The first definite olfactory rudiments, arising as two solid ingrowths of the deep layer of ectoderm on the under-side of the head, are to be seen in sections of Stage XXVII.

A section through the left olfactory rudiment of Stage XXVIII (fig. 1) shows the early structure of the organ clearly, with its ingrowing thickening of the deep ectodermal layer covered by the thin superficial ectoderm. The cells in both layers are heavily laden with yolk particles, as illustrated in the text-figure by black dots.

In the later stages, XXIX and XXX, the rudiment is similar; the cells near their outer ends are becoming progressively less yolky.

Those who have had experience of problems involving inbreeding or the correlation of relatives in a population with specified proportions of Mendelian genotypes will have realised what difficulties reside in the absence of a convenient notation. In the absence of such a notation difficulties arise more from the unwieldiness of the expressions obtained than the abstruseness of the problems encountered. The familiar chessboard diagrams of Mendelian hybridisation and the form of a contingency table for relatives alike suggest that a matrix notation might be generalised to take into account both sex-linked inheritance and the more usual type of transmission. A determinant form would meet some of the requirements of autosomal transmission; but is not adapted to the asymmetrical case, when the male sex cannot be heterozygous. In extending to more remote relationships a previous inquiry (3) into fraternal and filial correlations involving sex linkage, the writer has found it useful to employ a system of operations applicable to other types of inquiry. The former investigation is not yet complete. In the meantime it seemed possible that the method used might prove suggestive to other workers. A preliminary account of the fundamental operations is here given in so far as they are relevant to single gene substitutions.

A Male Sandy Ray (Raia circularis Loudon) caught in the North Sea, off Aberdeen, measured 47·5 cm. broad by 43 cm. long (excluding the tail) and exhibited an abnormality which is of very rare occurrence. This consisted of two well-developed fins situated ventrally and associated with the pectoral girdle and the last two branchial arches. The measurements of the fins in the formalin-hardened specimen were as follows: length, 8 cm.; breadth at base, 1·3 cm.; breadth at free end, 2 cm. (left), 3·2 cm. (right); thickness at base, 0·7 cm. The integument covering the lateral and medial borders of the proximal portion of each fin was continuous with that over the fourth and fifth gill clefts respectively. In addition, a strong fold of integument bound the proximal fourth of each fin to the ventral aspect of the body of the fish (fig. 1). A brownishred pigment was present on the ventral aspect of the terminal third of each, and a less marked pigmentation of the same colour on the dorsal aspect of the middle third. Small denticles, similar to those found elsewhere on the fins, were numerous on the distal third, but were more abundant on the dorsal aspect.

In a previous paper an expression was given by one of us for the Doppler effect in a de Sitter universe. In the present communication an attempt is made to obtain a more general expression for the Doppler effect, and for this purpose certain results have been derived relating to the properties of null geodesics in a general Riemannian space. In the paper referred to, the problem of defining spatial distance in general space was discussed and certain formulæ suggested which satisfy the essential conditions associated with the idea of spatial distance. This problem has now been approached from a different standpoint, and a new formula of more general form is now put forward. In the final section of the present paper a physical interpretation is given of the theorem relating to null geodesics which is proved in the first section. This involves a discussion of the concept of energy in general relativity, in order that a decision may be arrived at as to what law governs the transference of energy by radiation from one point to another. It appears that when the simplest and most natural assumptions are made for this purpose, then these are precisely the assumptions which are necessary to ensure the general consistency of relativity with the principles of quantum theory. This is illustrated by an application to a statical universe, and to the reddening of light in an expanding universe.

So far as we are aware, the only measurements that have been made of the linear extension of a colloidal medium due to gaseous sorption are those of Dr F. T. Meehan, published in 1927. Meehan studied the response of charcoal to carbon dioxide at atmospheric pressure and different temperatures, and found it to expand almost equally along and across the grain when it took in the gas. No tests of this kind appear to have been carried out with other rigid gels or with other gases, and none with any gas under pressure.

The problem of fitting a polynomial to data by Least Squares has engaged the attention of many writers. The methods of approach have been many and various. Continued fractions, determinants, the calculus of finite differences and sums, the method of moments, the linear combination of data, the use of the orthogonal polynomials of Legendre and Tchebychef, these and doubtless other instruments of analysis have been pressed into service. At the end of the present paper is given a selective bibliography, which we hope on a future occasion to complete and to supplement by adding brief indications of the standpoint and achievement of each investigator.

