In this paper the minimum fundamental gap of a kind of sub-elliptic operator is concerned, we deal with the existence and uniqueness of weak solution for that. We verify that the minimization fundamental gap problem can be achieved by some function, and characterize the optimal function by adopting the differential of eigenvalues.

The fact that certain bodies expand on solidifying, as in the case of water, has long been well known, and this property has been recognised in some of the metals, owing to their filling the mould in which they are cast so as to reproduce the finest lines. The fact of their so doing is, however, only known qualitatively—no accurate measurements having, so far as we are aware, been recorded. The property being of great interest to ironfounders, we have undertaken a series of experiments to determine its real value—the materials being put at our disposal by Messrs M'Dowall, Steven, & Co., to whom we tender our best thanks.

1. The morphology and anatomy of Trichomanes aphlebioides Christ, a species endemic to New Guinea, is shortly described.

2. The most outstanding feature of the species is the presence of aphlebioid leaves. These latter are shown to be the modified first fronds of the axillary branches.

3. The anatomy of the alphebioid leaves is described. They show great reduction in relation to the humid conditions of the high forests in which the plant grows.

4. The morphology of aphlebiæ and aphlebioid leaves is briefly discussed.

5. No experimental evidence is available with regard to the function of the aphlebioid leaves. Mention is made, however, of the various suggestions which have been put forward in relation to this question.

In this paper we give a multiplier theorem for one-sided Hardy spaces which generalizes the results given by Strömberg and Torchinsky for two-sided weights. Also we state the version with a Sawyer's weight ω.

FAM. EUPHROSYNIDÆ.—In Euphrosyne foliosa, Aud. and Ed., the separate nerve-cords are comparatively large, and lie quite within the body-wall, the oblique muscles, which generally bound the longitudinal ventral muscles, decussating beneath them.

AMPHINOMIDÆ.—The cords are somewhat small and flattened in Chloeia, and occupy an area bounded internally by a transverse band of fibres, and externally by the circular muscular layer and the hypodermic basement-tissue. The oblique muscles are attached at the outer border of each trunk.

The author commenced by stating that although these researches lead him to believe in the existence of two planets revolving in orbits external to that of Neptune, and although there was a great deal of evidence to show that he had actually determined the elements of the orbits, yet the latter point, being dependent on a coincidence of probabilities only, cannot be considered a certainty until the planets are observed.

The author accepts the theory of cometary orbits which supposes that these bodies, wandering through space, are attracted by the sun into the solar system so as to pursue parabolic orbits, and that some of these, in passing a planet, may have their velocity diminished, in which case they will afterwards describe an ellipse in a definite period.

The chapter on Jacobians and the chapter on Hessians were the only two for the period 1880–1900 that seemed to indicate that a slackening had begun in the interest taken in their subjects. We now find that, so far as Jacobians are concerned, any movement of the kind must have been temporary, the new period (1900–1920) showing the number of relevant writings listed for report to be almost if not quite double that of its predecessor.

This paper deals with determining in a constructive manner those members of a linear space of functions which are of integrable-square. The space considered is the set of solutions to an ordinary differential equation, and the solutions of integrable-square are delineated by way of initial conditions. Numerical procedures for implementing the construction are discussed, and application is made to the deficiency index problem. Results from some specific computations are given.

The problem in question is to determine the equation of the curve traced out by any point of an elliptic or hyperbolic disc which touches two fixed rectangular axes. Mechanically constructed figures of different forms of the curve have been given by Tait, who also showed that the same glissette can be traced either by means of an ellipse or a hyperbola. If p, q are the coordinates of the tracing point referred to the axes of the disc as axes of coordinates, the glissette is clearly the θ eliminant of

and Cayley stated that it would be found to be of order 8 in x, y. The actual elimination was first performed by Muir, who obtained in the first instance an equation of order 10. On dividing by an extraneous quadratic factor a lengthy equation of order 8 was obtained; and subsequently Lord M'Laren verified the accuracy of the terms of highest order in this equation.

