606 
DE, J. CASEY ON CYCLIDES AND SPHEEO-QT7AETICS. 
CHAPTEE IY. 
Spher o-quartics (continued) . 
43. In discussing the properties of sphero-quartics, we have hitherto considered a 
sphero-quartic as the intersection of a sphere and a cyclide. There is another mode of 
considering sphero-quartics, which offers many advantages for the investigation of these 
curves, namely, to consider a sphero-quartic as the curve of intersection of a sphere and 
a quartic cone tangential to the cyclide, the vertex of the cone being at the centre of 
the sphere, which we shall take as one of the spheres of inversion of the cyclide. This 
method of studying the sphero-quartic will give us an opportunity of showing the con- 
nexion which exists between the invariants and covariants of plane conics and of circles 
on any tangent plane to E, and take two points P, P' in opposite directions from T on OT so that 
OT 2 — TP 2 = OT 2 — TP' 2 = >- 2 , 
the locus of the points P, P' is the cyclide W ; hut denoting OT by p, and OP by o, this gives us 2yq = r 2 -f Y, or 
2 fa a 2 cos 2 a. + ¥ cos 2 (3 + c 2 cos 2 y — 2(f cos a + y cos (3 + h cos y) = r 2 + % 
cos a, cos /3, cos y being the direction cosines of OT. Hence, if the centre of the sphere he now taken as origin, 
we have the equation of the cyclide 
4 (fax 2 + bfa + ctr) = (or -\-y" + z 2 + 2 fa + 2<jy + 2 liz + r 2 ) 2 . 
II. The equation of the cyclide given in I. is the envelope of the quadric S + /xC+ju, 2 =0, where S represents 
the cone crar + bfa + c 2 z 2 , and C the sphere x 2 -f y 2 -f- z 2 + 2 fa -f 2gy -f 2hz + r 2 ; and the condition that this should 
represent a cone is the discriminant 
(a 2 + fa(¥ + fafa + fafar 2 + fa) - faffa + fafa + fa- fag 2 fa + fafa + fa- fair (a- + fa (IP +fa=0, 
or, as it may be written, 
y.f\ + pv 
d 2 + y b 2 + y c 2 + y 
, fa 1 " 2 , » 
If the five values of y in this equation he denoted by y v y 2 , y 3 , p 4 , y s , we have the equations of the five cones 
which have double contact with the cyclide (see art. 187), S+ yfap-y 2 , S-\- y 2 C+ y 2 , &c. ; and the vertices of 
these five cones are, by the same article, (the five centres of inversion of the cyclide. Since one value is obviously 
= 0 in the foregoing equation, we see that the cone whose vertex is the centre of the sphere of inversion 
fa -fy+(y—ff) 2 +(z~hy—r 2 =0 
will be,' when that centre is taken as origin, 
azx 2 + Iry 2 + <?z" = 0. 
Hence, if the other centres be taken respectively as origin, the equations of the other cones will be 
(a 2 + y 2 )x 2 + ( b 2 + y 2 )y 2 + fa + y 2 fa= 0 fa) 
( a " + 1 fax 2 + ( b 2 + y 3 )y 2 + fa + y 3 ) z ~— 0 fa) 
(a 2 + yjx 2 + (b 2 + yfay 2 + (c 2 + yfa 2 = 0 (y) 
( a " + i i o) x ~ + fa + Hs )y~ + fa + yfa '— ^ (*0 
Now, since the cone a 2 x 2 +b 2 y 2 + c 2 z 2 ~0 is the reciprocal of the asymptotic cone of the focal quadric 
o 9 2 
_j_5_ — i, infer that the cones (a'), fa), (y), fa) are the reciprocals of the asymptotic cones of the 
