§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

§ 1. Introductory. The formula to be proved is

where n is zero or a positive integer, R(m)>0, R(αs-l)>0, s = 1, 2, …, p, p≧q.

§ 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).

§ 1. Introductory. The formula to be proved is

where ρq+1 – αp+1 > 0, αr – ρq+1 > 0, r = 1,2, …, p, and p≧q + 1.

§ 1. Introductory. The formula to be established is

where m is a positive integer,

and the constants are such that the integral converges.

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

In Muir's Theory of Determinants, Vol. III, pp. 232–237, there will be found accounts of papers by H. Nägelsbach, J. Hammond and J. W. L. Glaisher, in which expressions for the Bernoulli numbers are obtained in terms of determinants. In the present paper an expression for Bn will be derived which appears to be new, but which is very like some of those mentioned by Muir.

§ 1. Introductory. The formula to be proved is

where b>0.

1. There are exceptional integrals of the total differential equation

in the case when it is not completely integrable, and so when the invariant

is not identically zero, which do not seem to be mentioned by any standard authorities such as Cartan, Goursat, de la Vallée Poussin, and Schouten and Kulk. These are integrals of (1) which do not reduce I to zero. They arise only when the first partial derivates of P, Q, R are not all continuous. A simple example is z = 0 as an integral of

§ 1. Introduction. A problem of some interest in mathematical statistics is that of determining conditions under which the randomisation distribution of a statistic and its normal theory distribution are asymptotically equivalent, as these two distributions are used in alternative approaches to the same inference problem.