DE. J. CASEY OX CYCLIDES AND SPHEEO-QUAETICS. 
671 
respect to U, and the eight planes are the reciprocals of the eight points of meeting ; 
hut these eight points of meeting lie on a generator of U , Y ,/ — TJ"V. Hence the propo- 
sition is proved. 
Cor. 1. The line LM is a generator of the reciprocal of JJ'Y" — U ,, V'. 
Cor. 2. The quadric U'V' — U^V' is the hyperboloid generated by the polar lines of 
LM with respect to all the quadrics of the pencil U+#V. In fact the polar lines form 
one system of generators, and the intersection of the polar planes of art. 222 the other 
system of generators. 
Cor. 3. The eight planes of art. 222 are homographic with the eight points in which 
the corresponding lines of A meet LM. 
224. From art. 161 it is evident that ^ has sixteen stationary points. These are the 
points of contact with N, N', N", N w of the common tangents of J and N, J', N'; J", N"; 
J'", W. Lienee it follows that A has sixteen stationary planes. Hence we have three 
of the characteristics of A ; for m evidently is equal to 4, and r= 8 from art. 222. There- 
fore we have 
m=4, a = 16, r=8. 
Hence by Cayley’s equation we get 
ii=12, /3 = 0, #=16, y=8, <7=38, h— 2. 
225. We can now show the connexion which exists between the singularities deter- 
mined in the last article and those of bicircular quartics. Let us suppose a cone whose 
vertex is any point on TJ, and which stands on UV ; then for such a cone we have the 
singularities (see art. 220) 
(Jj — 4:, *=8, o = 2, r=8, z=0, < = 12; 
but if we invert the sphere U into a plane, taking the vertex of the cone as centre of 
inversion, UV will be inverted into a bicircular quartic, which will be the curve of inter- 
section of the cone with the plane into which U inverts, and therefore having the same 
singularities as the cone. The numbers here determined are therefore the singularities 
of a bicircular quartic (see ‘ Bicircular Quartics,’ art. 46). 
226. Again, to show the connexion with the evolute of a bicircular, let us consider a 
plane section of A ; the characteristics are (see art. 221) 
p, = 8, *=12, o = 16, r=38, *=4, < = 16. 
Now the cone whose vertex is the pole with respect to U of the plane of sections, and 
which stands on the cuspidal edge of S, will be the reciprocal of the section ; and if the 
plane of section be a tangent plane to U, its pole will be a point on U ; therefore the 
singularities of this cone will be 
(jj— 12, *= 8, §=38, r=16, ^=16, <=4; 
but when the sphere is inverted into a plane as in the last article, it has been shown in 
art. 92 that the cone here considered, viz. the one standing on the cuspidal edge of X, 
intersects the plane into which the sphere inverts in the evolute of the bicircular into 
