10 
ON THE QUARTIC SURFACES (*]£ U, V, F) 2 =0. 
[487 
or, what is the same thing, 
X 2 (X 2 + Y 2 + Z 2 ) + \ [(a 2 - k 2 ) X 2 + (b 2 - k 2 ) Y 2 + (c 2 - k 2 ) Z 2 - W 2 ] +k 2 W 2 = 0, 
whence the equation is 
{(a 2 -k 2 )X 2 +(b 2 -k 2 ) Y 2 + (c 2 - k 2 ) Z 2 - W 2 } 2 - 4<k 2 W 2 (X 2 + Y 2 + Z 2 ) = 0, 
viz. this is a quartic having the nodal conic 
W = 0, (a 2 — k 2 ) X 2 + (b 2 — k 2 ) Y 2 + (c 2 — k 2 ) Z 2 = 0. 
The order of the reciprocal or parallel surface is thus 36 - 24, =12, as it should be. 
The nodal conic of the quartic surface is the reciprocal of a bitangent or node-couple 
quadric cone, vertex the centre, in the parallel surface: this cone is imaginary for the 
ellipsoid, but real for either of the hyperboloids, and its existence in the case of the 
hyperboloid is readily perceived. 
Reverting to the equation 
x 2 y 2 z 2 ( k 2 \ 
aF+D + WVo + tfTe~ v 1 + e) w ~°’ 
or say 
this is 
(a 2 + 0) (b 2 + 6) (c 2 + 6) (k 2 + 0) w 2 
- « 2 (ib 2 + 0) (c 2 + 0) 0 - y 2 (c 2 + 0) (a 2 + 0)0-z 2 (a 2 + 0) {b 2 + 0) 0 = 0, 
(A, B, G, D, E\0, 1) 4 = 0, 
where putting for shortness 
a = a 2 + b 2 + c 2 + k 2 , 
/3 = b 2 c 2 + c 2 a 2 + a 2 b 2 + k 2 {a 2 + b 2 + c 2 ), 
ry = a 2 b 2 c 2 + k 2 (b 2 c 2 + c 2 a 2 + a 2 b 2 ), 
8 = a 2 b 2 c 2 k 2 , 
and 
p = x 2 + y 2 -f- ^ 2 , 
q — (b 2 + c 2 ) x 2 + (c 2 + a 2 ) y 2 + (a 2 + b 2 ) z 2 , 
r — b 2 c 2 x 2 + c 2 a 2 y 2 + a 2 b 2 z 2 , 
we have 
A = 12w 2 , 
B = Saw 2 — 3p, 
C = 2 f3w 2 — 2 q, 
B = Syw 2 — Sr, 
E = 12 8w 2 . 
The equation of the parallel surface is of course 
{AE- 4>BD + 3GJ - 27 (A CE - AD 2 - B 2 E + 2BCD - C 3 ) 2 = 0.