§ 1. Introductory. The formula to be proved is

where b>0.

§ 1. Introductory. The integral

where b>0, was given by Hardy (1). It was proved by applying Mellin's inversion formula. An alternative proof, based on the differential equation

satisfied by Kn(x), has been given by the author (2).

In a recent paper J. L. Synge (1952) has shown that the theories of Einstein and Whitehead predict practically the same phenomena in the solar field. He finds that the rotation of perihelion, the deflexion of light rays and the red-shift in the sun's spectrum are the same in both theories. In this paper the motion of the centre of mass of a double star is investigated. It is found that Whitehead's theory predicts a secular acceleration in the direction of the major axis of the orbit towards periastron of the larger mass. The possibility of a binary star having such an acceleration has already been considered by Levi-Civita (1937), and he has given an example in which it may become detectable in less than a century. On the other hand, it has been shown by Eddington and Clark (1938) that there is no secular acceleration according to Einstein's theory.

Kuiper's recent theory of the origin of the solar system is criticised on several grounds. Firstly, it is pointed out that the empirical relation between the ratio of the masses of two consecutive planets (satellites) on the one hand and the ratio of their distances from the sun (primary) on the other hand is not the one discussed by Kuiper. Secondly, it is shown that the densities needed for a successful application of Kuiper's theory are probably not attained in the system considered by him. Finally, some other points are discussed which enter into most theories about the origin of the solar system.

The classical theorems of Vitali and Blaschke are shown to be simple consequences of an inequality of an interpolatory character due to J. M. Whittaker.

Theorems generalising one of Montel relating to functions bounded in a half-plane and tending to zero at a sequence of points are established by similar methods.

The object of this paper is not to produce a chart of co-tidal and co-range lines which is more accurate than an existing one, but to investigate methods of computing such charts on the supposition that there are no observations of tidal streams such as were used to produce the existing chart. Only coastal observations of tidal elevations are supposed to be known, for such conditions would exist in many parts of the world. The methods used are similar to the so-called “relaxation methods”, using finite differences in all variables and attempting to satisfy all the conditions of motion within the sea, proceeding by successive approximations. There are many difficulties, peculiar to the tidal problem, in the application of these methods, due to the very irregular coast-lines and depths, gaps in the coasts, shallow water near the coasts, frictional forces, and the very serious complication due to the fact that the tides are oscillating and thus require two phases to be investigated simultaneously owing to their reactions one upon the other. One very important point in testing the methods is that no use whatever should be made of existing charts in obtaining first approximations of heights to commence the processes, not even where there are wide entrances to the sea. The resulting chart is shown to be very closely the same as the existing chart, thus proving the validity of the method.

The relation between the maximum term and the maximum modulus of an entire function is exhibited by means of general theorems and specific examples. Functions of zero order and of infinite order are mainly considered.

An investigation is made of the motion of a one-dimensional finite gas cloud which is initially at rest and is allowed to expand into a vacuum in both directions. The density of the gas at rest is assumed to rise steadily and continuously from zero at the boundaries to a maximum in the interior of the cloud.

If the subsequent motion is continuous, it is completely specified by analytical solutions in seven different regions of the x-t plane joined together along characteristics. The motion of one of the boundaries is discussed, and conditions found for it to have (i) an initial stationary period or (ii) a final constant velocity of advance into the vacuum. The gas streams in both directions from a dividing point at zero velocity. This point ultimately tends to the mid-point of the initial distribution.

The possible breakdown of the continuity of the motion is discussed, and a condition on the initial density distribution found for shock-free flow to be maintained.

A sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.

If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete process

where Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under which

These new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.

§ 1. Introductory. In § 3 a generalisation of the formula [MacRobert, Phil. Mag., Ser. 7, XXXI, p. 258]

where αp+1 = ½m + ½n, αp+2 = ½m - ½n, R(m ± n) > 0, and x is real and positive, will be established. In the course of the proof Hardy's formula [Mess, of Maths., LVI, (1927), p. 190],

where R(b)>0, will be required. This was originally proved by an application of Mellin's Inversion Formula. An alternative proof is given in § 2, and some related formulae are deduced.

1. Introduction. Given a series we define , by the relations

The series Σan is said to be summable (C, k) to the sum s, if

as n→∞, and strongly summable (C, k), k>0, with index p, to the sum s, or summable [C; k, p] to the sum s, if

The theory of operational solutions of differential equations in applied mathematics suggests a method of developing the theory of Fourier and allied series that is simpler for ordinary applications than the classical development. It may be useful to those whose interests lie in such applications rather than in the deeper analytical processes associated with this subject.

The purpose of this paper is to present a proof of the following theorem of Maclagan-Wedderburn.*

Every finite skew-field† is a field.

The proof depends on group theory and on the properties of Galois fields. As an introduction, §§1–4 are devoted to a systematic and self-contained account of the theory of Galois fields.

The theory of quadratic congruences modulo an integer is dominated by the Quadratic Law of Reciprocity (see § 1), which makes it possible to decide in a very short time whether a quadratic congruence

is solvable or not. The law was first proved by Gauss.* It took him over a year to obtain his first proof, which depends on a tedious lemma in elementary number theory. He subsequently obtained seven further proofs, and today more than fifty proofs are known, most of them based on the ideas of Gauss. The object of the present paper is to present a proof which is a modernised version of Gauss's seventh proof, applying the ideas of that proof to a finite set of objects, the elements of a finite or Galois field.

Suggested by the analogy between the classical one-dimensional random-walk and the approximate (diffusion) theory of Brownian motion, a generalization of the random-walk is proposed to serve as a model for the more accurate description of the phenomenon. Using the methods of the calculus of finite differences, some general results are obtained concerning averages based on a time-varying bivariate discrete probability distribution in which the variates stand in the particular relation of “position” and “velocity.” These are applied to the special cases of Brownian motion from initial thermal equilibrium, and from arbitrary initial kinetic energy. In the latter case the model describes accurately quantized Brownian motion of two energy states, one of zero energy.

We consider a volume of material, divided into two regions 1 and 2. each of density ρ, by a moving surface S. On S a change of phase occurs, at a definite temperature (which we may take to be zero) and with absorption or liberation of a latent heat L per unit mass. If θl, kl, K1 are the temperature, thermal conductivity and diffusivity of phase 1, and θ2, k2, K2 corresponding quantities for phase 2, the surface S is the isothermal

and the boundary condition on this surface is

Subscript letters denote partial differentiation.