698 
DE. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
of cy elides have eight common generating spheres, the locus of the pole points of a fixed 
sphere is a circle. 
290. If a system of cy elides pass through the same eight pairs of inverse points, that 
is, if they have a common curve of intersection, the polar circles of a fixed circle generate 
a cy elide. 
Let the polar spheres of two fixed pairs of inverse points be P + XQ and P'+^Q' ; 
eliminating X, we get the cyclide PQ' — P'Q=0. 
291. j Reciprocally, if a system have eight common generating spheres, the polar circles 
of a fixed circle generate a cyclide. 
292. If a system of cy elides pass through cl common curve, the locus of the pole points 
of a fixed sphere is a torse curve of the sixth degree. 
Demonstration. Let the polar spheres of three pairs of inverse points lying in the fixed 
sphere be P+XQ, P'+XQ f , P" + aQ"; then, eliminating X, we get the system of deter- 
minants 
P, P', P", 
Q, Q', Q", 
which represents a twisted curve of the sixth degree. For the intersection of the 
cy elides PQ' — P'Q, PQ" — P"Q, each of which has the imaginary circle at infinity as a 
double line, is a twisted curve of the eighth degree ; but this includes the circle (PQ), 
which is not part of the intersection of the cy elides PQ" — P"Q, P'Q" — P"Q'; there is, 
therefore, only a curve of the sixth degree common to the three determinants of the 
matrix (175). 
Cor. The cone whose vertex is the centre of U, the common sphere of inversion of 
the cyclides, and which stands on the curve of the sixth degree, is only of the third 
degree. For any plane through the vertex of the cone meets the curve in six points; but 
these are inverse two by two, since the curve is evidently an anallagmatic, and therefore 
only three edges of the cone lie in the plane. 
293. Given seven pairs of inverse points of a cyclide, the polar spheres of a given inverse 
pair of points pass through a given pair of inverse points. 
For evidently the polar sphere of a given fixed pair of inverse points with respect to 
W-fXW'-J-^W" will be of the form P-j-XP'-f-^P", and will therefore pass through a 
given fixed pair of inverse points, namely, the two points common to the spheres P, P', P". 
Reciprocally, given seven generating spheres of a cyclide, the locus of the pole points 
of a fixed sphere is a fixed sphere. 
294. If W=(#)(«, j3, y,cf) 2 =0 be a given cyclide, we have seen that (fifX, g>, v, g) 2 =0 
is the tangential equation of the focal quadric ; but if the discriminant vanish of the 
equation of a quadric in tangential coordinates, it represents a conic in space, and the 
corresponding cyclide will be binodal. Hence we have the theorem, if the discriminant 
vanish of the equation of a cyclide, the cyclide will he a binodal cyclide. 
295. Since the discriminant contains the coefficients in the fourth degree, it follows 
(175) 
