570 
PROFESSOR CAYLEY ON SKEW SURFACES, OTHERWISE SCROLLS. 
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, Kg ' ; through 
the points of contact draw at pleasure the lines g?AC and gBD ; 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 C and D 
respectively, this will be the line V ; and, the equations x=-§, ;g = 0, z— 0, w = 0 
denoting as above, the equation of the surface will be x*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—iy, and in like manner z-\-iw, z—iw for z, w, then the equa- 
tion takes the form 
(x 2 —y*)z — 2 xyw = 0 , 
which is a cubic scroll S(l, 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 z, w have opposite signs, crunodal ; and for the remainder of 
its length, or where z, w have the same sign, acnodal. There are two different forms, 
according as the line is for the portion intermediate between the cuspidal points cru- 
nodal and for the extramediate portions acnodal, or as it is for the intermediate portion 
acnodal and for the extramediate portions crunodal. 
Cubic Scroll S(l, 1, 3). 
33. Starting from the equation 
(\x+t*y)(yw—xz) + (*X^> yY= 0, 
then putting w—[hz for w and Xz for z , this may be written 
(Xx+^iyw- z(Xx-\-py)} +(*X\x-\-py,y) 3 =0, 
or, what is the same thing, 
x(yw— xz)+(*$x, yy=z0 ; 
and then, if (#X^> y) 3 =(«, /3, y, vY-> this may be written 
x{y(w-\- fix + yy) —x(z— ax ) } +^/ 3 =0 ; 
or changing the values of w and z, we have 
x(yw — xz ) +?/ 3 =0 
for the equation of the scroll S(l, 1, 3)*. 
* It is somewhat more convenient to change the sign of z, and take cc{yw-]-xz)-\-'>/=0 as the canonical 
form. 
