C. V. 
28 
,S. [340 
340] A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 217 
the foregoing 
the elimination of a, /3, 6 from these four equations gives the required equation of 
the scroll. Writing for a moment 
>dal directrix 
>dal conic in 
© = aa 2 + 2/ia 4- /8, 
F = 9 a + / 
the order 4, 
M = la. + m , 
we find 
c/3 2 +2 F/3 + © = 0, 
(©y 2 + 2Myw + dw 2 ) ¡3 2 — 2 (®yz + Mwz) /3 + <3)z 2 = 0; 
or, introducing at this place the condition c = 0, the first equation gives /3 linearly, and 
we thence obtain 
© (©y 2 4- 2 Myw + dw 2 ) + 4 F (%yz 4- Mwz) + 4 F 2 z 2 = 0, 
or, what is the same thing, 
(@y + 2Fz) 2 + 2Mw (@y + 2Fz) + %dw 2 = 0 ; 
the equation 
conic; since 
irface U= 0; 
dal conic are 
t is, TJ must 
lly the term 
whence, observing that we have 
& _ ax 2 + 2hxy + by 2 F _gx +fy M~ lx + my 
y 2 ’ y ’ y ’ 
the equation of the scroll is 
(iax 2 + 2 hxy + by- + 2 gzx + 2/yz)' 2 
+ 2 (ax 2 + 2hxy + by 2 4- 2y^a; 4- 2/y^) (&c 4- wty) w 
4- (ax 2 + 2hxy 4- by 2 ) did 2 = 0. 
We see from the equation that the surface contains the line (x — 0,y = 0) as a 
double line, the conic 
w = 0, ax 2 4- 2hxy 4- by 2 4- 2y^ 4- %fyz = 0 
as a double curve, also the conic 
e x =0, y=0) 
z = 0, ax 2 4- 2lixy 4- by 2 4- 2lxw 4- 2myw 4- dw 2 = 0 
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 
3 nodal conic 
z = 0, ax 2 4- 2hxy 4- by 2 = 0. 
Quartic Scroll, Eighth Species, S (1, 3 2 ), with a directrix line, and a directrix skew 
cubic met twice by each generating line. 
47. We see, d priori, that the scroll is of the order 4, that is, a quartic scroll ; 
in fact for the quartic scroll $(1, m 2 ) the order is =[m] 2 + M (first memoir, p. 457 
[ante p. 172]), and we have here m— 3, M — h — \ [m] 2 = 1 — 3 = — 2 ; that is, order 
= 6-2, =4.