054 DK. J. CASEY ON CYCLIDES AND SPHEKO-QUARTICS. 
X, g>, v connected by an equation of the second degree. Hence the envelope is a conic 
section. 
2°. On the question, If a variable conic osculate one conic, and have double contact 
with another given conic, to find the envelope of the chord of contact. 
Let the osculated conic be 
cix 1 + by 2 + cz 2 = 0 , 
and the one of double contact 
x 2 +y 2 +z 2 = 0, 
then the variable conic must be of the form 
x 2 -\-if + z 2 — (Xx + jjsij + vzf = 0 . 
Now, if we want to describe a conic having double contact with x 2 -\-y 2 -\-z 2 , where 
Xx-\-py-\-vz cuts it and touching ax 2 -\-by 2 -\-cz 2 , the points of contact on ax 2 + b if -f- cz 2 
will be given as the points of intersection of ax 2 -\-by 2 -\-cz 2 with the Jacobian of 
ax i -\-by 2 A r cz 2 , x 2 -\-y 2 -\-z 2 , and Xx-\~ixy-\-vz’, that is, the points of contact will be the 
points of intersection of ax 2 + by 1 -j- cz 2 with the conic an q if two 
of these points of intersection coincide, the conic which has double contact with x 2 -\-y 2 -\-z 2 
will osculate ax 2J r by 2 -\-cz 2 ; hence we must form the condition that the conics touch 
ax 1 -yhf-^-cz 2 
This is easily found to be 
q X(6 — c) n(c — g) ^_ v(a — h) q 
ai(b - c)ix s + bi(c-a)fyi+d(a-b)M= 0 
( 120 ) 
Now, since this denotes a curve of the sixth class, and the former condition l°a, curve 
of the second, they will have twelve common tangents; hence v=12. 
Cor. In the same way it may be proved that ^ = 12. 
17G. We shall now return from our digression on bicirculars. 
At the points where the nodal conic N of the developable % (see art. 161) cuts J, the 
osculating circle of the sphero-quartic WIT cuts J orthogonally; and hence it is its own 
inverse with respect to the sphere a. Therefore the four points in which J cuts N are 
points of stationary osculation. Hence there are on a sphero-quartic sixteen points of 
stationary osculation. 
Cor. The cone of articles 170, 171 in this case breaks up into two planes; and since 
the poles of the planes of the osculating circles of WU form the cuspidal edge of the 
developable 2, we see that 2 has sixteen stationary points which lie four by four on the 
four nodal conics N, N', N ,f , N"', the four stationary points on N being the four points of 
contact of the common tangents of N and J; and similarly for N', N", N"'. 