The problem of defining the concept of spatial distance in a general riemannian space-time has been discussed by E. T. Whittaker, who showed how to translate into the terms of four-dimensional geometry the method adopted by astronomers in determining the distance of a star from the earth by means of a comparison of its absolute and apparent luminosities. The problem was further considered by the present writer, who derived a different formula for spatial distance by a method which was essentially one of partitioning the whole of space-time into space and time relative to the given observer. It was suggested that the new spatial distance might be that determined practically by parallax-measurements, though the evidence in support of this suggestion was perhaps hardly sufficient to carry conviction.

The application of the facts of criss-cross inheritance to commercial poultry breeding is widespread, and every year witnesses the hatching of thousands of chickens, the plumage-colour character of whose mothers was genetically silver, whilst that of their sires was genetically gold. Dominant silver and its allelomorph, recessive gold, form a pair of sexlinked characters, the genes corresponding to them being placed in the X-chromosome. Such evidence as exists points directly to the conclusion that in the fowl the male is the homogametic sex (XX), the female the heterogametic (XY or XO). The mating silver hen, (SX)Y, and gold cock, (sX) (sX), gives, in more or less equal numbers, gold daughters, (sX)Y, and silver sons, (SX) (sX), and, since these colours can readily be distinguished in the newly hatched chick, it is possible to identify males and females amongst day-olds by reference to their down colour. The diagnostic value of a sex-linked character depends upon the precision of the distributive mechanism of the sex-chromosomes.

The characteristics of social behaviour in man are conditioned by previous experience. What is observed is the product on the one hand of a certain genetic constitution and on the other of an intricate, prolonged, and at present largely obscure, process of training and physical environment, including both the environment of the fœtus and family influences, social and physical. The experimental methods for detecting differences due to single gene substitutions cannot be applied directly. Indeed, we can see no immediate prospect of applying to social behaviour methods of genetic analysis such as have led to the mapping of the chromosomes in animals and in plants. With methods available at present, genetic inquiry can undertake to detect whether any gene differences are associated with observed differences, and whether such gene differences are recognisable throughout a comparatively wide or narrow range of social and physical environment.

During recent years it has become increasingly possible to interpret unusual genetic phenomena by reference to peculiarities in chromosome behaviour. Examples of such cytological explanation are found in the studies of triploidy by Bridges and Anderson (1924) and Redfield (1930, 1932), and of translocations by Dobzhansky and Sturtevant (1930), Rhoades (1931), and Dobzhansky (1932a). But, thus far, the phenomenon of complete linkage in the male of Drosophila has not been explained satisfactorily by direct observation of spermatogenesis. Doubtless this has been due partly to technical difficulties, and partly to the fact that chromosome behaviour is only now beginning to be understood. However, the first difficulties have been eliminated by Metz (1926), Guyenot and Naville (1929), and Huettner (1930); and the second by Belar (1928), and more recently by Darlington (1932b).

The Dipnoi form an archaic group of Fish with so many unusual features that it is of importance to work out the development of all their organs in detail, and this investigation was therefore undertaken. The early stages were studied in the collection of slides in the Zoology Department of Glasgow University, and the adult organs were obtained from the material brought back from the Gran Chaco by Dr. G. S. Carter in 1927. The work was done in Glasgow under Professor J. Graham Kerr, F.R.S., to whom I am deeply indebted. The larval stages are numbered according to his system in Keibel's Normentafeln, Heft x (3).

In a communication to this Society in 1916 Professor E. T. Whittaker gave an illuminating exposition of the use of matrices in dealing with continued fractions. In particular he discussed the problem of differentiating the following function of x:—

the value of f′(x) being obtained in the first instance from the leading element in the square of a certain reciprocal matrix {M(x)}−1 (cf. formula (9), § 3, below) by a method which I have recapitulated at the opening of § 4. The expression for f′(x) was then given as the quotient of two determinants obtained in a simple and specific manner from the matrix M(x) itself. It was remarked that this treatment of continued fractions takes its rise in a work by Sylvester, who incidentally showed that the fraction f(x) could be expressed as the quotient of two determinants. It was clear that this quotient expressed the leading element in the reciprocal of the matrix M(x): and the question naturally arose, what were the values of the other elements of this reciprocal?

The occurrence in coal seams of planes of division, generally approximately at right angles to the stratification, has been known for centuries. Owing to the important part these planes play in the lay-out of workings, miners have found it necessary carefully to pay heed to their direction and character. They long ago observed, for example, that, in the majority of seams, there are main divisions or “cleats” and secondary divisions or “cross-cleats,” and that these systems intersect at an angle of generally 80 to 90 degrees. It was also seen that cleats are always best developed in the bright parts of bituminous coal, and that they are less distinct, more widely spaced, or even non-apparent in durain and in cannel. When the dull and bright parts are finely laminated, however, the cleat planes pass through the mass without interruption. Sparry inclusions (ankerites) are very common in the cleats of bright coal and equally rare in dull coal; occasionally the inclusions extend a little way into the roof, and then the existence of associated cleat in the roof is particularly easy to detect.

