In connection with their proof of the fundamental relation

Whittaker and Watson state that “several proofs of this important proposition have been given, but none are simple.”

§ 1. Consider the system of n first order linear differential equations:

together with the n boundary conditions

where aij, bij are constants and where we assume for simplicity of notation that gii = 0.

There are various methods in existence for the practical solution of a set of simultaneous equations

Some of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

Clebsch was the first to investigate the properties of the plane quartic curve. The object of the present paper is to study the properties of the plane trinodal quartic curve.

In a recent paper Dr G. C. McVittie discussed the solution with axial symmetry of Einstein's new field-equations in his Unified Field Theory of Gravitation and Electricity. Owing to an error in his calculation of the field equations, Dr McVittie did not obtain the general solution, which we discuss in the present paper.

A lemma used in the above paper can be extended to cover the case of a sequence Sn determined by a recurrence relation of the form

r being any positive integer.

For convenience denote everywhere. Let us suppose that an inequality of the form

holds for p = 2, 3, … ., n — 1. The constant K can be determined from the initial values of Sn, when kn and ln have been found.

In the Mathematische Annalen, 11 (1877), 440–444, E. Toeplitz has proved the following theorem: If three quadrics are polar quadrics of three points with respect to a cubic surface, then their (2, 2, 2) invariant vanishes; the invariant being of the second degree in the coefficients of each of the three quadrics.

§1. A series

has been defined by J. M. Whittaker to be absolutely summable (A), if

is convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.

for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

In der Theorie der Invarianten binärer Formen gilt der Satz, dass die projektiven Komitanten eines vollen Systems durch sukzessives Ueberschieben berechnet werden können. Hierauf beruht der Originalbeweis für die Endlichkeit, den. P. Gordan 1868 zuerst gegeben hat. Konstruktiv wird dieser Beweis aber erst dann, wenn man von vorneherein eine obere Grenze für den Grad der Komitanten eines vollen Systems in den Koeffizienten der Grundformen angeben kann. Also eine natürliche Zahl N, die die Sicherheit gibt, dass alle Komitanten eines vollen Systems gefunden sind, wenn man die Ueberschiebungen soweit berechnet hat, dass ihr Grad N nicht übersteigt.

It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.

The alternant with powers of variables for its elements

has often been investigated and its properties are well known. In the present note an “alternant” with factorial elements,

will be discussed. Here a, b, c, …, z, α, β, γ, …, ω are integers such that

and

with the convention that a(0) = 1, O(0) = 1, and that, where negative indices occur (in the reductions of §2), the factorial is to be regarded as zero.

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansion

where j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

The object of the present note is to show that a well-known theorem in the theory of non-linear partial differential equations, which is usually proved analytically, admits of a geometrical proof which exhibits the relations concerned in a more intuitive manner.

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formula

which is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted average

the linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.

§ 1. Copson and Ferrar have proved the following theorem:

Ifis a divergent series of positive terms, and (pn + qn) un−pnun−1 = ynalways, where yn → 0, then un → 0.