G94 
DR. J. CASEY ON CYCLIDES AND SPHERO- QUARTICS. 
Chasles, ‘ Sections Coniques,’ art. 136, also 4 Bicircular Quartics,’ arts. 153-155) ; but P is 
Hence, omitting the double accents, we see that the equation of the sphere of which 
oi, ft, ft, b' are the pole points is 
( 4 +^+ 4 +^) w =° ( i72 ) 
And we have evidently the following theorem : — If through a pair of inverse points 
(a!, ft, y, o') we describe any circle X cutting the polar sphere of {a! , ft, ft, l') in a pair of 
inverse points (oi', ft', ft’, l"), X will cut the cy elide in two other pairs of inverse points, such 
that the four segments made on X by the pairs of points (oi, ft, ft, o'), (a", (3", y", b") and 
by the cyclide are harmonic (see 4 Bicircular Quartics,’ art. 153). 
279. From the last article we see that, if (oi, 13', ft, b') be a pair of inverse points on the 
cyclide, the equation of the generating sphere which touches the cyclide at [pi, (3 1 , ft, ) is 
(4+0'4+4+^) w=o (173) 
This equation establishes a relation between the coordinates (a 1 , ft, ft, b’) of a pair of 
inverse points of the cyclide, and (a, (3, y, b) the sphero-coordinates of any other pair of 
inverse points on the generating sphere which touches the cyclide at [oi, ft, ft, b'); and since 
the relation is symmetrical with respect to [oi, ft, ft, b 1 ) and (a, ft y, b), we infer the 
following theorem : — If through any pair of inverse points we describe a generating 
sphere of the cyclide, the locus of all their points of contact is the sphero-quartic which 
is given as the curve of intersection of W with the sphere (173), or the polar sphere of 
[oi, ft, ft, b>). 
280. The discriminant of the equation (171) is 
YV'W"=P 2 ; 
and by omitting the double accents we see that the equation of the binodal cyclide 
which circumscribes W, and which has the pair of points (oi, ft, ft, b’) as nodes, is 
w '={(4+^4+T'4+ y aM* flM) 
281. Since a cyclide has five spheres of inversion, taking any point A, Ave get five 
points, namely, the inverses of A with respect to the five spheres of inversion of the 
cyclide. Let the inverse points be A 1? A 2 , A 3 , A 4 , A 5 ; and, with the five pairs of inverse 
points (AA,), (AA 2 ), (AA 3 ), (AA 4 ), (AA 5 ), we get by the last article five binodal cyclides 
circumscribed to W, and these binodals will have one common node, namely, the point A ; 
the other nodes of these circumscribed cyclides will be the five points A,, A 2 , A 3 , A 4 , A s . 
282. If Ave invert the cyclide W from the point A, the five binodal cyclides of the 
last article invert into five cones of the second degree, each having double contact Avith 
the inverse cyclide (see art. 187). Noav all the points of contact of the five double 
tangent cones lie on five concentric spheres. Hence Ave have the folloAving theorem : — 
