After giving a summary of recent literature on the subject, the author then proceeded to state the nature of the material which he had employed in the present investigation.

The bird selected was the common fowl (Gallus domest.), and he had examined a series of microscopic sections through the limbs from the fourth day of incubation to the day of hatching.

The mammalian embryos examined were mice and rabbits, and the fingers of the human fœtus from an embryo approaching the full period of uterogestation.

The preparation of adamantine carbon or diamond has exercised the genius of philosophers from the very earliest times; but it was not until the middle of the last century (1772) that Lavoisier established the diamond's true nature—notwithstanding the simplicity of the experiments required to demonstrate the fact—and showed it to consist of pure carbon in a crystallised state. Since that time very many attempts have been made to prepare it artificially, but until the recent and now famous experiments of Mr J. B. Hannay there has not been the slightest approach towards the solution of this problem. Great obstacles stood in the way of success, the chief being the complete insolubility of carbon in all known liquids, coupled with its non-volatility and infusibility; while the subject was rendered even more difficult and obscure, by ignorance of the conditions under which the diamond is produced in nature, its peculiar crystalline form, together with extreme rarity, indicating a probable very slow formation, and rare natural existence of the conditions necessary for its formation.

I have given the name of Radiant Spectrum to a phenomenon which I discovered in 1814, and which I described to this Society in the early part of that year.

It will be understood from fig. 1, which represents the brilliant radiation which surrounds a very small image of the sun, when it is formed either by reflection or refraction, or otherwise.

At the last meeting of the Society I stated that I had just procured a remarkable essay by Listing, part of which bears on the subject of knots, and that I had found in it an example of a change of form not producible by the modes of deformation I had employed.

It had for some time struck me as very singular that, thoug I could easily prove that (when nugatory intersections are removed) a knot in which the crossings are alternately over and under is not farther reducible, I could not prove all its possible deformations to be producible by inversions or projections of the kinds specified in my paper; but, as soon as I recognised the existence of amphicheiral forms, I saw that it was probable that they would furnish a key to my difficulty. I immediately set to work to classify the simpler of such forms; and while I was thus engaged I got the Göttinger Studien for 1847, in which is Listing's paper, with the title Vorstudien zur Topologie.

At Dr Murray's request, I have during the past winter continued the investigation of Messrs Irvine and Young on the solubility of carbonate of lime in its different forms in sea water (the results of which they submitted to this Society in May 1888); and the following notes of the work done and the results obtained by me, under Mr Irvine's guidance, in the laboratory of the Marine Station, Granton, may be of interest.

Dear Sir,—During the very delightful excursion in your “Medusa,” in which I had, with two of our friends, the pleasure of accompanying you through some of the western lochs of Scotland, we had not only many opportunities of admiring the special beauties of your country, but also many periods devoted to scientific work and to scientific discussions. You were good enough to demonstrate to us your ingenious methods of determining deep-sea temperatures. You dredged in our presence masses of beautiful shining Schizopods, and also before our eyes you formed the once so famous Bathybius Huxleyi. We conversed on general principles of natural philosophy, and on the different modes of regarding organic life and organic forms. You then invited me to give you a written explanation of my own morphological views—an invitation which I propose to aocept in the following pages.

1. The properties of a colloidal sol of cholesterol have been investigated, more particularly in relation to the effect of pH in the presence of protein (gelatin) of various concentrations.

2. The cataphoretic P.D. between the particles of the sol and the medium has been determined in the presence of gelatin at various concentrations and at various pH values.

3. The effect of the presence of gelatin in preventing the precipitation of a cholesterol sol by sodium chloride at various pH values has been investigated. It has been shown that the amount of gelatin required to protect is approximately that required to cover the particles with a layer of gelatin from one to two atoms thick.

4. The mechanisms of precipitation and protection are discussed in the light of these results.

The problem of the influence exerted by the ovary upon metabolism has been studied by a number of authors, but no definite conclusions have yet been reached. In recent years the rôle of the sex-cycle has been considered, and various studies of the relation between the œstrous cycle and metabolism have been undertaken. Apart from the indirect influence, which is due to the increase in the spontaneous activity (Slonaker, 1925), such as is observed in the mouse or rat during œstrus as an effect of ovarian hormone, an influence of the ovarian function upon basal metabolism appears to exist, but nothing definite is known. Since the secretion of “œstrin” from the ovary appears to be periodical (Wiesner, 1926), the question of whether the supposed influence upon metabolism is also of a periodical nature is of particular interest.

FOR the past two years I have been engaged in the investigation of the iron-water of Scotland, England, and Wales, and altogether have examined over a hundred samples. This publication is a preliminary notice of some new species of iron-bacteria which I have found in the course of this investigation. At present the number of known forms belonging to this class is six. In this paper I wish to outline the main characteristics of five new species, reserving for later papers the detailed accounts of their life-histories.

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

Stability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

In the framework of fixed point theory, many generalizations of the classical results due to Krasnosel'skii are known. One of these extensions consists in relaxing the conditions imposed on the mapping, working with k-set contractions instead of continuous and compact maps. The aim of this work if to study in detail some fixed point results of this type, and obtain a certain generalization by using star convex sets.

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

It is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.

In this note we prove the following

Theorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.

We study homogenization processes for the heat equation in multilayers with interlayer conduction, modelled by Neumann transmission conditions. We establish the homogenized equations for three kinds of dependency of the interlayer conduction magnitude upon the interlayer distance.

An acoustic line source is moving parallel to the vortex sheet dividing two fluids in relative motion and the lifetime of the source is finite. The resulting phenomena are compared with those created by a fixed pulsed source. It is found that the waves produced by the source in motion are not so singular as, though qualitatively similar to, those due to a source which remains in one position but Helmholtz instability still has a dominant part to play. This statement is true whether the source be moving subsonically or supersonically so long as it has non-zero velocity.

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.