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.

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—

The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

In 1900 von Ihering directed attention to the presence in certain Lamellibranchs of a special muscle lying in the posterior portion of the ventral mantle edge, close to the inner end of the inhalent siphon. This muscle consisted of two strands, each running diagonally from an origin on one valve to be inserted on the other, and crossing one another in the mid-ventral line so as to form a muscular apparatus with the appearance of a St Andrew's cross. He regarded this cruciform muscle as a specially differentiated group of fibres belonging to the pallial edge, acting as an accessory adductor muscle, a point of view in which he has been followed by all subsequent observers. To this von Ihering added the speculation that it had been by some similar process of specialisation of marginal pallial muscle fibres that the two other true adductor muscles of Lamellibranchs had originated.

Euchæta norvegica is one of the largest known copepods and has a wide range of distribution in northern waters. Sars (1903, p. 38) records it from the deep waters of the fjords along the Norwegian coast and mentions that it occurred in two places in the polar basin crossed by Nansen. With (1915, p. 162) records its presence in North Atlantic waters in various places visited by the Danish Ingolf Expedition. That it is generally distributed throughout the North Atlantic is shown by Bigelow (1924, p. 230). It occurs also in the deep waters of the Clyde sea-area, where at times it is abundant, particularly in Upper Loch Fyne.

Euchœta norvegica has been found to be an important constituent of the food of the American pollack (Willey, 1921), and Bigelow (1926) expresses the opinion that it may form an important part of the food of mackerel and herring in the Gulf of Maine. In Loch Fyne in the Clyde sea-area, where Euchœta occurs in abundance at certain times of the year, it has never been found in the stomachs of herring (Scott, 1907). This is surprising for, as Scott remarks, Euchœta is “rich in oily matter, and apparently is of as much value as Calanus for herring food.” Macdonald (1927), however, has found that adult Meganyctiphanes norvegica feed on Euchœta, so, indirectly at least, it contributes to the food of the herring.

The object of the present paper is to put upon record for the guinea-pig data which I have collected regarding (1) the relation between the number of young in a litter and the number of corpora lutea of pregnancy in the ovaries, (2) the distribution of corpora lutea between the two ovaries in two successive ovulations, and (3) the utilisation of the uterine horns in successive pregnancies. The data were obtained in the course of other researches and are arranged in tabular form (Tables I and II). The records from which they have been collected refer to 40 female guinea-pigs killed post partum, of which 16 were pregnant and 24 had not been reimpregnated. The size of the previous litter is known for 38 of these females; both ovaries were serially sectioned at 10 microns in the 40 animals, but in 4 the records of only one ovary were preserved.

Two races or physiological species (A and B) were found in Drosophila pseudo-obscura by Lancefield (1922), who described later (1925, 1929) the genetical behaviour of the interracial hybrids. The present writer (1931, 1932, a, b) has reported further investigations concerning the sterility of the male and the regional reduction of crossing-over in the female of the interracial hybrids. The results obtained suggested that there are genetic fertility factors, located in the sex chromosome complex, which are primarily responsible for the sterility of the F1 males. A similar conclusion has been reached more recently by Dobzhansky (1933 b) from his cytological studies of hybrid males.

The radioactivity of potassium was discovered by Campbell and Wood (1907). Since the β-ray activity shown by all the potassium salts investigated was very small, the suspicion naturally arose that this activity was due to a trace of radioactive impurity and not to the potassium itself. All tests for the presence of an emanation gave, however, negative results, and it became necessary to conclude that the observed emission of the β-radiation was indeed a property of the potassium atom. The subsequent investigations of Campbell (1907) and of Hoffman (1924), whose attempts to alter the activity by chemical operations were unsuccessful, supported this conclusion, as did also the work of Biltz and Marcus, who prepared specimens of potassium sulphate from potassium minerals of different geological ages and found that the activity of the preparations per atom of potassium was always the same.

The photoelectric threshold is a property which is characteristic of a surface state. Accordingly, with a view to a more complete investigation of a turned metallic surface, the apparent thresholds of some metal surfaces, examined immediately after being turned on a lathe, were determined.

