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.