II. STRONGER CONSTRAINTS ON THE YUKAWA-TYPE HYPOTHETICAL INTERACTION FROM THE MEASUREMENT OF THE LATERAL CASIMIR FORCE BETWEEN CORRUGATED SURFACES It is customary to normalize the Yukawa interaction potential between two neutral point masses m$_{1}$ and m$_{2}$ (atoms) at a separation r to the potential of Newtonian gravity and represent it in the form [21, 41] V $^{Yu}$( r ) V $^{N}$( r ) = α e − $^{r/λ}$, V $^{N}$( r ) = − Gm$_{1}$m$_{2}$ r . (1) Here, G is the Newtonian gravitational constant, α is a dimensionless constant characterizing the strength of the Yukawa interaction and λ is its interaction range. Specifically, if the effective Yukawa potential between atoms m$_{1}$ and m$_{2}$ is generated by the exchange of light bosons of mass m , one has λ = ℏ / ( mc ). The lateral Casimir force acting between the sinusoidally corrugated surfaces of a sphere and a plate was first experimentally demonstrated in Refs. [42, 43]. In Ref. [43] a measure of agreement between the data and the theoretically calculated lateral Casimir force was used to obtain constraints on the parameters of Yukawa interaction, namely, α and λ . This, however, did not lead to definitive constraints determined at a high confidence level because the determination of the lateral Casimir force between corrugated surfaces was not sufficiently precise. In a recent experiment [38] the lateral Casimir force was measured between two aligned sinusoidally corrugated surfaces of a grating and a sphere with equal corrugation periods Λ = 574 . 7 nm and corrugation amplitudes A$_{1}$ = 85 . 4 nm and A$_{2}$ = 13 . 7 nm, respectively. The grating was made of hard epoxy with density ρ$_{g}$ = 1 . 08 × 10 $^{3}$kg/m 3 on a 3 mm thick Pyrex substrate. The top of the grating was covered with a Δ$_{Au}$$_{,g}$ = 300 nm thick Au coating of density ρ$_{Au}$ = 19 . 28 × 10 $^{3}$kg/m$^{3}$. The sphere of R = 97 . 0 µ m radius was made of polystyrene of density ρ$_{s}$ = 1 . 06 × 10 $^{3}$kg/m 3 and uniformly coated with a Δ$_{Cr}$ = 10 nm layer of Cr ( ρ$_{Cr}$ = 7 . 14 × 10 $^{3}$kg/m$^{3}$) and then with a Δ$_{Au}$$_{,s}$ = 50 nm layer of Au in a thermal evaporator [38]. The lateral Casimir force was independently (without fit to any theory) measured as a function of the phase shift ϕ between corrugations on the sphere and on the grating over the region of separations, a , in the range from 120 to 190 nm. At each separation a$_{i}$ 5