340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 219 
or, developing, 
(a/3' — a'/3) (<p — 6) + (af — a'y) (<fy — 6 2 ) + (aS 7 — ol'8) ((f> 3 — 6 s ) 
+ (/3f — fi'y) (0(f> 2 — 6 2 (f>) + (/38' — /3'8) (6cf) 3 — 0 3 </>) + (78' — <y'8) (6-(f) 3 — 6 3 (fr) — 0 ; 
the several terms in (6, </>), each divided by <p — 6, give respectively 
1, <f> + 0, (<f) + 6y-(f>d, 6(f), 6(f) ((f) + 6), 6 2 (f)\ 
which are equal to 
(r\ — qr, q 2 —pr , pr, —pq , p 2 ); 
hence replacing also a/3' — a'/3, &c. by their values c, &c., we find 
(c, —b,f a, g, h)(r 2 , -qr, q 2 -pr, pr, -pq, p 2 ) = 0, 
or, what is the same thing, 
(h, f c, b, a-f -g\p, q, rf= 0, 
where the coefficients (a, b, c, f, g, li) satisfy the relation af+ bg + cli — 0; p, q, r stand 
respectively for yw — z 2 , yz — xw, xz — y‘\ 
Writing for greater convenience 
(h, f c, b, a-f, -g) = (a, b, c, 2f, 2g, 2h), 
or, what is the same thing, 
(a, b, c, f, g, h) = (b + 2g, 2f, c, b, - 2h, a), 
then we have 
af+ bg + cli = ac + b 2 + 2bg — 4fh = 0; 
and hence finally we have for the equation of the scroll S (1, 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 n th order; the intersection by an arbitrary plane, say 
the plane w = 0, is a curve of the nth order (*\x, y, x) n = 0; any point (x, y, z, 0) 
28—2