1. Dual integral equations of the form

where f(x) and g(x) are given and Ψ(x) is the unknown, have been increasingly studied in recent years; the first solutions were given for the case g(x) ≡ 0 by Titchmarsh [1] (for 0 < α < 2) and Busbridge [2] (for — 2 < α < 0). An interesting and much simpler method of solving the equations in the same case, g ≡ 0, was given by Gordon [3]. He also showed that the problem of solving the general equations (1) and (2) can be reduced to a problem in which g ≡ 0. He did not pursue this idea as far as finding and simplifying the solution of (1) and (2) but this has been done recently (see [4]) and Noble [5] used a similar idea in treating the case f ≡ 0.

There exist several different approaches to the problem of solving dual integral equations involving Bessel Functions [1, 2, 3, 4, 5, 6,7], and Erdelyi and Sneddon in a recent paper [8] have shown that the introduction of certain operators occurring in the theory of fractional integration enables the relationships between the various methods to be clearly demonstrated. For dual integral equations other than those involving Bessel Functions the operators introduced by Erdélyi and Sneddon are not always the appropriate ones to use and it seems to be of interest to consider this more general type of situation.

1. Introductory. In this paper certain infinite integrals involving products of four Bessel functions of different arguments are evaluated in terms of Appell's function F4 by the methods of the operational calculus. The results obtained are believed to be new.

As usual, the conventional notation will be used to denote the classical Laplace integral relation

In the proofs of the formulae the following results will be required [1, pp. 281, 284], [3, pp. 78, 79].

A fundamental problem in the theory of ordinals is the assignation of principal sequences to limit numbers of the second number class.

It is our main object here to show that a certain class of methods, which are a natural generalisation of those used in the solution of the corresponding problem for the real numbers (the description of which we omit), must fail to solve the problem. The methods are those which rest on the following assumption: the principal sequence assigned to any limit number of the second number class is determined once the first i terms of that sequence are known.

In a previous paper in this journal [1], I gave formulae for determining the coefficients in certain dual trigonometrical series. The derivation of these formulae involved rather sophisticated assumptions and some intricate manipulation of the hypergeometric function and relied heavily on the solution of Schlömilch's integral equation. I have now found a much simpler formal solution by using Mehler's integral representation of the Legendre polynomial and the final formulae for the coefficients can be given in a more attractive form. As the results of my previous work have had several recent applications to physical problems, it seems worth while to give some details of this improved solution.

A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where

for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.

If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form

and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

This paper is paper gives what appears to be a new Rodrigues’ formula for the Associated Legendre Polynomials defined by [5, p. 122]

with the restriction that m is an even positive integer, which helps to evaluate some integrals. Putting m = 2k in (1.1) and replacing Pn(x) by the Gegenbauer Polynomial and using [3, p. 176]

We obtain

Putting a =v–½ in the relation [4, p. 283]

We get