correction factor to the Casimir force due to stochastic roughness in the configuration of a sphere above a plate with no corrugations is given by Now we take into account that in the framework of the multiplicative approach the correction factor due to surface roughness is the same for real and ideal metals. Keeping in mind that the Casimir force between a perfectly shaped ideal metal sphere and a plate is F ( a ) ∼ 1 /a $^{3}$, we obtain the correction to the Casimir energy by integrating the Casimir force with the inclusion of roughness, η$_{F}$ ( a ) F ( a ), with respect to a Note that for the experimental variances (see Sec. II) the fourth order term in Eq. (39) is negligibly small. Thus, for the first set of measurements at the shortest separation a = 120 nm the second order term contributes 3% of the force, but the fourth order term only 0.2% of the force. For the second set of measurement with deeper corrugations on the sphere the contributions of the second and fourth order terms at separation a = 134 nm are 4% and 0.3%, respectively. To obtain the correction factor to the lateral Casimir force due to the surface roughness with account of the sinusoidal corrugations on the sphere and on the plate, we should replace a with separation z$_{s}$ − z$_{p}$ between the corrugated surfaces in Eqs. (3) and (5) and perform the averaging with respect to the phase shift and the period Calculating both integrals in the second equality in Eq. (40) and substituting the obtained result into the first equality, we arrive at where F ( α, β ; γ ; z ) is the hypergeometric function. 24 η$_{corr}$ ( a ) = 1 + 3 δ 2 1 + δ 2 a ( a + A$_{1}$ − A$_{2}$ ) 1 / $^{2}$( a + A$_{2}$ − A$_{1}$ ) 1 / 2 × F ( 1 2 , 1 2 ; 1; 4 A$_{1}$A$_{2}$ ( a + A$_{1}$ − A$_{2}$ )( a + A$_{2}$ − A$_{1}$ ) ) , (41) η$_{corr}$ ( a ) = 1 + 3( δ 2 1 + δ 2 $_{2}$) 〈 1 ( z$_{s}$ − z$_{p}$ ) 2 〉 , (40) 〈 1 ( z$_{s}$ − z$_{p}$ ) 2 〉 = 1 2 π Λ a 2 ∫ 2 π 0 dϕ ∫ Λ 0 dx 1 [1 + β cos(2 πx/ Λ − α )] $_{2}$. η$_{F}$ ( a ) = 1 + 6 δ 2 1 + δ 2 a 2 + 45 ( δ 2 1 + δ 2 $_{2}$) 2 a 4 . (38) η$_{E}$ ( a ) = 1 + 3 δ 2 1 + δ 2 a 2 + 15 ( δ 2 1 + δ 2 $_{2}$) 2 a 4 . (39)