The rapid progress which has been made in the last few years in the knowledge of the rotational, vibrational, and electronic energy of molecules has extended the range of chemical investigation, particularly in the region of chemical kinetics. This knowledge is derived chiefly from the study and interpretation of band spectra, and one of the most obvious cases to examine is that of “active” nitrogen which emits a characteristic spectrum and is capable of energetic chemical reaction with a wide range of substances. For most of the early work, including spectroscopic observations, we are indebted to the excellent and thorough investigations of Strutt.

The problems of the evolution and ancestry of the numerous breeds of domestic animals have not yet been finally solved. In many cases craniological methods of investigation have been used, and the results obtained have formed the basis of far-reaching conclusions. Such data are, however, open to serious criticism because the influence of environmental agencies during the life of the individual may be responsible for great variability of characterisation of the plastic animal body. The difficulty is increased by innumerable migrations of the ancestral stocks, by extensive hybridisation between distinct though closely related types, and the subsequent mixing of the various types in the modern breeds.

§ 1. Notation.—Suppose that a0, a1a2,… are real constants, such that, for some fixed p (≥1), the series is convergent. It is a consequence of the generalised Riesz-Fischer Theorem that there exists a function f(t) L1+p over ( – 1 , 1), of which

is the Fourier Series.

Several additional members of the series of normal saturated dicarboxylic acids have been investigated by X-rays, and the results are in agreement with the conclusions arrived at in an earlier paper. In the series of mono- and di-alkyl malonic acids, successive reflection planes are separated by the length of one molecule. The series of fatty acid amides closely resembles that of the fatty acids themselves. The cleavage planes are separated by the length of two molecules oriented in opposite directions.

My thanks are due to Mr N. K. Adam for the specimens of mono- and dioctadecyl malonic acid, and to Sir James Walker for his advice and helpful criticism.

A series of reflex adjustments of balance disturbed round the anteroposterior and horizontal axes are described in the domestic duck.

The chapter on Jacobians and the chapter on Hessians were the only two for the period 1880–1900 that seemed to indicate that a slackening had begun in the interest taken in their subjects. We now find that, so far as Jacobians are concerned, any movement of the kind must have been temporary, the new period (1900–1920) showing the number of relevant writings listed for report to be almost if not quite double that of its predecessor.

Up to the present time no sympathetic nervous system has been described in Lepidosiren paradoxa, and the following note is an attempt to fill this gap in our knowledge.

Projective and differential geometry are in close touch at two places, once because of the fundamental rôle played by a quaternary quadratic form in each,

and again through the quadratic in six associated variables,

where

The various definitions which have been adopted by one or more writers for Dnf(x), where n is real but not an integer, fall roughly into three classes:—

(1) Liouville —a method which assumes a convergent expansion

and defines Dnf(x) as ΣAαneax.

(2) Riemann, Grünwald, Laurent and others—methods which, however they begin, ultimately come to the “integral definition,”

the lower limit of integration being arbitrary:

where k is a positive integer, 0<ρ<1, and n=k – ρ.

(3) Pincherle5—a method which, seeking an operator with certain properties, gives an infinite series as a definition of Dnf(x)

Of a number of absorbing and emitting systems subjected to radiation, the proportions which suffer given sequences of absorptions and emissions are examined. The simple results which follow when the proportions so affected are small are applied to the photographic action on the assumption that in exposure the grain acts as a single system. They indicate that the grain requires two absorptions in succession to render it susceptible to “full” development, and three absorptions with one intermediate “emission” to render it susceptible to “partial” development, the terms “full” and “partial” being defined. These results are found to apply over the whole range of wave-length tested, from 4800 Å. to below 2000 Å.

Collections of scales from salmon of the Moisie River taken in 1922, 1923, and 1924 were the subjects of two papers read before the Society on previous occasions. The following report deals with material from the same source and is based on scale collections made in the years 1926 and 1927, the number of samples for each year being respectively 470 and 413. The period over which the collections have been made is from the end of the first week in June to the end of July, but the number of samples occurring outside the four-weeks period from mid-June to mid-July is extremely small.

1. Many systems of preferential voting have been proposed, differing both in the way in which the preferences are expressed and the method of dealing with them in order to arrive at a result. A useful account and examination of the principal methods was given by E. J. Nanson. This paper also contains a description of the method commonly known under his name. We shall consider only those systems in which the electors express their preferences with regard to all the candidates, and in which all these preferences are taken into account.

The actual performance that can be obtained from a clock under the most favourable circumstances is a matter of considerable scientific as well as technical interest, and this has been stimulated recently by the development of the Shortt clock. While there is no question that this clock can hold its rate in a way that no other type will do, there is evidence that exaggerated expectations are entertained with respect to its performance, even in well-informed quarters. This is my reason for publishing the following particulars, which are a digested account of the going of two of these clocks at the Royal Observatory, Edinburgh, during the year 1927.

A number of investigators have sought to extend existing knowledge regarding the mode of origin and structural characteristics of coal as revealed by X-rays, taking advantage of the ease with which the true coal may thereby be distinguished from the associated inorganic matter commonly termed“ ash.”

This paper is a direct continuation of the paper entitled “ The Invariant Theory of the Quaternary Quadratic Complex ” published in the Proceedings of the Royal Society of Edinburgh, Session 1927–28, 48, pp. 70–91. In that paper a prepared system for the quadratic complex has been obtained and tabulated there in § 13. We here deduce from that prepared system the complete system of concomitants. The notation throughout this paper is the same as that suggested in § 13 (loc. cit.), except that, for convenience, the symbol α has been used instead of the symbol 1.

1. A great deal of interest has of late years centred round the arithmetical function r(n), defined as the number of integer solutions of the equation

p2 + q2 = n,

so that, for example, r(1) = 4, the four solutions being (0, ±1) and (±1, 0).