176 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
that the formulae obtained are inapplicable to these special cases; for instance, as 
will immediately be seen, the number of the lines (7 (to, n, p, q) is obtained as the 
number of intersections of the surface S (to, n, p) by the curve q, — 2mnp x q = 2mnpq ; 
but if the curve q lie on the surface S(m, n, p), then (7 (to, n, p, q) is no longer 
= 2 mnpq. 
The G formulae, Articles 23 to 34. 
23. G (to, n, p, q).—Considering the scroll 8 (to, n, p) generated by a line which 
meets each of the curves to, n, p, this meets the curve q in q 8 {in, n, p) points 
through each of which there passes a line G (w, n, p, q) ; that is, we have 
G (to, n, p, q) = q S (to, 11, p). 
But from this equation we have 
8 (to, n, p) = G( 1, to, n, p) =p S(1, to, n) ; 
thence also 
$(1, w, ?z)=(7(l, 1, in, ii) = nS( 1, 1, n), 
and 
8(1, 1, m) = G( 1, 1, 1, to) — m 8(1, 1, 1); S{ 1, 1, 1) = (7(1, 1, 1, 1) = 2, 
since 2 is the number of lines which can be drawn meeting each of four given right 
lines. Hence ultimately 
G(m, ii, p, q) = innpqG( 1, 1, 1, l) = 2innpq. 
24. G(m 2 11, p).—In a precisely similar manner we find 
G (to 2 , n, p) — npG( 1, 1, to 2 ) = np 8 (1, to 2 ), 
and it is the same question to find G (1, 1, to 2 ) and to find 8(1, to 2 ). I investigate 
G (1, 1, to 2 ) by considering the particular case where the curve to is a plane curve 
having 11 double points. The plane of the curve meets the two lines 1, 1 in two 
points, and the line through these two points meets each of the lines 1, 1, and meets 
the curve in in points ; combining the last-mentioned in points two and two together, 
the line in question is to be considered as £ [to] 2 coincident lines, each of them 
meeting the lines 1, 1, and also meeting the curve to twice. But we may also 
through any double point of the curve draw a line meeting each of the lines 1, 1 ; 
such line, inasmuch as it passes through a double point, meets the curve twice ; and 
we have h such lines. This gives for the case in question (7(1, 1, to 2 ) = h + \ [to] 2 ; 
or, introducing in the place of h the quantity M(= h — \ [to] 2 ), so that h = \ [to] 2 + M, 
we have 
G (1, 1, to 2 ) = [to] 2 + M ; 
and, to the double points of the plane curve, there correspond in the general case 
the apparent double points of the curve to. Admitting the correctness of the result 
just obtained, we then have 
G (to 2 , ii, p) = np ([to] 2 4- ili).