The forces and moments on a moving body partially immersed in the surface of a deep ocean of heavy fluid are considered in the limit of small Froude number, F. Asymptotic expressions for velocity potential and free surface elevation are developed. The choice of the first terms of the asymptotic sequence is indicated by the behavior, at small F, of the classical results of "small disturbance theory" - analysis starting from the linearized free-surface boundary conditions. It is found that the leading terms depend on the local disturbance, which can be expanded as a power series in F. The wave pattern contributes higher-order terms which are not analytic about F = 0; only estimates of the order of these terms are obtained. Consequently the present work does not estimate drag but is confined to consideration of transverse forces and moments.Once the asymptotic sequence is assumed, perturbation of the exact equations and boundary conditions about F = 0 is straightforward. The zero-order potential is that of the "reflection-plane" model of Davidson. For a restricted class of shapes, the slender body theory is applied to the zero-order and first-order problems. A general method is developed using conformal mapping to solve the first order problem for sufficiently slender shapes of arbitrary cross-section. This method is applied to two particular shapes, viz. a wing of zero thickness and a half-submerged body of revolution, both in sideslip. The correction to the reflection plane model is found to be generally quite small in the range of F for which this theory is expected to apply.
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