Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1917"

On waves in an elastic plate

Abstract: The theory of waves in an infinitely long cylindrical rod was discussed by Pochhammer in 1876 in a well-known paper The somewhat simpler problem of two-dimensional waves in a solid bounded by parallel planes was considered by Lord Rayleigh and by the present writer‡ in 1889 The main object in these various investigations was to verify, or to ascertain small corrections to, the ordinary theory of the vibrations of thin rods or plates, and the wave-length was accordingly assumed in the end to be great in comparison with the thickness It occurred to me some time ago that a further examination of the two-dimensional problem was desirable for more than one reason In the first place, the number of cases in which the various types of vibration of a solid, none of whose dimensions is regarded as small, have been studied is so restricted that any addition to it would have some degree of interest, if merely as a contribution to elastic theory Again, modern seismology has suggested various questions relating to waves and vibrations in an elastic stratum imagined as resting on matter of a different elasticity and density These questions naturally present great mathematical difficulties, and it seemed unpromising to attempt any further discussion of them unless the comparatively simple problem which forms the subject of this paper should be found to admit of a practical solution In itself it has, of course, no bearing on the questions referred to

On the Dynamics of Revolving Fluids

Abstract: So much of meteorology depends ultimately upon the dynamics of revolving fluid that it is desirable to formulate as clearly as possible such simple conclusions as are within our reach, in the hope that they may assist our judgment when an exact analysis seems impracticable. An important contribution to this subject is that recently published by Dr. Aitken. It formed the starting point of part of the investigation which follows, but I ought perhaps to add that I do not share Dr. Aitken’s views in all respects. His paper should be studied by all interested in these questions. As regards the present contribution to the theory it may be well to premise that the limitation to symmetry round an axis is imposed throughout.

An application of the theory of probabilities to the study of a priori pathometry.—Part I

Abstract: Prefatory .—It is somewhat surprising that so little mathematical work should have been done on the subject of epidemics, and, indeed, on the distribution of diseases in general. Not only is the theme of immediate importance to humanity, but it is one which is fundamentally connected with numbers, while vast masses of statistics have long been awaiting proper examination. But, more than this, many and indeed the principal problems of epidemiology on which preventive measures largely depend, such as the rate of infection, the frequency of outbreaks, and the loss of immunity, can scarcely ever be resolved by any other methods than those of analysis. For example, infections diseases may perbaps be classified in three groups: (1) diseases such as leprosy, tuberculosis, and (?) cancer, which fluctuate comparatively little from month to month, though they may slowly increase or decrease in the course of years; (2) diseases such as measles, scarlatina, malaria, and dysentery, which, though constantly present in many countries, flare up in epidemics at frequent intervals; and (3) diseases such as plague or cholera, which disappear entirely after periods of acute epidemicity. To what are these differences due? Why, indeed, should epidemics occur at all, and why sbould not all infections diseases belong to the first group and always remain at an almost flat rate? Behind these phenomena there must be causes which are of profound importance to mankind and which probably can be ascertained only by those principles of careful computation which have yielded such brilliant results in astronomy, physics, and mechanics. Are the epidemics in the second class of diseases due (1) to a sudden and simultaneous increase of infectivity in the causative agents living in affected persons; or (2) to changes of environment which favour their dissemination from person to person; or (3) merely to the increase of suscep­tible material in a locality due to the gradual loss of acquired immunity in the population there; or to similar or other causes? And why should diseases of tbs third class disappear, as they undoubtedly do, and diseases of the first class remain so persistently?—all questions which immediately and obviously present themselves for examination.

Motion of Solids in Fluids When the Flow is Not Irrotational

Abstract: The chief interest in the results obtained in the following pages lies in the fact that a mathematical result has been obtained concerning the motion of solids in fluids which is verified accurately when recourse is had to experiment, with real solids moving in real fluids. This is so exceptional a circumstance that it is hoped that the interest which it gives to the mathematical work will serve to extenuate, to a certain extent, the clumsiness of the methods employed. The problem solved is two-dimensional. An infinite cylindrical body of any cross-section moves in a uniformly rotating fluid with its generators parallel to the axis of rotation. The stream lines and the reaction between the solid and the fluid are found.

