Extensive and intensive research on the Chermes group has been carried out on the Continent during the last ten years, but there has been no close study of this group in Britain. This paper is a record of introductory research in this country on this complex but interesting and important group.

Mr President, Ladies, and Gentlemen,—I desire first of all to express my deep gratification at the great honour you have shown me by inviting me to give an address to this renowned Society. Scotland and Norway are closely united by bonds of friendship and also of kinship. There are many Norwegians with Scotch blood in their veins, just as there must be a large number of Scotch people who are descended from the Vikings who came here, long ago, and found the Scotch lassies so charming that they never returned to their own country!

In spite of the great amount of attention which has been bestowed upon this group by many workers, it must still be said that our knowledge of its systematic position is very meagre and uncertain, and that there are points in its anatomy which require elucidation. Through the kindness of Professor M‘Intosh I have been enabled to examine some specimens of P. australis, and also a Phoronis, which appears to be a new species. I reserve a detailed account of my results for later publication, and here only refer very briefly to leading points. I may mention that Professor M‘Intosh has also allowed me the inspection of his serial sections of P. buskii. M‘Intosh, and after a careful comparison I have not the slightest doubt that this is a distinct species from P. australis—it has been usual to regard the distinction between these two forms as not of specific value.

Divisions of the Body.—It is usual, in describing the structure of Phoronis, to refer to two different parts of the body which lie before and behind the septum respectively. I propose, for reasons shown later, to emphasise the division of the body into—(1) The epistome, lying dorsally to the mouth, and having very definite, though somewhat involved, relationship to the next part; (2) the tentacular region, which I prefer to call the collar, consisting of a ring round the mouth region, the oral part being produced into two arms or processes which bear tentacles and are coiled, and the aboral end being limited superficially by the nerve ring and fundamentally by the so-called septum; (3) the trunk, including all the region behind the septum.

In a communication to the Mineralogical Society, published in September 1910 (Min. Mag., vol. xv. p. 390), I suggested that the relation between the component structures of twin crystals was similar to that existing in combinations of crystals of different substances with definite relative orientation (Barker, Min. Mag., vol. xiv., 1907, p. 235), and was determined by equality of molecular distances in the two structures in the plane of contact or composition.

It has been my purpose in the present paper to determine what are the possible, geometrical relations between crystal structures in which such equality exists in all or some of the molecular rows in the plane of contact.

The authors refer to Regnault's experiments, which have shown that aqueous vapour in the atmosphere has the same vapour density at ordinary temperatures as aqueous vapour above 100° C.; and they bring forward fresh experiments upon alcohol and ether to show that when mixed with hydrogen these vapors preserve their normal density at 20° or 30° C. below the boiling points of the liquids, and infer generally that vapours, when partially saturating a permanent gas, retain their normal densities at low temperatures.

We consider minimization problems involving the Dirichlet integral under an arbitrary number of volume constraints on the level sets and a generalized boundary condition. More precisely, given a bounded open domain Ω ⊂ Rn with smooth boundary, we study the problem of minimizing ∫Ω |∇u|2 among all those functions u ∈ H1 that simultaneously satisfy n-dimensional measure constraints on the level sets of the kind |{u = li}| = αi, i = 1,…, k, and a generalized boundary condition u ∈ K. Here, K is a closed convex subset of H1 such that ; the invariance of K under provides that the condition u ∈ K actually depends only on the trace of u along ∂Ω.

By a penalization approach, we prove the existence of minimizers and their Hölder continuity, generalizing previous results that are not applicable when a boundary condition is prescribed.

Finally, in the case of just two volume constraints, we investigate the Γ-convergence of the above (rescaled) functionals when the total measure of the two prescribed level sets tends to saturate the whole domain Ω. It turns out that the resulting Γ-limit functional can be split into two distinct parts: the perimeter of the interface due to the Dirichlet energy that concentrates along the jump, and a boundary integral term due to the constraint u ∈ K. In the particular case where K = H1 (i.e. when no boundary condition is prescribed), the boundary term vanishes and we recover a previous result due to Ambrosio et al.

Cows which develop the behaviour and conformation of the male are known as “bullers.” Information obtained from breeders, veterinary surgeons, and butchers would seem to indicate that the phenomenon of partial sex-transformation in the cow is by no means of rare occurrence. The appearance of this condition is recognised by breeders as indicative of the onset of functional sterility, and any animal showing signs of becoming a “buller” is usually separated from the breeding-stock and prepared for the butcher before the assumption of the male secondary characters has reached a marked degree of expression.

Experiments are described, the aim of which is the determination of the energy spent in producing one pair of ions when electrons of velocity corresponding to 50–270 volts are totally absorbed in air. It is found that this energy varies with the initial speed of the electron, but that it asymptotes to the value 37 ± 2 electron-volts when the energy of the electron is very great.

This value of the “volts per ion pair” is compared with those found by other experimenters. A correlation of the three electronic “constants,” the range of the electron, the ionization per unit path, and the total ionization is attempted. It is concluded that, unless in the case of very fast cathode rays or β-particles, the “ionization per unit path,” as measured by ionization chamber experiments, has no meaning.

The present paper continues the study, initiated in a previous paper, of methods of fitting polynomials to data by least squares, with the aim of systematising the numerical procedure. The case considered here is that of unequally weighted data. A further paper will deal with the still more general case of weighted and correlated data. The numerical methods put forward for the three cases are quite different, and the only common point in the theory is the use of orthogonal polynomials.

