DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
G77 
cusp ; and we see that there is no difficulty in completing the investigation — that is, being 
given the degree, the finite double points or cusps, and the double points or cusps at 
the circular points at infinity of a plane curve, to find the characteristics of the evolute. 
CHAPTER Nil. 
Spliero- Cartesians. 
235. If a Cartesian cy elide be intersected by any sphere, I shall call the curve of 
intersection a sphero-C artesian. It is evident, if the intersecting sphere become a plane, 
that the sphero-Cartesian will become a Cartesian oval. We have seen that, being given 
a sphere U and a quadric F, the cyclide which has U for a sphere of inversion and 
F for a focal quadric will intersect U in the same sphero-quartic as the reciprocal of F 
with respect to U intersects U. Now, when F is a sphere its reciprocal with respect to 
U is a quadric of revolution. Hence we have the following fundamental theorem: — 
A sphero-Cartesian is the curve of intersection of a sphere and a quadric of revolution. 
236. The focal sphero-conics of a sphero-Cartesian are circles. 
Demonstration, Let the sphero-Cartesian be the intersection of the sphere U and 
quadric Y. Then, since V is a quadric of revolution, the cones which can be described 
through (UV) have but one system of circular sections, and therefore the cones reci- 
procal to them have but one system of focal lines ; but the reciprocal cones with respect 
to U intersect U in the focal sphero-conics of UV ; therefore the focal sphero-conics of 
LTV are circles. 
237. One of the four cones through (UV) is a right cylinder on a parabolic base, the 
plane of the base being perpendicular to the planes of circular sections ofV. 
Demonstration. Let 
then 
U = 0 + af + (y + Pf + (z + yf - 
r 2 =0, 
-1 = 0 ; 
U — « 2 V == iax + 4(3y + (z + y f — %{z — yf -)- or — r = 0 , 
and this will be of the form 
a'z*+fy + d = 0 
by a change of axes, 
biquadratic 
Hence the proposition is proved ; 
v-/j. > v ^ 
« 2 + A~ r 6 2 + A~J~c 2 + A 
^ =l+? 
(see Salmon’s £ Geometry of Three Dimensions,’ page 146), whose roots are the values 
of for which 
(x-«Y+(y-PT+(z-y)'-t‘+*($- Hp+?-i)=o 
4 z 2 