Experiments on tribo-electricity. I.—The tribo-electric series

Abstract: I.- Historical . Little exact knowledge of tribo-electricity has yet been accumulated, and this subject has certainly not been raised to the dignity of a quantitative science. In the present paper will be found an account of experiments in which the conditions of the solid bodies rubbed together have been greatly varied. Thus, temperature has been changed both before and during friction; the surfaces used have been rubbed with considerable pressure while hot; they have, wherever possible, been rubbed together when flexed; and they have been prepared before rubbing by being ground and polished in various ways. Much information has thus come to light as to the electrical surface conditions of a variety of solids. Before setting forth the methods and results of this paper, let us note the information previously available. Good summaries of the subject will be found in Wiedemann's 'Electriciat' (2) and Chwolson's treatise. (1).

Experiments with mercury jets

Abstract: 1. Introductory .—Certain electro-chemical experiments, to be described in a subsequent communication, arose out of peculiarities which we noticed in the behaviour of a stream of mercury issuing from the lower, drawn-out, end of a vertical tube containing that liquid. The salient feature of the phenomena is the relation between the length of the continuous part of the liquid stream, called for convenience the “jet-length,” and the “head” under which the mercury is ejected. The experiments were made with orifices of diameters varying from about 0·25 mm. downwards to about 0·1 mm. It was found that the relation between the jet-length l and the head h depended appreciably upon the form of the nozzle through which the mercury escaped. For definiteness the nozzles were all constructed in the same way, namely by drawing out a piece of capillary tubing of about 1 mm. bore and severing it, as nearly as possible perpendicularly, at the centre of the constricted part. The nozzle so obtained tapered, in between 1 and 2 cm., from the original diameter of the tube to the diameter of the orifice and was very nearly cylindrical at the point of discharge. The values of the jet-length were determined by means of a cathetometer microscope.

Some cases of wave motion due to a submerged obstacle

Abstract: 1. As far as I am aware, only one case of wave motion caused by a submerged obstacle has been worked out in any detail, namely the two-dimensional motion due to a circular cylinder; for this case, Prof. Lamb has given a solution applicable when the cylinder is of small radius and is at a considerable depth. The method can be extended to bodies of different shape, and my object in this paper is to work out the simplest three-dimensional case, the motion of a submerged sphere. The problem I have considered specially is the wave resistance of the submerged body. In the two-dimensional case, this is calculated by considerations of energy and work applied to the train of regular waves. But for a moving sphere the wave system is more complicated, like the well-known wave pattern for a moving point disturbance, and similar methods are not so easily applied; I have therefore calculated directly the horizontal resultant of the fluid pressure on the sphere. Before working out this case, the analysis for the circular cylinder is repeated, because it is necessary to carry the approximation a stage further than in Prof. Lamb’s solution in order to verify that the resultant horizontal pressure on the cylinder is the same as the wave resistance obtained by the method of energy.

Absorption bands of atmospheric ozone in the spectra of sun and stars

Abstract: In a paper communicated to the Royal Society in 1890, Sir William Huggins gave an account of his discovery of a new group of lines in the photographic spectrum of Sirius, situated near the limit of atmospheric transmission. A photograph taken with a long exposure was described as showing no strong lines after the termination of the hydrogen series until about λ 3338 was reached, at which place there appeared the first of a group of at least six lines, all of which were nearly as broad as those of hydrogen. The third line of the group, about λ , 3278, appeared to be the broadest. The sixth line occurred almost at the limit of the photograph, where the spectrum was very faint, and it was not possible to determine whether this was the last member of the group. The wave-lengths of the lines, which were only considered to be roughly approximate, were given as 3338, 3311, 3278, 3254, 3226, and 3199. In 1915, a copy of an excellent photograph of the spectrum of Sirius, which had been taken with a small quartz prismatic camera at the Royal Observatory, Edinburgh, was kindly forwarded to us by Prof. Sampson. In this photograph the Huggins group is well shown (Plate 3, fig. 3), and it was at once suspected, from the general appearance of the spectrum, that a system of bands, and not a group of lines, was in question. From our knowledge of the physical conditions in the atmosphere of Sirius, as indicated by other parts of the spectrum, the existence of bands in this star seemed highly improbable, and it was an obvious inference that they might be produced by absorption in our terrestrial atmosphere. This view was strongly supported by photographs of the solar spectrum which we obtained with a small quartz spectrograph near sunset, as compared with similar photographs taken when the sun was at a considerable elevation. The spectrum of the low sun showed very decided indications of bands occupying about the same positions as those in the spectrum of Sirius, but, on account of the superposition of solar lines, the wave-lengths of the bands could not be very certainly derived.

