  We calculate the additional force due to two-axion exchange acting in a
sphere-disc geometry, used in experiments on measuring the gradient of the
Casimir force. With this result, stronger constraints on the pseudoscalar
coupling constants of an axion and axion-like particles to a proton and a
neutron are obtained over the wide range of axion masses from 0.03mV to 1eV.
Among the three experiments with Au-Au, Au-Ni and Ni-Ni boundary surfaces
performed by means of dynamic atomic force microscope, major improving is
achieved for the experiment with Au-Au test bodies. Here, the constraints
obtained are stronger up to a factor of 170, as compared to the previously
known ones. The largest strengthening holds for the axion mass 0.3eV.
