ON THE SKEW SURFACE OF THE THIRD ORDER. 
[From the Philosophical Magazine, vol. xxiv. (1862), pp. 514—519.] 
The skew surface of the third order, or “ cubic scroll ” (disregarding a certain 
special form), may be considered as generated by a line which always passes through 
three directrices; viz., a plane cubic having a node, and two lines, one of them 
meeting the cubic in the node, the other of them meeting the cubic in an ordinary 
point. The analytical investigation possesses some interest as an illustration of the 
analytical theory of skew surfaces in general. 
Take for the equation of the cubic 
(a 3 + /3 3 ) xy — (¿r 3 -f- y 3 ) afi = 0, 
which belongs to a cubic having a node at the origin, and passing through the point 
(a, /3); and for the equations of the two lines 
(x — mz = 0, y — nz — 0), 
(x- a = 0, y - /3 = 0). 
Then, (X, Y, Z) being current coordinates, the equations of the generating line will be 
X = x + AZ, 
Y — y + BZ; 
when this meets the line (X — mZ = 0, Y—nZ= 0), we have 
mZ = x + AZ, 
nZ = y + BZ, 
and thence 
x {n — B) — y (m — A) = 0 ;