The High-Frequency Resistance of Multiply-Stranded Insulated Wire

Abstract: Conductors which have to carry high-frequency currents are often made up of a large number of separately insulated fine wires stranded together, with the object of compelling the current to distribute itself over the whole cross-section of the conductor. This may be done for two different reasons: firstly, to decrease the variation of inductance and resistance with change of frequency; and, secondly, to decrease the effective resistance of the conductor at high frequencies. To be effective, it is necessary that every wire should occupy in turn the same relative position in the conductor, so that the electromotive force induced in each wire by the magnetic flux should have the same average value over the whole length of the wire. If every separate wire has the same resistance, the same applied P. D. and the same induced E. M. F., both in amplitude and phase, they will all necessarily carry the same current, and the total current will therefore be uniformly distributed between all the wires. The usual method of obtaining this similarity in the path of every strand is to make a conductor of three, four, or five wires twisted together, and then to twist three such conductors together, and so on until the resulting conductor contains the required number of wires. A large number of such multiple conductors are sometimes plaited or braided into a tubular con­ductor. Two of the individual wires of such a conductor may be in contact, except for the insulation, at a certain point, and then again further on at another point, one wire having followed an internal and the other an external path between the two points. From a knowledge of the current, frequency, and projected area of the loop formed by the two wires, measured normal to the magnetic flux, the E. M. F. induced in the loop can be calcu­lated; thus, if the multiple conductor has a diameter of 1 cm. and is wound into a coil with one turn per centimetre, and if the current is 100 amperes at a frequency of 106, the E. M. F. induced in a loop of 5 sq. cm. is about 20 volts, giving 10 volts between the wires at each point of contact. In circuits containing spark-gaps the rate of change of the current may reach very high values and thus cause much greater potential differences between the separate strands, which must therefore be efficiently insulated to prevent sparking between them.

On the Deflection of the Vertical by Tidal Loading of the Earth's Surface

Abstract: The earliest calculations on the above subject in which the flexure of the earth's surface is taken into account, as well as the direct attraction of the water, are contained in a well known paper by G. H. Darwin. It was there shown, in a particular case, that the tilting effect would conspire with the attraction, and would on certain assumptions be directly proportional to it. The latter remark was stated as due originally (in a more general form) to Sir W. Thomson. On inserting numerical values it was found that the apparent deflection due to tilting would considerably exceed that due to attraction. The whole subject has of late excited renewed attention, owing to its bearing on observations of lunar deflection of gravity, and in a recent paper by Terazawa the matter is specially considered from the latter point of view. The present paper, after discussing a few typical problems, goes on (in 4) to examine the effect of one or two considerations which have hitherto been disregarded in such calculations. It is true that the corrections involved are under some conditions negligible, but they are of theoretic interest, and it is found that at great distances from a load, and therefore in all cases of widely distributed load, they may attain considerable relative importance. In the first place, owing to the deformation of the earth’s surface and the altered distribution of density an additional horizontal component of force on the plumb-line is introduced; this tends to counteract the attraction of the water. A more important point is that the influence of gravity on the deformation has been ignored. In attempting to estimate the effect of gravity it has been found convenient, in order to avoid difficulties not altogether of a mathematical kind, to limit the investigation to the case of incompressibility. I have also neglected, in the first instance, the disturbance in the field of gravity itself, due to the load and the deformation, so far as this affects the strains. When subsequently the alteration of the field is taken into account a curious point arises. For mathematical simplicity the “earth” has been treated, as is usual in such investigations, as flat and infinitely extended. It appears that if this were the case the surface would be unstable, whatever the degree of rigidity, for disturbances exceeding a certain critical wave-length. This wave-length is, however, enormous, and reason is given for the view that inferences can still legitimately be drawn from our results as to the character of the effects actually produced.

