720 
BE. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
Cor. The condition that AY shall have the circles (a$), (ay), (S/3), (Sy) lying on its 
surface, while the four pairs of inverse points (a/3y), (a/3S), (ayS), (fiyh) shall lie on the 
surface of W', is the equation, reciprocal of the former, 
4 A'©' ( I> = @' 3 + 8 A' 2 © (238) 
357. We have seen if two cyclides W, AY' be reciprocals with respect to 
U 2 — a" -f- (3 2 -f- y 2 -f- S 2 , 
that their focal quadrics are reciprocals with respect to the sphere U. Hence it follows 
that if the focal quadric of W be a plane conic, the focal quadric of W' will be a cone. 
Hence we have the following theorem : — The reciprocal of a binodal cyclide is a sphero- 
quartic , and vice versa. 
358. If a cyclide W breaks up into two spheres , its reciprocal W' breaks up into two 
spheres. For if W breaks up into two spheres, its focal quadric is a quadric of revolu- 
tion circumscribed to U, and the reciprocal of the focal quadric with respect to U is 
another quadric of revolution circumscribed to U. 
This and the last article belong to the Chapter on reciprocation, but were accidentally 
omitted. 
359. If we form the discriminant of TW + Id W -f k ll 'W", the coefficients of the 
several powers of k, k', k" will be invariants of the system of cyclides. There are two 
invariants, however, of the system ZW + ^'W'+Z/W" which, as being combinants, 
deserve attention. These invariants we shall call I and J. 
The combinant I vanishes whenever any four of the eight generating spheres common to 
AY, AY', AY" are connected by a linear relation , that is, pass through the same two points. 
It is easy to see that this is equivalent to the statement that I vanishes for the values 
of k, k', k" which will make Z AY ffi/hAV'+^'W" represent two spheres. The equations of 
AY, W, AY", as having a common sphere of inversion, may plainly be written in the forms 
AY =aa 2 + b(3 2 -hey 2 +d V + ca 2 , 
AY' == a'cc 2 +Y/3 2 +c'y 2 +t^ 2 + Fa 2 , 
A V"==a"a? + b"(3 2 + c"y 2 +■ d"V+ e"s 2 , 
where a 2 -}-/3 2 -j-y 2 +^ 2 + r=0 identically; and it is clear that I is the product of the ten 
determinants (a, V , c"), &c. For (a, V , c")cT-\- (d, V , c"ft + (e, V , c")s 2 is evidently a cyclide 
of the system Z;AV+Z; , AV , d-/^ ,, AA ^,, ; and this reduces to two spheres if one of the determi- 
nants (a V c") vanishes. Hence I is the product of the ten determinants. 
Cor. The combinant I vanishes also whenever any four of the eight pairs of inverse 
points common to AY, W', AY" are homospheric. 
360. The combinant J vanishes whenever any two of the eight generating spheres common 
to AY, AY', AY" are coincident ; or again, when any two pairs of the eight pairs of common 
inverse points are coincident, so that J will be the tact-invariant of the three cyclides. 
If the generating sphere at a pair of inverse points common to the three surfaces pass 
