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DE. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
or the system of four circles, 
ct^ad{bd — b'c)±L$\/bbXcd—c'a)±v\/cc'{ab' — a!b)— 0. . . . (219) 
The section is therefore a sphero-quartic counted twice and four circles. 
Cor. The four circles are generators of the sjphero-quartic. 
336. We can show geometrically that a generating circle of the cy elide W on each of 
the eight generating spheres common to W, W', S', or to W, W', T', is also a generating 
circle of the cyclicle (212), and therefore that these eight circles form the locus which is the 
intersection of W with T' 2 — 4ATW (see art. 335). Since S' and W 7 are reciprocals with 
respect to W, it is evident that, on the eight spheres which are common generators of 
W, W', S', at the points of contact of W, W' with S' these spheres are coincident. Hence 
one of the generating circles of AV on each of these generating spheres is also a generating 
sphere of the cyclide (212) ; hence YV and (212) have eight common generating circles. 
337. The cyclide (212) is the same generalization of the developable circumscribed 
to two quadrics which a cyclide of the second degree in a, 0, y, § is of a quadric, — thus 
to the generating lines of one corresponding generating circles of the other, and to the 
nodal conics corresponding nodal sphero-quartics, and so on. Hence, by the system of 
substitutions established in Chapter XIII., we can get from the properties proved in 
Chapter VIII. of the developable A, theorems which hold for the cyclide (212). The 
following are a few illustrations : — 
1°. Eight lines of 2 meet any arbitrary line. Hence eight generating circles of (212) 
meet any arbitrary circle orthogonal to the sphere U. 
2°. The curves of taction of % divide homographically the lines of the system. Hence 
the curves of taction of (212) divide homographically the circles of the system. 
3°. The nodal lines of 2 divide equianharmonically the lines of the system. Hence 
the nodal sphero-quartics of { 212) divide equianharmonically the circles of the system. 
4°. Any line of 2) meet its curves of taction in points the tangents at which to the curves 
of taction envelope a plane conic. 
Hence any generating circle of (212) meet its curves of taction in points, the generating 
circles of the curves of taction through which are generators of a sphero-quartic. 
338. Since the surface (212) is the envelope of a variable sphere cutting U ortho- 
gonally, and whose centre moves along the twisted quartic (VV'), then (V V') is the 
deferente. From this generation we can also infer the properties of (212). Thus the 
cuspidal edge of { 212) is the locus of the limiting points composed of the sphere U and 
the osculating planes of (VV'). 
2°. There are sixteen pairs of stationary points on the cuspidal edge ; these correspond 
to the stationary planes of (YY’). 
3°. Any sphere cutting U orthogonally meets the cuspidal edge in twelve pairs of inverse 
points. This follows from n= 12 (see art. 224). 
4°. The cuspidal edge is an anallagmatic, U being the sphere of inversion. 
339. To find the locus of a pair of inverse points whose polar spheres with respect 
