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In section 1–2 a certain space-smoothing operation is defined and its usefulness in solving elliptic equations is demonstrated in the case of a Poisson equation. It leads to solutions in a closed form which possess the numerical simplicity of the ordinary iteration methods, but is converging more rapidly. The reverse operation of unsmoothing is also defined as far as it can be done, and it is mentioned that the combined processes of smoothing and unsmoothing are convenient tools for obtaining a spectral analysis of horizontal scalar fields, and also to remove systematical errors which are made when derivatives are taken as finite differences. In sections 3–6 the application of smoothing is shown in the barotropic forecasting problem. At first a general theorem is proved concerning trajectories of two-dimensional non-divergent flow. It states that if the streamfunction ψ1 of such a flow can be decomposed into two components ψ2, α of which α is individually conserved in the ψ1-motion, then the displacements of the fluid particles up to any time can be found by at first displacing in the stationary flow αt=o = const and then adding from the resulting positions the displacements in the flow with the streamfunction ψ2. The theorem is first applied to barotropic flow. In this case the first stationary field to displace in is the deviation between the actual and smoothed flow, while the second field to displace in is the smoothed flow. The space-smoothing is next applied to an equation expressing the individual conservation of a quantity s in a two-dimensional non-divergent flow. The Reynolds term belonging to the smoothed equation is studied and found to depend essentially upon the deformation properties of the velocity field. The role of deformation for the net spectral flow of energy in the s-field is studied. The smoothing is in particular applied to the vorticity equation to show how this possibly can be utilized in the integration problem. In sections 7–11 the baroclinic case is considered. In sections 7–8 is shown the fundamental role of deformation for the interchange of potential and kinetic energy. It is found that in the advective model there is direct proportionality between the change in total kinetic energy and total thermal wind energy, and also a direct proportionality between the change in kinetic energy for the vertically mean motion and the thermal wind energy. In section 9 is discussed the possible importance of non-linear interference for the understanding of the creation and local distribution of disturbances in the atmosphere. The integration problem is discussed in sections 10–12. At first an extension of the barotropic displacement rule is given for the vertically mean motion. The trajectory problem for levels other than the mean level is touched in section 12, and a simple non-advective model discussed shortly in section 11.DOI: 10.1111/j.2153-3490.1955.tb01185.x