112 PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS, 
and we may, by changing the values of z and w, make the term in (x, y) to be 
(* 30 » y) 4 +(<*%+Py)(*Xv> y) 3 +(yx+ty)(*' 30 , y)\ 
where the arbitrary constants a, (3, <y, l may be so determined as to reduce this to a 
monomial kx\ kx 3 y, or Jca?y*. 
56. The coefficient k may vanish, and the equation of the scroll then is 
2 (*X^ y) 3 +w(*'X>, #) 3 =o, 
or, what is the same thing, it is 
OXtf, y)\z, w)= 0, 
viz. the scroll has in this particular case the simple directrix line 2=0, w=0, thus 
reducing itself to the third species , S(l 3 , 1, 4), with a triple directrix line and a single 
directrix line. It is proper to exclude this, and consider the ninth species, S(l 3 ), as 
having a triple directrix line, but no simple directrix line. 
57. The scroll S(l 3 ) maybe considered as a scroll S (m,n,p) generated by a line 
which meets each of three given directrices ; viz. these may be taken to be the directrix 
line, and any two plane sections of the scroll. The section by any plane is a quartic 
curve having a triple point at the intersection with the directrix line ; moreover the 
sections by any two planes meet in four points the intersections of the scroll by the line 
of intersection of the two planes. Conversely, taking any line and two quartics related 
as above (that is, each quartic has a triple point at its intersection with the line, and 
the two quartics meet in four points lying in a line), the lines which meet the three 
curves generate a quartic scroll S(l 3 ). This appears from the formula 
S(ra, n, p)=2mnp—ot,m—fin—yp (Second Memoir, No. 5); 
we have in the present case 
m— 1, n= 4, p— 4, a=4, /3=3, y=3, 
and the order of the scroll is 32 — 4—12 — 12, =4, that is, the scroll is a quartic scroll; 
there is no difficulty in seeing that through each point of the line there pass three 
generating lines, but through each point of either of the plane quartics only a single 
generating line ; that is, that the line is a triple directrix line, but each of the plane 
quartics a simple directrix curve. 
58. We may instead of the section by any plane, consider the section by a plane 
through a generating line, or by a plane through two of the three generating lines 
which meet at any point of the directrix line ; if (to consider only the most simple case) 
each of the planes be thus a plane through two generating lines, the section by either 
of these planes is made up of the two generating lines, and of a conic passing through 
the directrix line ; the directrices are thus the line and two conics each of them meeting 
the line ; we have therefore in the foregoing formula 
m=l, n— 2, p— 2, a=0, /3=1, y=l, 
and the order of the scroll is 8— 2— 2, =4 as before. 
