the maximum magnitude of the lateral Casimir force was achieved at some phase shift ϕ$_{i}$ (keeping in mind that the dependence of the lateral force on ϕ is not strictly sinusoidal, ϕ$_{i}$ must not be multiples of π/ 2). The absolute errors, Δ$_{i}$, of the measured maximum magnitudes of the lateral Casimir force at different separations were determined at a 95% confidence level. The measurement data were compared with the exact theory taking into account a nonzero skin depth of the Au coating and treating the Casimir force due to the nontrivial geometry of sinusoidally corrugated surfaces in the framework of the Rayleigh scattering approach. It was found that the experimental data agree with the theory within the limits of the experimental errors Δ$_{i}$. This means that the magnitudes of any possible lateral Yukawa-type force that might arise between the corrugated surfaces of a sphere and a grating must satisfy the inequality with Δ$_{i}$ as defined above. Equation (2) follows from the fact that the measured force consisting of the lateral Casimir force and possible lateral Yukawa-type force agrees with the theoretical lateral Casimir force to within the experimental error of the force measurement (note that in the experiment under consideration the experimental errors Δ$_{i}$ are much larger than the errors in computations of the lateral Casimir force with the help of the scattering approach). To obtain constraints on the parameters of the Yukawa-type interaction α and λ , following from Eq. (2), we should substitute in this equation an explicit expression for the lateral Yukawa-type force. This can be found in the following way. We first consider a smooth homogeneous sphere of density ρ$_{s}$ placed at a distance a (in the z direction) above a plane plate of density ρ$_{g}$ with thickness D and linear dimension L . Under the conditions a, λ ≪ D, L, R which are satisfied in our case with large supply, one can consider a plate of infinite area and thickness (i.e., a semispace). The expression for the Yukawa-type interaction energy and force between a thick plate (semispace) and a sphere spaced at a separation a are obtained by the integration of V $^{Yu}$( r ) in Eq. (1) over both volumes and subsequently taking the negative differentiation with respect to a [30]: ∣ F Yu , lat ps, cor ( a$_{i}$, ϕ$_{i}$ ) ∣ ≤ Δ$_{i}$ , (2) E Yu ps ( a ) = − 4 π $^{2}$Gαλ$^{4}$ρ$_{g}$ ρ$_{s}$e − $^{a/λ}$Φ( R, λ ) , F Yu ps ( a ) = − 4 π $^{2}$Gαλ$^{3}$ρ$_{g}$ ρ$_{s}$e − $^{a/λ}$Φ( R, λ ) , (3) 6