In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integral

where f (x) is an analytic function of x, regular in the circle

where an are constants and qn (ω) = in+1Qn (iω).

In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved that

From (1)

follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

In the theory of ordinary linear differential equations with three regular singularities and in the theory of their special and limiting cases, integral representations of the solutions are known to be very important. It seems that there is no corresponding simple integral representation of the solutions of ordinary linear differential equations with four regular singularities (Heun's equation) or of particular (e.g. Lamé's equation) or limiting (e.g. Mathieu's equation) cases of such equations. It has been suggested (Whittaker 1915 c) that the theorems corresponding in these latter cases to integral representations of the hypergeometric functions involve integral equations of the second kind. Such integral equations have been discovered for Mathieu functions (Whittaker 1912, cf. also Whittaker and Watson 1927 pp. 407–409 and 426) as well as for Lame functions (Whittaker 1915 a and b, cf. also Whittaker and Watson 1927 pp. 564–567) and polynomial or “quasi-algebraic” solutions of Heun's equation (Lambe and Ward 1934). Ince (1921–22) investigated general integral equations connected with periodic solutions of linear differential equations.

1. A well known theorem in classical dynamics tells us that if Q, P, and are written as functions of q, p and t by means of the equations

where W (q, Q, t) is an arbitrary function, and if two functions F (Q, P, t) and f (q, p, t) are related by

then q and p are conjugate with respect to f(q, p, t) if Q and P are conjugate with respect to F (Q, P, t).

The generalised Whittaker vector is Λμ which is prevented from vanishing by rejection of the constancy of ω, previously assumed by all writers. It is shown that (1) the null divergence of Λμ is equivalent to Dirac's equation, (2) the length of Λμ measures the probability of occurrence of the electron (3) components of Λμ are connected with the Dirac wave functions and.possible transformations of x5 are probably related to the Uncertainty Principle.

Through four generally placed lines in space of four dimensions there passes a doubly infinite system of quadric primals, but through five lines there pass in general no quadrics. It therefore follows that there must exist some special relationship between five lines in order that they may be generators of a quadric. This problem has been discussed by Richmond,1 who gives a condition which is in a restricted sense an extension of Pascal's Theorem. The five lines being taken in order certain points may be obtained which lie in a space. In Section I we state Richmond's criterion and show that it is sufficient as well as necessary. Section II is concerned with the twelve spaces which arise if all the different possible orders of the lines are considered. They cut by pairs in six planes whose configuration is developed. In Section III other lines connected with the configuration are introduced. It is shown that by taking crossers of the lines of our original figure in a certain manner five further generators are obtained, and that the same entire configuration of generators arises whether we begin with the five original or the five final lines. Furthermore, though the twelve spaces analogous to Pascal lines obtained from the final five are new, yet the six planes, their intersections by pairs, and the configuration dependent from them, are the same as those constructed from the original five.

Several papers on the subject of spatial distance in General Relativity appeared a few years ago, and a simple extension of this idea to any pair of points in any Riemannian space was given by me in a thesis. A distance invariant was defined, and this was found to depend upon a certain two-point invariant which was first introduced by H. S. Ruse in a study of Laplace's Equation. This invariant, now written ρ and defined in (3), has lately re-appeared, and it may now be of interest to publish the results found earlier. These include a geometrical interpretation of ρ, a simple method of calculation, and an expansion as a power series in the geodesic arc. The dependence of ρ upon the geodesic arc is also considered.

The chain of circle theorems now associated with the name of de Longchamps may be stated as follows.

(i) Any two lines L (A), L (B) meet in a point (AB).

(ii) Any three lines L(A), L (B), L(C) determine three points (AB) (BC) (CA) which lie on a circle S (ABC) with centre (ABC),

(iii) Any four lines L(A), L(B), L(C), L(D) determine four circles S(ABC) S(ABD) S(ACD) S(BCD) which meet in a point P (ABCD) and four centres (ABC) (ABD) (ACD) (BCD) which lie on a circle S(ABCD) with centre (ABCD).

(iv) Any five lines L(A), L(B), L(C), L(D), L(E) determine five points like P (ABCD), which lie on a circle SM (ABCDE) (Miquel's Theorem), five circles like S (ABCD), which meet in a point P (ABCDE), and their five centres like (ABCD), which lie on a circle S (ABCDE) with centre (ABCDE).

(v) Any six lines L (A), … . , L (F) determine six circles like SM (ABCDE) which meet in a point PM (ABCDEF), six circles like S (ABCDE), which meet in a point P (ABCDEF) and six centres like (ABCDE) which lie on a circle S (ABCDEF) with centre (ABCDEF).

The functions f(t) and h(t) that occur in what follows are supposed to be integrable (L) in every finite interval in which they are defined; and the order of summability, which need not be an integer, is not negative.

In this note denotes a divergent series of positive, decreasing terms for which are real numbers (convergence factors) such that is convergent. We put

H. Rademacher has shown that

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

In an earlier paper1 the author investigated the relation existing between the induced matrices of a group of permutation matrices and the table of group characters of the irreducible representations of the corresponding symmetric group. It was found that the traces of a particular set of induced matrices sufficed to give, by a relatively simple transformation, the complete table of characters.It was remarked also that for n > 4 the set of compound matrices of permutation matrices, on the other hand, could at most provide only part of the table; for in fact the number of compounds, n + 1. is then less than P (n), the numbe'r of partitions of n. For this reason the subject was not pursued into further detail.

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.

This note exhibits some old results in what is possibly a new form.

The simple harmonic oscillator, whose kinetic and potential energies are given in terms of a single coordinate x by

where b is a positive constant, has the equation of motion

The solution, appropriate to the initial conditions at t = 0 is conveniently written

A linear algebra of order n, in general non-commutative and non-associative, may be regarded as being determined by the “cubic matrix ”consisting of its n3 constants of multiplication, and conversely. This requires that the n basis elements (units) of the algebra should be specified, and should be given in a definite order. Then the various “transpositions” of the cubic matrix induce corresponding “transpositions” of the algebra, for which a notation is given in §2.

In the paper by H. S. Ruse (this volume 144-152) in equation (1.5) read “Rαβ” instead of “Rαβ”; in equation (1.10) read “€αβ” and “€βα” respectively instead of “€αβ” and “€βα”.

In the paper by C. T. Rajagopal (this volume 162-167) on page 165 line 2 read “dt1” instead of “dt” on page 166 line 2 read “dt1” instead of dt” and in line 2 of the footnote on page 166 read “(16)” instead of “(15)”.

The six quaternary quartic forms

were first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.