The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.

The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.

A continuous arc joining two points (x1, y1), (x2, y2) is the aggregate of points given by

where f and φ are continuous functions, and where

A series is said to be summable by Riemann's method of order k to S, or summable (R, k) to S, if

where k is a positive integer, and where we suppose that

is convergent for values of h in a neighbourhood of the origin.

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.

The planes of a given S5 can be represented by the points of a locus V9 in space of nineteen dimensions. This locus is a double locus on a certain other manifold V14, and the tangent spaces of the V9 generate a W18. Segre has a memoir on the subject of these loci, in which he arrives at his results by a series of short steps, the argument being mainly geometrical.

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

A function of sets γ(E) absolutely additive is termed by Radonée Poussin continuous if γ(E) vanishes whenever E consists of a single point only. It is known that if a function of sets is not continuous there exists a countable set ε1 and a continuous function of sets such that

The present paper is a companion to one which appeared in these Proceedings some months ago. Since that time the characteristics of the Hyper-Octohedral group of order 2n.n! have been calculated by Dr Alfred Young, who has kindly allowed me to make use of his results.

Mr Ursell has given a proof in these Proceedings (vol. 24, p. 445, (1928)), that there exist non-combining groups of wave functions of an assembly of similar systems. His proof was not sufficient; but it can be amended so as to be rigorous. It will be necessary to suppose, however, that the wave function ψt is analytic for all real values of the time, for each set of space coordinates: q1, q2, q3, q4….

1. We consider a small cube of the solid, of edge l, in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (−u, −v, −w). Then the particles may be considered to have displacements (u, v, w) from a standard position; and the cube has strain components of the usual forms

The equation of state of an ionised gas has been discussed from time to time by various writers*. It is of importance chiefly in astrophysical applications. The simplest form of the problem, and the only form considered in this note, is to find the pressure in a gas of given temperature in which the numbers of atoms per unit volume at the different possible stages of ionisation are supposed known. In describing the ionisation an electron is counted as “free” when its total energy is positive. If this problem can be solved we should have a good estimate of the relative importance of the various physical factors at work, without tackling the more difficult, though more natural, problem of finding the ionisation for given temperature and pressure.

In previous communications to the Society the author has discussed the eddy current losses in the sheaths of cables supplying a polyphase alternating current system; and has given equations from which the loss could be computed in any case. The present communication deals with the particular case of the cables leading from the supply-transformers to the electrodes of a mercury vapour rectifier, and derives interest from the fact that the loss in the sheaths is found to be substantially independent of their nature.

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.