630 
DR. J. CASEY 0 N CYCLIDES AND SPIIERO-QTJARTICS. 
-1, 
cos (a/3), 
cos (ay), 
cos (ah'), 
a -h r' , 
cos (/3a), 
-1, 
cos (/3y), 
cos (f3h), 
[3 -hr" , 
cos (ya), 
cos (y/3), 
— 1) 
cos (yd), 
7 -hr"'. 
= 0. . . . (86) 
cos (Sa) , 
cos ((5/3), 
COS (Sy), 
-1, 
l -hr'"'. 
a-hr' , 
(3 -hr", 
7+"', 
l-hr"". 
0, 
It hence follows that every bicircular can be expressed in the form aa 2 + b(3 2 +cy 2 -\-dh' i = 0, 
where a, [3, y, e$ are four circles mutually orthogonal, and that for this system of circles 
the relation is an identical one, a 2 -]- /3 2 + y 2 -f c> 2 = 0. 
94. From the last article we see that the same bicircular can be written in the four 
following forms : 
( d — a)a 2 + (d — b )/ 3 2 -f- (d — c )y 2 = 0 ? 
(a— 5 )/ 3 2 + («— c)y 2 -\-(a— d)h 2 =0, 
(b — c)y 2 -\-(b — d)l 2 -\-(b — a)a 2 , 
( c — djh 2 -f-(c — a)a 2 + (c — 5)/3 2 , 
(87) 
and that consequently the tangential equations of the four focal conics of the quartic 
are given by the equations 
(d— a)X 2 -\-{d— b)f-\-(d— c)v 2 =0, 
(a— b)g?-\-(a — c)v 2 + («— d)f =0, 
(b-cy +(b-d)f + (d-a)x 2 =0, | 
(c —d)f J r (c-a)7d-\-(c — b)y = 0. J 
( 88 ) 
CHAPTER YI. 
Projection of Sjphero-quartics. 
95. If a sphero-quartic be projected on one of the planes of a circular section of any 
quadric passing through it by lines parallel to the greatest or least axis of the quadric , 
the projection will be a bicircular quartic whose centres of inversion will be the projections 
of the centres of inversion of the sphero-quartic. 
Demonstration. Let U be the sphere given by the equation U 2 ; (a 2 -)-/3 2 + y 2 H-§ 2 ) = 0, 
and Y the quadric which intersects U in the sphero-quartic, then the centres of a, (3, y, d 
will be the vertices of the four cones which can be drawn through the sphero-quartic, 
that is they will be its centres of inversion. Let Kg. 3. 
KPP' be an edge of the cone passing through two 
points of the quartic, K being the centre of a; then, 
since a is a sphere of inversion of the quartic, the 
rectangle KP KP' is constant. Hence if O be the 
centre of the quadric V, the radius vector OK of the 
quadric parallel to KP' is constant ; therefore the 
locus of K is a sphero-conic : and if the point Rj 
5 j 7 
