In this paper the authors communicate the first results of an attempt to discover the relation which must exist between the chemical constitution and the physiological action of a substance. As the chemical constitution of the majority of physiologically active substances is unknown, they investigate the subject by examining the physiological action of a substance before and after the performance upon it of a definite chemical operation, introducing a known change into its constitution. The question is thus reduced to a problem in what may be called a chemico-physical calculus of finite variations.

1. Alloys were prepared containing 14, 16, 18, 30, 38, and 48 per cent, of tin, and the remainder copper and manganese in the ratio 7 : 3.

2. Tests were carried out at different temperatures on the materials as cast, normalised, annealed, and quenched.

3. At 15° C in the normalised condition the six alloys gave I respectively equal to 55, 77, 82, 0·4, 96, and 1, for H = 100.

4. When cooled to −190° C the three alloys poor in tin were much more magnetic.

5. Annealing produced no decided improvement.

6. The susceptibility diminishes with increase of temperature, and the critical temperatures range from 225° to 275−190° C.

8. Reannealing tends to restore the initial susceptibility of the quenched material, but has little influence on the coercive force.

9. The magnetic properties are probably due to the formation of solid solutions of certain definite concentrations.

In man, in the rabbit, and some other animals the trachea is built up of a series of cartilaginous rings incomplete behind; the rings being completed in this position by the trachealis muscle.

The mucous membrane forms a smooth cylindrical lining for the whole tube, and is covered by a ciliated epithelium.

In the cat and dog the cartilage rings completely encircle the trachea, and overlap posteriorly, and the trachealis muscle, which is well developed, is placed outside the cartilage, and has a powerful action in varying the diameter of the tube.

When this muscle contracts the overlapping ends of the cartilage ride one upon another, and the projecting ends form a vertical ridge down the inside of the trachea, which can be readily seen on slitting the organ open. This ridge is separated by a deep groove from the other end of the plate.

To the foregoing Professor Cayley has kindly made the following additions:—

The investigation may be carried further: writing for shortness u3, u4 &c., in place of ψ(3), ψ(4), &c., the equations are

and hence assuming

we have

It is shown that exactly six non-isomorphic distributively generated near rings can be defined on the infinite dihedral group. One has the null multiplication, two have trivial multiplications in which the group of left annihilators has index two, and three are distributive and have their product sets equal to a subgroup of order 2.

The strong* topology s*(X) of a Banach space X is defined as the locally convex topology generated by the seminorms x ↦ ‖Sx‖ for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterized by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y. The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C*-algebras and, more generally, all JB*-triples, exhibit this behaviour.

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation containing a small positive factor ε in front of the second order part of the differential operator. Pointwise a priori estimates for the solution are established by means of an energy integral method. With the help of these estimates it is shown that the solution admits an asymptotic expansion into powers of ε which is uniformly valid in compact subsets of ℝ2. All results are obtained under the assumption that the characteristics of the first order part of the differential operator satisfy a so-called “timelike” condition. A discussion of the concepts “timelike” and “spacelike” is given.

Uniqueness of positive solutions of the following equation:

has been obtained, where B ⊂ ℝn is the ball of unit radius and λ > 0. Moreover, if n = 2, then the solutions are nondegenerate.

Models which treat molecules as graphs provide the first approximation to practically all chemical and physical theories of polymers. This paper, worded so as to aim at intelligibility by both chemists and mathematicians, deals with statistical effects of correlations on distributions of molecules in their graph-like states. The correlations are formally equivalent to fertility correlations in family trees, and the theory covers the range of correlation to the point where a man's fertility expectations depend on how many brothers he has. Chemically, this describes the ‘second shell substitution effect’ which includes certain steric hindrance situations; mathematically, the problem is solved by restoring the Markovian nature of a conventional Galton-Watson process. Long-range correlations are an inevitable and troublesome aspect of the structure of gels and this work represents a step towards improving models for real gels.

Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.

In this paper we study the equilibrium equations for axisymmetric deformations of isotropic circular plates in tension. We give results on the global multiplicity of solutions and study the stability of the trivial homogeneous solution for large displacements.

The problem discussed is that of determining the sequence {an} such that

where (λn) is the sequence of positive zeros of the function λJν(λ) + HJν(λ), arranged in order of increasing magnitude, þ, ν and H are real constants (−I <þ < I, ν > −½) and f1(ρ), f2(ρ) are prescribed. By expressing the sequence {cn} in terms of a sequence of integrals involving a function g(t) the problem is reduced to the solution of a non-singular Fredholm integral of the second kind for g(t).

We consider the second order, linear differential equation

in the case where, roughly, q ∊ L1 [0, ∞). We devise a representation for the spectral function, τ(t), associated with (*) which is valid for t sufficiently large. Our results are the best possible.

In this paper we study the asymptotic behaviour of reaction–diffusion systems with a small parameter by using the n-dimensional Feynman–Kac formula and large deviation theory. The generalised solutions are introduced in Section 2. We obtain the travelling wave joining an unstable steady state and an asymptotically stable steady state of a diffusionless dynamical system in a reaction–diffusion system with nonlinear ergodic interactions, and a special case with nonlinear reducible interactions.

The paper discusses the nature of the singularities at any finite point t = t0 of real solutions x(t), y(t) of the system of differential equations x′ = P(x, y), y′ = Q(x, y) in which P, Q are polynomials in both x and y. Its main interest is in cases when the leading terms of P, Q are of the same degree n. Conditions are given under which the only possible singularities are algebraic poles and pseudo-poles. Conditions are also given under which the only possible singularities are of wildly oscillatory type.

We consider the nonlinear Sturm–Liouville problem with two parameters on the general level set

We establish asymptotic formulae of the n-th variational eigenvalue λ = λn(μ, α) as α→∞ and α↓(nπ)2.

The question of “correctness” of cardinal interpolation with shifted three-directional box splines is solved for arbitrary orders of the directional vectors. It is shown that the corresponding symbol can be viewed as a collection of curves with certain properties (convexity, increasing argument, etc.) which are investigated in detail. The method of proof involves an induction argument which is based on properties of the exponential Euler splines (studied in [6]).

This paper, although self-contained, is a continuation of the work done in [8], where the motion of a viscous, incompressible fluid is considered in conjunction with the rotation of a rigid body which is immersed in the fluid. The resulting mathematical model is a Navier-Stokes problem with dynamic boundary conditions. In [8] a unique L2,3 solution is constructed under certain regularity assumptions on the initial states. In this paper we consider the Navier-Stokes problem with dynamic boundary conditions in the Lebesgue spaces Lr,3 (3<r<∞) and prove the existence of a unique solution, local in time, without imposing any regularity conditions on the initial states.

We answer some questions about the semicontinuity in L∞ and in Lp of functionals defined on W1,n (Ωℝn) as integrals of non-convex functions of the gradient matrix, and we give a conjecture for the general case.

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.