590 
DR. J. CASEY OX CYCLIDES AND SPHEKO-QUARTICS. 
Thus, for example, equation (20) developed is 
Sb 
x , 
y, 
z , 
w , 
1, 
a. 
a!, 
a", 
a'", 
1, 
l >, 
v. 
b". 
V", 
1, 
— c. 
-c\ 
—c", 
-c'\ 
1, 
cl, 
d', 
d", 
d'", 
12. Denoting by J) . . . J 8 the eight orthogonal quadrics (17) ... (24), and remembering 
that X, (a, v, g are the coordinates of the pole of the plane of contact of one of these 
surfaces with S, since these coordinates satisfy the first four equations of art. 11, we 
see easily that they belong to the point common to the system of six planes represented 
by the system of six equations, 
±A=+B= + C= + D, 
in which the arrangement of the signs correspond to the quadric which we consider. 
We have then the following theorem : — 
The poles of contact of the eight orthogonal quadrics Jj . . . J 8 are the eight radical 
centres of the four quadrics , S— A 2 , S— B 2 , S— C 2 , S— D 2 . 
13. The polar of the point x, g>, v, £, with respect to S — A 2 , is 
x(„r — A«) + [fy — A a!) + v{z Aa") + g(w — A a" 1 ), 
and this reduces, in virtue of the first equation of art. 11, to 
+ vz -f- qw = Ip A ; 
and eliminating X, q, v, § from this and the four equations of the same article, we get 
+ 
A, 
x , 
y. 
z , 
w , 
± 
1 , 
a, 
a', 
a", 
a'". 
± 
1 , 
h, 
v. 
b", 
V ", 
± 
1 , 
c, 
d , 
c", 
d", 
± 
1 , 
d, 
d', 
d 
d m , 
(25) 
where the choice of signs depends on the quadric J. This is evidently the plane of con- 
tact of one of the conics of intersection of J and S — A 2 . We have then the following 
construction for the eight orthogonal quadrics: — 
Let us imagine tangent cones whose vertices are the eight radical centres , and the re- 
quired quadrics pass through the conics of contact. 
14. The equations of J,, J 3 , &c. take a very simple form when referred to the tetra- 
hedron which has for vertices the poles of the planes A, B, C, D with respect to S in 
te tr ahedral coordinates. 
Let x', y, z ', w 1 be the new coordinates of the point {x,y, z , w), the poles being a, a', a", 
