PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
125 
that is, 
1Q= Y 5 W(YW - Z 2 ) +X 5 X(XZ — ' Y 2 ) + 2 Y 3 Z 3 XW 
= — Y 2 Z 2 (XZ — Y 2 ) ( Y W — Z 2 ) — Y 2 Z 2 ( Y W — Z 2 )(XZ — Y 2 ) -j- 2 Y 3 Z 3 XW 
= - 2Y 2 Z 2 { (XZ - Y 2 )(YW— Z 2 ) - XYZW} 
= — 2 Y 2 Z 2 { Y 2 Z 2 — Y 3 W — Z 3 X } 
=0, by the equation of the scroll ; 
and we thus see that the equation of the reciprocal scroll is 
(yw — z 2 ) ( xz—y 2 ) —{yz—xw) 2 — 0, 
or say 
q 2 — pr=0, 
viz. it is a scroll S(l, 3 2 ) generated by a line meeting the line #=0, w— 0, and the cubic 
curve p=0, q=Q, r=0 twice. The equation is obviously included in the general equa- 
tion 
(H, F, C, B, A-F, -G^p, q, r) 2 =0, 
where AF+BG + CH=0; viz. writing A=B=C=G=H=0, this becomes F(q 2 — pr)=0 
82. Beturning to the general case of the scroll, eighth species, S(l, 3 2 ), it is proper 
to show geometrically how it is that the reciprocal is a scroll, ninth species , S(l 3 ). 
Consider in the scroll S(l, 3 2 ) any plane through the directrix line ; this contains three 
generating lines of the scroll, viz. these are the sides of the triangle formed by the three 
points of intersection of the plane with the skew cubic : hence in the reciprocal figure 
we have a directrix line such that at each point of it there are three generating lines ; 
that is, we have a scroll S(l 3 ) with a triple directrix line. Conversely, starting with the 
scroll S(l 3 ), each plane through the triple directrix line meets the scroll in this line 
three times, and in a single generating line ; whence there is in the reciprocal scroll a 
simple directrix line ; but in order to show that it is a scroll, S(l, 3 2 ), we have yet to 
show that there is, as a nodal directrix, a skew cubic met twice by each generating line ; 
this implies that, reciprocally, in the scroll 8(1 3 ) each generating line is the intersection 
of two osculating planes of a skew cubic (tangent planes of a quartic torse), each such 
plane containing two generating lines of the scroll — a geometrical property which is far 
from obvious; and similarly in the scroll, ninth species, S(3 2 ), where the reciprocal scroll 
is of the same form, the property that each generating line is a line joining two points 
of a skew cubic leads to the property that each line is also the intersection of two oscu- 
lating planes of a skew cubic (or, what is the same thing, two tangent,, planes of a quartic 
torse). 
Addition, May 18, 1869. 
Since the foregoing Memoir was written I received from Professor Cremona a letter 
dated Milan, November 20, 1868, in which (besides the ninth and tenth species consi- 
dered above) he refers to two other species of quartic scrolls. He remarks that there is 