In this paper the canonical form of matrix pencils will be discussed which are based on a pair of direct product matrices (Zehfuss matrices), compound matrices, or Schläflian matrices derived from given pencils whose canonical forms are known.

When all pencils concerned are non-singular (i.e. when their determinants do not vanish identically), the problem is equivalent to finding the elementary divisors of the pencil. This has been solved by Aitken (1935), Littlewood (1935), and Roth (1934). In the singular case, however, the so-called minimal indices or Kronecker Invariants have to be determined in addition to the elementary divisors (Turnbull and Aitken, 1932, chap. ix). The solution of this problem is the subject of the following investigation.

Two theories have been put forward in order to explain the formation of manganese deposits in marine muds, and more particularly with regard to manganese nodules: one by Murray, in a paper read before this Society in 1876, the other by Buchanan in 1888. Murray assumes the gradual oxidation of carbonate of manganese resulting ultimately in the formation of hydrated peroxide of manganese.

The following experiments were performed with normal human gastric secretion. After thoroughly washing out the stomach of a healthy man, several ounces of a dilute solution of Caffyn's liquor carnis were introduced into it. After the lapse of an hour the contents of the stomach were drawn off, filtered, and used instead of the pure acid solutions in former experiments. The acidity of the gastric fluid was due to inorganic acid and amounted to 0·15 per cent. hydrochloric acid.

This experiment shows that, in the stomach, with an acidity of the contents less than that even normally present in the gastric secretion, the action of ptyalin is wholly restrained.

Was the ferment merely inhibited from action by the acid, or was it destroyed ?

To determine this, I took equal volumes of this gastric fluid, 1 per cent, starch solution, and saliva, and having mixed them, carefully neutralised the mixture with a solution of caustic potash, using very delicate test-papers to show the neutral point.

On examining the mixture shortly after neutralisation, the whole of the starch was found to have undergone conversion. It reduced Fehling's solution strongly, and contained 0·22 per cent. of reducing substance. This demonstrates that, with an acidity equal to 0·05 per ′cent, hydrochloric acid, the action of ptyalin is restrained.

I performed similar experiments with the gastric fluid from a case of chronic gastric catarrh, the acidity of which was equal to 0·067 per cent. hydrochloric acid.

Let p be an odd prime and let G be a non-abelian finite p-group of exponent p2 with three distinct characteristic subgroups, namely 1, Gp and G. The quotient group G/Gp gives rise to an anti-commutative 𝔽p-algebra L such that the action of Aut (L) is irreducible on L; we call such an algebra IAC. This paper establishes a duality G ↔ L between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the m-th symmetric power of the natural module of SL(2, 𝔽).

This work was commenced at the suggestion of Prof. Ewart, F.R.S., to whose kindness I am indebted for the material used, and for much assistance and advice during the progress of the investigation. Prof. Ewart placed two specimens at my disposal—a male and a female, both of which were about 43 cm. long, excluding, of course, the lash. As, however, the male had been partly dissected for special points in connection with another investigation, it was only available for the study of the IXth and Xth cranial nerves; the main bulk of the present work, therefore, having been carried out on a single specimen. Hence there are several points on which my work will require confirmation.

As the dissection of the cranial nerves of Chimæra elucidated facts of a more interesting character than I had anticipated, it was thought desirable to publish a preliminary statement containing a brief account of the facts, and to leave the full description with figures and morphological deductions for a future communication, which I hope to lay before the Eoyal Society in a few weeks' time. The present paper, therefore, pretends to be a brief review of the more important facts and nothing more.

Having nothing new to describe with respect to the olfactory and optic nerves, I shall commence with the oculo-motor:—

Third Nerve.—This is a large nerve, and arises from the crus cerebri by two principal roots just behind the pituitary body. On emerging from its foramen in the orbit, it immediately gives off a branch to the superior rectus muscle of the eye, and divides into two large dorsal and ventral branches running respectively over and under the optic nerve.

In this paper, the general ordinary quasidifferential expression M of nth order, with complex coefficients, and its formal adjoint M− are considered. It is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in (the limit-circle case) that all well-posed extensions of the minimal operator T0(M) have resolvents which are Hilbert Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all of the standard essential spectra to be empty. These results extend those for the formally symmetric expression M studied in [1] and [14], and also extend those proved in [8] for one singular endpoint.

We give sharp conditions under which the composition of two homeomorphisms of finite distortion is of finite distortion and has integrable distortion. As an application, we obtain a generalization of the classical uniqueness theorem of homeomorphic solution to the measurable Riemann mapping problem.

This note develops a connection formula for spectral functions given in a recent paper by Gilbert and Harris.

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.

Let A be a strong independence algebra of infinite rank m. Let ℒ(A) be the inverse monoid of all local automorphisms of A. The Baer–Levi semigroup B over the algebra A is defined to be the subsemigroup of ℒ(A) consisting of all the elements α with dom α = A and corank im α = m. Let Km be the subsemigroup of ℒ(A) generated by B−1B. Then Km is inverse and it is generated (as a semigroup) by N2, the subset of ℒ(A) consisting of all nilpotent elements of index 2. The 2-nilpotent depth, Δ2(Km) of Km is defined to be the smallest positive integer t such that Km = N2∪…∪(N2)t. In fact, Δ2(Km) is either 2 or 3 and a criterion is found which distinguishes between the two cases.

If N denotes the set of all nilpotents in ℒ(A), then the subsemigroup generated by N is also Km. In fact, Km is proved to be exactly N2.