Many papers have been written on transformations of sequences which can be written as

where is a function of a finite or infinite number of variables for fixed m. If the number of variables is finite it becomes large with m.

Almost all these papers are on linear transformations of the sr, e.g. Cesàro and Abel means, but there are obvious transformations of non-linear form which are regular, and it is a theorem on these which is considered here.

In the first paper of this series (L.Q.I)1 we have shown that the logarithmetic LQ of a finite quasigroup Q is a quasigroup with respect to addition and that it is a subdirect union of the logarithmetics of the elements of Q.

In this second part we shall discuss further the structure of LQ in its additive aspect, and obtain results concerning the order N of LQ. For plain quasigroups (§3) the structure of LQ(-\-) is studied in more detail and it is shown that N is a power of n, the order of Q.

The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.

We consider a medium in which the equation satisfied by a disturbance ø(x,t) is capable of solutions of the form sin(vx—wt), with v and ω real but not necessarily of the same sign. If the phase velocity U = ω/v is not of constant magnitude the medium is said to be dispersive, and the group velocity V may conveniently be defined as dω/v, an expression easily re-written in other familiar forms. From this definition the two physical interpretations of V may easily be seen. In one interpretation we consider a superposition of two harmonic waves with slightly different v (and ω); the velocity of advance of the group form, being the velocity of advance of a point at which the two waves have a constant phase difference, is dω/v.

In his book on Hydrodynamics, Lamb obtained a solution for the potential flow of an incompressible fluid through a circular hole in a plane wall. More recently Sneddon (Fourier Transforms, New York, 1951) obtained Lamb's solution by an elegant application of Hankel transforms.

Since the streamlines in this solution are symmetric about the wall, it is not of particular physical interest. In this note, Sneddon's method is used to give a solution in which the fluid is infinite in extent on one side of the aperture but issues as a jet of finite diameter on the other side.

If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,

If we now regard λ as a parameter and z as the independent variable then (1.1) can be written

Supposing, therefore, that a function is of the form then

The question of the reflection of a wave by a cylindrical mirror is of interest in a number of fields. It is a problem in which it is difficult to obtain an expression for the reflected or scattered field without recourse to physical assumptions which are sometimes somewhat dubious. An attempt was made by Sommerfeld to solve the problem of a plane wave incident upon such a perfectly conducting mirror by means of what he termed the “Non-Final Determination of Coefficients”. Unfortunately, a close examination of the problem renders it doubtful whether the method can be legitimately employed. It is possible, however, to solve the problem by expressing the scattered field in terms of the currents produced in the mirror, and finding the current generated in the mirror by an arbitrary incident field. The problem which we shall consider is the following two- dimensional one.

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the series

When |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).

Suppose that un ≥ un+1 > 0, for n = 1, 2, … and that e1, e2, … are factors which make convergent.

Let where all the a's are positive, diverges, and Define En by the equation

For the case where an = 1 it has been shewn by Fuchs1 and Karamata1 that, under various conditions, En = 0(1) The object of this note is to extend some of their results.