On Multiple Integrals

Abstract: 1. In the modern theory of absolutely convergent integrals, as distinct from the older Riemann theory, the difference between multiple and repeated integration falls to the ground. Every multiple integral is equal to the corresponding repeated integrals, and the formulae involving such multiple integrals, even when expressed without the repeated integral notation, can be obtained by means of the repeated integration process. To really grasp the distinction between multiple and repeated integrals we have to take as independent variables functions of bounded variation; more precisely, we have to replace the ensemble of variables ( x , y , z , ...) by a single function of bounded variation, which, indeed, takes the place of the product of the variables xyz ..., the more general integration reducing to the ordinary when this product replaces the function in question. I have already had occasion elsewhere to explain what is meant by a function of bounded variation of any number of variables, and to define integration of any bounded function, and absolutely convergent integration of unbounded functions, with regard to such functions of bounded variation It will be sufficient here to call special attention to the fact that a function of bounded variation, for simplicity, let us say, of two variables, x and y , is expressible, to a value of the function pres , as the difference of two functions, which are monotone increasing in x , in y , and in ( x , y ), representing the positive and negative variations in the plane, increased by the differences of the variations along the axis of x and along the axis of y respectively. There is, in consequence of the definition, a unique limit at each point for each of the four standard modes of approach, these four limits coinciding, except at points lying on a countable set of lines parallel to the axes at most. The modes of approach here considered correspond to the four quadrants when the point is the origin, the lines corresponding and parallel to the axes being excluded. The limits along these excluded lines may be different from one another and from the other four.

The lateral vibrations of bars of variable section

Abstract: The present investigation, though strictly mathematical in character, arose in connection with a suggestion, put forward by Prof. A. Dendy and the present author in another paper communicated to the Society, that the siliceous deposits found on certain sponge spicules occurred at nodes of the spicules, regarded as vibrating rods. These vibrations, being set up and maintained by the impact of currents of water on the spicules, are necessarily of the lateral type. For the detailed examination of such a suggestion, it is necessary to obtain a comprehensive account of the positions of the funda­mental nodes on a free-free bar, as dependent on the law of variation of its cross-section. The present paper contains, in fact, the formal analysis whose results were quoted without proof in the other paper. This analysis is of considerable generality, as will appear, and the particular examples selected for purposes of illustration, together with the manner in which the variable cross-section is dealt with, have been determined by the requirements of the biological application already mentioned. One general problem is in view throughout the work, and it may be stated as follows

On the Ordinary Convergence of Restricted Fourier Series

Abstract: 1. The necessary and sufficient condition that a trigonometrical series should be a Fourier series is that the integrated series should converge to an integral throughout the closed interval of periodicity, and should be the Courier series, accordingly, of an integral. Conversely, starting with the Courier series of an integral and differentiating it term by term, we obtain the Courier series of the most general type, namely, one associated with any function possessing an absolutely convergent integral. If the Courier series which is differentiated is not the Courier series of an integral, but of a function which fails to be an integral, at even a single point, the derived series will not lie a Courier series.

Spectroscopic observations on the active modification of nitrogen. —V

Abstract: The spectrum of the nitrogen afterglow, as ordinarily obtained, consists of three groups of bands, which we designated as the α, β , and γ groups. The α -group is in the visual region, and consists of a selection of the first positive bands of nitrogen. The β -group is chiefly in the ultra-violet, but extends into the violet. It can be observed in the discharge, but is much better developed in the afterglow. The γ -group is wholly in the ultra-violet, and identical with Deslandres’ third positive nitrogen group. I have not much to add to the results of the former paper as regards the measurement and classification of these various groups of bands. The circumstances under which they appear have, however, been reinvestigated, and the conclusions of the earlier paper modified in some important respects.

