The theories of fractional calculus and of the Hankel transform developed in for the spaces F′p, μ of generalised functions are used to study distributional analogues of dual integral equations of Titchmarsh type. These are shown to have infinitely many solutions in F′p, μ under very general conditions on the parameters involved. These results are used to study the corresponding classical problem in weighted Lp spaces. Existence and uniqueness of classical solutions are investigated and examples given of both uniqueness and non-uniqueness for the classical problem.

In the general theory of non-selfadjoint elliptic boundary value problems involving an indefinite weight function, there arises the problem of obtaining a priori estimates for solutions about points of discontinuity of the weight function. Here we deal with this problem for the case where the weight function vanishes on a set of positive measure.

We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.

This paper gives sufficient conditions ensuring that a non-linear control system of the form

is controllable by means of control functions u(t), such that each ui(t) only takes two values, with a finite number of switches. It is assumed that the ‘unperturbed’ system ẋ = A(t)x + B(t)u is controllable in the usual sense, i.e. by measurable and bounded controls.

We first study the Poisson equation Δu = f in Ώω, and , where Ωω = {(r cos θ, r sin θ): 0<r<1, θ ∈(0,ω)} is a sector in ℝ2, ω ∈ (0, 2π), Г0 = {(cos θ, sin θ): θ ∈ (0, ω)} and Г1 = ∂Ωω − Г0,b and λ are in ℝ1. We obtain Schauder-type estimates and Fredholm alternative theory for the problem. We then study the symmetry breaking problem for the Gel'fand equation Δu + λeu = 0 in Ωω and obtain a complete picture about the relationships among three parameters λ, b, and ω in the problem.

An E-unitary inverse semigroup, S, has the property that, if x=S, and e2 = e=S, then (xe)2 = xe implies that x2 = x. As a consequence of this, we can see that S is an extension of its semilattice of idempotents, E, by its maximal group morphic image, G. Thus, following McAlister (1974), we attempt to describe S in terms of E and G. If we extend the semilattice E to a larger semilattice F, we are able to describe S in terms of a semi-direct product of F and G, giving a new interpretation to the approach of Schein (1975).

The radioactivity of potassium was discovered by Campbell and Wood (1907). Since the β-ray activity shown by all the potassium salts investigated was very small, the suspicion naturally arose that this activity was due to a trace of radioactive impurity and not to the potassium itself. All tests for the presence of an emanation gave, however, negative results, and it became necessary to conclude that the observed emission of the β-radiation was indeed a property of the potassium atom. The subsequent investigations of Campbell (1907) and of Hoffman (1924), whose attempts to alter the activity by chemical operations were unsuccessful, supported this conclusion, as did also the work of Biltz and Marcus, who prepared specimens of potassium sulphate from potassium minerals of different geological ages and found that the activity of the preparations per atom of potassium was always the same.

At the beginning of the paper some observations made on the water particles in clouds on the Rigi on the 21st of May last are described. Previous observations with the fog-particle counter had shown that there is a relation between the density of a cloud and the number of water particles observed. On the occasion above referred to the number was very much greater than corresponded with the density. It is pointed out that the number of dust particles in the air which become centres of condensation depends on the rate at which the condensation is taking place, quick condensation causing a large number of particles to become active, slow condensation causing a small number; and that after the condensation has ceased a process of differentiation takes place, the larger particles, robbing the smaller ones of their water, owing to the vapour-pressure at the surface of drops of large curvature being less than at the surface of drops of smaller curvature. The particles in a cloud are by this process reduced in number, those remaining becoming larger and falling quicker, the cloud thus tending to become thinner by the reduction of the number of particles and by the falling of some of them. It is shown that the exceptional readings above referred to, obtained on the Rigi, were owing to the observations then made being taken in a new and rapidly-formed cloud, due to the strong wind blowing at the time causing a quick ascent and rapid cooling and condensation, the result being the formation of a large number of very small water particles. Though the number was very great, the particles were so small they were only just visible with great care with the magnifying power used in the instrument. Previous observations on cloud particles had been made in slowly-formed or in old clouds after the process of differentiation had been in play for some time, and after the drops had been reduced in number and increased in size.

The isochronism of a pendulum is only true to a first approximation. If we make the arc large, the period increases, reaching infinity just before the pendulum makes complete circuits. Assuming the full value of the Circular Error, the actual calculation is given in No. 2 of this series (Proc. Roy. Soc. Edin., vol. xxxviii, 1918, p. 169) for the first few degrees, where it is seen that even with variations that may occur in practice, the change of daily rate may easily amount to several seconds. The defect was recognized before regulation of pressure and temperature were considered feasible, as early, for instance, as in 1673, when Huygens published his Horologium Oscillatorium. In fact Huygens invented the cycloid to prevent it, and gave an ingenious device to make the pendulum describe true cycloids—a device that has been given up for various reasons, among others, that it involves disturbance of the pendulum which it is very desirable to avoid.