The introduction of standardised methods for describing that characteristic of social behaviour denoted by the word intelligent is of comparatively recent date. Therefore it is not feasible to undertake a study of the intellectual resemblance of relatives along the ancestral line. In two classes of collateral relatives, however, such resemblance can be made the subject of inquiry. These are (a) members of the same fraternity— twins or ordinary sibs—and (b) first cousins. This investigation, which was prompted by a previous inquiry into the intellectual resemblance of twins carried out in the Department of Social Biology, is confined to a study of ordinary sibs and first cousins. Though resemblance between both sibs and first cousins has already been investigated, there are in each case special reasons which justify further inquiry.

This communication is a sequel to a former work (Proc. Roy. Soc. Edin., vol. liii, 1933, pp. 151–163). An explicit, and apparently new, form is given for the rational reduction of a matrix to diagonal form, applicable to symmetric matrices and also to continuants. An account follows which co-ordinates several existent theories of generalized continued fractions, and concludes with a few determinantal theorems as corollaries.

In the course of an investigation on the effects of intravitam staining with trypan blue on the uterus and ovaries of the guinea-pig at different periods of the œstrous cycle in virgin animals and after parturition, it was found necessary to have definite information with regard to possible variations in the lengths of the cycles. Stockard and Papanicolaou (1917 and 1919) reviewed the earlier work on the œstrous cycle in this rodent, and introduced the vaginal smear technique by which successive cycles could be studied in the living animal. The results of their researches seemed to indicate that the non-pregnant female guinea-pig comes into “heat” regularly about every sixteen days, and that there are only slight time-variations in the periodic rhythm of a given female, and that among different individuals there is a limited variation in the lengths of the cycles. Practically similar results were obtained by Selle (1922), though Voss (1927) indicated that his cases did not confirm the very regular periodicity.

The theoretical treatment of consanguineous unions in human populations has been the subject of several publications, notably by Dahlberg (1927), by Haldane and Moshinsky (1933), and by the present writer (1933). In these attention has been directed to the concentration of discrete characters among the offspring of related parents in a community mating at random. This communication is a contribution to the more general problem of variability which can be expressed in metrical terms. Its theoretical interest resides in the fact that deductions from the genetical theory of inbreeding are open to little, if any, ambiguity, when used as an instrument for detecting gene differences whose manifestation is considerably modified by differences of environment prevailing in human societies. Two criteria will be examined. The first is the effect of consanguineous parentage upon the correlation of their offspring both when autosomal and sex-linked transmission occur. The second is the comparison of the variance of a group of individuals with consanguineous parents with that-of a group of individuals with unrelated parents taken from the same social environment. Since first cousin marriages constitute the type of relatively close inbreeding most commonly encountered in practice, the inquiry which follows is confined to the treatment of unions between first cousins.

In a recent paper on the fæcal pellets of the Anomura, pellets were described from a series of Galatheidea, Paguridea and Thalassinidea, but at that time no Hippidea were available. Since then, through Professor J. H. Orton, I have obtained specimens of Hippa asiatica from Madras and have prepared sections of their pellets. As the latter form an interesting addition to the series of curious pellets, apparently peculiar to the Anomura, a special description of their structure seems to be justified.

In a previous communication (Proc. Roy. Soc. Edin., vol. lii, 1932, p. 337) a visual method of measuring the diffusion coefficients of Br2 : H2, Br2 : N2, Br2 : O2, and Br2 : CO2 was described. The investigation has been extended to include similar measurements for bromine diffusing into argon, nitrous oxide, methane, and hydrogen chloride. Preliminary experiments in which a photoelectric cell replaces the human eye have also been made and confirmatory results have been obtained.

1. The product moment distribution in the general case of p normal variates, obtained in 1928 (1), and again in 1933 (2), has been awaiting further analysis. Some indication has already been given (Wishart, 1928) that new results might be expected from it; in the particular case of two variates obtained previously by Fisher (3), it has been used to deduce the distributions of the correlation coefficient (3), co-variance (4), and regression coefficient (5). In the general case, it has been used by Wilks (6) to furnish a proof of Fisher's distribution of the multiple correlation coefficient (7), and also in connection with his idea of a generalized variance (8). Further analysis appears to be most fruitful in studying statistical regression in general. It is shown in Part I of this paper that the product moment distribution can be split up into a chain of independent factors. Most of the known distributions related to regression or partial correlation are simply obtained, in a manner which clearly indicates the relations they bear to one another; the distribution of a partial regression coefficient of any order is also readily derived.