9 
487] ON THE QUARTIC SURFACES (*$77, V, Tf) 2 = 0. 
in regard to the variable parameter 6, viz. the equation is 
(—■+ ~ + ~) (a 2 X 2 + b 2 Y 2 + c 2 Z 2 - W 2 ) - (X 2 + Y 2 + ZJ = 0, 
(see Salmon, [Ed. 2], p. 144 [Ed. 4, p. 172]). It hence at once appears, that the quartic 
surface has 12 nodes, viz. these are the four angles of the fundamental tetrahedron 
(XYZW), and the eight points 
X 2 F 2 Z^_ 
a 2 + b 2 + c 2 
" X 2 + Y 2 + Z 2 = 0, 
, a 2 X 2 + b 2 Y 2 + c 2 Z 2 - If 2 = 0, 
or writing as it is convenient to do 
(a, /3, 7) = (6 2 — c 2 , c 2 — a 2 , a 2 — 6 2 ); 
and therefore 
a + /3 + 7 = 0, a 2 a + b 2 /3 + c 2 7 = 0, a 4 a + b 4 /3 + c 4 7 = — a/3y ; 
these are the eight points 
X 2 _ _ a 2 F 2 __^ X__ c[_ 
if 2_ ~/V If 2 ~ ~~ 7a ’ If 2 ~ ~ a/3 ’ 
the order of the reciprocal of the quartic surface is thus 36 — 2.12, =12, which is in 
fact the order of the surface of centres. 
The equation of the centro-surface is given, Salmon, [Ed. 2], p. 151, and Quart. 
Math. Jour., t. 11. (1858), p. 220, in the form 
(«. /3, 7> 6 (£> V, £ w) 12 = 0, 
where £, 77, £, <u stand for a«, %, C2, iw; it is therefore of the degree 18 in regard to 
a, b, c. 
Parallel surface of the ellipsoid. 
This is given, Salmon, [Ed. 2], p. 148 [Ed. 4, p. 176], as the envelope of the quadric 
surface 
x 2 
a 2 + 6 + 
y 2 z 2 
b 2 + 6 + c 2 + 0 
w 2 = 0. 
The reciprocal quartic is thus the envelope of 
6>If 2 
(a 2 + e)X 2 + (b 2 + d)Y 2 + (c 2 + 0) Z 2 -~- d = o, 
or writing k 2 + 6 = X, this is 
(a 2 —¿ 2 + \)X 2 + (6 2 -& 2 + A) Y 2 + (c 2 -k 2 + \)Z 2 - j^l - If 2 = 0, 
C. VIII. 
2