216 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. [340 
I refrain on the present occasion from a more particular discussion of the foregoing 
six species of quartic scrolls. I establish two other species, as follows: 
Quartic Scroll, Seventh Species, S( 1, 2, 2), with nodal directrix line, and nodal directrix 
conic which meet, and with a simple directrix conic which meets the nodal conic in 
tivo points. 
44. We see, d priori, that the scroll generated as above will be of the order 4, 
that is, a quartic scroll. In fact using the formula (ante, No. 5), 
Order = 2 mnp — cum — (in — 7p, 
Simple conic, n = 2, /3 = 1, 
Line , p = l, 7=2, 
45. Take (x = 0, y = 0) for the equations of the directrix line, z = 0 for the equation 
of the plane of the simple conic, w= 0 for that of the plane of the nodal conic; since 
the conics intersect in two points, they lie on a quadric surface, say the surface TJ= 0; 
the equations of the simple conic thus are z = 0, TJ = 0; those of the nodal conic are 
w — 0, U = 0. The directrix line x = 0, y = 0 meets the nodal conic; that is, U must 
vanish identically for x = 0, y = 0, w = 0; and this will be the case if only the term 
in z 2 is wanting; that is, we must have 
U = (a, h, 0, d, f, g, h, l, m, n\x, y, z, w) 2 . 
But we may in the first instance omit the condition in question, and write 
U = (a, h, c, d, f, g, h, l, m, n\x, y, z, wf ; 
this would lead to a sextic instead of a quartic scroll. 
46. The equations of a generating line (since it meets the directrix line x=0, y=0) 
may be taken to be 
x= ay, 
the condition in order to the intersection of the generating line with the nodal conic 
is at once found to be 
aa 2 + 2ha + h + 2/3 (f+ga) + c/3 2 = 0, 
and that for its intersection with the simple conic 
aa 2 4- 2ha + h + 26 (m + la) 4- dO 2 = 0 ; 
and writing the equations of the generating line in the form