Substituting the values of A$_{i}$ and δ$_{i}$ from the first set of measurements (see Sec. II) in Eq. (41), we find that at separations a = 120, 150, and 200 nm the correction factor due to surface roughness takes the values η$_{corr}$ = 1 . 054, 1.029, and 1.015, respectively. For the second set of measurements (i.e., using ˜ A$_{2}$ and ˜ δ$_{2}$ in Sec. II) the correction factor in Eq. (41) takes the values ˜ η$_{corr}$ = 1 . 060, 1.041, and 1.021 at separations a = 130, 150, and 200 nm, respectively. It is seen that even for the second set of our measurements, where the surface roughness on the sphere is larger than in the first one, the roughness correction is rather moderate and does not exceed 6% of the lateral Casimir force. In Fig. 8 (the lower dashed line) we present the computational results for the maximum values of the lateral Casimir force as a function of separation in the framework of the PFA [i.e., using Eqs. (33) and (37)] for the first set of our measurements. This line does not take surface roughness into account. The upper dashed line in Fig. 8 presents the results for max | F C $_{lat}$| η$_{corr}$ as a function of separation, i.e., taking into account the correction due to surface roughness. Keeping in mind that the multiplicative approach provides the means to estimate the effect of roughness rather than to calculate it precisely on the basis of the fundamental theory, we consider the band between the dashed lines in Fig. 8 as a theoretical prediction given by the PFA approach. As can be seen in Fig. 8, the experimental data indicated as crosses are inconsistent with the prediction of the PFA approach, thus revealing the role of diffraction-type effects on the corrugations of relatively small period used in this experiment. In Fig. 10 a similar comparison between the experimental data and theory using the PFA approach is done for the second set of our measurements with corrugations of larger ampli- tude. The lower dashed line is computed using Eqs. (33) and (37) with the surface roughness disregarded. The upper dashed line is obtained by the multiplication of the computation results by an additional factor ˜ η$_{corr}$ computed using Eq. (41). Again the theoretical band between the dashed lines is inconsistent with the experimental data shown as crosses due to the neglect of diffraction-like effects. In the next section we present a more fundamen- tal theoretical approach taking the diffraction effects into account which brings theory into agreement with the experimental data. 25