The quantum-mechanical Hamiltonian function, $H$, for the general vibrating-rotating polyatomic molecule of $N$ atoms has been expanded by the method of Wilson and Howard to second order of approximation. It has been practicable to obtain solutions of the Schrödinger equation $\mathrm{}(H-E)\mathrm{}\psi =0\mathrm{}$ for asymmetric molecules, axially symmetric molecules and linear molecules. Expressions for the anharmonic terms occurring in the vibration energy, the effective moments of inertia, the amplitudes of the internal angular momentum of oscillation and the centrifugal distortion coefficients are derived so that when the normal coordinates are known the vibration-rotation energies of a polyatomic molecule may be calculated from the results given. Tetrahedrally symmetric molecules and models in which internal rotation occurs are regarded as anomalous cases and are not treated.

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