DR. J. CASEY OR CYCLIDES AND SPHERO-QTJARTXCS. 
593 
19. Let the two lines in which the tangent plane P of the last article cuts F be denoted 
by L, L' ; now all the generating spheres whose centres are on L have a common circle 
of intersection ; if this circle be called C, and the corresponding circle for L' be called C', 
then these circles are evidently homospheric, and the two points common to them are 
plainly points on the cyclide. I say moreover that the circles C, O lie altogether on 
the cyclide. For through L draw another tangent plane to F intersecting F in another 
line L" ; then corresponding to this line we have another circle, C", which is also homo- 
spheric with C, and their points of intersection are points on the cyclide ; hence the circle 
C lies altogether in the cyclide, and so do the circles C', C", &c. Hence we infer the 
following method of generating cyclides analogous to the rectilinear generation of 
quadrics : — 
Being given three circles, C, C', C", cutting U orthogonally , the intersection of their 
planes being the centre of U, then the envelope of a variable circle whose motion is directed 
by cutting each of these circles twice is a cyclide. 
Cor. 1. Every generating sphere of a cyclide intersects it in the two generating circles 
passing through its points of contact with the cyclide. 
Cor. 2. The generating spheres touch but do not intersect the cyclide if their focal 
quadric be not a ruled surface. 
20. If the sphere U reduce to a point, which will happen when the four spheres of 
reference a, (3, y, h (see art. 4) pass through a common point, the method of generating 
cyclides given in art. 18 becomes simplified as follows: — 
Being given a quadric F and a point U, then the locus of the reflection of U made by 
any tangent plane to F will be a cyclide. This is plainly equivalent to the following: — 
The pedal of a quadric is a cyclide ; or again, the inverse of a quadric with respect to 
any arbitrary point is a cyclide. 
Or we may state the whole matter thus : — Being given a quadric F and a point U, 
from U draw a perpendicular UT on any tangent plane to F, and on UT take P, P' in 
opposite directions such that UT 2 — TP 2 =UT 2 — TP ,2 =F 2 , where k is a constant, then 
the locus of P, P' is a cyclide. 
There are three cases to be considered. 
1°. If k 2 be positive the sphere U is real. 
2°. If k 2 be negative the sphere U is imaginary ; this will happen when the radical 
centre of the spheres of reference, a, [3, y, c), is internal to these spheres. 
3°. If k 2 vanish, U reduces to a point. The cyclide is in this case the inverse of a 
quadric. 
The point U is a nodal point on the cyclide. The tangent planes to the cyclide at 
the node U form a cone, which is reciprocal to the cone whose vertex is at U and which 
circumscribes F. Hence the point U will be a conic node when F is either an ellipsoid 
or hyperboloid. We shall examine more minutely the species of the node in this case 
when we come to the Chapter on the Inversion of Cyclides. 
21. If the focal quadric F be a cone the cyclide becomes modified in a remarkable way, 
