G92 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
have four common generating circles ; for the two conics which lie on the polar plane 
of the common centre of inversion have four common tangents. 
Cor. 4. If W=aa 2 -{-bj3 2 -\-C'y‘ 2 -\-db 2 =0, W' + b'(3 2 -j- e'y 2 + d'b 2 = 0 he two cyclides, 
the sphero-quartics (WU) unci (W'U)have sixteen common generating circles; for they 
have four centres of inversion common, namely, the centres of the spheres «, j3, y, S. 
CHAPTER XIY. 
Section I . — Poles and Polars. 
Observation. All the spheres which we shall have occasion to use in this and the 
following chapter will be of the form xoc.-\-yfi-\-zy-\-wh, where x, y, z, w are numerical 
coefficients. 
274. If (a, b, c, d, l, to, n,p, q, rju, (5, y, e>) 2 =0 he a cyclide, and 
Xi<*+yiP +«,y+ Wx^S„ x 2 u -\-y 2 fi -pz 2 y-\- w 2 l = S 2 
be two spheres, then the condition that XSi + ^S 2 =0 may be a generating sphere of the 
cyclide is given by the determinant 
=0. (167) 
This determinant may be written -j- 2 /.ap -j- — 0 , and we have a quadratic for deter- 
mining the ratio X: [a. Now if <p= 0 we shall have the two values X: \a equal, but with 
contrary signs. Hence <p = 0 is the condition that the spheres S,, S 2 and the two gene- 
rating spheres of the cyclide whose centres are collinear with their centres, or, in other 
words, the two generating spheres which are coaxal with them, should form an harmonic 
system of spheres. 
Def. An harmonic system of spheres is a system passing through a common circle and 
whose centres form an harmonic row of points ; this system possesses the property that 
four tangent planes through any common tangent line form an harmonic system; or 
again, that the segments which these spheres intercept on the line of collinearity of 
their centres may be an harmonic system of segments on that axis. See Chasles, 
‘Sections Coniques,’ art. 136. 
275. The equation <p=0 is the determinant 
a, 
n, 
TO, 
P> 
X i i + ^a 2 , 
n , 
l , 
<h 
} pi +W« 
TO, 
h 
c , 
r. 
Xz x T - l^z 2 , 
P’ 
<b 
r , 
d, 
XW l +[^W 2 , 
Xll j -f- (AX 2 , 
Xz i -j - 2 5 
Xw l y*w 2 ) 
o, 
a, 
n, 
TO, 
P > 
Xi, 
n, 
K 
i , 
2 > 
y» 
TO, 
l , 
c , 
r , 
-0. 
P> 
2 > 
r , 
d , 
$21 
y v 
z 2 , 
W 2 i 
o, 
( 168 ) 
