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Using the values of k$_{ben}$ and z$_{0}$ obtained through the electrostatic calibration (see Sec. III) the measurement data for the deflection signal S C lat were converted into values of the lateral Casimir force at every separation a as: 

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The resulting lateral Casimir force for a sphere with the corrugation amplitude A$_{2}$ = 13 . 7 nm over four corrugation periods is shown in Fig. 7(a-c) as dots versus the normalized lateral displacement x/ Λ between the corrugated surfaces at separations a = 124 . 7, 128.6, and 149.8 nm, respectively. Similar results were obtained at all other a listed above. As can be seen in Fig. 7, the lateral Casimir force is a periodic function of the phase shift. The maximum values of the lateral force, f ≡ max | F C $_{lat}$| , decrease with the increase of a . The important characteristic feature of the periodic curves in Fig. 7(a-c) is that they are asym- metric , i.e., the dependence of F C lat on x/ Λ is not strictly sinusoidal (note that the sinusoidal dependence of the lateral force on the phase shift holds only if the calculation is restricted to the lowest order in corrugation amplitudes ∼ A$_{1}$A$_{2}$ ; see Sec. V), The true asymmetry of the measured lateral force is obvious even without the theory curve. For example, in Fig. 7(a) the average shift of each maximum point from the midpoint of two adjacent min- ima is (0 . 12 ± 0 . 2)Λ. With the increase of separation, the relative contribution of higher perturbation orders in a small parameter A$_{1}$A$_{2}$/a 2 decreases and the dependence of the lat- eral force on the phase shift becomes more close to sinusoidal. The solid lines in Fig. 7 are related to the exact theoretical computations. They are discussed in Sec. VI. 

The experimental values of the max | F C $_{lat}$| versus separation a are shown in Fig. 8 as crosses. The arms of the crosses indicate the total experimental errors determined at a 95% confidence level using the following procedure. From the data of the phase curves in Fig. 7 (and related curves at all the other separations measured) we have calculated the mean values ¯ f and the variances s ¯ f of the quantity f = max | F C $_{lat}$| at each a (see columns 2 and 3 in Table 1). For the first set of measurements under consideration, the averaging was performed over the four periods. The respective Student coefficient is t$_{(1+}$$_{γ}$$_{)}$$_{/}$$_{2}$ (3) = 3 . 18 where γ = 0 . 95. Thus, the experimental random error is obtained from the data in column 3 of Table 1 as 

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The main sources of systematic errors in the quantity f are the errors due to the uncertainty 

