DE. J. CASET ON CYCLIDES AND SPHEEO-QTT AETICS . 
605 
are mutually orthogonal, and its equation will be of the form «a 2 -f 6/3 2 +cy 2 + f/£ 2 =0 ; 
but a 2 + 1 3 2 +y 2 + ^ 2 =0 is an identical relation. Hence, eliminating each of the variables 
a 2 , /3 2 , y 2 , o 2 in succession, we see that the same sphero-quartic may be expressed by either 
of the four equations : 
(«-^)/3 2 +(a-c)y 2 + («-tZF = 0, | 
{b _ay + (b-cW+(b-d)V = 0 , | _ (44) 
(c — a)cd-\-(c — b)fi 2J r (c — d)b 2 = 0, J 
(cl-ay+(d~b)(3 2 -l-(d-c)y 2 =0, j 
and by the last article we see that the sphero-quartic has four focal sphero-conics, whose 
tangential equations are : 
(a — b)y?-\-{a— c)v~ d)f=0, 
(b- a )\ 2 +(b-cy +(b-d)f=Q, 
( c — a)k 2 + (<? — b)y?-\- (c — d)f=0, 
(d—a)\~ -\-(d—b)fjij 2 -\-(d—cy= 0. 
Cor. Sphero-quartics maybe generated in four different ways as the envelope of a vari- 
able circle which cuts a given circle orthogonally, and whose centre moves along a given 
sphero-conic. 
42. If W=(«, b, c, d, l , m, n,p , q, rfa, (3, y, ^) 2 =0 be the general equation of a 
cyclicle, and U the sphere orthogonal to a, j 3, y, c), then it is easy to see that the results 
of substituting the coordinates of any point P of the sphero-quartic (WIJ) in the equations 
of a, /3, y, b are proportional to the perpendiculars from the centres of a, (3 , y, b on the 
tangent plane to U at the point P ; but if these be perpendiculars to X, v, g, we see 
that the surface whose tangential equation is 
(a, b, c, d, l, m, n, p , q, rfX, p, v, gf 
is inscribed in the developable formed by the tangent planes to U along the sphero-quartic 
WU, but this tangential equation is that of the focal quadric of W. Hence we have the 
following theorem: — The developable circumscribed about the focal quadric of a cyclide 
and the corresponding sphere of inversion U touches the sphere along the sphero-quartic 
(WU), and the cones whose vertices are at the centre of U, and which stand on the nodal 
conics of the developable , intersect U in the focal sphero-conics of the sphero-quartic WU. 
The latter part of the theorem is evident by writing the equation of the cyclide in terms 
of four spheres mutually orthogonal, and from the equations (45) of the last article*. 
* [We have given in art. 33 the equations in tangential coordinates of the five focal quadrics of a cyclide; 
the following investigation gives, being given the equations of a focal quadric and the corresponding sphere of 
inversion in Cartesian coordinates, the equations in Cartesian coordinates of the four remaining focal quadrics. 
I. Let U + 1 = 0 be the focal quadric E of a cyclide W, and (.r— f)' + (y— [/)"+(.- — ti) 2 — r"=0 be 
Ci“ 0“ c 
the corresponding sphere of inversion ; then if from the centre 0 of the sphere we let fall a perpendicular OT 
4 0 2 
(45) 
