62 
ON A QUARTIC SURFACE WITH TWELVE NODES. 
[650 
viz. the surface having the 12 nodes is the original surface 
where 
p(YZ+XW) 2 + q(ZX + YW) 2 + r(XY+ZWy, 
p + q + r = 0. 
The Jacobian of the quadrics 
YZ+XW = 0, ZX + YW = 0, XY + ZW = 0, 
w, 
z, 
Y, 
X 
= 
z, 
W, 
x, 
Y 
Y, 
X, 
W, 
Z 
viz. the equations are 
X s — X (F 2 -v Z 2 + If 2 ) + 2FZF = 0, 
Y 3 — Y (Z 2 + X 2 + W 2 )+ 2ZXW=0, 
Z 3 -Z (X 2 + F 2 + W 2 ) + 2XYW = 0, 
Tf 3 - TT(X 2 + F 2 + Z 2 ) + 2XYW = 0, 
each of which is satisfied in virtue of any one of the pairs of equations 
(Y-Z= 0, X-W = 0) 
(Z-X = 0, Y- W = 0) 
(X —Y=0, Z-W = 0) 
(Y+Z= 0, X + W = 0), 
(Z +X = 0, Y + TF = 0), 
(X+ F = 0, w = 0), 
so that the Jacobian curve is, in fact, the six lines represented by these equations. 
Any two of the three tetrads form an octad, the 8 points of intersection of 
three quadric surfaces: a figure representing the relation of the 12 points to each 
other may be constructed without difficulty. 
Each tetrad is a sibi-conjugate tetrad quoad the quadric X 2 +Y 2 + Z 2 +W 2 = 0. 
The three tetrads are not on the same quadric surface.