ON SKEW SURFACES, OTHERWISE SCROLLS. 
181 
[339 
339] 
-ion of the G 
remarked that 
irve m in m 
sach of them 
the lines in 
is, as already 
the arbitrary 
fore-mentioned 
1 be. This is 
nally obtained 
the curve m 
— pq, and the 
1 ïï № + è [?P 
m = to, M = 0), 
hyperboloids 
any point of 
der m of an 
we have 
40. In the case of a curve m, which is the complete intersection of two surfaces 
oi the orders p and q respectively, or say a complete (p x q) intersection, we have 
m —pq, (h = \pq(p-l)(q-l) and .*.) M = - \pq (p + q- 2); 
and we find 
S (m 3 ) = ipq (pq — 2) (2pq —3p — ‘3q + 4), 
= £/3(/3-2)(2/3-3a + 4) 
if a.=pq, f3 —p + q. The mode of obtaining this result by a direct investigation was 
pointed out to me by Dr Salmon; see post, Annex No. 2. 
Particular cases of the formula for G (m 4 ), Articles 41 & 42. 
41. In the case of a (p, q) curve on the hyperboloid, putting as before m=p + q, 
M = —pq, we find 
G (m 4 ) = T V [p + qf +p + q- pq (| [p + qf - 2 (p + q) + -\f) + \ p-q 1 , 
which is 
= T2 ([^P + M 4 ) ~mP~ !] 3 - %p [q - l] 3 , 
vanishing if p, q are neither of them greater than 3 : this is as it should be, since 
there is then no line which meets the curve four times. The curves for which the 
condition is satisfied are (1, 1) the conic, (1, 2) the cubic, (2, 2) the quadriquadric, 
(1, 3) the excubo-quartic, (2, 3) the excubo-quintic (viz. the quintic curve, which is the 
partial intersection of a quadric surface and a cubic surface having a line in common), 
and (3, 3) the quadri-cubic, or complete intersection of a quadric surface and a cubic 
surface. If either p or q exceeds 3, we have the case of a curve through every 
point whereof there can be drawn a line or lines through four or more points, and 
the formula is inapplicable. 
42. In the case of a complete (p x q) intersection, we have as before m = pq, 
M = — \pq(p 4- q — 2), and the formula for G(m 4 ) becomes 
G (m 4 ) = ( 
— 66a +144 
+ ß (3a 2 +18a- 26) 
I +/3 2 . —6a 
1 + ^.2, 
a formula the direct verification whereof is due to Dr Salmon ; 
see post, Annex No. 3. 
The formulai for NR (1, m, n) and NR (1, m 2 ), Articles 43 to 46. 
43. NR( 1, m, n).—Through the line 1 take any plane meeting the curve m in rn 
points and the curve n in n points; then if vi 1} m. 2 be any two of the m points, and 
Wj, n. 2 any two of the n points, the lines m Y n 2 and m 2 n 2 are generating lines of the