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The study of the vibrations of polyatomic molecules has recently turned into one of the most widespread and powerful methods of studying molecular structure. These vibrations ap- pear directly in the infrared absorption spectra and Raman spectra of gases, liquids, and solids. A measurement of the number of bands in addition to their positions (frequencies or wavelengths) offers the possibility of obtaining a great deal of important information regarding the geometric and mechanical properties of the molecules, the types of chemical bonds, and so forth. It is now quite difficult to list the vast number of specific problems solved by measuring vibrational fre- quencies. As a result of the successful development of research methods and the widespread applica- tion of vibrational spectra in analyzing the structures of molecules and the constitution of ma- terials, it now becomes necessary to develop the theory of molecular vibrations further. Existing theory, of course, is based on the assumption of the harmonicity of molecular vi- brations, which, strictly speaking, is not justified experimentally. The anharmonicity of the molecular vibrations has therefore to be taken into account by introducing appropriate approxi- mations. Thus, in carrying out calculations on the vibrations of polyatomic molecules, one uses the force constants calculated from the observed frequency values. However, as a result of the anharmonicity of the vibrations, the values of the observed frequencies differ from the harmonic values, and the force constants used therefore differ from the true ones, i. e.