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Substituting Eq. (27) into Eq. (28) we can find the relative frequency shift due to the Yukawa-type interaction defined in the same way as in Eq. (25) 

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Calculating both integrals on the right-hand side of Eq. (30), one obtains [44] 

where 

The constraints on the parameters of Yukawa-type interaction can be now obtained from the inequality 

where the left-hand side is defined in Eq. (31). The computational results are presented by the solid line in Fig. 5. Within the interaction range λ ≤ 2 µ m the best constraints given by the solid line were obtained in an equilibrium situation when the temperatures of the plate and of the environment are equal to 310 K. In so doing the magnitude of the absolute error Δ$_{1}$ γ$_{z}$ = 3 . 06 × 10 − 5 was used which occurs at the separation distance a$_{1}$ = 6 . 88 µ m [39]. Within the interaction range λ ≥ 5 µ m the best constraints are also obtained in an equilibrium situation. Here, the measurement at the separation a$_{2}$ = 9 . 95 µ m was used with the respective absolute error equal to Δ$_{2}$ γ$_{z}$ = 1 . 41 × 10 − 5 [39]. In an intermediate interaction range of λ from 2 to 5 µ m the best constraints follow from out of equilibrium measurement data, where the temperature of an environment was the same as before, but the temperature of the plate was equal to 479 K. The best constraints were found from the measurement performed at a$_{3}$ = 7 . 44 µ m with Δ$_{3}$ γ$_{z}$ = 2 . 35 × 10 − 5 [39]. 

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For comparison purposes, the long-dashed line in Fig. 5 shows the constraints obtained in Ref. [24] from the torsion pendulum experiment of Ref. [23]. It is seen that constraints obtained from the torsion pendulum experiment are from about one to two orders of mag- nitude stronger (depending on the interaction range) than the constraints given by the solid line. It is important to bear in mind, however, that the constraints from the measurement of the Casimir-Polder force (the solid line) were determined at a 67% confidence level, whereas 

