410] A THIRD MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 327 
that is 
¿il = Y 5 W (YW-Z 2 ) + X 5 X (XZ- F 2 ) + 2 Y 3 Z 3 X W, 
= - Y 2 Z 2 (XZ — F 2 ) (YW-Z 2 ) - Y 2 Z 2 (YW — Z 2 ) (XZ- F 2 ) + 2Y 3 Z S X W, 
= — 2Y 2 Z 2 {(XZ— Y 2 ) (YW—Z 2 ) — XYZWj, 
= - 2Y 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 $ (1, 3 2 ) generated by a line meeting the line 
x = 0, w = 0, and the cubic curve p = 0, q = 0, r = 0 twice. The equation is obviously 
included in the general equation 
(H, F, C, B, A-F, q, r) 2 = 0, 
where AF+ BG + GH = 0 ; viz. writing A=B = G=G = H=0, this becomes F (q 2 — pr) = 0. 
82. Returning to the general case of the scroll, eighth species, >8(1, 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 (1, 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 >8(1 3 ) with a triple directrix line. 
Conversely, starting with the scroll >8(1 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 >8(1, 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 (l 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 osculating 
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 
considered above) he refers to two other species of quartic scrolls. He remarks that