  Stronger constraints on the pseudoscalar coupling constants of an axion to a
proton and a neutron are obtained from an indirect measurement of the effective
Casimir pressure between two Au-coated plates by means of micromechanical
torsional oscillator. For this purpose, the additional effective pressure due
to two-axion exchange is calculated. The role of boundary effects and the
validity region of the proximity force approximation in application to forces
of axion origin are determined. The obtained constraints are up to factors of
380 and 3.15 stronger than those found recently from other laboratory
experiments and are relevant to axion masses from $10^{-3}$eV to 15eV.
