We start with the expression for the Yukawa-type pressure between two parallel plates made of Si of density ρ$_{Si}$ = 2 . 33 × 10 $^{3}$kg/m 3 and covered with chromium layers of thickness Δ$_{Cr}$ = 50 nm, as in the experiment of Ref. [40]. Keeping in mind that the experimental parameters satisfy the conditions a, λ ≪ D$_{i}$, L$_{i}$ , where D$_{1}$$_{,}$$_{2}$ and L$_{1}$$_{,}$$_{2}$ are the thicknesses and linear dimensions of both plates, one can consider these plates as semispaces. The integration of the Yukawa-type potential (1) over the volumes of both plates leads to [21, 50] E $^{Yu}$( a ) = − 2 πGαλ $^{3}$ρ2$_{Si}$e − $^{a/λ}$. (20) E Yu l ( a ) = − 2 πGαλ $^{3}$e − a/λ [ ρ$_{Cr}$ − ( ρ$_{Cr}$ − ρ$_{Si}$ ) e − Δ Cr $^{/λ}$] 2 . (21) P Yu l ( a ) = − ∂E Yu l ( a ) ∂a = − 2 πGαλ $^{2}$e − a/λ [ ρ$_{Cr}$ − ( ρ$_{Cr}$ − ρ$_{Si}$ ) e − Δ Cr $^{/λ}$] 2 (22) ∣ l ( a ) ∂a ∣ = 2 πG | α | λe − a/λ [ ρ$_{Cr}$ − ( ρ$_{Cr}$ − ρ$_{Si}$ ) e − Δ Cr $^{/λ}$] 2 . (23) Applying this equation to the plates covered with chromium layers one obtains This leads to the following expression for the Yukawa pressure and for the magnitude of its gradient The constraints on the Yukawa-type interaction can be obtained from the inequality 14 determined in Ref. [40] at different separations a$_{i}$ where measurements were performed at a 67% confidence level. Now we substitute Eq. (23) into Eq. (24) and determine the allowed and prohibited regions of the parameters α and λ . These regions are separated by the solid line in Fig. 4. The best constraints shown by the solid line were obtained using the experimental data at different separation distances. Thus, at λ ≤ 0 . 2 µ m the value Δ$_{1}$( ν 2 − ν 2 $_{0}$) = 67 . 1 Hz 2 that was used occurs at a$_{1}$ = 0 . 553 µ m. For interaction ranges from λ = 0 . 2 to 0 . 35 µ m and λ ≥ 0 . 35 µ m we have used the values Δ$_{2}$( ν 2 − ν 2 $_{0}$) = 58 . 4 Hz 2 and Δ$_{3}$( ν 2 − ν 2 $_{0}$) = 19 . 2 Hz $^{2}$, which occur at separations a$_{2}$ = 0 . 574 µ m and a$_{3}$ = 0 . 8805 µ m, respectively [40]. For comparison purposes, in Fig. 4 we present also the constraints on the where Δ$_{i}$( ν 2 − ν 2 $_{0}$) is the absolute error in the measurement of the frequency shift ν 2 − ν 2 0 β ∣ l ( a$_{i}$ ) ∂a ∣ ≤ Δ$_{i}$( ν 2 − ν 2 $_{0}$) , (24)