PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
573 
Quartic Scroll , Fifth Species , S'(l 2 , 1 2 , 4), with a twofold (2(+2)tuple) generating line, 
and with a double generator. 
42. Let the equations of the double generator be x-\-y= 0, z-\-w= 0 ; then the line in 
question must be a double line on the surface represented by the last-mentioned equa- 
tion, and this will be the case if only the second and third terms contain the factors 
{x-\-y) and {x-\-y) 2 respectively. The equation for the fifth species consequently is 
{yw-xzy+Z(yw-xz)(x+y)(x, y)+(x+yf(x, yf= 0. 
Quartic Scroll, Sixth Species, S(l 3 , 1, 4), with a twofold (3(+l)tuple) generating line. 
43. Taking ( x =0,y=0) for the equations of the directrix line, z=0 for the equation, 
of a plane section, and assuming that the plane y = 0 passes through the tangent which 
is the line of approach, and that the plane through the directrix line and the corre- 
sponding point on this line are respectively given by the equations x=-"Ky and z—\w, 
the equation of the scroll is 
(yw-xz)(x, y) 2 +(x, y) 4 = 0. 
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(l, 2, 2), with nodal directrix line , and nodal directrix 
conic which meet, and with a simple directrix conic which meets the nodal conic in 
two points. 
44. We see, a 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 — am — fin — yp, 
we have here 
Nodal conic, 
m= 2 , 
a =0. 
Simple conic, 
n =2, 
0=1, 
Line , 
P= 1, 
7=2, 
and hence 
Order =8-2-2, =4. 
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 U= 0 ; the 
equations of the simple conic thus are 2=0, U=0 ; those of the nodal conic are w=0, 
U=0. The directrix line #=0, y=0 meets the nodal conic ; that is, U must vanish iden- 
I tically for x=0, J/=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, b, 0, d,f g, h, l, m, njx, y, z, w)\ 
But we may in the first instance omit the condition in question, and write 
U ={a, b, c, d,f g, h, l, m, nfx, y, z, w) 2 ; 
\ this would lead to a sextic instead of a quartic scroll. 
apcccLxiv. 4 g 
