[339 
339] ON SKEW SURFACES, OTHERWISE SCROLLS. 183 
30 the Nodal 
j lines of the 
nodal residue NR; but in like manner the lines to 3 and to 2 to 4 are generating lines 
of the scroll, and their intersection is a point of NR; and so the lines m x to 4 and 
w 2 to 3 are generating lines of the scroll, and their intersection is a point of NR. We 
have thus (3 x [to] 4 =) | [to] 4 points of NR on the arbitrary plane through the line 1. 
But there are besides the points of NR which lie on the line 1; and if the number 
: in \ [top [>i] 2 
ine 1, and if 
of these be (a), then 
NR (1, to 2 ) = ^ [to] 4 + a. 
the line 1. 
h, as we have 
of the form 
e degree mn. 
] 2 ; but these 
46. The points (a) are included among the cuspidal points of the scroll lying on 
the line 1. Supposing for a moment that x = 0, y = 0 are the equations of the line 1, 
then this line being a (| [to] 2 + Jf)tuple line on the scroll, the equation of the scroll 
is of the form {A, ...'fyjx, y)hW+M — q, where A, ... are functions of the coordinates of 
the degree ^ [to] 2 : the number of cuspidal points on the line 1 is thus 
(2.1 [to] 2 (i [to] 2 - 1 + M) =) [to] 2 (i [to] 2 - 1 + if). 
But these include cuspidal points of several kinds, viz. we have 
[to] 2 [to] 2 — 1 + M) = 2a + 3/3 + R', 
if (a) be the number of points in which the line 1 meets NR, 
» $ >> » >> » ^3 [to 3 ), 
„ R' „ „ „ „ Torse (to 2 ), 
nerated by a 
question is 
where Torse (to 2 ) denotes the developable surface or Torse generated by a line which 
meets the curve to twice. The order of the Torse in question is 
R' — — 2 (m — 3) M 
(see post, Annex No. 5); and then since /3 = S (to 3 ) = ^ [to] 3 + M (to — 2), we find 
2a = [to] 2 [to] 2 — 1 + M) — 3 [?7i] 3 + M (-m — 2)) + 2M (to — 3), 
= [to] 4 + [to] 3 + if ([to] 2 — to), 
and thence 
NR (1, to 2 ) = | [to] 4 + \ [to] 3 + if (| [to] 2 — \rn). 
But I have not succeeded in finding by a like direct investigation the values of 
NR (to, n, p), NR (to 2 , n), NR (to 3 ). 
ig the curve 
to 2 and m 3 to 4 
point of the 
Formulce for NT (I, to, n), NT {1, to 2 ), Articles 47 and 48. 
47. We have 
NT( 1, to, n) — NG (1, to, n) — mn (to + n — 2) + mNnM 
+ NR (1, to, n) + \ mn {mn + m + n — 3) 
+ NR (1, to, n) + f [to] 2 [a] 2 ,