628 
mi. J. CASEY OX CYCLIDES AXE SPHEBO-QUAETICS. 
mined, together with the poles of a, will he the eight centres of inversion, and the three 
circles cutting a orthogonally, and having the first three pairs of points as poles, will be 
the circles of inversion. 
85. If two of the points coincide in which a cuts F, there will be then only six centres 
and three circles of inversion ; but since the coincident circles whose centres are the 
point of contact of a and F reduce to a point, that point will not be a centre of inversion 
in the ordinary sense, that is, the centre of a circle which inverts the sphero-quartic into 
itself, but it will be the centre of a circle which inverts the sphero-quartic into a sphero- 
conic. Hence in this case there are only four ordinary centres of inversion. 
If three of the points coincide, there will be then only two ordinary centres of inver- 
sion, namely, the poles of a ; but the point of osculation and the point diametrically 
opposite to it will be the poles of a circle, which inverts the quartic into a sphero-conic, 
and « will be the ordinary circle of inversion. 
86. The following proposition is not difficult to be proved. 
If § be the radius vector from any origin on the surface of U to a sphero-conic and, 
measured in the same direction from the same origin, an arc be determined by the 
condition that 
2 tan ^ §'= tan \ £, (85) 
the locus of the extremity of o' is a sphero-quartic. 
87. Besides the method of inversion hitherto used in this section, namely by the equa- 
tion (83), by which any curve on the surface of a sphere is inverted into another curve 
on the surface of the same sphere, there is another, the more ordinary method, by which 
it can be inverted into a curve on the surface of another sphere or on a plane. Thus a 
sphero-quartic being the intersection of a cyclide W and a sphere U, and since if both 
be inverted from any point in space they invert into surfaces of the same kind, we see 
that a sphero-quartic inverts into another sphero-quartic, or into a bicircular quartic, if 
the arbitrary point be taken on the surface of U. 
88. Since a sphero-quartic is the intersection of a sphere and a quadric, four cones 
can be described through it ; then it is plain that the vertices of these four cones are four 
points in space, which are centres of self-inversion of the quartic, that is, the quartic is 
an anallagmatic curve, and it has four centres of self-inversion. 
89. Let us consider the sphero-quartic WU, where W is the cyclide aod -f- bp 2 -f- cy 2 + db 2 , 
and U its sphere of inversion given by the equation U 2 ^« 2 -j-^ 2 -j-y 2 -|-ci 2 =:0, then the 
centres of a, (3, y, ^ are the four vertices of the four cones through WU, that is, they are 
the four centres of inversion. lienee we have the following theorem : — If 
W =«a 2 -f- bj3 2 -f cy 2 + dl 2 -b ee 2 = 0 
be a cyclide given in its canonical form, then the sphero-quartic, which is the intersection 
of W and any of its spheres of inversion, has the centres of the four remaining spheres as 
centres of inversion. 
90. If the arbitrary point we invert from be any point on the sphero-quartic WU 