We consider the problem $-\text{div}(p(x)\nabla u)=\lambda{u}+\alpha|u|^{r-1}u$ in $\varOmega$, $\partial u/\partial\nu=Q(x)|u|^{q-2}u$ on $\partial\varOmega$, where $\varOmega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $q=2(N-1)/(N-2)$ and $2<r<q$. Under some conditions on $\partial\varOmega$, $p$, $Q$, $\lambda$, $\alpha$ and the mean curvature at some point $x_0$, we prove the existence of solutions of the above problem. We use variational arguments, namely the concentration–compactness principle, min–max principle and the mountain-pass theorem.

Solution branches of a semilinear elliptic problem with Neumann boundary conditions are studied at its corank-2 bifurcation points. It is shown that generally there are exactly four different nontrivial solution branches passing through a corank-2 bifurcation point. The bifurcating solution branches are parametrised via a nonsingular enlarged problem. Branch switching at bifurcation points is incorporated with a continuation method.

The author states that the results of the Trigonometrical Survey of Great Britain are nownearly ready for publication, and that he has deduced from them the most probable measures which they afford of the length of a meridian, and the figure of the earth.

After determining the most probable spheroid from all the astronomical and geodetical operations in Great Britain, it has been found that the plumb-line is sensibly deflected at several of the trigonometrical stations; but in almost every case the physical cause of such irregularity may be with probability inferred.

In this method the liquid of which the latent heat is to be determined is contained in a glass vessel which is suspended by a fine wire from the hook of a balance, and the amount of electrical energy is measured which is required for the evaporation of an observed mass. The electrical energy is supplied by a large current passing through a small resistance in the liquid itself. In order to reduce heat losses as much as possible, the vessel carrying the liquid and the resistance is completely surrounded by a doublewalled shield, which is filled by the saturated vapour of the liquid itself. The estimation of the electrical energy supplied necessitates tne measurement of the current, C (amperes), the resistance R (ohms), of the heating coil, and the time in seconds, T, during which the current flows. It is expressed by C2RT joules, and if M be the mass evaporated in gms., the latent heat expressed in joules per gm. becomes C2RT/M.

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

Exact solutions recently discovered for non-conservative three-wave resonance are here related to the ‘one-lump’ solutions obtained by Backlund transformation in conservative cases.

Professor Allman exhibited a portion of a pelvis of a seal, which had been recently obtained from the Pleistocene deposits in the neighbourhood of Kirkaldy, and sent to him for determination by Mr Martin Eigney of that place.

From a note received from Mr Rigney, it appears that the bone was found in the Tyrie clay field, about two miles west of Kirkaldy, and about a quarter of a mile from the shore of the Firth, and that it lay 18 or 19 feet below the surface of the soil, and about 30 feet above the present level of high-water. It was unaccompanied by any other remains.

While the occurrence of the condition termed Cyclopia cannot be considered as altogether a rare event, the obscurity in which its pathology is still involved makes it desirable that a full description of every case should be put on record. The specimen in my possession was that of a well-formed female embryo which had apparently reached the seventh month. With the exception of the malformation to be specially considered, there was no abnormality either in its external appearances, or in the structure and disposition of any of its viscera.

The paper contains some curious applications of the principle of dissipation to the conduction of heat, the connection of heat and electricity, thermo-electric currents, the electric convection of heat, &c. But in this abstract we confine ourselves to one very simple case of the conduction of heat, as the hypothesis on which it is investigated is fundamentally assumed in all the other applications.

The following facts appear to have been conclusively established by universal experience:—

1. Nearly all, if not all, solid substances become self-luminous at the same temperature.

2. The red rays are the first to become visible, and, on increasing the temperature, colours of less wave-lengths are successively added in the order of their wave-lengths.

3. While colours of shorter wave-lengths are being added, those which were previously visible become more intense.

The elliptic cylinder functions, or solutions of the Mathieu equation

that have the period π or 2π, are of four distinct species. These are denoted respectively by ce2n(x, θ), se2n+1 (x, θ), ce2n+1 (x, θ), se2n+2(x, θ), where n indicates the number of zeros in the open interval o < x < ½π. The object of this paper is to show that a function of any type can be expressed in terms of functions of other types, or of their derivatives. This provides the analogies to such trigonometrical relations as

but the single terms on the right-hand side of this and similar relations are replaced, in the results to be proved, by infinite series. These results are quite distinct from Whittaker's recurrence-relations (Journ. London Math. Soc., vol. iv, 1929, pp. 88–96); in particular, they do not involve the non-periodic second solutions.