DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
697 
that we have a biquadratic to solve to determine X, in order that W+XW' may represent 
a binodal cyclide. Hence, through the curve of intersection of two cyclides, four binodal 
cvclides may be described. The binodes of these binodals are thus determined. If we 
denote by W 1? W 2 , W 3 , W 4 the differentials of W with respect to a, (3, y, b respectively, 
and by X n X 2 , X 3 , X 4 the four roots of the biquadratic in X, then any three of the four 
spheres 
w.+^w', w 2 +x,w' 2 , w 3 +x 1 w;, w 4 +aX 
will determine by their mutual intersections the binodes of one of the binodals ; and the 
binodes of the remaining binodals are got from these by using the remaining roots of 
the biquadratic in place of X 1? namely, X 2 , X 3 , X 4 respectively. 
296. There are four spheres whose pole points are the same with respect to all the 
cyclides passing through a common curve of intersection of tic o cyclides , namely , the polar 
spheres of the four pairs of nodes of the four binodals of the last article. For to express 
the condition that 
«W'+j3W'+yW,+&W', 
«x' +/3x; + y x' 3 +sx' 4 
should represent the same spheres, we find the following set of determinants : 
W', 
W', 
W'„ 
w', 
X', 
X', 
X', 
X' , 
and these satisfy by the last article the binodes of the binodal cyclides. 
297. The four spheres are such that the two points common to any three are the pole 
points of the fourth ; and , conversely , the four pairs of binodal points are such that the 
sphere determined by any three is the polar sphere of the fourth pair. Thus, if the two 
cyclides be W and W', their equations in terms of the four spheres will be of the form 
aX\+bY 2 +eZ 2 +dV 2 , 
a'X 2 + 6' Y 2 + FZ 2 + d'Y 2 , 
and the nodes of the binodal cyclides are the four pairs of points 
(XYZ), (XYV), (XZV), (YZV). 
It is to be remembered that the spheres X, Y, Z, V are not mutually orthogonal. 
298. If the cyclide W' break up into two spheres, the form W-j-XW' becomes 
W+XLM. In general the intersection of two cyclides is a twisted curve of the eighth 
degree ; but if one of the two cyclides reduce to two spheres, the intersection becomes 
two sphero-quartics. Any pair of inverse points on the circle LM has the same polar 
sphere with respect to all the cyclides of the system W + XLM ; and in particular all the 
cyclides of the system have the same generating spheres at the four points where W is 
met by the circle LM. 
Lastly, all the cyclides of the system are enveloped by four binodal cyclides. For if 
