег 4
are generating lines of the scroll /8(1, тег 2 ), and their intersection is a point of the
