8 
ON THE QUARTIC SURFACES (*JU, V, W) 2 = 0. [487 
Envelope of the planes through the points of an ellipsoid at right angles to the 
radius vectors from the centre. 
This is given in my paper “ Sur la surface &c.” in the Annali di Matematica, 
t. H. (1859), [250], as the envelope of the quadric surface 
a? 
+ 
2T 
2 — - 
b 2 
+ 
— 6w 2 = 0. 
The reciprocal quartic surface is thus the envelope of 
( 2 -s) z,+ M) 7,+ ( 2 -s)*-s 1p - 0 ’ 
or, what is the same thing, 
/X 2 Y 2 Z 2 \ 1 
(, (^ + F + ^)- 2(Xl+F! + ^ ) + 5 lf2 = 0 ’ 
viz. this is 
/X 2 Y 2 Z 2 \ 
W ” (f + F + f) “ (X! + V1 + =°’ 
which is in fact the inverse surface 
X 
X 2 + Y 2 + Z 2 ’ X 2 + Y 2 + Z 2 ’ X 2 + Y 2 + Z 2 
for X, F, Z 
X 2 Y 2 Z 2 
of the ellipsoid „ + —- = 1; this is obvious geometrically inasmuch as the reci- 
CAj 0 C 
procal of the variable plane is the inverse of the point on the ellipsoid. 
The quartic surface has the nodal conic 
W=0, X 2 + Y 2 + Z 2 = 0 ; 
and also the node X = 0, F=0, Z= 0; there is consequently in the order of the 
reciprocal surface a reduction 24 + 2 = 26, or the order of the reciprocal surface is = 10. 
Centro-surface of the ellipsoid. 
Writing the equation of the ellipsoid in the form — 2 + ^ — w 2 = 0, the centro- 
surface is given as the envelope of the quadric surface 
a?x 2 b 2 y 2 c 2 z 2 2 _ „ 
feVa 2 ) 2 + oe+b 2 ) 2 + (tf+c 2 ) 2 “ w ~ 5 
(Salmon, [Ed. 2], p. 400, [Ed. 4, p. 179]), and hence the reciprocal quartic surface is the 
envelope of 
{ a+ ^ x * + { b + t) 7 ’ + ( D+ t) Zl - w ^ 0 ’