DR. J. CASEY ON CYCLIDES AND SPHERO-QUAETICS. 
693 
And if we suppose x x , y x , z x , w x constant while x 2 , y 2 , z 2 , w 2 vary, removing the suffixes 
from the lower row in the determinant, we see that if the centre of a variable sphere 
S=0 moves along the plane 
(169) 
a , 
01 , 
on. 
Pi 
#1, 
01 , 
l , 
y» 
on, 
l, 
c , 
r , 
Z\1 
V’ 
r , 
d, 
x , 
z , 
w, 
o, 
- x^a 
J r , Jift J r z 
,y + w : 
spheres form an harmonic pencil of spheres. N ow the sphere S= =xu -f- yft + zy wb, whose 
centre moves in the plane (169), evidently passes through two fixed points, namely, the 
two limiting points of the Jacobian sphere U of a, /3, y, and of the plane (169) ; I 
shall call these points the pole points of the sphere x^u-^y^-^z^ft-ioft. 
Cor. The plane (169) is the polar plane of the centre of with respect to the focal 
quadric of the cyclide. 
276. If two spheres be such that one of them, A, passes through the pole points of the 
other, B, then, conversely, B passes through the pole points of A. This is evident from the 
determinants of the last article, from which it appears that the relation between the 
spheres is reciprocal. I shall extend the known terms of conics and quadrics, and call 
two such spheres conjugate spheres, and their two pairs of pole points conjugate pairs 
of pole points. 
277. If two circles in space be such that the pole points of any sphere passing through 
one lie on the other, then, conversely, the pole points of any sphere passing through the 
latter lie on the former. This is analogous to the theorem in quadrics, that if two 
lines A and B he such that the polar plane of any point of A passes through B, then, 
conversely, the polar plane of any point of B passes through A, and may be derived from 
it by the substitution explained in the last chapter. 
278. If W=( # \u, ft> y , c)) 2 =0 be a cyclide, and cx!, ft, y , S' the sphero-coordinates 
of a pair of inverse points of W, that is, the pair of points given as common to the 
matrix 
^ 5 ft 5 7 ’ ^ 1 
ft ', y'i o', 
and a ", (3", y", b" the sphero-coordinates of another pair of inverse points, then Xu' -{-pa" 
&c. will be the sphero-coordinates of a pair of points concyclic with a', ft, y', b' and 
a", ft', y", I ' 1 ; and if these satisfy the equation of W, we shall have 
X 2 W , + 2^P+^W"=0 (171) 
Now if P=0, the circle through a!, ft, y, ; a!', ft 1 , y", b" meets the cyclide in two pairs of 
inverse points, which are harmonic conjugates to the two pairs a!,ft, y', b ' ; ol", ft", y", o" (see 
5 b 2 
(170) 
