638 
DB. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
118. The method of forming the reciprocal of one cyclide with respect to another 
will be given in a subsequent Chapter; in this we shall anticipate so much of the 
results as to say that it is identical with the method of quadrics. This being premised, 
if we form the reciprocal of the cyclides 
rta 2 + bfir -f cy 2 -f- (Ih 2 = 0, 
tfa 2 bfi 2 C7 2 dh 2 _ 
a + k' b + lc' c + k^ d+ k 
with respect to U 2 =a 2 -b/3 2 +y 2 + o 2 =0, we get 
?-V-aV -- 0 
[a + k)cd , {b + k)/3 2 (c + &) 7 2 {d+k)l- 
a + b + c + d 
and from the forms of these reciprocals it is plain that they have double contact along 
the whole sphero-quartic, in which each is intersected by the common sphere of inver- 
sion U. 
119. The three confocals to a given cyclide which can he drawn through any given point 
are mutually orthogonal. 
Definition. Confocal cyclides are cyclides having a common sphero-quartic of foci. 
Demonstration. The focal quadrics of a confocal system of cyclides pass through a 
common curve of intersection ; this is the sphero-quartic, which is their common line of 
foci. Now let P be the point through which the cyclides pass, and taking P' the inverse 
of P with respect to U, then the plane which bisects P P' perpendicularly forms with 
P, P', and U a coaxial system, and the three quadrics touching this plane and passing 
through the common line of foci will be the focal quadrics of three confocal cyclides 
passing through P, P' and cutting each other orthogonally. For if X, Y, Z be the points 
of contact of the quadrics with the plane, it is evident that the spheres whose centres are 
X, Y, Z, and which cut U orthogonally, are themselves mutually orthogonal. Hence the 
proposition is evident. 
120. The cyclides in the last article are not only orthogonal at P, P', but each pair of 
them are orthogonal throughout their whole intersection*. 
Demonstration. Let us consider the two cyclides whose focal quadrics touch the 
plane at X, Y, and let us consider any edge of the developable which circumscribes the 
focal quadrics. This edge will be divided in involution by the system of quadrics 
passing through the common focal sphero-quartic. The double points of the involution 
will be the points of contact of the edge with the two quadrics which the developable 
circumscribes. Hence the spheres having these points as centres and cutting U ortho- 
gonally will themselves cut orthogonally, and hence it follows that the cyclides of which 
they are generators will cut orthogonally in a curve which has double contact with the 
circle of intersection of their generating spheres. 
* Hence it follows, by Dunn’s theorem, that these cyclides intersect each other in lines of curvature. 
