608 DB. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
Now if we put S for the point sphere x 1J r y- J r z i , the equations of the two circles may 
be written 
sec£=0 (47) 
It is clear that these equations (47) may also he interpreted as denoting separately the 
two sheets of the cone (46) — that is S*+L sec % represents one sheet of it, and S^ — L sec 
the other. Hence we infer from this article and from article 38 that the equation 
(a, b, c, d, l, m, n, p, q, r%a, /3, y, S) 2 =0 
will represent a cyclide , a sphero-quartic, or a tangent cone to the cy elide , whose vertex is 
IV. When S + gQ-\-g 2 (see II.) represents a cone, the coordinates of the vertex are, by the usual process, 
—pf —pa — M 
a ~ + g + g & r 
if referred to the centre of J as origin, or 
erf b 2 g cVi 
or g b" g C“ g 
if referred to the centre of E as origin. Hence we have the following theorem : — If E =— -j-?b A-—— 1=0, 
a 2 b 2 c 2 
and J = (.t— ff + (y— g) 2 -\-(p— tef— r 3 =0 be a corresponding focal quadric and sphere of inversion of a 
cy elide, and if g L , g 2 , g 3 , g i he the four roots of the biquadratic in g, which is the discriminant of /rF+ J, then 
the coordinates of the centres of the other four spheres of inversion are : 
« 2 / 
by 
cVi 
er + g* 
b 2 +p L 
c 2 + g 1 
erf 
bfl 
cVi 
a + Pi 
v+rl 
(z + gh 
eij 
bp 
cVi 
al +ph 
v+p; 
c2 +P3 
erf 
by 
e?h 
a '+Pt 
^' + Pi 
v+g± 
Cor. These values satisfy the system of determinants 
cc y z 
o’ 7TJ’ “o’ 
« h ~ c " =0, 
(*-/)» (y~9)> (~- 7 0> 
01 Cx by ClZ 
E 
’ 5 '• ’=0. 
; > j J? 
Hence we have the following theorem : — If F and J be a corresponding focal quadric and sphere of inversion 
of a cy elide, then the five centres of inversion of the cyclide lie on the Jacobian curve of J and E (see Cayxey, 
“Memoir on Quartic Surfaces,” Proceedings of the London Mathematical Society). 
Y. Being given 
+ = J^(x-ff+(y-9fHz-W-r 2 =0, 
the equation of the cyclide is 
4(aV+&y +c 2 z 2 )— (x 2 +y 2 +z' ! +2fa+2gy+2hz+r 2 y= 0 (A) 
