Taylor and Fisk have discussed the internal conversion of magnetic multipole radiation by K electrons and have found internal conversion coefficients which are much larger than those for electric multipoles, and which increase with the order of the multipole. The object of the present note is to determine whether this state of affairs holds also for pair production by internal conversion of magnetic multipole radiation.

A period of a function f(z) is defined to be a number ω (≠ 0) such that

is identically zero; and it can be shown that an integral function may either have no periods or else a single sequence kλ (k = ± 1, ± 2, …).

1. Let ξ, η denote the rectangular Cartesian coordinates of a point in a plane. Let J (ξ, η) denote a harmonic function which is positive in the half-plane η > 0. In this paper, we first show (Theorem I) that every such function J determines a non-negative number d, and a bounded non-diminishing function G(x), such that

The group of order 168 discovered by Klein, which it is now known can be generated by two operations E, of order 7, and ϑ, of order 2, which satisfy the relations E7 = 1, ϑ2 = 1, (Eϑ)3 = 1, (E4ϑ)4 = 1, has a vast literature. But for the most part each author pursues the matter from his own point of view; and it seems it may be useful to present a simplified approach to the theory which takes account of various possible aspects, in particular the geometrical. This is the object of the present note; for most of its contents I have found it necessary to do fresh work, so that the paper is by no means a transcript of what is already available.

We have examined the two-dimensional gliding of a semi-infinite plate on the surface of a stream of finite depth. We have calculated the lift L in terms of the ratio D/H, where H is the depth of the stream at infinity up-stream, and D is the height of the trailing edge of the plate above the surface of the stream.

In particular we have found that the trailing edge cannot be at a height of more than 0·07H approximately, above the up-stream fluid surface.

In conclusion, the writer would like to thank Prof. G. I. Taylor, F.R.S., for suggesting this problem and for his continued interest in it.

Suppose that a, c and A are real, and that

Let n (X) be the number of zeros of

in ; then it is not difficult to prove that

for some K = K (A, a, c, θ). The problem in this paper is to determine the function K explicitly in terms of its variables.

The probability of excitation of helium by electron impact to the more important doubly excited levels is calculated, and comparison made with the experimental results of Priestley and Whiddington who have observed two lines due to double excitation. The calculated relative intensities and angular distributions are found to be in qualitative agreement with experiment, and reasons are given for the lack of quantitative agreement.

Buri's method of solution of the equations of turbulent motion in a boundary layer has been used extensively in a previous paper. Since the results of that investigation have not been compared with experiment, nor has any other application of Buri's method been checked experimentally, it seems advisable to apply this method to a problem for which the experimental results are known.

The successful development in recent years of the topology of general closed sets is largely due to the application of combinatory methods, which have led to an elaborate theory of approximation of closed sets by infinite cycles, to the generalization of duality theorems for closed sets, and to a geometrical theory of dimensions. Corresponding to the combinatory invariants involved the main results of these theories concern for the most part properties of the set as a whole, which cannot possibly express fully its internal and in particular its local structure. Still less is our knowledge of the eventual relations between the local structure of a set and its properties in the large.

This problem, in the particular case of the hydrogen molecular ion, is so well known that a detailed reference to the large number of papers dealing with it is unnecessary. The investigations to which chief reference is made will be found at the end of this paper.

All the light elements up to aluminium and some heavier ones have been examined for disintegration by slow neutrons. Large effects have been found in lithium and boron and a small effect in nitrogen, the reactions being

and probably

The charged particles emitted in the disintegration of lithium and boron afford a convenient and sensitive indicator for slow neutrons.

One of us (M. G.) desires to acknowledge the financial assistance received from the International Student Service, London, and from Magdalene College, Cambridge.

In a paper which is to be published shortly formulae have been obtained for the virtual characters of the system of surfaces cut out on a primal of S4, having ordinary singularities, by adjoint primals of any order; an account of the principal results has been given elsewhere. As a first application of these formulae we propose in the present paper to investigate the canonical systems belonging to various threefolds and thence to construct the canonical models which correspond to them.

In previous communications* the author has discussed the recombination of gaseous ions, subsequently extending the theory to include large ions† and also to apply to the coagulation of colloids and of smokes‡. The predictions of the original theory were in excellent agreement with experiment, but discrepancies appeared when it was generalized. The author is now indebted to Dr F. J. W. Whipple for the discovery of a numerical error in the first paper, the correction of which leads on the whole to a much more satisfactory agreement with experiment.

Verblunsky has recently proved the following theorem:

Theorem A. Suppose that

for every positive σ. Write

and suppose that J (σ, x) is bounded in σ, x for σ > 0 and all x. Then

exists almost everywhere, and

almost everywhere.

The temperature dependence of the photo-electric effect with the image field is discussed, the result obtained agreeing with Fowler's T2 law near the threshold. Fowler's law is discussed, and shown to rest upon a mistake in the normalization of the free wave functions.

The two methods of analysing experimental results due to Fowler and Du Bridge are also discussed, and it is shown that in general the latter should give the more reliable results.

The author wishes to repeat his thanks to Prof. R. H. Fowler and to Mr A. H. Wilson for their constant interest in the work.

If a simple calculation is made of the amount of heat liberated when one body is slid over another, and of the possible rate of escape of the heat by conduction and emission, it shows that the temperature of the surface may reach a high value.

For material stratified in parallel planes in local thermodynamical equilibrium we have Milne's well-known result that

where πF is the constant net integrated flux of radiation, τ is the optical depth and B is the “ergiebigkeit” which is related to the temperature T by the relation

σ being the Stefan-Boltzmann constant. If we define the effective temperature by the relation

we have from (1)

where T1 is the temperature ar τ = 1. In this note we establish a similar result for extended photospheres where the curvature of the outer layers is properly taken into account.