[340 
340] 
A SECOND MEMOIR ON SKEW SURFACES, OTHERWISE SCROLLS. 
213 
be noticed 
te equation 
3 equations 
<T). The 
die equation 
two points, 
which there 
then, if the 
nes through 
w = 0 will 
aving a node, 
Kq ; through 
tie node draw 
the line 1; 
its G and D 
z — 0, w = 0 
eetion of the 
at the point 
of 
or z, 
necessity 
iv, then 
2fth crunodal; 
le is for that 
he remainder 
two different 
spidal points 
intermediate 
Cubic Scroll S ( 1, 1, 3). 
33. Starting from the equation 
{Xx + fiy) {yw — xz) + (* \x, yf = 0, 
then putting iv — yz for w and Xz for z, this may be written 
(\x + fiy) [yw -z{Xw + /zy)} + {*^Xx + yy, yf = 0, 
or, what is the same thing, 
x {yw — xz) + (* ffx, y) 3 = 0 ; 
and then, if (*$#?, yf = {a, ¡3, y, 8\x, yf, this may be written 
x {y (w + ¡3x -f yy) — x{z — a«)} + 8y 3 = 0 ; 
or changing the values of to and £, we have 
x (ytv — xz) + y 3 = 0 
for the equation of the scroll S(l, 1, 3)(*). 
34. The Hessian of the form is x i , and it thus appears that the plane x = 0 is 
a determinate plane through the double line. But y — 0 is not a determinate plane ; 
in fact, if for y we write y + Xx, the equation is 
— x?z + xiv (y + Xx) +(y + Xx) 3 = 0, 
that is 
— x 2 (z — Xw — 3X 2 y — X 3 x) + xy (w + SXx) + y 3 = 0, 
which, changing z and w, is still of the form x {yw — xz) + y 3 = 0. 
The planes z= 0, w = 0 will alter with the plane y — 0, but they are not determined 
even when the plane y = 0 is determined ; in fact we may, without altering the equation, 
change w, z into w + Oy, z + 6x respectively. 
35. In the equation x {yiv — xz) +y 3 = 0, writing y — Xx, we find for the equations 
of a generating line, y = Xx, z = Xtv + X 3 «. Considering the section by the plane 
ax + fiy + yz + 8w = 0, we have 
x : y : z : w = — yX — 3 : — yX 2 — SX : — SX 3 + /3X 2 + aX : yX 3 + /3X + a 
for the coordinates of the point where the generating line meets the section. 
The generating line meets the nodal line at the intersection of the nodal line by 
the plane z = Xtv ; that is, the points £ = Xw on the nodal line correspond to the planes 
y = Xx through the nodal line. In particular the point w = 0 on the nodal line corre 
sponds to the plane x = 0 through the nodal line : the point <yz + 8w = 0 on the nodal 
line (that is, the point where this line is met by the plane ax + (3y + yz + Sw = 0) 
corresponds to the plane yx + 8y = 0 through the nodal line ; the intersections of the 
plane ax + fiy + yz + 8w = 0 by this plane yx + 8y = 0, and by the plane x = 0, are the 
tangents of the section at the node. 
1 It is somewhat more convenient to change the sign of z, and take x(yw+xz) + ij 3 —0 as the canonical 
form.