172 
OX SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
NR (m 2 , n) = n (§ [mp + ilf (-| [mp — 2m + 3)) 
+ № (2 V m Y +1 \j n J + i m f + M ([m] 2 — I) + if 2 . £), 
NT (m 2 , n) = ^ S 2 — S + nM(m — f) 4- N (■£ [m] 2 + M), 
= n (| [m] 4 + 2 [mp + M ([m] 2 + m — -|) + ilf 2 . ^) 
+ [ n f (2 [w&P + 2 [m] 3 + [m] 2 + ilf. [m] 2 + M‘ 1 . -|) + N [m] 2 + ilf); 
included in which we have 
*9 (1, m 2 ) = [mp + ilf, 
AD (1, m 2 ) = [mp + [m] 3 + ilf (| [m] 2 — -|) + M 2 . 
NG (1, m 2 ) = [m] 3 + M. 3 (m — 2), 
AD! (1, m 2 ) = | [mp + ilf (|- [mp — 2m 4- 3), 
iVT(l, m 2 ) = |- $ 2 — S 4- ilf (m — |), 
= ^ [m] 4 + 2 [m] 3 4- ilf ([m] 2 4- m — -1) + ilf 2 . ; 
and finally 
$ (m 3 ) = 1 [m] 3 + (m — 2) M, 
ND (m 3 ) = -i [m] 6 4- ^ [mp 4- \ [m] 3 4- ilf (p [m] 3 4- ^ [m]) + ilf 2 . ^m, 
NG (m 3 ) = [m] 4 4- 6m 4- M (3 [m] 2 — 12m 4 33) + ilf 2 .3, 
NR (m 3 ) = t \t [m] 6 4- f [m] 5 — P [mp 4- 3m 
4- ilf [m] 4 — ^ [m] 3 — f [m] 2 4- 8m — 20) 4- ilf 2 (|- [mp — 2m), 
NT (m 3 ) = ^ S 2 — S + 3m + ilf [mp — |m 4-11)4- ilf 2 , 
= is t w ] 6 + 2 [ m ] 3 + [ m P + 3m 
+ ilf [m] 4 + J- [mp 4- p [m 2 ] — \m 4- 13) 4- M 2 [m] 2 — fm + 3). 
The formulae are investigated in the following order, ND, G, NG, S, NR, and NT. 
The ND formulce, Articles 11 to 13. 
11. ND (m, n, p).—Taking any point on the curve m, this is the vertex of two 
cones passing through the curves n, p respectively ; the cones are of the orders n, p 
respectively, and they intersect therefore in np lines, which are the generating lines 
through the point on the curve m; hence this curve is an np-tuple line on the 
scroll S (m, n, p), and we have thus the term m. £ [np~f of ND. Hence 
ND (m, n, p) — m [np] 2 + n.\ [mp] 2 + p. ^ [map, 
= \mnp (mn + mp + np — 3).