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D R. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
U along the sphero-quartic, then the cone whose vertex is at the centre of U, and which 
stands on this double line, intersects U in the corresponding focal sphero-conic ; and it 
is plain that the four foci on this sphero-conic are the four points where the double line 
of the developable intersects U. Hence the sixteen foci of a sphero-quartic are the six- 
teen points in which the double lines of the developable which circumscribes the sphere 
along the sphero-quartic intersect the sphere. 
126. In Chapter IV. we have shown that the equation of a sphero-quartic may be so 
interpreted as to represent a quartic cone, namely, by regarding the circles a, /3, y, <i, 
which enter into the equation of a sphero-quartic, as single sheets of a cone, whose 
vertex is at the centre of the sphere. Again, the equation of a sphero-quartic may be 
interpreted so as to represent a cyclide, that is, by regarding a, (3, y, b as spheres cutting 
U orthogonally, and the quartic cone given by the former interpretation will be a tangent 
cone to the cyclide given by the latter. Hence we have, from article 123, the following 
theorem : — 
The puartic cone which circumscribes a cyclide , and whose vertex is at the centre of a 
sphere of inversion of the cyclide, has sixteen focal lines, which are four by four the edges 
of four confocal cones. 
127. The four confocal cones of the last article possess another important property; 
to demonstrate it we must prove some properties of binodal cyclides. 
Let us consider the sphero-quartic WU, W being the cyclide ad 2 b(3 2 + Cy 2 -j- db 2 , and 
U 2 =a 2 + /3 2 -)-y 2 +^ 2 , then WU is the intersection of U with any of the four binodals got 
by eliminating a, (3, y, b successively between W and U 2 . Now each of these binodals 
has three focal quadrics and one focal conic, which focal conic is also a focal conic of the 
three confocal quadrics of the cyclide to which it belongs. Eliminating a, we get the 
binodal (a — b)(3 2 -\-(a — c)y 1J r {a — dfr, and the focal conic of this is one of the double 
lines of the developable 2 circumscribed about U along WU. Let the four double lines 
of 2 be the conics C, G, C", G" ; if C be the focal conic of (a — b)(3 2 + (« — c)f -\-{a—dfr, 
then 2 is the developable circumscribed about C and U. Hence, by art. 34, the three 
focal quadrics will be the three quadrics described through the conics G, C", G" re- 
spectively, and having C for a focal conic. 
128. Since the four cones standing on C, G, G', G" are confocal, whose vertex is at 
the centre of U, and the four cones are confocal which have the same vertex and 
of which one stands on C and the remaining three are circumscribed to the three 
quadrics of the last article, hence we have the theorem, that the cones which stand on 
G, C", G" are circumscribed to the focal quadrics of the binodal which has C for a focal 
conic. 
129. From the theorems of the two last articles we infer at once’the following, which 
is the one referred to at the commencement of art. 127 : — If WU be a sphero-quartic , 
the four cones having the centre of U for a common vertex, cmd standing on the confocal 
sphero-conics of WU, pass respectively through the focal conics of the four binodals o/WU, 
and each is circumscribed to a focal quadric of each of three of these binodals. 
