572 PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
the section has, besides the tacnode, no other double point ; that is, the scroll has no 
nodal generator : the section may however have a third double point, and the scroll 
has then a nodal (double) generator. In the case a=3, /3=1 the section has a triple 
point, and the line of approach is the tangent at one of the branches at the triple point; 
the scroll has a twofold, say a 3( + l) tuple directrix line: as the section admits of no 
further singularity, this is the only case. The foregoing enumeration gives three species 
of quartic scrolls S(l, 1, 4), and three species of quartic scrolls S(l, 1, 4), together six 
species, viz. these are as follows : — 
Quartic Scroll, First Species , S(l 2 , 1 2 , 4), with two double directrix lines , 
and without a nodal generator. 
38. Taking (#=0, y=0) and (5=0, w= 0) for the equations of the two directrix 
lines respectively, the equation of the scroll is 
MOj y)*fa w ) 2 =o. 
Quartic Scroll , Second Species, S'(l 2 , 1 2 , 4), with two double directrix lines, \ 
and with a double generator. 
39. This is in fact a specialized form of the first species, the difference being that 
there is a nodal (double) generator. Supposing as before that the equations of the 
directrix lines are (#=0, y—0) and (5=0, w=0) respectively; let the equations of the 
nodal generator be (x-\-y—§, 5+w = 0) ; then, observing that for the first species the 
equation may be written {*fx, y)\z, 5+w) 2 =0, it is clear that if the terms in 5 2 and 
z{z-\-w ) are divisible by {x-\-yY and (x-\-y) respectively, the surface will have as a new 
double line the line (#-f-?/=0, 5-J-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 
(0+y)!> (%+y)(x, y\ {x,y) 2 Jz,z+wy= o. 
Quartic Scroll, Third Species, S(l 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 (5=0, w= 0) 
for the equations of the single directrix line, the equation is 
(*3> 5 y )\ z > ®)=o. 
Quartic Scroll, Fourth Species, S(l 2 , 1 2 , 4), with a twofold (2(+2)tuple) directrix line, 
and without a nodal generator. 
41. Taking (#=0, y— 0) for the equations of the directrix line, 5=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=Xw, the equation of the scroll is 
(yw-xzf+(yw-xz)(x, yf+(x, yf= 0. 
