212 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
[340 
or changing the values of (x, y) and of (z, w), the equation is 
arz + yhu = 0, 
which may be considered as the canonical form of the equation. It may be noticed 
that the Hessian of the form is x?y 2 . 
29. We may of course establish the theory of the surface from the equation 
a?z + y 2 w = 0 ; the equation is satisfied by x = Xy, w = — X 2 z, which are the equations 
of a line meeting the line (« = 0, y = 0) (1) and the line (z = 0, w— 0) (1'). The 
generating line meets also any plane section of the surface; in fact, if the equation 
of the plane of the section be ax + ¡3y + jz + 8iv — 0, then we have at once 
x : y : z : w = 8X 3 — yA. : 8X 2 — y : aX + /3 : — a\ 3 — j3X 2 
for the coordinates of the point of intersection. 
30. The form of the equation shows that there are on the line 1 two points, 
viz. the points (as = 0, y = 0, z= 0) and (oc = 0, y = 0, w = 0), through each of which there 
passes a pair of coincident generating lines: calling these A and B, then, if the 
coincident lines through A meet the line 1' in C, and the coincident lines through 
B meet the line 1' in D, it is easy to see that x = 0, y — 0, z = 0, and w = 0 will 
denote the equations of the planes BAC, BAD, BCD, and ACD respectively. 
31. We obtain also the following construction : take a cubic curve having a node, 
and from any point K on the curve draw to the curve the tangents Kp, Kq ; through 
the points of contact draw at pleasure the lines pAC and qBD; through the node draw 
a line meeting these two lines in the points A, B respectively, this will be the line 1; 
and through the point K a line meeting the same two lines in the points G and D 
respectively, this will be the line T; and, the equations x = 0, y — 0, z — 0, w = 0 
denoting as above, the equation of the surface will be x 2 z + y 2 w — 0. 
The points A and B are cuspidal points on the nodal line; any section of the 
scroll by a plane through one of these points is a cubic curve having at the point 
in question a cusp. 
32. It is to be noticed however that the cuspidal points are not of necessity 
real; if for x, y we write x + iy, x — ty, and in like manner z + liv, z — nv for z, w, then 
the equation takes the form 
(x 2 — y 2 ) z — 2 xyw = 0, 
which is a cubic scroll S (1, 1, 3) with the cuspidal points imaginary. 
In the last-mentioned case the nodal line is throughout its whole length crunodal; 
in the case first considered, where the equation is x 2 z + y 2 w = 0, the nodal line is for that 
part of its length for which 0, w have opposite signs, crunodal; and for the remainder 
of its length, or where z, w have the same sign, acnoclal. There are two different 
forms, according as the line is for the portion intermediate between the cuspidal points 
crunodal and for the extramediate portions acnodal, or as it is for the intermediate 
portion acnodal and for the extramediate portions crunodal.