The following paper is one member of a series designed to elucidate the circumstances of correct time-keeping, the first two of which were published in 1918.

The problem has in the meantime gained appreciably in interest. It was known that time-determinations from night to night at the same observatory were liable to discrepancies, which were presumed to have an accidental character and would be eliminated by grouping. Now, comparisons by the aid of wireless telegraphy of determinations made at different observatories, and grouped with the intention of eliminating accidental error, exhibit like discrepancies, of a more systematic character, with a range certainly not less than 0s·3. The search for the cause of this fault is engaging the attention of astronomers. It is a larger fault than is found in any other region of their science; and as its source is obscure, additional value attaches to any collateral investigation bearing upon it.

In 1862 I communicated to the Society a notice of the ravages of the Limnoria terebrans on timber employed in engineering structures exposed to the action of the sea. In that communication I stated that African, English, and American oaks, mahogany, teak, beech, ash, elm, and the different varieties of pine, were found sooner or later to become a prey to the Limnoria. The special object of the notice was, however, to show that timber subjected to preservative processes did not long resist the attacks of the Limnoria, and, more especially, that thoroughly creosoted timber is readily perforated by it, and subsequent experience has fully shown that these statements were correct.

Pour faire un usage intelligent destables des fonctions irrationelles en général, et des tables de logarithmes en particulier, il est indispensable de conaître les formules fondamentales de l'interpolation, et de savoir se rendre compte du degré d'approximation que l'on peut obtenir dans les différénts cas. Je traite dans ce mémoire ces deux points qui sont à peine indiqués dans la plupart des ouvrages élémentaires, et qui sont complètement omis dans les introductions les plus développées à des tables d'ailleurs très estimables.

Dana, in his Report on the Crustacea of the North Atlantic Exploring Expedition, described a number of genera founded on larvae of various Crustacea. Of these genera Calyptopis, Furcilia and Cyrtopia were shown by Claus in 1863 to represent three stages in the development of the Euphausiidas. Metschnikoff, in 1869, described a still earlier stage under the name of Metanauplius; and in 1871 the same investigator was enabled to show that the young Euphausiidse are hatched as true Nauplii, having a rounded body and three pairs of swimming appendages.

This instrument consists of a close spiral of insulated copper wire bent into the form of an anchor ring, so as to form an endless solenoid. The spiral is placed in a rectangular groove turned on the edge of a wooden or brass ring of suitable thickness and diameter. Short lengths of wire at both ends of the copper spiral are left straight. These, after being well insulated, are twisted together and led to two terminals, which serve as electrodes. The ring containing the spiral is fixed on a base board with its plane vertical, and at right angles to the magnetic meridian, when the instrument is in use. A short magnet, rigidly attached at right angles to the lower end of a stiff wire, is suspended from a silk fibre, so that its centre is in the circular centre line of the anchor ring.

In continuation of former investigations of the alteration of compressibility of water, which is produced by dissolving various salts in it, I was led to imagine that some instructive results might be furnished by solutions such as those of sugar, whose bulk is nearly the sum of the bulks of their constituents:—for, in them, we might expect little change in compressibility from that of water itself; i.e. in accordance with my hypothetical formula, little change in the term regarded as representing the molecular pressure.

The following preliminary results have recently been obtained for me by Mr Shand, Nichol Foundationer, who employed the Fraser gun and the Amagat gauge procured for my “Challenger” work:—and a new set of piezometers of the same (Ford's) glass as that whose compressibility I had determined to be 0·0000026. These have been carefully gauged, but have not as yet been directly compared with those formerly employed.

The solutions experimented on were prepared, in Dr Crum Brown's Laboratory, by Mr W. W. Taylor, M.A., B.Sc., and contained respectively 5, 10, 15, 20 parts, by weight, of sugar to 100 of water. The temperature varied but slightly from 12°·4 C. during the whole course of the experiments.

It is contended that Quaternions (as a method) are more comprehensive and less artificial than—and, in fact, in every way far superior to—Coordinates. Thus Professor Tait, in the Preface to his Elementary Treatise of Quaternions (1867), reproduced in the second and third editions (1873 and 1890), writes—“It must always be remembered that Cartesian methods are mere particular cases of quaternions where most of the distinctive features have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the quaternion solution becomes identical with the Cartesian one. Nothing, therefore, is ever lost, though much is generally gained, by employing quaternions in place of ordinary methods. In fact, even when quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to choice of particular coordinate axes.”

I.—Nous avons vu, récemment, que les Macrwridœ, dans l'état actuel de nos connaissances, sont représentés par une espèce à l'intérieur du Cercle Polaire Arctique et par deux espèces à l'intèrieur du Cercle Polaire Antarctique.

We extend several known properties of the Dirichlet index to the case of minimal conditions and prove that the index is invariant under positive t bounded perturbations of the associated quadratic form. The index is also shown to be minimal for fourth order operators with certain growth conditions on some of their coefficients.

We study the extreme points of two classes of polynomials of degree at most n:

It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.

For a large class of functions f, we consider the nonlinear biharmonic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. The discussion is based on the degree theory for C2 proper Fredholm maps developed by Fitzpatrick, Pejsachowicz and Rabier.

We study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod 2 cohomology (as a graded vector space, a ring and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct ‘exotic’ conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.

The steady-state diffusion problem is considered in a thin plate perforated periodically by many cylindrical holes of critical sizes. The plate is scaled to a plate of thickness 1. The asymptotic behaviour of the solution to the resulting rescaled equation is studied when the thickness of the original plate, the holes' size and period converge to 0. The phenomenon of dimension reduction occurs, i.e the limiting equation is posed in the cross-section only. The equation contains a linear term which describes the sink effect of the holes. This term depends on the relationship between the thickness, the period and the size of the perforations.

In this paper, we study under what boundary conditions the inequality

$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

We are interested in non-negative non-trivial solutions of

where 1 < p and Ω is a bounded smooth domain in ℝN with 3 ⩽ N ⩽ 9. We show that given a non-negative integer M there is some large p(M,Ω) such that the only non-negative solution u, of Morse index at most M, is u = 0.

The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.