DE. J. CASEY ON CYCLIDES AND SPHERO-QTJARTICS. 
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we knew otherwise, it being the surface generated by the revolution of a Cartesian 
oval about the axis passing through the three collinear single foci. 
206. In order to find the equation of the cone whose vertex is any point x\ y', z! , and 
which stands on the circle of intersection of the sphere (142) and the plane (143), let 
us suppose x" , y", z" to be any point on a radius vector from x', y\ z! to any point of the- 
circle; then, if the circle divides the distance between these points in the ratio l:m, we- 
. . . T , , , Ix' + mx " ly' + mii" W + mz u „ , 
must substitute, by J oachimstal s method, ■ — j +ni " lor x -> y , z m (142) and 
(143) ; the results will be of the form 
l 2 S r + 2 IrriP + m 2 S" = 0, and HJ+mU'=Q; 
hence by eliminating I : m and suppressing the double accents, we get the required 
equation after restoring the values of S', P, &c. : — 
(2 ox + r 2 + (2[/j + y/)R 2 ) 2 (P 2 + y'~ + z! 2 + 2 ax' + r 2 + 2 ^R 2 ) 
— 2 ( 2 ax + r 2 +(2^4- f) R 2 ) (2 ax' -j- r 2 + (2^ + ft 2 ) R 2 ) (xx’+yy 1 ft-zz 1 ft- ax + ax' -f r 2 + 2 g,R 2 ) >(144) 
-f (2 ax' 4- r 2 -\- (2^ 4-^ 2 )R 2 )(^ 2 + y*+z 2 + 2ax + r 2 + 2y,R 2 ) = 0. 
207. Since the equation (144) involves the undetermined [x in the fourth degree, its 
discriminant with respect to g will involve x, y, z in the twelfth degree, -and this discri- 
minant will be the equation of the tangent cone ; but this will contain as a factor the- 
cube of the imaginary cone from (x 1 , y z') to the imaginary circle at infinity (see 
Salmon’s ‘ Geometry of Three Dimensions,’ art. 521). Hence the reduced degree is six ; 
and it can be shown, as in art. 94, that the reduced cone has no cuspidal edges. 
We can show otherwise that the reduced degree is six; for any section of the cy elide 
made by a plane through the vertex of the cone is a Cartesian oval, and the class of a 
Cartesian oval is six ; the degree of the cone therefore is six. 
208. Class of Tangent Cone . — Let us take the equation S 2 =5 3 L and find the polar 
cubic of the point x', y 1 , z'. This will be of the form 
and eliminating b 3 between this and the equation S 2 =5 3 L, we get 
L=2LU'- 1 - ]d 
'dx+Vdy^ 
and since the operation x 'gg c J ry , jy J r z '-^ performed upon L reduces it to a constant, and 
performed upon S reduces it to a plane, this equation represents a quadric. Hence it 
is easy to see that the points of contact of tangent planes drawn through a line to the 
tangent cone are the intersections of three surfaces of the degrees 3, 2, 2 ; hence the 
class of the tangent cone is 12 ; and we have shown that its degree is six, and that it has 
no cuspidal edges. Hence all the singularities are determined. 
