PROFESSOR CATLET ON CUBIC SURFACES. 
309 
Section XIV=12-B 5 -C 2 . 
Equation WXZ+Y 2 Z+ YX 2 =0. Article Nos. 165 to 171. 
165. The diagram of the lines and planes is 
N M W 
II II 
K) ESI Kj 
II II II 
Planes are 
Z=0 
Y=0 
2— Ej — 0 2 . 
X 
if 
iSI » 
X 
tn 
Cl 
X 
o 
II 
o 
X 
o 
II 
o 
1x15=15 
• • 
Torsal biplane. 
1X20=20 
• 
Ordinary biplane. 
1x10=10 
3 45 
• 
Plane through axis and 
the two rays. 
Cnicnodal ray. 
Biplanar ray in 
torsal biplane. 
Edge. 
t 
where the equations of the planes and lines are shown in the margins. 
166. The edge is a facultative line, as will appear from the discussion of the reci- 
procal surface : J=h'=l ; £'=0. 
167. Hessian surface. The equation is 
WXZ 2 +Y 2 Z 2 -3X 2 YZ+X 4 =0. 
The complete intersection with the surface is made up of the line X=0, Y=0 (the axis) 
5 times, the line X=0, Z=0 (the edge) 4 times, and a skew cubic, the equations of 
which may be written 
X, Y, W | =0. 
4Z, X, — 5Y I 
In fact from the equations U=0, H=0 we deduce H— ZU=X 2 (X 2 — 4YZ) = 0; and if 
in U=0 we write X 2 =4YZ, it becomes Z(XW+5Y 2 )=0 ; and then in 5U=0, writing 
5Y 2 =— XW, we have 
5WXZ+Z(-XW)-f5X 2 Y=X(5XY+4ZW) = 0. 
168. I say that the spinode curve is made up of the edge X=0, Z=0 once, and of 
the cubic curve; and therefore <r'=4. 
