It was shown by Sheffer that all functions of the two-valued propositional calculus can be denned in terms of a single primitive. This result has been extended by Post (3) and Webb (5) to the functionally complete m-valued propositional calculus. We shall consider here similar extensions for the extended propositional calculus and for the (functionally incomplete) systems of Łukasiewicz (1).

If A is a subset of a given metric space , and F is a function which correlates with each point x of A a family of non-vacuous subsets of , having the property that there exist members of the family of arbitrarily small diameter, shrinking down upon x, then F is said to be a blanket with domain A. It is well known that if a blanket is endowed with suitable covering properties, then it may be used for the purpose of differentiating one set function with respect to another. It is our intention to consider, in § 3, a type of blanket whose covering families are permitted to overlap in a controlled fashion, and to prove that these may be used for the differentiation of certain set functions with respect to a given one. The paper concludes with the construction, in § 4, of an example which shows that the results achieved are essentially the best possible under the hypotheses.

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

1. Let f(x, y) = ax3 + bx2y + cxy2 + dy3 be a binary cubic form with real coefficients. According as the discriminant D = b2c2+ 18abcd − 4ac3 − 4db3 − 27a2d2 satisfies D ≥ 0 or D < 0, the form f(x, y) can be factorized into either three real, or one real and two conjugate complex, linear factors. A form f1(x, y) is said to be equivalent to f2(x, y) if there is an integral unimodular substitution on the variables x, y transforming f1(x, y) into f2(x, y). Thus f2(x, y) is equivalent to f1(x, y) and, in fact, the set of all forms with given discriminant can be divided into classes of equivalent forms. Davenport (3) has specified a member of each class which satisfies

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled

‘Properties of p(n) and τ(n) defined by the relations

and was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties of p(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

In this paper the principles for the choice of a pair of standard solutions of a homogeneous linear differential equation of the second order, described in an earlier paper (2), are applied to Weber's equation

1. Published tables facilitating the numerical evaluation of the definite integral

from given numerical values f(xi) of the integrand in the form of a weighted sum

essentially deal with the following situations:

(a) The arguments xi are equidistant, i.e. xi = x0+iw; i=integer and the weight coefficients ci are tabulated for all arguments xi occurring in the summation. Among these tables the most important are:

(a1) panels in which the end-points of the integration a and b are two of the xi, and in particular

(a11) panels confined to the range of the integration in which

(a12) panels extending beyond the range of integration which usually have the same number of arguments, k, below a and beyond b, so that

(a2) panels in which a and b are at distance ½w from one of the xi, and in particular

(a21) panels confined to the range of integration in which

(a22) panels extending beyond the range of integration which usually have the same number of (k) points below a and beyond b, so that

(b) The n arguments xi are not equidistant but are specially chosen so that for a given n the sum S approximates to I with highest precision. Gauss's formula (see, for example, Milne-Thomson (4)) and various formulae of approximate product integration (see, for example, Beard (l)) are examples of these. In Chebyshev's integration formula (see, for example, Runge and König (5)) the xi required for equal weights ctare used, resulting in an important formula when the ordinates f(xi) are affeected by observational errors of equal variance.

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.

A method is given for finding, at the end of a sequential sampling procedure, a nearly unbiased estimate of the population mean.

In a contribution (3) to these Proceedings the author considered stochastic problems in physics where one had to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and where the distribution varied with another parameter t which might be continuous or discrete (if t represents time or thickness, it is of course continuous). The author introduced the concept of product densities and derived some general results relating to the functions representing these densities. Recently, Janossy(2), using a certain mathematical model for a nuclear cascade, introduced certain functions which bear a close relation to the product-density functions. The object of this note is to establish a complete correspondence between these two sets of functions and apply them to the specific problem of the development of a nucleon cascade. The diffusion equations involving product densities can be derived from the diffusion equations involving Janossy's functions.

The conditions under which bound states associated with atoms in the surface of a metal may exist are investigated, using the tight-binding approximation. These states arise as a result of modifications in the parameters of certain atoms. The modifications required to produce (a) bound states associated with all the atoms in the surface (surface states) and (b) bound states associated with particular small groups of atoms are found for the simple cubic lattice. It is also shown that most of the simpler crystal structures do not exhibit surface states without such modifications; in the graphite and diamond lattices, however, surface states exist solely by virtue of the lattice geometry.

A detailed account is given of the classification of the electromagnetic fields arising from a vibrating charge-current distribution. The work owes much to that of Berestetzky and Fierz, but differs from their treatments in that the exact expansions are taken as a starting point. Hence the general forms of the potentials and fields and their approximate forms are obtained directly. Formulae appropriate to large distances are given in §§2·1 and 3·1. The relation to the approach of Condon and Shortley (4) is discussed.

Problems involving the addition of heat to a flowing gas have received considerable attention in recent years. The practical interest lies in the application to ram jets and similar engines, and the presence of entropy gradients makes the problem of theoretical interest.

The object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.

The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.

Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.

In the course of some calculations on the lifetimes of excited states of mercury and potassium(9) and photo-ionization of excited potassium, wave functions were derived for various excited states of these atoms. Tables of these wave functions are presented in this paper.

J. S. R. Chisholm, Calculation of S-matrix elements

In equation (1b), and should be replaced by and . This results in similar changes in the expression (29) for , and in the line following equation (29) the factor -2 should be omited from the expression