678 DE. J. CASEY ON CYCLIDES AND SPHERO-QUAKTICS. 
represents a cone, reduces to a cubic when a=b, showing that in this case there are only 
three cones. 
Again, the equation 
\ctu 2 
1 +«A 
which is the biquadratic for the paraboloid ax 2 -\-by 2 -\-2rz=0 and the sphere 
{x — a,y -f- (y — /3) 2 + (z — y) 2 — f, 
becomes a cubic when a=b. 
238. If a sphero-Cartesian he projected on the plane of circular section of V by lines 
parallel to the axis of revolution, the projection will be a Cartesian oval. 
Demonstration. Let M~x' l ~\-y l -\-z 2 — r 2 =0, 
v _(»— «) 2 , ( y - 0) 2 
V = a 2 + a 2 
0. 
Now, putting x 2 +f— r 2 s: — S and <? 2 ( 1 
{x- 
■*) 2 +(y-/3) 2 
= S', these equations are equi- 
valent to z — S*=0, z — y — S'*=0. Hence, eliminating z, we get the equation of a Carte- 
sian oval. 
239. If a plane parallel to the planes of circular section of V intersect U and V in 
two circles u and v, the locus of the radical axis of u and v will be the cylinder on the 
parabolic base. 
Demonstration. Put z — k in the equations of U and V, and we have their sections by 
the plane z=Jc; thus 
u =x 2J r f + Id — r 2 = 0, 
>-«) 2 , { y —\ 3) 2 , (^- y) s 
1 = 0 . 
Hence the radical axis of u and v is u — azv — 0 ; therefore the same value of X for which 
U-j-bV becomes a parabolic cylinder reduces u-\-Xv to the radical axis of u and v, and the 
proposition is proved. 
Cor. 1. The curve of intersection of a sphere with a cylinder on a parabolic base is a 
sphero-Cartesian. 
Cor. 2. From recent articles we infer the following method of generating sphero- 
Cartesians. 
Let J be a circle and F a parabola in the same plane (say, in the plane of the paper) ; 
then from any point P in F erect two perpendiculars in opposite directions to the plane 
of the paper and equal respectively to ±T\/ — I, where T is the length of the tangent 
drawn from P to J ; then the locus of the extremities of the perpendicular will be a 
sphero-Cartesian. 
240. Since one of the four cones passing through the sphero-Cartesian (UV) is a 
parabolic cylinder, it follows that one of the nodal conics of the developable X formed 
by tangent planes to U along (UV) will pass through the centre of U. Hence we have 
the following theorem : — 
