s. [340 
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
215 
. is a quartic 
is, we have 
1, a quartic 
z., in general 
th two nodal 
nay however 
nerator. For 
we have a 
ic curve with 
3, /3 = 1. In 
point where 
} points: say 
tacnode. We 
ases: viz., in 
is, the scroll 
oint, and the 
i section has 
inches at the 
ls the section 
enumeration 
uartic scrolls 
lines, 
two directrix 
lines 
ce being that 
lions of the 
lations of the 
;t species the 
■ms in z 2 and 
double line the line (os + y = 0, z + w = 0), which will be a double generator; and we 
thus arrive at the equation of the second species of quartic scrolls, viz. this is 
( (« + V)\ O + V) O, y), O, y)%z, z + w) 2 = 0. 
Quartic Scroll, Third Species, S (1 3 , 1, 4), with a triple directrix line 
and a single directrix line. 
40. Taking (» = 0, y = 0) for the equations of the triple directrix line, and 
(z — 0, w = 0) for the equations of the single directrix line, the equation is 
(*$>, 2/) 3 0, w)= 0. 
Quartic Scroll, Fourth Species, S (1 2 , 1 2 , 4), with a twofold (2 (+ 2) tuple) directrix line, 
and without a nodal generator. 
41. Taking (x = 0, y = 0) for the equations of the directrix line, z — 0 for that 
of a plane section of the scroll, y — 0 for the equation of a plane through the tangent 
at the tacnode of the section, and supposing (see ante, No. 22) that the plane through 
the directrix line and the corresponding point on this line are respectively given by 
the equations x=\y and z = \w, the equation of the scroll is. 
(yw — xz) 2 4- (yw — xz) (x, y) 2 + (x, y) 4, — 0. 
Quartic Scroll, Fifth Species, S'(1 2 , ] 2 , 4), with a ttuofold (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 equation, 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 — xz) 2 + 2 (yw — xz) (x + y) (x, y) + (x -f y) 2 (x, y) 2 = 0. 
Quartic Scroll, Sixth Species, S( 1 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 corresponding point on this line are respectively given by the equations x = \y 
and z = \w, the equation of the scroll is 
(yw - xz) (x, y) 2 + (x, yY = 0. 
ave as a new