680]
ON THE HESSIAN OF A QUARTIC SURFACE.
277
we have
q^.2 / l
H + 12& 2 © JJ = 9k 6 w 4 P \ w 2 + 2
a 2 b 2 c 2
b-c 1 " r c 2 a 2 a 2 6 2
^ . ,x 2 y- z-
e “ 4, .i‘ + i + 5
Hence, recollecting that U = & 2 w 2 P — Q 2 , the Hessian curve of the order 32 breaks
up into
U = 0, w 4 = 0, that is, Q 2 = 0, w i = 0, or the nodal conic,
w — 0, Q— 0, 8 times (order 16),
U = 0, P = 0, that is, Q 2 = 0, P = 0, or the quadriquadric,
P = 0, Q = 0, 2 times (order 8),
and into a curve (order 8) which is
k 2 w 2 P -Q 2 = 0,
3 k 2
a 2 b 2 c 2
Q-hS + 's+S) = o.
a 4 6 4
viz. this, the intersection of the surface with a quadric surface, is the proper Hessian
curve.