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