In this ancient and honourable Royal Society we have an association of persons with a common motive, namely to assist the advance of natural knowledge. The chief functions of such a Society are (1) to provide facilities for intercourse, personal and formal; (2) to provide a library—and we have a great library of ever-increasing value ; (3) to provide the means of publication. Records of scientific research are not a readily marketable commodity. They would fare badly if left to the mercy of the ordinary laws of supply and demand. So we meet the cost of publication, and they go out, after some winnowing of chaff from wheat, with our imprimatur. Looking back through the published volumes of our Proceedings and Transactions, you will find papers by Kelvin and other Fellows which may be said without exaggeration to mark epochs in the development of scientific thought.

The cytological evidence concerning the pig is that the sex of the future individual is determined at the time of fertilisation, and that in respect of the elements of the sex-chromosome sex-determining mechanism the male is digametic. If this is so, if the two sorts of sperm are produced in equal numbers, if each kind is equally viable and functional, and if fertilisation is at random, then a primary sex-ratio (that which obtains at the time of conception) of equality must follow. If male and female zygotes are equally viable, then the secondary sex-ratio (that which obtains at the time of gestation) will also be equality. A secondary sex-ratio other than equality must be the reflection of an unequal primary sex-ratio, the result of a differential production of the two sorts of gametes elaborated by the digametic sex, of a selective fertilisation by these, or of a sexually selective mortality among the embryos or/and fœtuses. A secondary sex-ratio of equality can follow a pronounced inequality in the primary sex-ratio if a sexually selective mortality operates prenatally.

The following paper is a description of experiments carried out to determine the coefficient of diffusion of alkali salt vapours in a bunsen flame.

The colouration of the flame when a bead of salt is heated in it seems to be due to the presence of neutral atoms of the metallic vapour, and the intensity of the colour at any part of the flame depends on the concentration of the metallic vapour at the particular point.

The characteristic numbers of the Mathieu equation

are those values of a for which, when q is given, the equation admits of a solution of period π or 2π. The periodic solutions, or Mathieu functions, may be developed as a Fourier-series convergent for all values of q,

multiplied by one or other of the factors

In a previous paper the author has employed certain formulæ of Dr Dougall's connecting the Associated Legendre Functions Pnm, where m is a positive integer and n is not integral, with the polynomials Ppm in which p is a positive integer, to deduce the Addition Theorem for the Legendre Functions of the first kind from the corresponding theorem for the Legendre Polynomials.

The present paper is intended to be a supplement to two previous papers on the same subject by Professor E. T. Whittaker. For information regarding the meaning of the term “graduation” and the new standpoint from which Professor Whittaker views the subject we need only refer to the papers themselves. Our concern here is with the practical solution of the problem, for which we suggest an alternative method.

The number of writings on continuants belonging to the twenty-year period now reached is 44, being 15 more than for the immediately preceding period 1880–1900. The period bears also three other marks which serve to give it some little additional distinction, (1) the exceptionally full consideration given to the subject of factorisation (factorisation of continuants), (2) the first appearance of an entirely fresh form for special study, the “block continuant” of Simandl, (3) the fact that one of the said 44 writings is a text-book of very considerable extent devoted exclusively to continuants and continued fractions.

In a previous communication the electrosynthesis of n-dotriacontanedi-carboxylic acid was described. This work has been continued, and three more acids of this series have been prepared, namely, eicosanedicarboxylic acid, HOOC. (CH2)20. COOH; tetracosanedicarboxylic acid, HOOC. (CH2)24. COOH; and octacosanedicarboxylic acid, HOOC. (CH2)28. COOH.

1. During spectroscopic investigations of the phosphorus trioxide glow a method was found for maintaining a stationary glow for twelve to twenty-four hours.

2. Experiments were performed illustrating the necessity of water-vapour for the glow phenomena.

3. The effect of water-vapour on phosphorus trioxide in evacuated bulbs was systematically studied at 25°. Pressures of water-vapour ranging from 1–23 mm. were maintained by the use of certain salt hydrate mixtures, and of water.

4. In every case pressure increased owing to the production of phosphine.

5. When the pressure of water-vapour was 2·5 or 5 mm. a liquid, in addition to gas, was produced in sufficient quantity, after the elapse of several hours, to cause a rapid pressure rise to a maximum. Its vapour-pressure at different temperatures showed that it was liquid phosphorus hydride. A rapid pressure fall from the maximum was observed, and was due to the interaction of hydrides of phosphorus and phosphorus trioxide.

This investigation was proposed by Sir James Walker, F.R.S., to whom the author is deeply indebted for many valuable suggestions, as well as for his constant interest and kindly criticism. She has also to express her thanks to Messrs Brunner, Mond & Co., Ltd., for a grant which defrayed the cost of a Cenco Hyvac Pump and other apparatus.

The estimation of the percentage of firedamp (methane) present in the atmosphere of a mine is usually performed by observing the height of the flame cap over the lowered flame of a safety lamp. In ordinary circumstances the safety-lamp flame is reduced in size and luminosity to render the cap visible, and the estimation of the amount of inflammable gas is made by observing the height of the cap. It was pointed out by Chesneau (1892) that the visibility of the cap was increased if certain organic chlorides and copper salts were added to the fuel. Also it was shown by Briggs (1912) that strongly coloured caps were obtained if a small quantity of carbon tetrachloride was added to the fuel and a loop of copper inserted in the lamp flame itself.

