274 
[680 
680. 
ON THE HESSIAN OF A QUARTIC SURFACE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 141—144.] 
The surface considered is 
U = kW [f 2 + ¥ + ¥^~ ^ + y2 + = °’ 
or say 
U = k 2 w 2 P — Q 2 = 0, 
viz. this may be considered as the central inverse of the ellipsoid 
1 = 0. 
a 2 b 2 c 2 
The values of the second derived functions, or terms of the Hessian determinant 
are 
a, 
h, 
g> 
l 
y 
h, 
b, 
f> 
m 
g> 
f, 
C, 
n 
i, 
m, 
n, 
d 
—„ w 2 — 2Q — 4x 2 , 
a 2 
- 
4 xy 
y 
- 4xz , 
2 k 2 
— wx 
a 2 
-4 xy 
k 2 
¥ w 
2Q- 
4y 2 , 
- 4 yz 
2k 2 
~¥ wy 
— 4 xz , 
- 
4<yz 
y 
k 2 
¥ 
w 2 - 
- 2 Q - 4* 2 , 
2k 2 
~^r wz 
c 2 
2 k 2 
ff WX ’ 
2k 2 
-tf w y > 
2 k 2 
WZ ’ 
k 2 P