Numerous authors have been concerned with phenomena and problems of senescence, duration of life, order of dying-out, and related subjects. With the exception of studies on Drosophila (Pearl and Parker, 1921; Pearl, 1922), other insects (for literature see Korschelt, 1924) and, in some respects, human subjects (see actuarial literature), most of the research of previous authors was embarrassed by the lack of sufficiently large and sufficiently homogeneous material; many conclusions rest on an insecure basis, since they refer only to isolated specimens or to experiments carried out on small numbers of animals.

During a visit to Scotland this autumn I had the opportunity of meeting Dr John B. Simpson and of discussing with him the desirability of definitively dating, by varve-measuring, the interesting ice-oscillation which he had described from the southern slope of the Scottish Highlands (1933).

Beyond the ternary and quaternary cases considered by Clebsch (1891) and Weitzenböck (1910), there appears to be no writing on the bilinear form in cogredient variables—the correlation—from the point of view of the projective invariant theory. In the present work a complete system of concomitants is found for the single bilinear ground form. Incidentally it is shown that such a system belongs not to the single form but to the simultaneous system of two mutually conjugate forms. Symbolic methods are used, and the processes are analogous to those employed in the cases of two quadratic forms (Turnbull and Williamson, 1929), and also of one bilinear form (Turnbull, 1932) in contragredient variables. In contrast with these previous results it is remarkable that the present system involves no forms of weight greater than two (§ 3, § 6).

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–

(i) The data being u1, u2, …, un, the representation is to be given by linear combinations

(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than the kth.

(iii) The sum of squared coefficients which measures the mean square error of yi, is to be a minimum for each value of i.

In a recent paper (Sandeman, 1933) the question of the mathematical representation of the energy levels of the hydrogen molecule by methods based on the old mechanics has been considered. In view of the fact that the wave mechanics has supplanted the older theory, the further application of newer methods to this spectrum is desirable. Richardson and Davidson (1929 b, p. 45) worked out the potential energy function U(ξ) (ξ is denned on p. 51) for the state IsσЗpπ3Π on the old quantum mechanics and on the wave mechanics with the following results: U(ξ) = 5·50ξ2(1 – 1·556ξ + 1·58ξ2 – 1·56ξ3) volts on the old quantum mechanics and U(ξ) = 5·71ξ2(1 – 1·55ξ + 1·58ξ2 – 1·59ξ3) volts on the wave mechanics. They also (loc. cit., pp. 30 and 31) gave the U(ξ)'s for the states Isσ2pσ1∑, Isσ2sσ3∑, and IsσЗdπ1Πb, and on p. 45 they gave the formulæ, based on Fues's (1926) results, for transcribing the old quantum mechanics into the wave mechanics series. Richardson and Das (1929, pt. ii) gave similar data for the state IsσЗpσ3∑.

The mapping of atomic arrangement by X-ray analysis, first applied only to crystalline solids, but more recently extended to amorphous solids, liquids, and gases, has now been pursued for more than twenty years. In the present paper an attempt is made to review the position which has been reached and to forecast the probable lines of development in the future. Examples are chosen from inorganic chemistry, organic chemistry, mineralogy, metallurgy, and biochemistry, which are as far as possible representative of the problems to which X-ray analysis can be effectively applied. It is perhaps not generally realised how extensive these applications are and how much they are influencing other branches of science. A knowledge of the actual atomic arrangement introduces an important precision, simplicity, and novelty of outlook, and a plea might well be made for a wider use of the new knowledge.

In a recent communication (Sandeman, 1935) some problems arising out of the attempt to obtain accurate values of the molecular constants in applying the results of the wave mechanics to the secondary spectrum of hydrogen have been discussed, and an analysis of the two molecular states Is2s3∑ and Is2p1∑ on the basis of Dunham's (1932) solution of the Schrödinger equation for the molecular rotating vibrator has been carried out and compared with the previous calculations of Richardson and Davidson and Richardson and Das. In the present paper an analysis of the ground state (IsIs1∑) of the hydrogen molecule—made possible by recent advances in the measurement and allocation of the band lines —has been attempted. For comparison a similar analysis of the ground state of the molecular ion H2+(Is2∑) is included. Although the spectrum of the molecular ion is not known, the fact that the variables in the Schrödinger equation are separable enables an analysis to be built up from theoretical considerations.

§ 1. Introductory—In a former paper (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144) the author discussed the evaluation of a number of integrals of Associated Legendre Functions, regarded as functions of their degrees. The methods employed depended mainly on contour integration, and most of the integrals were evaluated in terms of infinite series of Associated Legendre Functions. In the present paper the methods employed are of a more elementary character, depending mainly on the use of Dirichlet Integrals; the results obtained are more general; and the integrals and the corresponding series are evaluated in simpler forms. The same notation is employed as in the previous paper. The Mehler-Dirichlet Integral

where o < θ < π, μ > – ½, is used throughout.

In several papers by Chapman (1918, p. 271; 1919, p. 113, etc.) the effect of the moon on the atmospheric pressure has been analysed; the chief inequality observed is semi-diurnal, being, for example, 0·0120 sin (2θ + 114°) millibar at Greenwich, 0·083 sin (2θ + 68°) millibar at Batavia, and 0·060 sin (2θ + 60°) millibar at Hongkong; θ being measured from upper lunar transit.

Charles Darwin in his Autobiography, written in 1876, gives in half a dozen pages (Life and Letters, vol. i, pp. 36–42) an account of his two academic sessions as a medical student in Edinburgh. This account includes brief references to his naturalist friends and acquaintances, a statement that he collected specimens in the tidal pools on the shore of the Firth of Forth and by going out with the “trawlers,” and that he made new observations on the “so-called ova of Flustra,” which “were in fact larvæ,” and on the egg-cases of Pontobdella muricata.

When each individual of a population of N members can be classified as being either A or A, and at the same time either B or B, the relative proportions, or probabilities, of the four types AB, AB, AB and AB can be set out in the familiar “fourfold table.”

Human populations have afforded favourite material for analysis by statisticians, and others, interested in mathematical theories of population-growth. Experimentally, however, human beings are not ideal biological material, so that other animals, such as protozoa, insects, and mammals, have been used. On account of their great reproductive capacity, rapid response to changes in their environment, and ease of handling in the laboratory, certain insects are very suitable for this type of work. The enormous reproductive capacity of insects has been the subject of comment by many biologists, but we are still comparatively ignorant of the exact rôle of the various factors involved in the limitation of their numbers.