DK. J. CASEY ON CYCLIDES AND SPHEEO-QTIAETICS. 
685 
to any plane L related to the quadric will correspond a pair of inverse points having a 
correlative reference to the cyclide, and these inverse points will be the limiting points 
of the sphere U (the Jacobian of «, j3, y, 5) and the plane L. 
258. We have determined in art. 5 the condition that the sphere xa-\-yi3-\-zy-Twb 
should be a generating sphere of W to be given by the determinant (7), and that this 
determinant in tetrahedral coordinates is the equation of the focal quadric F of W. 
Now since for any system of values of x, y, z , w which satisfies the determinant (7) we 
get a point on F, we see that to any point on F will correspond a generating sphere of 
W, and generally to any point P having any special relation to F will correspond a sphere 
Q having a similar relation to W ; in fact the sphere Q will have the point P for centre, 
and will be orthogonal to U. 
259. Since the tetrahedral coordinates of the centre of x x a -{-yf -\-z x y-\-W$ are 
# 15 y x , z x , and if four spheres orthogonal to U pass through the same pair of inverse 
points, with respect to U we know that their centres are complanar. Hence we have 
the following theorem : — 
The condition that the four spheres 
x x a -\-yfi +s 1 y+Wi&, x 2 a -\-y 2 fi +■ z 2 y -f wyi, &c. 
should pass through the same pair of inverse points is the vanishing of the value of the 
determinant 
X x , 
y» 
W 15 
a 2 . 
^25 
W 2 , 
X3 , 
y 3 , 
■^35 
w 3 , 
iT 4 , 
z 4 , 
tv 4 . 
260. In art. 257 it is proved that the pair of inverse points given by the matrix (164) 
correspond to a plane, and in art. 27 it is shown that the perpendiculars from the centres of 
a, /3, y, b on the plane are proportional to x x , ?/,, z x , w x of the matrix, that is, in other words, 
the coordinates of the plane are x x , y x , z x , w,. Hence we infer the following theorem 
The four pairs of inverse points given by the matrices 
a , /3, y, ^ , 
a, (3, y, b , 
« , /3, y, & , 
7 1 & 1 
U'n y 
> 
*^25 U'lt ^25 ^25 
y ^3? W31 
5 
^45 y 45 "45 W 4 , 
are homospheric , if vanishes the determinant 
^1, 
y 15 
a’ 2 . 
y« 
^25 
w a , 
x 3 , 
3/35 
^3? 
w 3 , 
X 4 , 
y» 
w 4 . 
(166) 
261. Since to any system of complanar points corresponds a system of spheres passing 
through a pair of inverse points, to a plane conic on the focal quadric of a cyclide W will 
5 a 2 
