PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 569 
m= 2+1? there is only one kind of cubic scroll (1, 1, 3), and we may say simpliciter 
that the scroll in question is the cubic scroll S(l, 1, 3). The conclusion therefore is 
that for cubic scrolls we have only the two kinds, S(l, 1, 3) and S(l, 1, 3). The fore- 
going equations of these scrolls admit however of simplification; and I will further 
consider the two kinds respectively. 
The Cubic Scroll S(l, 1, 3). 
28. Starting from the equation 
(*Jx,yf(z,w)=0, 
or, writing it at full length, 
z(a, b, cjx, y) 2 -\-w(a!, V, djx, yf= 0, 
we may find 0 X , 0 2 so that 
(a, b, cjx, yf+O^a’, b', djx, y) a =(ihff+2tf) s , 
(a, b, cjx, yf-\-6 2 {a\ b', djx, yf={p^-\-q 2 y) 2 , 
and 0 2 being unequal, since by hypothesis (a, b, c\x, y) 2 and (a 1 , V, d^x, y) 2 have no 
common factor. This gives 
(a , b , c ^) 2=a (i ? i^+M) 2 +/3(^+M) 2 5 
(a', b', djx, yY=y{PiX-\-q>yy+ ^(p.x+q.y) 2 ; 
or the equation becomes 
(uz + yw)(p,x + J +(fi z+ lw){p 2 x + q,yf = 0 ; 
or changing the values of ( x , y) and of (z, tv), the equation is 
x 2 z-\-y 2 w= 0, 
which may be considered as the canonical form of the equation. It may be noticed 
that the Hessian of the form is x 2 y 2 . 
29. We may of course establish the theory of the surface from the equation 
x 2 z-\-y 2 w = 0; the equation is satisfied by x=Xy, w = —X 2 z, which are the equations 
of a line meeting the line (x—0, y= 0) (1) and the line (z= 0, w=0) (1'). The gene- 
rating line meets also any plane section of the surface ; in fact, if the equation of the 
plane of the section be ux-\-fiy-{-yz-\-'hw = 0, then we have at once 
x:y:z: w='bX 3 — yX : bx 2 — y : aX-{-/3 : — aX 3 — fix 2 
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I for the coordinates of the point of intersection. 
30. The form of the equation shows that there are on the line 1 two points, viz. the 
points (#=0, y— 0, 2=0) and (#=0, y— 0, w=0), through each of which there passes 
a pair of coincident generating lines : calling these A and B, then, if the coincident 
lines through A meet the line 1' in C, and the coincident lines through B meet the line 
1' in D, it is easy to see that #=0, y— 0, 2=0, and iv= 0 will denote the equations of 
the planes BAC, BAD, BCD, and ACD respectively. 