The Initial Wave Resistance of a Moving Surface Pressure

Abstract: 1. The study of the water waves produced by the motion of an assigned pressure distribution over the surface has hitherto been limited to the steady state attained when the system has been moving with uniform velocity for a very long time. In his latest series of papers on water waves, Lord Kelvin made an elaborate graphical and numerical study of cognate problems, and expressed the hope of applying his methods to calculate the initiation and continued growth of canal ship-waves due to the sudden commencement and continued application of a moving, steady surface pressure. In the following paper, I have not attempted any analysis of the surface elevation itself, but I have proceeded directly to the calculation of the corre ­sponding wave resistance. At present the wave resistance is known only for the steady state for certain localised pressure systems in uniform motion, and it seems desirable to attempt some estimate of the time taken to attain this state when we take into account the beginnings of the motion. One might examine the effect of initial acceleration, but I have limited the problem by considering only the case of a system which is suddenly established, and is at the same instant set in motion with uniform velocity. The work is arranged in the following order: a general expression for the wave resistance as a function of the time, an exact solution for a certain waveless system, a comparison of this solution and the group approximation, and an approximate solution for certain systems which leave regular waves in their rear.

X-ray analysis of the crystal-structure of rutile and cassiterite

Abstract: The present paper deals with the results obtained in the investigation of the atomic structure of rutile and cassiterite by the X-ray spectrometer. A detailed account of the method has been given by Prof. Bragg and his son, W. L. Bragg, in a series of papers communicated to the Royal Society. It consists essentially in allowing a narrow beam of monochromatic X-rays—in this case the rhodium rays—to fall on the face of the crystals, mounted on a spectrometer table, the axis of rotation of which passes through the face of the crystal. The beam is “reflected” by the atom planes parallel to this face, and thence passes into an ionisation chamber, containing methyl bromide in order to increase the ionisation current. The setting of crystal and chamber with regard to the incident beam corresponds to that for which ordinary light is reflected.

A contribution to the study of the magnetic properties of manganese and of some special manganese steels

Abstract: The object of this research has been to find a connection between the magnetic susceptibility per unit mass (called the mass susceptibility) of different manganese steels and their chemical constitution. All measurements were made with the help of the magnetic balance of P. Curie and C. Cheneveau. In this instrument the body to be tested is attached to one arm of a torsion balance. A horseshoe magnet can be moved horizontally, so that the direction of motion is at right angles to the line joining the poles, and the body is gradually brought into a stronger and stronger field. The observed deflection of the torsion balance increases at first, then diminishes, and becomes zero when the body lies between the poles of the magnet. If the magnet be moved still further, a deflection in the opposite direction is observed, which again passes through a maximum at the point where the variation of the field of force is greatest. Fig. 1 illustrates the deflections observed in a particular case. The difference between the readings of the torsion balance when the deflections are greatest on either side is proportional to the mass susceptibility, so that this quantity may be determined by comparison with a specimen whose susceptibility is known. Throughout the paper the mass susceptibility is denoted by the symbol X .

The complete photo-electric emission from the alloy of sodium and potassium

Abstract: The experimental work described in the present paper suggested itself to the writer in connection with an earlier investigation on the law governing the temperature variation of the complete photo-electric emission from a hot body, i. e . the photo-electric emission from a body in equilibrium with the full (black body) radiation corresponding to its temperature. By making use of hypotheses contained in the quantum theory, the writer obtained the following expression for the current per unit area C = AT (1+2 k T/ Ф +2 k 2T2/ Ф 2) e -Ф / k t, where Ф is the work done in removing an electron from the hot body, and is equal to hv, v being the lowest frequency of the radiation capable of producing a photo-electric emission, and h being Planck’s constant. The quantity k is the "gas constant” reckoned for one molecule, and A is a quantity independent of T, and characteristic of the substance. As the expression inside the brackets in the above formula does not differ appreciably from unity, the latter is substantially of the same type as Richardson’s equation C = ATλ e -Ф / k T, (1) for the thermionic emission. Richardson* has also shown that it follows, from thermodynamic considerations, that this law governs the complete photo-electric emission. There is reason to believe that the thermionic emission is not wholly photo-electric in origin, but it is clear that some portion of it is the complete auto-photo-electric emission of the substance concerned, and that the law governing its temperature variation should be the same as that for the whole thermionic emission. We are thus led to expect that, when a body is exposed to an external source of full radiation, the same law will govern the variation of its complete photo-electric emission with the temperature of the source of the radiation. This expectation has been confirmed by experiments on the alloy of sodium and potassium, the description of which constitutes the subject of the present paper.

