So far as mathematical writers have as yet noted, a set of equations of the type

had not made its appearance in mathematical work prior to the year 1846: and it is almost absolutely certain that before that year the determinant of such a set had never been considered. It is not at all unlikely, however, that the expression

which is the case of the determinant for n equal to 3 had more than once turned up in other connections, and that its divisibility by a + b + c had been noted: but of this, too, there is no record.

About 1876 the author had occasion to devise a method of indicating the rate of flow of a liquid through a pipe system, without interfering with the flow. For this purpose he used a ball, heavier than the liquid, in a slightly tapered glass tube, through which the liquid flowed from the narrower to the wider end. In the first instance the tube was set vertical, with the wider part above. The ball fitted more or less closely at the lower end: it was carried up by the current till the clearance round it became such as to suit the particular rate of flow. Its height was read off on a scale fixed alongside of the tube, and served to indicate the rate of the flow, once the relation of scale-reading to flow had been experimentally determined.

These notes deal mainly—

(1) With the structure of the genital organs of Stylodrilus Gabreteœ (Vejd.), and with its possible identity with Bathynomus Lemani (Grube).

(2) A new species of oligochæt (Stylaria Lomondi).

I owe my opportunity of working on these forms to Sir John Murray, whom I must again thank for his great kindness.

Le deuxiéme Poisson abyssal de la Scotia dont je désire entretenir la Société Royale d'Edimbourg appartient au genre Neobythites, qui fut découvert, en réalité, par le Challenger, en 1875, près du Japon, par 1875 fathoms, bien que le nom n'ait été crée qu'en 1886, sur des matériaux de l'Albatross, comme nous le verrons plus loin.

Et ce deuxième Poisson est encore une espèce nouvelle, à laquelle je donnerai le nom de Neobythites Brucei, en l'honneur de M. W. S. Bruce, Leader de l'Expédition Antarctique Rationale Ecossaise, comme un témoignage de reconnaissance pour les services rendus à la Science au cours de soil importante Exploration.

In communications to this Society in 1893 and in 1912, and in other papers, I have shown that some of the nuclei in the atmosphere have a power of condensing water vapour even when the air is unsaturated, owing to the material of which they are composed having an affinity for water. All kinds of nuclei have an affinity for water vapour, and by their surface action condense some water at all degrees of humidity, so causing increase in their size and hazing effect; but the particles having an affinity for water condense very much more and cause fogs in. unsaturated air. In the present communication I intend to deal with the more common form of nuclei—that is, those which have little or no chemical affinity for water—and with those nuclei which require supersaturation to cause condensation; the reason for this investigation being some statements which have appeared in scientific journals in this country and abroad regarding the nuclei counted by the condensation method. It has been definitely stated that the greater number of the nuclei counted are not dust particles but only ions, or aggregations of them. It therefore seemed desirable that further investigation should be made into the subject to see if there was anything to support this ionic theory of the condensation nuclei in the air, and at the same time to inquire into their origin.

Every science has two aspects or two stages in its development. In the first, the inductive stage, observations are made and compiled, and axioms or laws are laid down. In the second or deductive stage the laws are applied by syllogistic reasoning, mathematical or otherwise, to elicit conclusions which either disclose new facts or show the inevitable connection between facts already known, and, in either case, complete the claim of the study to the rank of a science.

1. We have determined the relative viscosity of aqueous solutions of KCl, KBr, KI, HCl, HBr at 0°, 15°, and 25° C.; and at concentrations of 1 mol., 2 mol., and 3 mol. per litre. Also the equivalent conductivity of the same solutions at 0° C.

2. The change of viscosity with change of temperature diminishes from Cl-Br-I.

3. The effect of concentration on the viscosity depends on the temperature: it may affect the viscosity in opposite directions at different temperatures.

4. There are considerable differences in viscosity of chloride, bromide, and iodide, and especially in the effect of changes in concentration and temperature.

It is narrated that a physician, having given cogent reasons to show that there could be no recovery, and somewhat disconcerted by the return of sound health, yet maintained the cogency of his pathological arguments. The present may be a new edition of the same story; however, we shall advance the arguments, and leave it to futurity to tell whether there be any patient at all.

This form, which in many respects is a most interesting intermediate type, was dredged at Station 212, south of the Philippine Islands, on the 30th January 1875, in lat. 6° 55′ N., long. 122° 51′ E., at a depth of 10–20 fathoms, on a sandy bottom. The total length is about 52 mm., with a variable diameter of about 2 mm. at the blackish anterior region, and 4 or 5 mm. at the posterior. The tentacular or branchial region has a length of 6 or 7 mm.

It is necessary for a clear understanding of the chemical changes which occur on treating morphine in various ways, and for comparing its physiological action with that of the bodies so formed, that we should summarise at some length the exact position of our present knowledge regarding its chemistry and pharmacology.