In his original paper on the theory of complex spectra Slater (1929) calculated, as one of his examples, the relative positions of the multiplets 3P, 1D, 1S, arising from a configuration of two equivalent p electrons p2. The intervals between these multiplets, in the order written, were found to be in the ratio of 2:3. The same result was shown to hold for p4. Slater's method depends on setting up wave-functions for the atom by combining suitably the wave-functions of single electrons in a central field. With these atomic wave-functions the mean values of the electrostatic energy are calculated for the various multiplets. These mean values involve integration over angular co-ordinates, which can be performed, as well as integrations containing the unknown radial functions, which appear as parameters in the final result. The same method was subsequently applied (Condon and Shortley, 1931) to the configurations p2s, p4s, and extended (Johnson, 1932) to include in addition to the electrostatic energy the energy of the magnetic interaction between the orbits and spins, with which Slater was not concerned.

In two previous communications to the Society (Briggs, 1931, 1932), the graphical method, based on the ultimate analysis, was applied to the classification of carbonaceous minerals, and particularly to the elucidation of the process of “development” of those minerals—a process that involves the progressive histolysis of the primitive plant offal, and a progressive enrichment in carbon and impoverishment in oxygen of the mineral substances produced from it. Almost entirely, attention in those papers was directed to the solid minerals, namely, to the constituents of common coals and to the species severally known as torbanites, oil shales, cannels, and sub-cannels. On the present occasion, however, interest is concentrated on the petroleums. My purpose is to ascertain to what extent the graphical method is able to throw light on the origin and manner of development of the oils.

The discharge of gas from a coal seam or oil pool is at once the sign of a continuing change of composition in the fuel and an indication of the nature of the change. The process of change, which has been termed autometamorphism, is accelerated by warmth and pressure, and the chief by-products are carbon dioxide, methane, and water (Briggs, 1931). Other gases are found in fire-damp, but in proportions that are relatively so insignificant that, in the present connection, they may be altogether disregarded. The mixture known as “natural gas,” occurring in association with petroleum, usually contains other hydrocarbons besides methane; indeed, the “wet” gas set free from oil when the natural pressure is released may consist almost entirely of members of the paraffin series heavier than methane. Nevertheless the natural gas obtained from oilfields is chiefly CH4. The mean of 22 analyses of natural gas from oilfields in various parts of the world gives CH4, 86·3; CO2, 3·0; N2, 6·2 per cent. (Bacon and Hamor, 1916). In a generalised inquiry, then, it will be sufficiently accurate to reckon the whole of the inflammable ingredient of both natural gas and fire-damp as methane.

Little is known regarding the integration of Legendre Functions with respect to their degrees. In this paper several such integrals are evaluated, three different methods being employed. In § 2 proofs are given of a number of formulae which are required later. In § 3 an example is given of the evaluation of an integral by contour integration. The following section contains the proof of a formula of the Fourier Integral type, a special case of which was given in a previous paper (Proc. Roy. Soc. Edin., vol. li, 1931, p. 123). In § 5 an integral is evaluated by employing Fourier's Integral Theorem; while in § 6 other integrals are evaluated by. means of expansions in series.

Fifty years ago, when this Royal Society was a hundred years old, it fell to Lord Moncrieff, the President, to speak to the occasion. He sketched the history of the Society through its hundred years, and dwelt most on the days when it was born. He told how the Founders made their way one winter's night from homes in Lawnmarket or Canongate to the College Library (as Johnson and Boswell had lately done), with a few flickering oil-lamps to light the way and tallow-candles which dimly lit the room.

In the first decade of the present century Bloomer reported in a series of concise papers (for which see the list of references) the results of his investigations into the structure of the majority of the lamellibranchs which were then classified in the family Solenidæ. In addition to a general description of the external anatomy of these bivalves he paid especial attention to the details of the musculature, to the course of the alimentary canal and the structure of the stomach, while he gave rather generalised accounts of the nervous system and of the arrangement of the circulation, in which he almost entirely followed the previous work of Ménégaux (1890). Bloomer made little attempt to view as a whole the information which had been gathered together as a result of these years of work; the only papers which he published that can be regarded as dealing with his results from a general standpoint being a short one (1903b) on the classification of the British species of the genus Solen, and a second (1903c), equally brief, on the origin and function of the small fourth pallial aperture which is a marked feature of the anatomy of certain members of the family. This he proved to be a separated portion of the pedal gape, not homologous with the similar structure lying in a more posterior position in some of the Anatinacea and in Lutraria.

The material which forms the subject of this present paper was found in Coal Measure Shales exposed in the bank of the River Nethan, near Crossford, Lanarkshire, Scotland. The geological age of these beds is the Productive Coal Measures, Upper Carboniferous. The outcrop is near the position of the Kiltongue Musselband, about 8 fathoms above the Kiltongue Coal.