Gravitational Instability and the Figure of the Earth

Abstract: 1. In a paper published in 1902, “On the Vibrations and Stability of a Gravitating Planet,” I attempted to examine whether, owing to gravitational instability, an arrangement in concentric spherical shells might not be unstable for a planet of the size of our earth. It was not found possible to get exact results; progress only seemed possible by supposing that the problem would in its essentials be similar to a highly artificial problem in which gravitation was exactly annulled in the symmetrical configuration by the superposition of a countervailing field, so that in this state the matter was of uniform density and unstressed. The only possible justification for such an assumption was that I found myself unable to make any progress, either physically or mathematically, without it. Subject to this assumption, I found that, for a mass of the structure of our earth in its present state, the symmetrical configuration would undoubtedly be stable. It seemed likely, however, that in a past age in which the rigidity of the earth was considerably, although not enormously, less than it now is, the symmetrical configuration might have been unstable; the stable configuration would have been an unsymmetrical one, in which the surfaces of equal density were spherical but not concentric, so that the point of maximum density coincided neither with the centre of gravity nor the centre of figure of the earth’s surface. It was suggested that traces of this unsymmetrical configuration might still be found in the arrangement of oceans and plateaux on the earth’s surface. In 1906, Lord Rayleigh, in a paper “On the Dilatational Stability of the Earth,” drew attention to the wide departure from actuality implied in my assumptions, and suggested a set of assumptions which had more justification in nature. Lord Rayleigh’s suggestion was, in brief, that we might regard the symmetrical configuration as one in which the gravitational forces were balanced by hydrostatic pressures only. When disturbed, there is an additional stress superposed on to the hydrostatic pressure, and Lord Rayleigh suggested that this might be supposed connected with the additional strain— i. e ., the strain above and beyond that in the initial configuration by the ordinary elastic solid relations.

Duration of Luminosity of Electric Discharge in Gases and Vapours: Further Studies

Abstract: In several previous papers, I have described experiments showing that the luminosity of the spark or arc discharge through gases and metallic vapours lasts for an appreciable length of time after the discharge has ceased to pass, and that the luminous vapour can be removed from the region where the electric current is passing, and still remain luminous, showing the same spectrum as before, though in certain cases some lines of this spectrum fade out sooner than others. The present paper is a continuation of these studies. 1. The Luminous Jet at Very Low Pressures. Effect of a Transverse Electric Field . The first experiments to be described were designed to observe the effects at very low gaseous pressures, and with small currents. The advantage of this is that the luminous particles in the jet are then comparatively free to move under the action of an external force, and their motion in an electrostatic field can be conveniently examined. Moreover, since the ions are few, they do not disturb the uniformity of the electric field applied for this purpose.

Characteristic frequency and atomic number

Abstract: Recent work on the specific heat of chemical elements in the solid state has led to important conclusions with regard to the values of the atomic heat at constant volume. The curves showing the variation of atomic heat with temperature are all of the same form, and any given curve can be transformed into any other merely by altering the scale on which the temperature is plotted. This implies that the atomic heat C v is a function of the temperature T, dependent on a single parameter, the function being the same for different elements. Thus C v = F (Θ/T), where Θ is a certain temperature characteristic of the element in question. According to the quantum theory Θ = βv = hv/k , where v is the characteristic atomic frequency, h is Planck’s constant, 6·558 x 10-27, k is the gas constant for a single molecule, 1·372 x 10-16.

On the Order of Magnitude of the Coefficients of a Fourier Series

Abstract: § 1. Riemann’s theorem that the coefficients of a Fourier series converge to zero was shown by Lebesgue to still hold when integration is understood to be in the general sense now employed, absolutely convergent, or Lebesgue integration. Little progress has, however, been made in the determination of the order of magnitude of the coefficients. It has, indeed, been proved that, when the function has bounded variation, na n and nb n are bounded functions of n , and that, when the function is a continuous function of such a type as satisfies a condition of Lipschitz, n q a n and n q b n converge to zero, where q is a positive number not greater than unity, depending on the particular Lipschitz condition satisfied by the function. As regards the second of these results, involving the satisfying of a condition of Lipschitz, it is to be remarked that, in well-known series of the type Σ n -q cos nx and Σ n -q sin nx , the functions of which they are the Fourier series do not, in any interval containing the origin, satisfy any condition of Lipschitz, being, indeed, unbounded. In the present communication I obtain a number of theorems corresponding to each of these two results, including them as particular cases, and, at the same time, leading to the known properties of the simple sine and cosine series above referred to.