In this paper it is supposed that the effects produced by electric forces are intermittent and consist in the addition of finite increments of momentum occurring after finite intervals of time, and not, as is usually assumed, of a continuous flow of momentum. On the latter view, a body in the time δt receives an increment of momentum Fδt if F is the force; on the view discussed in this paper, it is a chance whether or not the body receives any increment; this chance is equal to δt/T, where T is a time, called the time interval of the force: it depends on the strength of the field and is equal to the average time between two increments. If I is the value of each increment of momentum, the expectation of increase of momentum in time δt is Iδt/T. Over intervals which are long compared with the time T the increase is the same as that under a continuous force equal to I/T, and for effects which are spread over such intervals the results of the intermittent theory agree with those of the usual theory. It seems possible, however, that in effects occurring in atoms or rays of light the times involved may be small enough to be comparable with T; in these cases the classical theory will no longer hold and an entirely different treatment is required. The paper discusses some of the effects which may be expected under these conditions : these include the spontaneous dissociation of electronic systems, the production of Röntgen rays. It follows, too, from the theory that there would be no radiation of energy from an electron describing a circular orbit; the theory, too, has an important application to the propagation of light; it shows that light cannot be propagated as divergent waves where the electric force falls off as the waves recede from the source; it requires that light should be corpuscular in character, i.e. must consist of units whose energy does not change as they travel through space.

In many recent publications Sir J. J. Thomson has suggested a “corpuscular” theory of light; that is, a theory which supposes that the energy of the light is done up into parcels or quanta which retain their energy undiminished as they travel out into space without the unlimited spreading which takes place with ordinary diverging waves. A corpuscular theory of light in this sense has indeed been advocated by Einstein and by many other writers: it is in fact evident that the wave-theory of optics, in its classical form, must be modified in some way in order to be capable of explaining the relations between light and electricity which have been discovered in the last thirty years. Thus, if cathode rays fall on a metallic plate and generate X-rays, which fall on a second metallic plate, electrons are ejected from the second plate whose energy is of the same order as that of the electrons in the primary cathode rays and is independent of the distance of the plates. This seems to indicate that the energy of the X-rays is retained in a compact parcel during the transmission between the plates.

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

Within comparatively recent times a large amount of work has been done on the cathode fall of potential. As a result it is now regarded by many investigators that no sudden fall of potential occurs close to the cathode, at least in the case of low voltage discharges. A comprehensive treatment of recent work on this subject is to be found in a paper by Stücklen. Further, F. W. Aston, in his investigations on the cathode fall, came to the conclusion that practically the whole fall occurred in the dark space between the cathode and the negative glow, and that if there were any force in the negative glow it was negligible compared with that in the dark space. The potentials used by Aston across his discharge tubes were comparatively small, however, never exceeding 1000 volts. With voltages of this order he showed further that the electric force in the dark space was very nearly in linear relation with the distance from the negative glow.

The present memoir is devoted to the determination of criteria for the possession of maxima and minima by an integral which involves one independent variable, one dependent variable, and the first and second derivatives of the latter. All the variations, leading to tests for the existence of maxima and minima, are of the class called “weak”; not only are the variations themselves small, but their derivatives of all orders are small also.

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

It is well known that two problems of harmonic analysis in ordinary three-dimensional space can be solved by Mathieu's functions, namely, (a) harmonic analysis for an orthogonal system of elliptic (or hyperbolic) cylinders,

(b) harmonic analysis for a system of confocal paraboloïds,

It is well known that the Plücker coordinates of a straight line in ordinary space satisfy a quadratic identity

which may also be considered as the equation of a point-quadric in five dimensions, if the six coordinates Pij are treated as six homogeneous coordinates of a point. Projective properties of line geometry may therefore be treated as projective properties of point geometry in five dimensions. This suggests that certain algebraic theories of quaternary forms (corresponding to the geometry of ordinary space) can best be treated as algebraic theories of senary forms: that is, forms in six homogeneous variables.

The Clinophone, an instrument devised by the writer several years ago, was introduced for surveying boreholes, and, in particular, the boreholes used in the freezing process of sinking shafts through heavily-watered or “running” strata. It has recently been employed in this connection at the new sinking at Seaham Harbour, Co. Durham. The freezing holes are drilled round the shaft, and each contains tubes conveying the cold solution which forms the protective ice-wall. The holes must be as vertical as possible. There is, indeed, no other application of the art of boring in which such precision in drilling is needed and where so small a maximum of deviation from the vertical has to be stipulated. An essential to success with the freezing process is that each of the boreholes be accurately surveyed so that its path becomes known from top to bottom.

In the case of the fowl—

(1) Fertile eggs can be expected within 24–48 hours after the introduction of the male, though the onset of fertility varies with different matings.

(2) The length of life of the sperm within the body of the female is about 15–20 days, though in exceptional cases it may be greater.

(3) Though fertility endures 15–20 days after the removal of the male, the eggs laid after the first week commonly fail to complete their development.

(4) If after the removal of the male a second male is introduced, the influence of the first sire is removed by the 7th to 10th day, the exact time differing in different cases. There is a relation between the general “vigour” of the individual and the fertilising power of the sperm it elaborates.