in k$_{ben}$ and due to the resolution of data. According to Eq. (21), Δ k$_{ben}$/k$_{ben}$ = 0 . 047. From Eq. (23) this leads to the systematic error Δ syst 1 f ( a ) = 0 . 047 f ( a ). The systematic error resulting from the resolution of data does not depend on separation Δ syst 2 f ( a ) = 0 . 035 pN. Combining these two errors at a 95% confidence level as random quantities characterized by a uniform distribution [2, 52, 53], we obtain the values of the total systematic error given in column 4 of Table 1. Using the rule for the combination of random and systematic errors at the same confidence level [2, 52, 53], we obtain the total experimental error Δ $^{tot}$f as a function of separation presented in the fifth column of Table 1. As can be seen in Table 1, all absolute errors are larger at shorter separations and decrease with increasing separation. This might be explained by local deviations of the shape of corrugations from perfect sinusoidal form which are more influential at shorter separations. The total relative error of max | F C $_{lat}$| at a = 121 . 1 nm is equal to 22.5% (at a 95% confidence level). The relative errors at separations a = 124 . 7, 128.6, and 137.3 nm are equal to 13%, 25.5%, and 14%, respectively. The error in the measurement of the absolute separations, Δ a , is a combination of the error in the determination of z$_{0}$ , Δ z$_{0}$ = 3 nm, indicated in Eq. (21) and the error in the quantity z$_{piezo}$ + S$_{nor}$m$_{tor}$ in Eq. (12). The latter is equal to the half of the first step of the z -piezo, Δ$_{p}$ z ≈ 2 nm. Combining these two errors at a 95% confidence level with the help of the same rule, as applied above for the systematic errors [2, 52, 53], one obtains Δ a = 4 nm. This is a factor of eight improvement as compared with Ref. [38]. Note that the bands between the dashed and the solid lines in Fig. 8 are related to theoretical predictions using the PFA and the exact theory, respectively. They are discussed in Secs. V and VI. The same measurement procedure, as described above, was applied in the second set of measurements for a sphere covered with deeper corrugations of an amplitude ˜ A$_{2}$ . The resulting lateral Casimir force over five or six corrugation periods is shown in Fig. 9(a-c) as dots versus the lateral displacement at separations a = 134, 156.5 and 179 nm, respectively. Similar results were obtained also at separations a = 145 . 2, 201.6, 224, and 246.6 nm. In the same way, as in Fig. 7, the lateral Casimir force at each a is a periodic function which is asymmetric (nonsinusoidal). The asymmetry is more pronounced at shorter separations. The experimental values of the max | F C $_{lat}$| versus separation are plotted in Fig. 10 as crosses. Similar to Fig. 8 the arms of the crosses show the total experimental errors of force and separation measurements determined at a 95% confidence level. Equation (22) and the 19