790] 
GEOMETRY. 
577 
having the common axis Oz; and the section by any plane z = 7 parallel to that of 
xy is the ellipse 
7 = 
x- if 
2a + 26 ; 
so that the surface is generated by a variable ellipse moving parallel to itself along 
the parabolas as directrices. 
In the hyperbolic paraboloid (fig. 21), the sections by the planes of zx, zy are 
the parabolas 
cc 2 y- 
z = z = — ■Y . 
Fig. 21. 
z 
having the opposite axes Oz, Oz'; and the section by a plane z = y parallel to that 
of xy is the hyperbola 
X 2, y 2 
^ 2a 2b ’ 
which has its transverse axis parallel to Ox or Oy according as 7 is positive or 
negative. The surface is thus generated by a variable hyperbola moving parallel to 
Fig. 22. 
itself along the parabolas as directrices. The form is best seen from fig. 22, which 
represents the sections by planes parallel to the plane of xy, or say the contour lines: 
c. xi. 73