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

606 ANALYTIC GEOMETRY
desired form by rotating two of the axes about the third
through an angle of 45~.
13. A quadric surface is symmetric in two planes which
are parallel to or identical with two coordinate planes. Show
that either the terms in yz, zx, xy do not appear in the equation
of the surface or the surface itself consists of two planes of
the type described. Prove that in the latter case the terms
in yz, zx, xy can be removed from the equation by rotating two
of the axes about the third through an angle of 45~.
14. Show that, if a quadric surface is symmetric in a coordinate plane or in a plane parallel to a coordinate plane, its
equation contains, in general, at most one of the three terms
in yz, zx, xy. When does the exception occur?
15. Prove that the conclusion of the previous exercise follows if the surface is symmetric in a coordinate axis or in a
line parallel to a coordinate axis. When does the exception
occur in this case?