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DR. J. CASEY OX CYCLIDES AXD SPHERO-QUARTICS. 
of the third degree,” follows as an immediate inference from the theorem of this article ; 
for the point in the spliero-quartic will be the vertex of a cone of the third degree 
which stands on the quartie, and then, from what we have proved, it follows that if through 
any edge of a cone of the third degree four tangent planes be drawn to the cone their 
anharmonic ratio is constant. 
CHAPTER VII.— POCI AXD FOCAL CURVES. 
Section I. — Foci of Cy elides. 
104. The conception of a focus which I shall use in this memoir is, in the case of 
surfaces, an infinitely small sphere having imaginary double contact with the surface ; 
and for curves, that of an infinitely small circle, having imaginary double contact with 
the curve. This being premised, let us take the cyclide given by its canonical form, 
W aar + bfi 1 + cy' 2 + dh 2 -j- cs 2 = 0 . 
We see from art. 33 that this cyclide is the envelope in five different ways of a variable 
sphere whose centre moves on a given quadric, and which cuts a given fixed sphere 
orthogonally. Thus, taking the quadric E of art. 33, the tangential equation to F is 
(a — l))[F -\-(a — c)v' 2J r(a— d)f-\-(a—ey 2 =0 ; and corresponding to this we have the sphere 
a, which is the one which the variable sphere cuts orthogonally while its centre moves 
on F. Now let a developable be circumscribed to a and F, then the curve of taction of 
the developable and F, and the curve of intersection of a and F, divide the surface of F 
into three regions, which possess the following properties : — In the first region every 
point is such that any sphere having it as centre and cutting a orthogonally is real, and 
moreover such that this sphere meets the consecutive one in a real curve ; in the second 
region every point is such that the spheres are real, but do not intersect the consecutive 
ones in real points ; while in the third region the orthogonal spheres are altogether 
imaginary. Hence it follows that every point in the spliero-quartic («F) is an infinitely 
small sphere having imaginary double contact with W, or, in other words, every point on 
(«F) is a focus of W. 
In like manner every point on each of the four sphero-quartics (/ 3F'), (y F"), (SF m ), (sF iv ) 
is a focus of W, so that a cyclide has in general five focal sphero-quartics. 
From this proposition it is evident that the name focal quadric which I have employed 
for the directing quadrics F, F', &c. is appropriate as suggestive of an important pro- 
perty of these surfaces ; had I followed M. De la Goueneeie I should have called them 
deferentes. In the next proposition we shall see an additional reason in favour of the 
name I have given. 
105. Definition. When the points of contact of a focus with the cyclide are points on 
the imaginary circle at infinity, I shall, following Dr. Salmon, call the focus a “ double 
focus” (see Salmon’s £ Higher Curves Professor Cayley, in his memoir on “Poly- 
zonal Curves,” uses the term “ nodo-focus ” to express the same idea ; and M. De la 
