PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
575 
as a simple curve on the surface, — the complete intersection by the plane z — 0 being in 
fact the last-mentioned conic, and the pair of lines 
2=0, ax 2 -\-2hxy-\-by 2 =0. 
Quartic Scroll , Eighth Species , S(l, 3 2 ), with a directrix line , and a directrix skew 
cubic met twice by each generating line. 
47. We see, a priori, that the scroll is of the order 4, that is, a quartic scroll ; in fact 
for the scroll S(l, m 2 ) the order is = [m] 2 +M (first memoir, p. 457), and we have here 
m— 3, M = A— i[m] 2 = 1 — 3 = — 2 ; that is, order =6—2, =4. 
48. The equations of the cubic curve may be taken to be 
x, y , 2 
y, 2, w 
= 0 , 
or, what is the same thing, 
xz—y 2 = 0, xw—yz= 0, yw—z 2 — 0; 
those of the directrix line may be represented by 
ax +/3 y -j-yz =0, 
ot!x-\-fiy-\-^z-\-'tiw=Q ; 
or, what is the same thing, if 
j3y' — /3 'y=a, al' — a'h =f, 
ya—y'a=b, 'b=g, 
aj3'—a'(5=c, yb' — y l 'h=h 
(and therefore identically af-\-bg-\-ch= 0), the line is defined by means of its “ six coor- 
dinates ” (a, b, c,f, g, h). 
49. The equations of the cubic curve are satisfied by writing therein 
x:y:z:w=l:t:t 2 :f, 
and therefore the coordinates of any two points on the curve may be represented by 
(1, 0, Q 2 , 0 3 ) and (1, <p, <p 2 , <p 3 ); hence, if x, y, z, w are the coordinates of a point in the 
line joining the last mentioned two points, we have 
x:y:z: w=l-\-m: : l6 2 -\-m<p 2 : l6 3 -\-m<p 3 , 
which equations, treating therein l, m as indeterminate parameters, give the equations 
of the line in question. And putting moreover 
p—yw—z 2 , q=yz—xw, r—xz—y 2 , 
we have identically 
p : g : r—Q<p : — ($+<p) : 1. 
50. In order that the line in question may meet the directrix line, we must have 
l(a+p0+y0 2 +l0 3 )+m(u+p<p+y? 2 +b<p 3 )=O, 
l(u'+p'0 + yd 2 + l ! P) +m(a'-(- (3 '<p + y'f + l'<p 3 ) = 0 ; 
