An asymptotic expression is found for the lift distribution on a long, narrow, laminar wing, at incidence in a supersonic stream. The approximations of the linearized potential theory are used.

This sequence of papers is concerned with a group of theorems on Fourier transforms, and with the corresponding results for Fourier series. The transform theorems deal with the equations

when two of the functions concerned are monotonic; and the series theorems, with the equations

when two of the functions or sequences of coefficients are monotonic. A general survey of the problem was given in two earlier papers, which we shall call M.F. (I) and M.F. (II), and the results obtained were tabulated in M.F. (II). There is thus no need to restate the position in detail here. We shall merely recall as briefly as possible what has been proved already, so as to show the scope of the present paper.

1. A steady motion of viscous liquid of constant density is considered in §§ 2–7; full allowance is made for the inertia of the motion, and it is assumed that the stream-function can be expanded in a double series in the coordinates x, y.

In this paper two collision cases and four cases of ships grounding will be examined and the circumstances analysed with a view to drawing some useful conclusions from them. In the case of the collisions no general theory can be advanced but they both have a navigational interest from the point of view of the International Regulations Jor Preventing Collisions at Sea and a certain amount of useful comment can be made on them.

In an earlier paper (1), which will be referred to as A, the present author has demonstrated the relativistic invariance, for general transformations of coordinates, of the Einstein-Bose and Fermi-Dirac quantizations of linear field equations derived from higher order Lagrangians. The proof consisted of the identification of the commutation relations with the generalized Poisson brackets introduced by Weiss (2) and proving the invariance of the latter.

When n points are distributed at random (equally likely anywhere) over a large area or volume, they may form small clusters or aggregates. In this paper a k-aggregate is defined more precisely as k points which can be covered by a small figure of given size, shape and orientation, but whose position is not otherwise specified. Exact formulae for the expected number of such k-aggregates are given in some cases, while a good approximate formula (accurate in most practical cases) and limits are stated, covering all cases.

Let

be w sequences of differentiable functions defined in the ranges

where aj, bj may be − ∞, + ∞ respectively. For each j and each m ≠ n let

be monotonic and let there be a constant K independent of m, n, j, θj such that

The counterfeit coin problems consist in detecting and locating the anomalous weights of counterfeit coins amongst a set of genuine coins. Suppose that altogether there are n coins, of which at least n − k are known to be genuine. We do not know which coins are genuine; but we do know that all genuine coins have the same (unknown) weight. The weights of the remaining k coins are unspecified and possibly all different.