GEOMETRY. 
578 
[790 
the continuous lines are the sections above the plane of xy, and the dotted lines the 
sections below this plane. The form is, in fact, that of a saddle. 
In the ellipsoid (fig. 23), the sections by the planes of zx, zxj, and xy are each 
of them an ellipse, and the section by any parallel plane is also an ellipse. The 
Fig. 23. 
surface may be considered as generated by an ellipse moving parallel to itself along 
two ellipses as directrices. 
In the hyperboloid of one sheet (fig. 24), the sections by the planes of zx, zy 
are the hyperbolas 
1, 
having a common conjugate axis zOz ; the section by the plane of xy, and that by 
Fig. 24. 
z 
any parallel plane, is an ellipse; and the surface may be considered as generated by 
a variable ellipse moving parallel to itself along the two hyperbolas as directrices. 
In 
are the 
the hyperboloid of two sheets (fig. 25), the sections by the planes of zx and zy 
hyperbolas