The Dissipation of Energy in the Tides in Connection with the Acceleration of the Moon's Mean Motion

Abstract: 1. In the following paper an expression for the rate of dissipation of kinetic energy in a rotating sea is obtained in the form of an integral of a function of the surface-current velocities. This expression has been used to find an approximate value for the rate of dissipation in the Irish Sea, for which much tidal information is available, and the order of magnitude of the result obtained suggests that this direct effect of viscosity may account for at least an appreciable part of the earth’s secular retardation. The water is supposed of uniform density, and at the bed of the ocean to be relatively at rest. The motion, which, as usual, is regarded as small, is supposed non-turbulent, and simple-harmonic with respect to the time with a period of 12 hours. In actual tidal motion eddies are present, so that the result here obtained for the rate of dissipation may be regarded as a lower limit.

On permanent periodicity in sunspots

Abstract: The most important index to the dynamical constitution of the Sun lies in its dark spots, which have been known, ever since their discovery by Galileo, to recur periodically. Numerical data for the numbers of spots simultaneously present, and later and more precise data for the extent of spotted area, are in existence for about two centuries ; and it is natural that efforts should be made, from all points of view, to extract the knowledge which they contain. The discussion of the fundamental question, whether there is permanent unbroken periodicity, due either to planetary influence or (as seems much more probable) to a period of dynamical oscillation belonging to the Sun itself, was taken up statistically by Dr. Simon Newcomb in a paper “On the Period of the Solar Spots.”* His criterion was to examine whether the more precise phases were equidistant throughout the record, or on the other hand their deviations increased continually (as √ n ) according to the law of errors. The phases chosen for scrutiny were four, those of maxima and minima and the two more definite intermediate times of mean spottedness. An analysis by the method of least squares led him to an unbroken period, of 11·13 ± 0·02 years, and thus of great definiteness. His conclusion is that “underlying the periodic variations of spot activity there is a uniform cycle unchanging from time to time and determining the general mean of the activity.” But to get this very remarkable degree of precision he had to reject the records belonging to the two decades around the year 1780, which showed violent irregularity in the phases. “I was at first disposed to think that these perturbations of the period might be real, but on more mature consideration I think they are to be regarded as errors arising from imperfection of the record. The derivation of any exact epoch requires a fairly continuous series of derivations made on a uniform plan. If we compare and combine the results of observa­tions made in an irregular or sporadic way, it may well be that the actual changes are masked by the apparent changes due only to these imperfections.” And, again, “it would seem from what precedes that a revision of the conclusions to be drawn from the observations of sunspots during the interval of 1775-1790 is very desirable.”

Revolving fluid in the atmosphere

Abstract: In a paper read on December 8, 1916, Lord Rayleigh makes an important contribution to the dynamics of revolving fluids, taking as “the starting point of part of his investigation,” the paper by Dr. Aitken on “The Dynamics of Cyclones and Anticyclones.” After setting out the general equations of motion of an inviscid fluid Lord Rayleigh says “for the present purpose we assume symmetry with respect to the axis of z so that u, v, w and P (assumed to be single valued) are independent of θ .” I take that to be the definition of a revolving fluid for the purpose of the subject under consideration, and it is that form of motion, that is to say, motion which is symmetrical with regard to a vertical axis, that Lord Rayleigh had in mind when he wrote the opening sentence of the paper: “So much of meteorology depends ultimately upon the dynamics of revolving fluid that it is desirable to formulate as clearly as possible such simple conclusions as are within our reach.” For most ordinary readers, meteorology in its dynamical aspect is quite rightly regarded merely as a useful synonym for “Cyclones and Anticyclones.” Dr. Aitken illustrates his view's of the nature of cyclones and anticyclones by many interesting experiments in the dynamics of revolving masses, and when Lord Rayleigh is moved by Dr. Aitken’s paper to set down clearly the conclusions that can be drawn from the theory of revolving fluids it is apparently with the hope that those conclusions may find their application in the phenomena exhibited by cyclones and anticyclones.

