► 8 
AND SEXTIC SURFACE CONNECTED THEREWITH. 
95 
398] 
le 
8 > 
We thus see that the equations of the nodal curve are 
wxyz = = = z 2 (\x* - yf) 
y — v v—\ X — y ’ 
the nodal curve is thus the partial intersection of the two cubic scrolls (skew surfaces) 
{y — v) wyz = x (yy 2 — vz 2 ), (v — X) wzx = y (vz 2 — Xv 2 ), 
viz. taking A, B, G, D to be the summits of the tetrahedron the faces whereof are 
x = 0, y = 0, z — 0, w = 0, the first of these has AD for a nodal directrix, BC for a 
single directrix, BD, CD for generators; the second has BD for a nodal directrix, AG 
for a single directrix, AD, CD for generators; the surfaces intersect in the line AD 
twice, the line BD twice, and the line CD; the order of the residual curve, or nodal 
curve of the developable, is thus 9 - (2 + 2 +1), =4 as it should be. 
I remark that the equation 
(y — v) y 2 z 2 + (v — X) z 2 x 2 + (X — y) x 2 y 2 = 0, 
is the equation of the cone having its vertex at the point D, (x=Q, y = 0, z = 0), and 
passing through the nodal curve; the lines DA, DB, DG are each of them a nodal line 
of the cone, or “line through two points” of the curve; for an excubo-quartic curve 
the number of lines through two points which pass through a given point not on the 
curve is in fact = 3. 
It remains to introduce the coordinates (x, y, z, w) into the equation of the 
developable. We have 
4«' = w + x — y — z, 
and thence 
giving 
and similarly 
and 
4>y' = w — x + y — z, 
4 z' =w — x — y + z, 
4 w' = w+ x + y + z, 
16y'zf = (w - xf - {y - z) 2 , 
16icV= (w + x) 2 ~(y + zf, 
8 (x'w + y'z') = w 2 + x 2 -y 2 -z 2 
8 (y'w' + zx ) = w 2 — x 2 + y 2 — z 2 , 
8 (z'w + xy) =w 2 -x 2 -y 2 + z 2 . 
16 (y'z + z'x + x'y') = (w -xf-(y-z f 
+ (w- yf ~(z - xf 
+ (w- zf -{x- yf, 
= 3w 2 - 2w (x + y + z) 
-x 2 -y 2 -z 2 -\- 2yz + %zx + 2xy. 
Moreover