90 
one of equality, the surface has circular sections parallel to 
the given plane ; when it is a ratio of equality, we get the 
hyperbolic paraboloid. All other things remaining the same, 
the focus and directrix may be changed without changing 
the surface described. If we confine ourselves to the cen- 
tral surfaces, the locus of the foci for a given surface will be 
an ellipse, which may be called the focal ellipse, each focus 
having a corresponding directrix perpendicular to the plane 
of this ellipse. 
If the focal ellipse be made the base of a cone, whose 
vertex is at any point v on the surface, the normal at this 
point will be one of the principal axes of the cone. But, 
as three surfaces, confocal to each other, and therefore 
having the same focal ellipse, may be described through a 
given point, if we suppose two other such surfaces to pass 
through the point v, the normals to these surfaces will be 
the other two principal axes of the cone. And if a system 
of surfaces, confocal to these three, be circumscribed by 
cones having a common vertex at v, the principal axes of 
all these cones will be the same as those of the cone which 
has the focal ellipse for its base. Indeed, the focal ellipse 
(which lies in the plane of the greatest and the middle axes 
of the ellipsoids) may, in the confocal system, be considered 
as the limit between the ellipsoids and the hyperboloids of 
one sheet. There is also, in the plane of the greatest and 
least axes of the ellipsoids, a focal hyperbola which is the 
limit between the confocal hyperboloids of one and of two 
sheets ; and of course, the cone which has this hyperbola 
for its base, and v for its vertex, has the same principal 
axes as the cones already mentioned. Right lines which 
pass, at the same time, through the focal ellipse and the 
focal hyperbola, possess remarkable properties. 
The foregoing are the leading propositions in Mr. 
Mac Cullagh’s paper. There are besides many particular 
theorems which could not be noticed within the compass of 
an abstract, 
