116 
PEOFESSOE CAYLEY ON SKEW SUEFACES, OTHERWISE SCROLLS. 
Moreover the coordinates of any point on the line in question are given by 
x : y : z : w=l-\-m : l0-\-m<p : l0 1 2 -\-m<p 2 : W-\-m<p 3 ; 
and writing as above p, q, r=yw—z 2 , yz—xw, xz—y 2 , we thence find, omitting the 
common factor (3— <p) 2 , 
p : q : r=0<p : — (0+<p) : 1 ; 
and eliminating Q<p, 0-f<p, we at once obtain 
(A, B, C, F, G, HXW’ i r -> r * > f—R r -> — i >2 )=0, 
or, what is the same thing, 
(H, F, C, B, A-F, -GXp, q, r ) 2 = 0 
as the equation of the scroll generated by the line in involution which meets the given 
skew cubic twice. 
Reciprocal of the Quartic Scroll S(3 2 ). 
67. I propose to reciprocate in regard to the quadric surface x 2 -\-y 2 -\-z 2 -\-t 2 =- 0 the 
foregoing scroll 
(H, F, C, B, A-F, -GXp, q, rf= 0. 
If the coordinates (a, b , c, f, g, h) of a line satisfy the condition of involution 
(A, B, C, F, G, H£«, b, c,f, g, A)= 0, 
then the coordinates (a, b, c, f, g, h) of the reciprocal curve will satisfy the condition of 
involution 
(F, G, H, A, B, C fa, b, c,f, g, h)= 0. 
The reciprocal of the before-mentioned skew cubic x : y : z : w=l : t : tf : f is the quartic 
torse having for its edge of regression the skew cubic 3XZ— Y 2 =0, YZ — 9XW=0, 
3YW— Z 2 =0 ; or, what is the same thing, the skew cubic X : Y : Z : W=1 : 3£ : 3£ 2 : f; 
see my paper “ On the Reciprocation of a Quartic Developable,” Quart. Math. Journ. 
t. vii. (1866) pp. 87-92. 
68. Hence the reciprocal of the quartic scroll is the scroll generated by a line 
(a, b , c, f g , h) the coordinates of which satisfy the condition of involution 
(F, G, H, A, B, CX®, b, c,f g, h)= 0, 
and which is moreover the intersection of two osculating planes of the skew cubic 
X : Y : Z : W=1 : 2>t : 3£ 2 : f. For the point the parameter whereof is t, the equation 
of the osculating plane is 
X, Y, Z , W 
1, St, St 2 , t 3 
= 0 , 
1 , 2 1, t 2 
1 , t 
or, what is the same thing, the equation is 
(t 3 , -t 2 , t, -1XX, Y, Z, W) = 0. 
