664 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
we see that if a Cartesian cyclide be intersected by any plane the curve of intersection 
will be a Cartesian oval ; for the equation will be of the form S 2 =h 3 l, where S and l denote 
the circle and line in which the sphere S and the plane L are intersected by the plane. 
203. From the equation (138) we see that the Cartesian cyclide is the envelope of 
the variable sphere 
a 2 -\-y 2 + £ 2 + y{x 2 -\-y 2 + 2 2 + 2ax -f r 2 ) -f gill 2 ; 
and if we form the discriminant of this, we get 
(l+p) 2 {(l+^r 2 4-^R 2 )-«> 2 } = 0. . (139) 
Now the factor (1-fq-y) 2 — 0 gives g^= — 1; and for this value of (m the variable sphere 
becomes a plane, namely the tangent plane which touches the cyclide along a circle; 
the remaining factor, 
(l+^)(^r 2 H-^ 2 R 2 )-a> 2 =0, (140) 
gives three values of gj, for each of which the variable sphere becomes an imaginary cone 
(that is, a point sphere), showing that there are three collinear single foci along the axis 
of the cyclide. The value im = 0 shows that the origin is a focus, which we knew before ; 
and the values giving the other foci are the roots of the quadratic 
f jfR 2 +(A(R*+r , -a i ) + r 2 =0 (141) 
204. Since the equation 
a 2 4- y 1 + ~ 2 + [f x 2 + y 2 + ~ 2 + 2 ax + r 2 ) -f- ^ 2 R 2 = 0 
is that of a sphere into whose equation an arbitrary constant enters in the second degree, 
its inverse with respect to any point will be a sphere into whose equation an arbitrary 
constant enters in the second degree ; that is, the inverse of a Cartesian cyclide will he a 
cyclide generated as the envelope of a variable sphere whose centre moves along a plane 
conic. It will therefore he a hinodal cyclide. 
This also appears from the fact that the inverse of a focus is a focus; and since the 
Cartesian cyclide has three collinear single foci, the inverse surface will have four con- 
cyclic single foci, namely the inverses of the three collinear foci and the centre of 
inversion. 
205. If we differentiate the equation (138) of the generating sphere with respect to g., 
we get 
x 2 -\-y 2 -1- z 2 p2ax-{-r 2J r 2^R 2 = 0 ; 
and if from (l-j-^) times this result we subtract the equation (138), we get 
2 ax + r ~ + + ^ 2 )R 2 = 0. 
Hence the Cartesian cyclide is generated as the locus of the curve of intersection of the 
sphere 
tf+yt+z^+Zax+r 2 ^^ 2 . (142) 
with the plane 
2ax+r 2 +(2y+f)n 2 =Q (143) 
From this it follows that a Cartesian cyclide is a surface of revolution — a fact which 
