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The Casimir force is calculated in the configuration of a spherical lens anda disc of finite radius covered by $Cu$ and $Au$ thin layers which was used ina recent experiment. The correction to the Casimir force due to finiteness ofthe disc radius is shown to be negligible. Also the corrections are discusseddue to the finite conductivity, large-scale and short-scale deviations from theperfect shape of the bounding surfaces and the temperature correction. Theywere found to be essential when confronting the theoretical results withexperimental data. Both Yukawa-type and power-law hypothetical forces arecomputed which may act in the configuration under consideration due to theexchange of light and/or massless elementary particles between the atoms of thelens and the disc. New constraints on the constants of these forces aredetermined which follow from the fact that they were not observed within thelimits of experimental errors. For Yukawa-type forces the new constraints areup to 30 times stronger than the best ones known up today. A possibleimprovement of experimental parameters is proposed which gives the possibilityto strengthen constraints on Yukawa-type interactions up to $10^4$ times and onpower-law interactions up to several hundred times.Comment: 15 pages, 3 figures, subm. to Phys. Rev.

Constraints for hypothetical interactions from a recent demonstration of the Casimir force and some possible improvements