The theory of the rotating vibrator has been developed by the late J. L. Dunham (1932). The essential step in Dunham's treatment of this question is his replacement of the potential expression occurring in the Schrödinger equation for the diatomic molecule by an arbitrary function in terms of the nuclear separation. When this replacement is made, the Schrödinger equation can be solved by methods developed by Wentzel (1926), Brillouin (1926), and Kramers (1926), and the energy of the rotating vibrator can be expressed as a power series in the quantum numbers in a form convenient for application to spectral data.

The early years of the nineteenth century saw the final establishment, with regard to the ordinary phenomena of light, of the wave theory. And, in those years, the man, Thomas Young (1807), whose work had been mainly effective towards the final acceptance of that theory, put forward a tentative hypothesis regarding the actions and phenomena of colour vision. It stated essentially that these phenomena were explainable most simply on the assumption that all colour sensations that we can experience are due to the existence of three fundamental colour sensations which can be called into play in various proportions by the action of external luminous stimuli falling upon the retina.

In previous papers by Desai and collaborators (Desai and Borkar, 1933 a; Desai and Desai, 1933 b; Mankodi, Barve, and Desai, 1936 a; and Joshi, Barve, and Desai, 1936 b) it has been shown that the changes which are produced in the charge on the colloidal particles during dialysis of ferric hydroxide, thorium hydroxide, prussian blue, and arsenious sulphide sols are not so simple as is usually supposed, and that data for viscosity and stability towards electrolytes do not necessarily give direct information with regard to the charge on the colloidal particles. In the present paper measurements of cataphoretic speed in the presence and absence of electrolytes, stability towards electrolytes, and conductivity of colloidal solutions of gold and vanadium pentoxide which have been dialysed, diluted, allowed to age, and exposed to sunlight to different extents, have been presented and the interpretations given in the previous papers amplified.

The collections of Eocene Ostracoda described in the following paper were made by Lieut.-Colonel L. M. Davies, R.A., and by Mr Pinfold of the Attock Oil Company. The specimens are of particular interest and importance owing to the fact that very few fossil entomostraca had previously been collected in India.

The differential equation

where θ denotes the operator xd/dx has p + 1 solutions valid for | x | < 1.

These may be denoted by

and

where r = 1, 2, …, p, and the asterisk signifies that the expression ρr − ρr + 1 is omitted.

The main object of the experiments was to obtain an indication of the manner in which fatigue of the eye by exposure to illumination of a given intensity, for various periods of time, affected the response of the eye to a rapidly intermittent illumination of constant strength. Expressed otherwise, the object may be stated to be that of determining the effect of fatigue upon the threshold values as made apparent in the flicker test.

In a previous communication (Barclay and Kermack, 1938) it has been shown that the specific legitimate fertility rates of Sweden and Denmark, viewed broadly over several decades, exhibit well-marked regularities, briefly described as conformity to a “diagonal law.” The procedure is to express the specific fertility rate, observed for any particular age-group at a certain period, as a percentage of the rate for the same age-group at a time preceding the decline of the birth-rate (“standard rate”). If now these percentages are plotted as contours on a graph, in which the abscissæ represent calendar years and the ordinates women's age, it is found that, when the age-group 15–20 is excluded, the constant percentage curves are approximately straight lines, running parallel to the diagonals in such a sense that, along any line, increase in calendar years corresponds to decrease in women's age. In Finland, the same general effect is apparent, though it is somewhat obscured by minor disturbances. It is found that, in the case of England and Wales, predictions made on the basis of the law give a reasonable agreement with estimated fertility rates calculated by the Registrar-General on the basis of census data.

1. In a previous communication (Robb and Tannahill, 1935) the lunar atmospheric pressure inequalities at Glasgow for days whose range of pressure was less than or equal to o·1 in. of mercury were investigated. It was found that a large first harmonic was present and that the second harmonic or semi-diurnal component had a phase angle of 285° in contradiction to the phase angle of 90° as predicted by tidal theory.

At times when processes of great ontogenetic importance are taking place, the interaction of heredity and environment presents problems of considerable interest. Laying as it does the foundations of future independent existence, prenatal development must be the concern of all those interested in the mental and physical suitability of human beings for the environment in which they have to live. Owing to lack of information, pregnancy from the point of view of the fœtus has perforce been largely neglected by the physician and sociologist, but the need to remedy this defect in our knowledge had been clearly indicated by Hogben (1931). As an of example of a field of research that would repay much attention, the work of Pearson (1914) and S. Hansen (1920) may be mentioned. These author wrote suggestively of the handicap of the first-born, using extensive data on the incidence of disease, mental defect, and the physical qualities of the new-born child in relation to birth order.

Heisenberg and Pauli (1929) developed a general scheme for the quantization of a field, if the field equations can be derived from a variation principle,

Here za denotes the field variables, L is the Lagrangian, and

The scheme of Heisenberg and Pauli is known to be Lorentz invariant. It is the purpose of this paper to show that it is also invariant with regard to all co-ordinate transformations allowed by the general theory of relativity. The method adopted to prove this is that used by Infeld (1937) and Pryce (1937) to prove the invariance of the “New Field Equations” against Lorentz transformations.