Morphine has the composition represented by the formula C17H19NO3. It is generally supposed to crystallise with one molecule of water, but there is reason to believe that the composition of the hydrated base is really 8 C17H19NO3, 9 H2O, the water being present in the proportion of 1⅛ molecule. Morphine is a monacid base, forming neutral and readily soluble salts with all the stronger acids. Like all alkaloids, it is built up on the ammonia (NH3) type, and is a tertiary amine, all the hydrogen atoms having been replaced by more or less complex radicals.

A number of investigators have sought to extend existing knowledge regarding the mode of origin and structural characteristics of coal as revealed by X-rays, taking advantage of the ease with which the true coal may thereby be distinguished from the associated inorganic matter commonly termed“ ash.”

A few months ago I received from Mr G. H. Knott three specimens of Sigillaria from the Halifax Hard Bed of Yorkshire having the structure of their steles and outer cortex well preserved.

The attempt to find reciprocal wave functions made in Part II led to some difficulties. We believe that these were due to an incorrect definition of the relativistic scalar product and Fourier coefficient. We shall show in this paper that a modification of these definitions leads to simpler equations, which allow us to determine the reciprocal wave functions and to find a transcendental equation for μ = aε/k, as was envisaged in Part I.

The family Molgulidœ was established by Lacaze-Duthiers in 1877, and distinguished from the Cynthiadœ, with which it had formerly been united by Heller.

Lacaze-Duthiers divides the species known to him into four genera,—Anurella, Molgula, Ctenicella, and Eugyra,—of which Anurella forms a sub-family or section the Molgulidœ anurœ, the other three constituting the Molgulidœ urodelœ. The character by which Anurella is distinguished from the other genera is the tail-less amoeboid larva; but this most interesting developmental difference does not seem to be accompanied by any structural features in the adult animal, and consequently it becomes impossible to distinguish the genus unless one examines the animal alive and at the breeding season.

WHILE studying the fossil Osmundaceœ several interesting points illustrating the origin of the leaf-trace were brought under our notice, which are exhibited with such exceptional clearness in Thamnopteris Schlechtendalii, Eichwald, sp., that they deserve especial recognition. The full discussion of the subject, however, is reserved until the publication of the next part of our memoir on the fossil Osmundaceœ.

The departure of the leaf-trace in Thamnopteris is typically protostelic. That is to say, the xylem of the leaf-trace first of all appears as a protuberance on the surface of the xylem of the stem (fig. 1).

It is well known, and fundamentally important in the justification of the current mode of deriving Student's t-distribution, that the mean m of n values xi all obeying the same normal law of probability, and the associated estimate of variance, namely

are statistically independent.

The authors prove results on uniqueness and global existence of initial and boundary value problems for the nonlinear pseudoparabolic equation

with nonhomogeneous boundary conditions. A salient feature of the paper is that F and its partial derivatives are allowed to be unbounded. In the special case b(x, t)= α2 (a positive constant), it is proved that the corresponding solution uα, under appropriate conditions on the data (which are satisfied, for example, by the Benjamin–Bona–Mahony equation), uα→ u0 the solution corresponding to β = 0, on sufficiently small time interval. A result on the asymptotic behaviour of the solution is given for t → ∞.

In this paper we study the direct and inverse scattering problem on the phase space for a classical particle moving under the influence of a conservative force. We provide a formula for the scattering operator in the one-dimensional case and we settle the properties of the potential that can be deduced from it. We also study the question of recovering the shape of the barriers which can be seen from −∞ and ∞. An example is given showing that these barriers are not uniquely determined by the scattering operator.

Let denote the semigroup, under composition, of all entire functions of a complex variable and for , let Γ(f) denote the Schützenberger group of the ℋ-class of which contains f. The main results of this paper completely describe Γ(P) where P is a polynomial function. Γ(P) is trivial if P is a constant function. If Deg (P) (the degree of P) is one, then Γ(P) is isomorphic to the group of all matrices of the form where a and b are complex numbers with a ≠ 0. If Deg (P) > 1, then Γ(P) is isomorphic to ℂM, the multiplicative group of nonzero complex numbers if and only if P(z) = α(z +β)n + γ where α ≠ 0. If P is not of this form and Deg (P) > 1, then Γ (P)is a finite cyclic group and its order can be determined by defining (where αn−1 and α n are the coefficients in P of zn−1 and zn respectively) and then inspecting the coefficients of Q.

In the general theory of ordinary linear quasi-differential equations, the set of Shin–Zettl matrices plays an important role. This paper displays certain properties of these matrices and their behaviour under a special form of transformation. Essentially, the problems can be considered within the framework of linear algebra.