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

QUADRIC SURFACES 565
It can be proved in the same way that the hyperboloid of
one sheet
+y2 Y2
(2) + — 1I a > b,
a2 b2c a>bc2
contains just two sets of circular sections. Hence it follows,
by ~ 5, Th. 3, that this is true also of the cone and the hyperboloid of two sheets. It is to be noted, however, that there
are no circular sections of the cone by planes through the
vertex and none of the hyperboloid of two sheets by planes
through the center.
The results obtained we now consolidate into a theorem.
THEOREM. An ellipsoid, a hyperboloid, or a cone, which is
not a surface of revolution, contains just two sets of circular
sections.
If, in Fig. 9, b approaches a as its limit, the planes KOB
and LOB both approach as their limits the (x, y)-plane. Consequently, an ellipsoid of revolution has but one set of circular sections. This is true also of the hyperboloids and cones
of revolution.
The circles in which the planes KOB, LOB intersect the
ellipsoid evidently lie on the sphere whose center is at 0 and
whose radius is b:
~(3) wX2 y2 z2
b2 b2 b2
They therefore lie on the surface whose equation results from
subtracting (1) from (3):
12 - + W 2 (1 — =0,
\b2 a2 b2 c2
or c2 (a2 b-) x2 - a2 (b2 - c2) z2 = 0.
But this surface consists of the two planes
(4) c/a2- b2x ~ a-/b2 - c2z = 0.
Consequently, these are the equations of the planes KOB,
LOB.