ON THE CYCLIDE. 
65 
[569 
569] 
matics, vol. xn. (1873), 
;he envelope of a variable 
the envelope of a variable 
touch three given spheres, 
four series, the spheres of 
ider the plane through the 
3 circles which touch the 
form four pairs of points, 
Is which are the transverse 
triable sphere is, that its 
the centre shall be on a 
be replaced first by the 
ng its centre on a given 
bre will be a conic in the 
)jection on the giveu plane 
t, moreover, follows, that if 
construct a conic having 
s-conic, then the conic so 
se given to it by Darboux, and 
Phil. Trans. 1871, pp. 582—721. 
iide. 
Two conics related in the manner just mentioned are the flat-surfaces of a system 
of confocal quadric surfaces; they may for convenience be termed anti-conics (fig. 1); one 
of them is always an ellipse and the other a hyperbola; and the property of them is 
that, taking any two fixed points on the two branches, or on the same branch of the 
Fig. l. 
hyperbola, and considering their distances from a variable point of the ellipse: in the 
first case the sum, in the second case the difference, of these two distances is constant. 
And similarly taking any two fixed points on the ellipse, and considering their distances 
from a variable point of the hyperbola, then the difference, first distance less second 
distance is a constant, + a for one branch, — a for the other branch of the hyperbola. 
And we thus arrive at a third, and simplified definition of the cyclide, viz. con 
sidering any two anti-conics, the cyclide is the envelope of a variable sphere having 
its centre on the first anti-conic, and touching a given sphere whose centre is on the 
second anti-conic. 
And it is to be added, that the same cyclide will be the envelope of a variable 
sphere having its centre on the second anti-conic and touching a given sphere whose 
centre is on the first anti-conic, such given sphere being in fact any particular sphere 
of the first series of variable spheres. And, moreover, the section of the surface by the 
plane of either of the anti-conics is a pair of circles, the surface being thus (as will 
further appear) of the fourth order. 
In the series of variable spheres the intersection of any two consecutive spheres 
is a circle, the centre of which is in the plane of the locus-anti-conic, and its plane 
perpendicular to that of the locus-anti-conic, this variable circle having for its diameter 
in the plane of the locus-anti-conic a line terminated by the two fixed circles in that 
plane. The cyclide is thus in two different ways the locus of a variable circle; and 
investigating this mode of generation, we arrive at a fourth definition as follows:— 
Consider in a plane any two circles, and through either of the centres of symmetry 
draw a secant cutting the two circles; in the perpendicular plane through the secant, 
draw circles having for their diameters the chords formed by the two pairs of anti 
parallel points on the secant (viz. each pair consists of two points, one on each circle, 
such that the tangents at the two points are not parallel to each other): the locus 
of the two variable circles is the cyclide. 
Before going further it will be convenient to establish the definition of “ skew anti 
points”: viz. if we have the points K lf K 2 (fig. 2), mid-point R, and L lf Z 2 , mid-point 
S, such that K 1 K 2 , RS and L X L 2 are respectively at right angles to each other, and 
c. ix. 9