  We prove an isoperimetric inequality for the second non-zero eigenvalue of
the Laplace-Beltrami operator on the real projective plane. For a metric of the
unit area this eigenvalue is not greater than 20\pi. This value is attained in
the limit by a sequence of metrics of area one on the projective plane. The
limiting metric is singular and could be realized as a union of the projective
plane and the sphere touching at a point, with standard metrics and the ratio
of the areas 3:2. It is also proven that the multiplicity of the second
non-zero eigenvalue on the projective plane is at most 6.
