626 
DE. J. CASEY ON CYCLIDES AND SPHEEO-QTTAKTICS. 
three explicitly and twelve implicitly, each of the circles a, (3, y, h containing three ; but 
six equations of condition are implied by the fact of the four circles being mutually 
orthogonal, and the identical relation a 2 -j-^ 2 +y 2 +^ 2 =0 (for constants are incorporated) 
is equivalent to one condition. Hence the given form contains eight independent 
constants ; and this is the number which determines a sphero-quartic. 
83. Since the circle a is orthogonal to the circles /3, y, h, if we take either of the points 
in which a diameter of U perpendicular to the plane of «, called the polar line of a (see 
Salmons ‘ Geometry of Three Dimensions,’ art. 358), cuts U, and tan 2 of half the cor- 
and. the centres of these two pairs of circles are the foci respectively of the sections of the ellipsoid in whose 
planes they lie ; also the radical axis of each pair is the line joining the pair of umbilics which lies in its plane. 
YII. In the equation of the cyclide given in the last article, if we put y — 0, z— 0, we get 
x' - 2x\a- + 7r + 7<r) + Salilx + (a 4 + h' + l A - 2aVr - 2/rP - 2 Zra 2 ) = 0 ; 
and the four roots being denoted as before by x x , x v x v x 4 , we see, by the system of values given in article V., 
that a 2 , 7 r, P are the roots of Euler’s reducing cubic for this quartic in x ; and if we denote by A, y , v the 
roots of Simpson’s reducing cubic for the same quartic, we get 
\=—(a?—h- — Jc 2 ), y = — (P-f-P — 7r), v= — (a 2 + 7r— P). 
Hence by these values we can write the equation of the quadrinodal cyclide in the following form, due to Pro- 
fessor Cayeet, who arrived at it by a mode of investigation altogether different from that used here : 
+ -7 + 2x \y" + z ")+yy 2 + vz\x - ay )(x—x 2 )(x - x 3 )(x - x 4 ) = 0. 
Till. The centre of similitude of the first pair of circles of article YI., that is, of the pair of circles 
(x + k ) 2 + z 2 = (a + 7t) 2 , ( x - lef + z" = {a — lif, 
is easily seen to be the point whose coordinates are 
that is, the middle point of the line joining two 
umbilics of P ; or, in other words, the middle point of the line joining two nodes of the cyclide, and the centre 
of similitude of the other pair of circles, is the middle point of the line joining the other pair of nodes. 
IX. The equation of the cyclide of article IY., that is, 4(a 2 .r 2 +7> 2 y 2 + cV) — C 2 =0, is the envelope of the 
quadric a 2 P+& 2 y 2 -(-c 2 z 2 +p C + p 2 =0, or, by restoring the value of C, of the quadric 
(a- y)X“ (h- y)y~ y)z- 0;-\-y--\-L— — ; 
and the condition that this should represent a cone is given by the equation 
(«" + y)(b- -j- p)(<- + ji~) ^ — Qj- + y)(c" + y^y (^— j- ^ = 0. 
This equation is satisfied by four values of y, showing there are four cones, each having double contact with the 
cyclide ; and the vertex of each cone is a centre of inversion. Now one value of y is evidently — b 2 , and the 
corresponding cone is 
(a?-V)J + X-V)f J ^+v(‘ZzS )= 0 ; 
a \ a- J 
and when this is transferred to the centre of P as origin, it becomes 
( hx—alcf—(lc 2 — 7r)z"=(), 
which represents a pair of planes whose line of intersection passes through a pair of umbilics of P ; any point 
on this line may therefore be called a centre of inversion of the cyclide, that is, any point on the line joining a 
corresponding pair of nodes of the cyclide is a centre of inversion of the cyclide. In like manner, from the root 
