[339 
339] 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
187 
L, to 2 ), we find 
have therefore 
necessary to increase the value of </> (m 3 ) by ymM + 8M 2 ; this is easily seen by writing 
down the values 
(f> (to 3 ) = ymM + 8ilf 2 + aM 3 , 
</> (to 2 , on') = ymM' + 8MM' + 3aM 2 M', 
</> (to, on' 2 ) = ymM + 8M3T + 3aMM' 2 , 
</> (on' 3 ) = yon'31'+ 8M' 2 + aM' 3 , 
the sum of which is 
= y(m + to') (M + M') + 8 (M+ MJ + a (31+ MJ, 
the corresponding term of (jo (to 3 ) ; hence the value of cfo (to 3 ) being correct without the 
foregoing addition, we must have 7 = 0, 8 = 0, a = 0; which confirms the foregoing 
values of </> (to, n, p), </> (:m 2 , oi). 
dues, we have 
56. The equation 
NT (to 3 ) = \ S 2 — S + cj) (to 3 ), 
gives 
NT (to 3 ) = -i S 2 — S + 3m + 31 (l [to] 2 — f to +11) + M 2 , 
— T8 [ W ] 6 + J [to] 5 + [to] 4 + 3to 
+ 31 (a [to] 4 + i ['TO] 3 +1 [to] 2 — £ ??i + 13) 
+ ilf 2 (£ [to] 8 — f to + 3). 
57. We have 
iV r A (to 2 , ?i) = AT (to 2 , %) — AT (to 2 , ft) — NG (m 2 , n), 
— n (f [to] 4 + If (£ [to] 2 — 2to + 3)) 
+ [w] 2 (i [to] 4 + | [fti] 3 + [to] 2 + ilf ([?ft] 2 — |) + if 2 . |). 
id (m=p + q, 
[g] 3 ; that is, 
1 q are each 
mula gives 
58. And moreover 
NR(m 3 ) = NT (to 3 ) — ND (to 3 ) — NG (to 3 ), 
= ^ [to] 6 +1 [to] 5 — | [to] 3 + 3to 
+ M (i [to] 4 — i [to] 3 — | [??i] 2 + 8?ft — 20) 4- ilf 2 (i [??i] 2 — 2on): 
and the investigation of the series of results given in the Table is thus concluded. 
(1, 1), (1, 2), 
(to, n, p) had 
ise the value 
the foregoing 
d have been 
Intersections of a generating line with the Nodal Total, Articles 59 to 63. 
59. We may for the scrolls S (1, on, oi) and /8(1, on 2 ) verify the theorem that each 
generating line meets the Nodal Total in a number of points = S — 2. 
In fact for the scroll /8(1, to, n), the directrix curves are respectively multiple 
curves of the orders mn, n, to, and a generating line meets each of these in a single 
24—2