ON THE CYCLIDE *. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xit. (1873), 
pp. 148—165.] 
The Cyclide, according to the original definition, is the envelope of a variable 
sphere which touches three given spheres, or, more accurately, the envelope of a variable 
sphere belonging to one of the four series of spheres which touch three given spheres. 
In fact, the spheres which touch three given spheres form four series, the spheres of 
each series having their centres on a conic; viz. if we consider the plane through the 
centres of the given spheres, and in this plane the eight circles which touch the 
sections of the given spheres, the centres of these circles form four pairs of points, 
or joining the points of the same pair, we have four chords which are the transverse 
axes of the four conics in question. 
It thus appears, that one condition imposed on the variable sphere is, that its 
centre shall be in a plane; and a second condition, that the centre shall be on a 
conic in this plane; so that the original definition may be replaced first by the 
following one, viz.: 
The cyclide is the envelope of a variable sphere having its centre on a given 
plane, and touching two given spheres. 
Starting herefrom, it follows that the locus of the centre will be a conic in the 
given plane: the transverse axis of the conic being the projection on the given plane 
of the line joining the centres of the given spheres; and it, moreover, follows, that if 
in the perpendicular plane through the transverse axis we construct a conic having 
for vertices the foci, and for foci the vertices, of the locus-conic, then the conic so 
constructed will pass through the centres of the given spheres. 
* I use the term in its original sense, and not in the extended sense given to it by Darboux, and 
employed by Casey in his recent memoir “On Cyclides and Spheroquartics,” Phil. Trans. 1871, pp. 582—721. 
With these authors the Cyclide here spoken of is a Dupin’s or tetranodal Cyclide.