Torsional Hysteresis of Mild Steel

Abstract: The stress-strain curve from no load to fracture for mild steel as usually obtained consists of three parts: (1) A straight line, followed by a part deviating only slightly from this straight portion; (2) a sharp bend, followed by a part approximately parallel to the strain axis; and (3) a curved rising part, leading ultimately to the breaking point. It is generally assumed that Hooke’s Law holds throughout the part (1), and is immediately followed by the sharply defined bend which constitutes the yield point. For mild steel first stressed in tension and then in compression, or subjected to positive and then negative torsional stresses, the stress-strain curve within a considerable range of stress is also supposed to be a straight line. It is further well known that if mild steel is stressed in tension beyond the yield point the elastic limit is raised, but only at the expense of lowering it in compression; or, if it is twisted beyond the yield point in one direction, its elastic limit is raised for stresses in that direction, but lowered for those in the opposite direction. Attempts have been made to relate the range of stress through which the stress-strain curve is a straight line with that through which a material, such as mild steel, can be stressed an infinite number of times without fracture. This is expressed by the well known Bauschinger’s Law, which, as stated by Mr. Leonard Bairstow, is as follows:—“The superior limit of elasticity can be raised or lowered by cyclical variations of stress, and at the inferior limit of elasticity will be raised or lowered by a definite, but not necessarily the same, amount. The range of stress between the two elastic limits has therefore a value which depends only on the material and the stress at the inferior limit of elasticity. This elastic range of stress is the same in magnitude as the maximum range of stress, which can be repeatedly applied to a bar without causing fracture, no matter how great the number of repetitions.”

The Selective Properties of the Copper Ferrocyanide Membrane

Abstract: In a previous research dealing with the structure of semi-permeable membranes, it has been shown that the copper ferrocyanide membrane is built up from colloidal particles having diameters ranging from 50 to 400 μμ ; also that the pores enclosed by these particles have an average diameter which, although large enough to allow passage to any non-colloidal molecule, is small enough to be entirely within the range of capillary effects. On the basis of these facts the hypothesis was advanced that the selective properties of colloidal membranes are due to selective adsorption; a copper ferrocyanide membrane immersed in cane-sugar solution, for instance, being permeable to the water and not to the cane sugar, because the water, rather than the sugar, is selectively adsorbed or “condensed” on to the membrane surface and into its capillaries. The present research is an experimental investigation of this view. The measurements made herein consist in a determination of the change in concentration brought about by immersing more or less dry colloidal copper ferrocyanide in cane-sugar solutions of various strengths. The results obtained tend to confirm the adsorption hypothesis for this case at least.

Note on the Specific Heat of Water

Abstract: In a paper on the “Specific Heat of Water and the Mechanical Equivalent of the Calorie” were given the figures from 0° to 80°C. which resulted from our investigation. These figures agreed very fairly with those of some other observers, but differed considerably from the corrected figures of Callendar and Barnes. In our curve there was a minimum value of the specific heat at about 25°C., and a rapid rise afterwards from 25° to 60° of nearly the same magnitude as the fall from 0° to 25°. In the curve of Callendar and Barnes the minimum was not reached till 38°C., and thereafter the rise was much slower. Callendar soon after devised a novel method of “continuous mixture” by means of an ingeniously devised heat exchanger, and applied this apparatus to test his former results in this range between 60° and 100°. Water at about 100°C. from a heater was passed through the heat exchanger against water from a cooler at from 25° to 35°C. The result of these experiments indicated that the ratio of the mean specific heat from 69° to 100°C. to the mean specific heat from 25° to 56°C. agreed within 1 in 5000 with the ratio derived from Callendar’s formula based on the figures of Callendar and and Barnes. These results of this new method were published in Callendar’s Bakerian Lecture in 1912, and were made the basis of a criticism of our results and those of other observers.

On Magnetic Inertia

Abstract: In the course of some investigations with regard to the Earth’s magnetism, it occurred to me that a magnetised body must possess magnetic inertia in virtue of its magnetisation, just as an electrified body possesses electric inertia. I am not aware that this has been considered before, nor have I seen any calculation of the amount to be expected. The method of dimensions shows that the magnetic inertia of a body of magnetic moment m should be a numerical multiple of m 2 a -3C-2, where C is the velocity of radiation and a a linear dimension of the body ( e. g ., the radius, in the case of a sphere).