576 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS', 
that is, eliminating l and m, we must have 
a +/3 ^+7 0 2 +^ 3 , a -\-p<p-\-y <p 2 -\-}><p 3 
= 0 , 
a'+/3^+7^ 2 +^^ 3 , a'4-/3'?)+7'<p 2 +By 
or, developing, 
(fif/3' — «'/?)(<£ — 0) + (a/ — a'y)( ( P 2 — ^ 2 ) + ( a ^ — ct!b)(<p 3 — $ 3 ) 
the several terms in (0, <p ), each divided by <p — $, give respectively 
i, <m (^+^) 2 -^, 0?, 0p(p+t), ey, 
which are equal to 
(V 2 , — yr, q 2 —pr,pr, —pq, _p 2 ); 
hence replacing also aj3' — a'/3, &c. by their values c, See., we find 
(c, - b,f ; a, y, A)(r 2 , -yr, q 2 —p>r,pr, -q>q, f)= 0, 
or, what is the same thing, 
(A,/, c, b , «-/, y, r) 2 =0, 
where the coefficients («, b, c,f y, A) satisfy the relation af-\-bg-\-ch= 0; y>, y, r stand 
respectively for 
yw—z 2 , yz—ccw , xz—y 2 . 
Writing for greater convenience 
(A,/, c, b, a — /, — y)=(a, b, c, 2f, 2g, 2h), 
or, what is the same thing, 
(a, b, c,f, y, A)=(b+2g, 2f, c, b, — 2h, a), 
then we have 
y/’+Jy+cA=ac+b 2 +2bg— 4fh=0. 
And hence finally we have for the equation of the scroll S(l, 3 2 ), 
(a, b, c, f, g, h \yw-z 2 , yz—xw , xz—y 2 ) 2 — 0, 
where the coefficients satisfy the relation 
ac+b 2 +2bg— 4fh=0. 
The equations of the directrix cubic are of course 
yw—z 2 = 0, yz—xw= 0, xz—y 2 = 0; 
and the directrix line is given by its six coordinates, 
(b+2g, 2f, c, b, — 2h, a) 
On the general Theory of Scrolls, Article Nos. 51 to 53. 
51. I annex in conclusion the following considerations on the general theory of 
scrolls. Consider a scroll of the nt\i order ; the intersection by an arbitrary plane, say 
the plane w= 0, is a curve of the nt\i order {*fx, y, z ) n = 0 ; any point (x, y, z , 0) where 
( x , y, z) satisfy the foregoing equation, is the foot of a generating line ; and we may 
