ON SKEW SURFACES, OTHERWISE SCROLLS.
171
[339
339] ON SKEW SURFACES, OTHERWISE SCROLLS. 171
else the orders
lg the curves
stem of lines,
irface or Scroll
it denotes the
10. I remark that the formulae are best exhibited in an order different from
that in which they are in the sequel obtained, viz. I collect them in the following
Table.
G (in, n, p, q) = 2mnpq,.
G (to 2 , ii, p) = np ([to]' 2 + M),
5 the surfaces
)ir, I give the
enerated by a
generated by
i 3 ) the surface
¡ntioned, these
G (to 2 , ft 2 ) = i [to] 2 [ft] 2 +M.i [ft] 2 + N. 1- [to] 2 + MN,
G (in 3 , n) =n (^ [to] 3 M (to — 2)),
G (to 4 ) = ~G [to] 4 + to + M (G [to] 2 — 2in + V-) + M 2 . G
S (in, n, p) = 2ninp,
NB (to, n, p) = | mnp (mn + rup + np — 3),
NG (in, n, p) = mnp (to + n + p — 3) + Mnp + iVmp + Finn,
t, p) are nodal
for n and p.
>eing reckoned
n and p, the
NR (to, 11, p) — mnp (4mnp — (mn + mp + np) — 2 (m + n + p) + 5),
(G NT (to, n, p) = ! S 2 — S + Mnp + Nmp + Fmn,
— 2mnp (mnp — 1) + Mnp + Nmp + Fmn ;
included in which we have
ss G (to 2 , n, p),
reckoned once
S (1, 1, to) = 2to,
ND( 1, 1, m) = [to] 2 ,
NG( 1,1, to) =[m] 2 + M,
nerating lines
¡, three times,
e double line
ft 2 ). And so
h of them a
i 3 ) = 6G (in 4 ).
NR (1, 1, to) = 0,
NT(1, 1, m) =%S 2 -S+ M,
= 2 [to] 2 + M,
and
/Sf(l, to, 11) =2mn,
NB (1, to, n) =\ 11111 (mn + m + n — 3),
curves m, ii, p
a remaining
non-coincident
rectrix curves.
S (in 2 , n) and
md such curve
lal Residue is
NG( 1, to, 11) = 11111(111 +11 — 2) + M11 + N111,
NR (1, to, 11) =| [to] 2 [n] 2 ,
NT (1, in, n) —^S 2 — S+ M11 + Nm,
= 2 [tow] 2 + Mn + Nm.
Moreover
S (in 2 , 11) =11 ( [to] 2 + M),
Residue of the
have
NB (to 2 , ft) = 11 (i ['/ft] 4 + [to] 3 + M (i [to] 2 - IG + M-. i) + [ft]* (i [fti] 3 + £ [to] 2 ),
iVir (to 2 , ft) = ft ( [fti] 3 + if. 3 (to — 2)) + [ft] 2 Q- [to] 2 + -¿if) + A (| [?ft] 2 + M),
1 In the first of the two expressions for NT (m, n, p), S stands for S (m, 11, p); and so in the first of
the two expressions for NT (m 2 , n), &c., S stands for S n), &c.
22—2
