706 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTIARTICS. 
Races. 
X, 
v , 
Angles. 
b" , 
< 3 (VI) , 
V’ 
6'", 
b Ciy) , 
z : 
r, 
s (v) , 
W, 
Q" , 
&" ; 
ancl then the discriminant of the invariant equation of the two quadrics is 
sin 2 b'. sin 2 b" . sin 2 b'" . sin 2 9 (IV) . sin 2 b (v) . sin 2 b (v, \ 
which, as in art. 317, is the product of twelve anharmonic ratios. Hence the tact- 
invariant of two quadrics is the product of twelve anharmonic ratios, and the vanishing 
of any one of these ratios is the condition of contact of the two quadrics. 
Cor. It follows from art. 317 that the condition of double contact of two bicircular or 
two sphero-quartics is expressible as the product of six anharmonic ratios , and, from the 
present article , of twelve anharmonic ratios, for the double contact of two cyclides. 
321. We now return from this digression (articles 315-320). If the cyclide W' (see 
art. 313) be a binodal cyclide, we have A' = 0 ; and we proceed to examine the meaning 
in this case of ©, 0, ©'. Let us take the nodes of W' as the points common to three 
of the spheres of reference a, ft, y, then in the equation of W' (see art. 313) _p', q', r', d' 
all vanish, and we get 0' =d(a'b'c' -\-2l'm'n ' — a'l 12 —b'm 12 — c'nt 2 ), or 0' vanishes if W' break 
up into two spheres, or if the nodes of W' be on the surface of W. Let the binodal 
cyclide which circumscribes W, and whose nodes coincide with those of W', viz. 
d (acd -f- bftr + cy 2 + 2 1 fly 2m<y a -j- 2n a/3) — ( pa -j- qft -j- ry ) 2 = 0 , 
be written 
a" a 2 + b"l 3 2 + c"f + 2 1" fly + 2m" y a + 2 n"aft = 0, 
then <E> may be written 
a"(b'c' - If -f b"(c'a! - m! 2 ) + c"(a'b' - nf + 2 1" (mint - a' l') + 2 mfn'l 1 - b'm ') + 2 n"(l'm' - c'n'). (196) 
Hence, by the theory of bicircular quartics (art. 174), <P vanishes when the intersections 
of three harmonic spheres of W 1 are three circles having double contact with W. In like 
manner 
de=a , (b l I"-lf+V(c l, a' , -inf+d(a%"-nf+2l\m''n! , -a''l")\ 
+ 2 m'(n”l" — b"m") + 2 n'(l"m" — d'n"), J 
or 0 vanishes when the generators of W are harmonic spheres of W' (see ‘ Bicircular 
Quartics,’ art. 218). 
When W' breaks up into two spheres, both A' and © f vanish. Let the two spheres 
be a, ft, then W' reduces to no, ft, and L reduces to n! 2 (r 2 —cd), or <J> will vanish when 
the intersection of the two spheres is a circle having double contact with W. In like 
manner 0 vanishes when the two spheres are conjugate spheres with respect to W. 
The condition will be satisfied, 0 2 =r4AL, if either of the two spheres be a generating 
sphere of W. 
322. Given nine cyclides, W 1? W 2 .... W 9 , it is possible in an infinity of ways to deter- 
