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. 



