I. Various writers (Ferrar, 1927) have started out with different definitions of generalised derivatives. Essentially, the problem of the generalised derivative is a problem in interpolation. The values of the derivatives are known for all integer values of n; for all positive integers, being the ordinary derivatives; for zero, being the function itself; for negative integers, being repeated integrals. Any function of n which has the above values at the integers (i.e. any cotabular function) is a solution of the problem. Out of the infinite number of cotabular functions, there exists one discovered by E. T. Whittaker (Whittaker, 1915; Ferrar, 1925; Whittaker, 1935), called the cardinal function, possessing rather remarkable properties. In particular, if the cardinal series defining the cardinal function is convergent, then it is equivalent to the Newton-Gauss formula of interpolation.

1. Formulæ are given in a convenient form for calculating the numbers of animals likely to be required in a controlled experiment for testing the effect of a treatment, when a rough knowledge of the possible maximum efficiency of the treatment is available, and the level of mortality amongst the controls is approximately known. For most purposes formula (8) is sufficiently accurate.

2. A formula (11) is deduced for calculating the limits between which the real advantage of the treated over the control is likely to lie.

3. The method has been extended to the case where various treatments are compared amongst themselves. A formula (15) is deduced for the calculation of the limits within which the real superiority (or inferiority) of any one treatment over the aggregate of all the treatments is likely to lie.

Ditylum Brightwelli, a marine plankton diatom, shows very peculiar osmotic relations which have recently been investigated by Gross (1939). The cells (fig. 1) consist of a central mass of protoplasm containing the nucleus and connected by a few protoplasmic strands with the peripheral plasma membrane in which are embedded numerous chromatophores. The remainder of the space inside the cell is filled with cell sap. As in other diatoms, the protoplast is enclosed by a siliceous cell wall. Gross found that Ditylum plasmolyses rapidly in isotonic and hypotonic NaCl and sugar solutions, being reduced in a few seconds to a small spherical body about one-third to one-twentieth of its original volume, similar in all respects to a resting spore. In a solution of NaCl + CaCl2, and in unbuffered artificial sea-water, plasmolysis occurs in all cells, but the process is considerably delayed and completed only after several hours. In artificial sea-water, the pH of which is adjusted to about 8, the cells remain unchanged.

Suppose that in a given non-negative definite symmetric matrix R = [rij] of order n the diagonal entries are replaced by arbitrary quantities x1, x2,…, xn so that the matrix assumes the form

Matrices of this type are met with in the statistical technique known as Factorial Analysis (Thurstone, 1935; Thomson, 1939); there the nondiagonal elements rij(i≠j) are the correlation coefficients of certain tests and are given by observation. The diagonal entries of the “correlational matrix,” which is always non-negative definite, are originally all equal to unity, but, on the hypothesis which underlies the process of Factorial Analysis, it is permissible to diminish the diagonal entries arbitrarily provided the modified matrix is still non-negative definite, it being assumed that the amount which has been deducted from the diagonal cells is due to the variance of the specific factors, the investigation of which is not the primary aim of the theory.

Many studies on the chromosomes of the sheep have been published in the last decade, but their results are unsatisfactory and contradictory, and hence it was found necessary and justifiable to undertake a further analysis, and the present paper reports the data obtained.

In the past two methods have been applied to the study of differential fertility. One may be called the absolute method, recently applied to Scottish data by the writer. The other may be called the comparative method, subsequently applied to the same statistics by Mr Barclay and Dr Kermack. Since the conclusions obtained by the two methods are not always consonant, it is important to evaluate the scope and reliability of each.

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.

This paper is the continuation of the papers by Born (1939, Part I) and by Born and myself (1940, Parts II and III). Here the theory is extended to the case of spinor wave functions satisfying the Dirac equation.

The pioneer work of Stockard and Papanicolaou (1917, 1919) forms the basis of most subsequent observations on the œstrous cycle of the guinea-pig, and their results are, in the main, accepted by later workers—Ishii (1920), Selle (1922), Nicol (1932), etc.

The intertidal and shallow-water sands of the coasts of the British Isles contain, in many cases as the dominant amphipod fauna, species of the three genera Haustorius, Urothoë, and Bathyporeia of the family Haustoriidæ, often coexistent in the same habitat and obtainable in the same samples. The swimming and burrowing mechanisms of Haustorius arenarius have been described by Dennell (1933), and I have described (1939) the mechanisms in some species of the genus Bathyporeia. This paper deals mainly with Urothoë marina; no reference is made to the other species of the genus, although U. brevicornis has been examined and found to be similar to U. marina. Since all species of the genus Urothoë are very similar morphologically and live in similar habitats, it may be assumed that the following description of U. marina will apply to them also, but I have had no opportunity of examining the more southern species U. grimaldii or U. pulchella. Crawford (1937) states that the burrowing habits of U. brevicornis and U. grimaldii var. poseidonis are very similar to those of Haustorius.

The theory has often been put forward that cancer is a somatic mutation. The changes which differentiate the cancerous cell from the normal are reproduced in many or all of the cells derived from the original malignant cell, i.e. they show the properties by which we define a mutation. Thus, to a geneticist, the possibility which most readily suggests itself is (1) that of a mutation in the genetic material of the somatic cell. Other possible mechanisms that might simulate the results of (1) are (2a) the introduction into the cell of a malignant virus; (2b) the “activation” of some such virus pre-existing in the cell in an inactive state, a situation which would imply mutation in the virus since the change becomes reproduced; (3) the production of some autocatalytic substance other than gene or virus, inducing malignant changes in the cell which contains a certain amount of it.

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.

1. In a previous note (McCrea, 1936) the following problem was enunciated: A rectangular lattice is given; a particle P moves from one lattice-point to another in such a way that, when it is at any interior point, it is equally likely to move to any of its four neighbouring points. P is liberated at any given lattice-point and it is required to find the probability that it will ultimately reach any stated point in the boundary of the lattice, assuming that on arrival at a boundary point its movement ceases.

The fundamental impulse determining the migration of birds is connected with the reproductive cycle; but when that has been recognised it must also be admitted that there are secondary conditions of environment which regulate the actual movements of which a migration is compounded. Of these weather has long been looked upon as one of the most important. There is still, however, much divergence of opinion as to the extent and nature of the influence of weather, and indeed in recent years statistical analysis has led workers, such as Bretscher (1915), to doubt whether any satisfactory connection between meteorological conditions and migration can be demonstrated, and Lincoln to announce boldly that “the state of the weather at any point has little if anything to do with the time of arrival of migratory birds” (1935, p. 59).

Any solution of the Lamé equation

in which sn u has the modulus k, may be termed a Lamé function, but this paper is devoted mainly to functions of real periods 2K or 4K. We suppose that k2 ≤ 1; the number n is real, but not necessarily an integer, and it is sufficient to take n ≥ − ½. These periodic solutions exist for certain characteristic values of h; a table of such values is included in this paper.

Cytological studies carried out by McClung (1905, 1914) on various species of Orthoptera have shown that the male is the heterogametic sex. The male has only one X-chromosome, whereas the female has two. During spermatogenesis two kinds of gametes are produced, one with the X-chromosome and the other without it. It was also found that the segregation of the single sex chromosome takes place at first meiotic anaphase. The present paper describes the sex chromosome of the male Hexacentrus mundus Walker, from India. During the meiotic division this chromosome exhibits peculiarities which it is believed have not hitherto been seen in any species of the Orthopteræ.