Fifty years ago, when this Royal Society was a hundred years old, it fell to Lord Moncrieff, the President, to speak to the occasion. He sketched the history of the Society through its hundred years, and dwelt most on the days when it was born. He told how the Founders made their way one winter's night from homes in Lawnmarket or Canongate to the College Library (as Johnson and Boswell had lately done), with a few flickering oil-lamps to light the way and tallow-candles which dimly lit the room.

The object of the present investigation is to throw further light upon the differences between the disturbances in the chemical processes in the body in fever caused, on the one hand, by the mere rise of body temperature, and on the other by the action of the micro-organismal poisons on the tissues of the body.

The influence of such micro-organismal poisons has been already studied by several investigators-among others, by Noël Paton, Dunlop, and Macadam in the case of diphtheria (Journ. of Phys., vol. xxiv. p. 331,1899; see also chapter “Fever and Infection,” by Kraus, Von Noorden's Metabolism and Practical Medicine, vol. ii. p. 90); but so far few detailed studies of the metabolism in high temperature due to non-infectious processes have been recorded.

This investigation was begun as a result of certain observations made by Dr T. Graham Brown, on reflexes in the guinea-pig (45). In various experiments he removed parts of the cerebral cortex and found that little if any disturbance, either motor or sensory, ensued. It was thereupon suggested that an examination should be made of the cortex histologically and by electrical stimulation in order to locate the position of the various areas.

THE subject of the “negative” viscosity of solutions has of late attracted considerable attention. Since the publication of the paper by one of us with Ranken in the Transactions of the Royal Society of Edinburgh, there have appeared a paper by Jones and Veazey, and two by Getman.

Since the latter of the papers by Getman contains a comprehensive review of the older literature on the subject, it is unnecessary for us to do more than indicate those points which are of fundamental importance. First noticed by Hübner in 1873, “negative” viscosity was observed by Sprung in the course of his extensive researches to occur in solutions of several potassium and ammonium salts, while corresponding sodium salts do not cause it. Later investigations added to the salts which are known to give rise to it; mention may be made of hydrobromic acid, in which it occurs at low temperatures but not at higher temperatures, whilst it does not occur at all in the case of hydrochloric acid.

My title is purposely ambiguous, because it has to represent two things—I intend to treat not only of what a quaternion really is, but also of its self-containedness, or independence.

Professor Cayley has just stated that “while co-ordinates are applicable to the whole science of geometry, and are the natural and appropriate basis and method in the science, Quaternions seem to me a particular and very artificial method for treating such parts of the science of three-dimensional Geometry as are most naturally discussed by means of the rectangular co-ordinates x, y, z.”

In January and February 1879 Mr J. Y. Buchanan communicated to this society two papers, on the distribution of temperature under the ice in Linlithgow Loch. In these papers he gives a most interesting and valuable series of temperature observations made by him, of the water, at different points, and at different depths, in the loch while it was covered with ice. He also gives the temperatures as taken by him under similar circumstances in Loch Lomond. These observations by Mr Buchanan disclose a somewhat unexpected thermal condition of the water.

In the microphone transmitter, as usually employed in circuit with a battery and a Bell telephone, we have essentially two pieces of carbon resting lightly against each other, through which the current passes. That the instrument may work effectively two things are requisite,—first, that the carbons be always in contact, or at least sufficiently near for the current to pass between them; and, secondly, that they be not pressed together so tightly as to prevent any motion of the one relatively to the other. This state of things is sufficiently well described by the term “loose contact,” first used, I believe, by Professor Stokes.

This communication is a sequel to a former work (Proc. Roy. Soc. Edin., vol. liii, 1933, pp. 151–163). An explicit, and apparently new, form is given for the rational reduction of a matrix to diagonal form, applicable to symmetric matrices and also to continuants. An account follows which co-ordinates several existent theories of generalized continued fractions, and concludes with a few determinantal theorems as corollaries.

The object of this paper is to describe a new thermometer screen, and to give the results of some trials made this autumn and winter with a Stevenson screen as generally used, and one modified in the way described in a previous part of this investigation. Also to give comparative readings taken with those screens and with the new one.

A chief difficulty in the study of photo-chemical action lies in its variety, for it gives rise not merely to changes of a chemically simple kind, but also to the innumerable complex changes of plant-life and photography.

The difficulty is further increased by the extremely small proportion of material substance, chemically altered in a limited time by the action of those vibrations of the ether which we call light. To this, perhaps, more than to any other cause, must be attributed our continued inability to ascertain the true nature of photo-chemical changes so constantly observed as the decomposition of silver haloids by light.

Repeated attempts have been, made to connect particular chemical effects, such as oxidation and reduction, with greater or smaller wave-length; but all such attempts have ended in self-contradiction (1, 2).

Now, if there be any single characteristic associating all photochemical actions together, this characteristic must be present in the simplest as well as in the most complex cases, and the consideration of the former seems, therefore, most likely to lead to definite conclusions.

Perhaps the most striking and the most simple case that presents itself is the increased electric conductivity imparted to crystalline selenium on exposure to light.

Monkmann has shown that pure soluble sulphur also acquires increased conductivity when exposed to light, though to a less degree than selenium (3).

Ultra-violet light (4), and also the Röntgen rays, impart electric conductivity to the air.

In these eases the change is a temporary one.