The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with remarkable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important rôle in Nature's scheme.

In Part I (1938) an application of the theoretical work of the late J. L. Dunham (1932) to the Isσ2pσ1∑ and Isσ2sσ3∑ states of D2 has been carried out, and the constants of these states have been calculated and compared with those for the corresponding states of H2. This work shows that the corresponding isotopic states are much more dissimilar than we should have expected.

In a recent paper (Proc. Roy. Soc. Edin., vol. lix, 1939, pp. 49–54) expressions in multiple series with argument 1 – z were found for Generalised Hypergeometric Functions of the type

where p > 1. These formulæ are generalisations of known formulæ for the ordinary hypergeometric function F(α, β; ρ; z), and they are established by induction. The same method will be here employed to obtain generalisations of other known formulæ, these generalisations being in the form of multiple series.

The morphological characters of mammalian sperm cells taken from the ductuli efferentes differ only slightly from the sperm derived from the vas deferens. However, it is known that spermatozoa from the caput epididymis, when kept in physiological salt solution, quickly become immotile, whereas those from the cauda epididymis retain their motility for a long time (Moore, 1928). During the slow passage through the epididymis the spermatozoa undergo a physiological process of maturation, which is said to occur under the influence of the epithelium of the epididymis and to result in a lesser susceptibility on the part of the spermatozoa to extraneous influences (Braus and Redenz, 1924; Redenz, 1926; and Lanz, 1929). Other authors maintain that this maturation of the spermatozoon is not conditioned by environmental influences (Young, 1931). In any case the spermatozoa achieve full functional ability only after they have reached the cauda epididymis and the vas deferens. These are the spermatozoa which enter the female genital tract at copulation, and thus it follows that spermatozoa for artificial insemination in the mouse must be taken from the vas deferens and cauda epididymis.

Numerous combinatory problems arise in connection with a set of elements subject to a non-associative process of composition—let us say of multiplication—commutative or non-commutative.

Non-associative products may be classified according to their shape. By the shape of a product I mean the manner of association of its factors without regard to their identity. Shapes will be called commutative or non-commutative according to the type of multiplication under consideration. Thus if multiplication is non-commutative, the products (AB.C)D and (BA.C)D are distinct but have the same shape, while D(AB.C) has a different shape. The three expressions, however, have the same commutative shape. I confine attention to products (like these) in which the factors are combined only two at a time.

Now that it is accepted that the phenomena observed in genetical experimentation are repercussions of events which have occurred previously in the chromosomes, it is the rule confidently to appeal, whenever possible, to the evidence derived from cytological studies for final explanations of genetical behaviour. This being so, it follows that in those instances in which, for any reason, experimental breeding work is difficult, studies of the chromosomes during the division cycles may be expected to yield significant information concerning the peculiarities of hereditary transmission that might be expected were such genetical investigation undertaken. Further, the larger mammals of economic importance are, under present conditions, unsatisfactory genetical material; they are expensive, and commonly the details of structure and function which they present for examination are not easily or precisely definable and seem to obey no simple rule of inheritance. For these reasons it seems desirable that thorough cytological studies of these forms should be undertaken, for at least these are not costly and they may be expected to provide explanations of the apparently complicated genetic behaviour which these forms exhibit.

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.

The calculation of the partial regression of different, but correlated, dependent variates on the same set of independent variates is of common occurrence. Frequently, also, the differences between the resultant regression coefficients are of interest, and tests of significance for these differences are consequently required. In the classical wheat experiments on Broadbalk field at Rothamsted, for example, the influence of rainfall on yield has been investigated for the various plots, which receive different manurial treatments (Fisher, 1924). An extension of the method of partial regressions was used, which analytically amounted to the evaluation, for each plot separately, of the regression of yield on quantities representing the amount and distribution of rain in each year, these quantities being the same for all plots, since the plots were situated on the same field. One of the most interesting aspects of these results was the differences which were revealed between the effects of rainfall on the differently manured plots. (See also Cochran, 1935.)

In a recent paper (Frazer, Jones, and Skan, 1937) some methods are discussed for the approximate representation of functions by means of polynomials. The coefficients in the polynomials are determined by the equation

where c is the column of coefficients, h is a column of known constants, and M is a matrix which depends on the method of representation adopted. The present paper shows how the reciprocal matrix M-1 can be computed rapidly and simply in the two cases where M is a moment matrix or an alternant matrix.

The method of artificial insemination in the mouse, which has been used successfully only once previously (Mark and Long, 1911), has made it possible to collect exact data on the duration of life and the fertilising capacity of spermatozoa in the female genital tract. Earlier results concerning the conditions under which spermatozoa from the male genital tract attain their maximum activity (Merton, II) and the exact knowledge of the time of parturition (Merton, I) were helpful in carrying out artificial insemination during the following œstrous period.