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DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
the four chords of contact being the four great circles 
x tan g'+y tan g"dcz tan §"' = 0 (66) 
These great circles are the four axes of similitude of the three circles L, S^— M, 
S’— N, and the four sphero-conics are the four director or focal conics of the four pairs 
of circles (regarded as sphero-quartics) which are tangential to the three circles S*— L, 
S*— M, S^— N. (See my memoir “ On the Equations of Circles.”) 
GO. Each of the four sphero-conics (62) to (65) inscribed in J touches four other 
sphero-conics inscribed in J ; their equations are 
J={x tan §' cos (B±C )-\-y tan 4 g" cos (CrhA) + 2 tan cos (AiB) 2 }. . (67) 
By means of these sphero-conics can be proved Dr. Hart’s extension of Feuerbach’s 
theorem, and the equations of Dr. Hart’s circles can be found. (See my memoir “ On 
the Equations of Circles.”) 
61. We now return, after this long digression, to the sphero-quartic. 
Let us consider the function of the second degree, 
(a, b, c,f, g, L, S 1 — M, S 1 — N) 3 (68) 
This will represent a sphero-quartic on the surface of the sphere U, or a quartic cone 
having its vertex at the centre of U, according as we interpret S 1 — L &c. as circles, or 
as single sheets of cones (see art. 44). Now the equation (68) is the envelope of 
*(S*_L)+|£4S*-M)-KS*-N). If the condition be fulfilled, 
(A, B, C, F, G, H)> p 0, (69) 
where A, B, &c. denote as usual bc—f 2 , ca—g 2 , &c. ; but the circle 
x(S*-L)+/a(S*-M)+i(S*-N)=0 
has the plane xL+|O<M+j/N = 0 perpendicular to its axis, that is, perpendicular to the 
radius of U passing through its centre, and in virtue of the condition (64) the envelope 
of the plane is the cone 
(a, b, c,f ; g, hX L, M, N) 2 =0, (70) 
and therefore the locus of the centres of the generating circles of the sphero-quartic (68) 
is the sphero-conic, in which the cone reciprocal to (70) intersects the sphere U. 
62. If we regard the equation (68) as representing a quartic cone, we see that the locus 
of the axis of its generating right cone is a cone of the second degree, namely, the reci- 
procal of the cone (70), and its generating right cone cuts orthogonally another right 
cone, namely, the right cone which is orthogonal to the three cones S’ — L, S J — M, S- — N; 
the equation of the orthogonal cone is given, art. 51, equation (54). 
Cor. If we suppose any plane to cut the quartic cone of this article, it will intersect it 
in a quartic curve having two double points ; it will also intersect the generating right 
cone in a conic, which will be the generating conic of the quartic curve ; and finally it 
will intersect the directing cone of the quartic cone in a conic, which will be the directing 
conic of the quartic curve. It was in this manner I was first led to the discovery of the 
