Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

QUADRIC SURFACES

581

centers of these sections. Find their coordinates in the case
of the ellipsoid (1), ~ 1.
SIMILAR QUADRICS
Definition. Two central quadrics are said to be similar if
the principal sections of one are similar, respectively, to the
principal sections of the other.
6. Prove that the ellipsoids defined by the equation
a2 b2 c2
where X is a parameter, are similar.
7. Show that, of the hyperboloids represented by the
equation
x2 y2 z2
(2) + -.X, X A — O,
those for which X is positive are all similar, and that this is
true also of those for which X is negative. Prove that all the
hyperboloids have the same asymptotic cone.
8. Prove that all the ellipsoids (1) have the same pairs
of conjugate diameters and diametral planes.
9. The same for the hyperboloids (2).
Definition. Two paraboloids of the same type are similar,
if the principal sections of one are proportional in scale
(Ch. VI, ~ 1) to the principal sections of the other.
10. Prove in each case that the paraboloids defined by the
given equation are all similar:
X2 y2_ x9 y2.
(a) 2+b=2Xz, X#O; (b) — b-=2 z, XA O.
RULED SURFACES
11. Show that the pencil of planes through the ruling Vo of
the hyperboloid (1), ~ 4, cuts the surface in the set of rulings
U and that the pencil of planes through UO cuts it in the
rulings V.