700 
DR. J. CASEY ON CYCLIDES AND SPHERO - QE AETICS . 
2°. Every central quadric has two systems of circular sections. Hence every quartic 
cy elide has two systems circumscribed to it of binodal Cartesian cy elides, and the locus of 
their nodes is two right lines respectively perpendicular to the directions of the planes of 
circular sections of the focal quadric. 
Section II. — Poles and Polars. Sphero-quartics. 
308. The investigation of the polar properties of sphero-quartics is analogous to that 
employed in the last section for cyclides. 
Thus if (i a , b, c,f g, hfa, 0, -y) 2 =0, where a, 0, y are circles on a sphere U, be the 
equation of a sphero-quartic, and if A 1 a+^ 1 04-i' 1 ‘y=C 1 and A 2 a+ | «- 2 0+J / 2 y=C 2 be two 
circles, then the condition that ZC 1 +mC 2 =0 should be a generating circle is given by 
the determinant 
a , A, g , ZAj — 1— 
hi b, f lq 1 -\-m(A 2 , 
g, f, c , lv l -\-mv 2 , 
lykpnVkv l[h x -\-ni[h^ Iv^mv 2 , 0. 
This determinant may be written in the form 
7 2 2'+27mp+m 2 S"=:0 (179) 
Hence <£>=0 is the condition that the circles C,, C 2 , and the two generating circles whose 
centres lie on the same great circle with their centres, should form an harmonic pencil of 
circles, or it is the condition that their centres should form an harmonic row of points; 
or, again, it is the condition that their diameters should form an harmonic system of 
segments on the same great circle of U. 
309. The equation Q = 0 is the determinant 
(178) 
a , 
h, 
P 
x. 
h, g, 
f, 
/> 
P 2 h. 
0, 
= 0 . 
(180) 
This is the condition that the circles Cj and C 2 may be conjugate circles with respect to 
the quartic ; if the suffixes be removed from the lower row, we see that, if the centre of 
a variable circle C^ha. -\- p[3 vy=0 move along the great circle 
a, 
h, 
9 > 
a, 
9 > 
p, 
pi, 
h, 
0 , 
=o, 
(181) 
then C=0, C 1 =A 1 a+|M» 1 0+j' l y, and the two generating circles whose centres are on the 
same great circle with their centres form an harmonic pencil ; but if a variable circle 
