The Royal Society

In the course of an investigation of the effect of surface scratches on the mechanical strength of solids, some general conclusions were reached which appear to have a direct bearing on the problem of rupture, from an engineering standpoint, and also on the larger question of the nature of intermolecular cohesion. The original object of the work, which was carried out at the Royal Aircraft Estab­lishment, was the discovery of the effect of surface treatment—such as, for instance, filing, grinding or polishing—on the strength of metallic machine parts subjected to alternating or repeated loads. In the case of steel, and some other metals in common use, the results of fatigue tests indicated that the range of alternating stress which could be permanently sustained by the material was smaller than the range within which it was sensibly elastic, after being subjected to a great number of reversals. Hence it was inferred that the safe range of loading of a part, having a scratched or grooved surface of a given type, should be capable of estimation with the help of one of the two hypotheses of rupture commonly used for solids which are elastic to fracture. According to these hypotheses rupture may be expected if (a) the maximum tensile stress, ( b ) the maximum extension, exceeds a certain critical value. Moreover, as the behaviour of the materials under consideration, within the safe range of alternating stress, shows very little departure from Hooke’s law, it was thought that the necessary stress and strain calculations could be performed by means of the mathematical theory of elasticity.

The present paper contains a discussion of some optical properties of a medium containing minute metal spheres. The discussion is divided into two Parts: the first Part dealing with colours in metal glasses, in which the proportion of volume occupied by metal is small; the second Part dealing with metal films, in which this proportion may have any value from zero to unity. In Part I. the observations of Siedentopf and Zsigmondy beyond the limit of microscopic vision (‘Ann. der Phys.,’ January, 1903) are discussed. It is shown that the particles seen in a gold ruby glass are particles of gold which, when their diameters are less than 0.1μ, are accurately spherical. I have endeavoured to show that the presence of many of these minute spheres to a wave-length of light in the glass will account for all the optical properties of “regular” gold ruby glass, and that the irregularities in colour and in polarisation effects sometimes exhibited by gold glass are due to excessive distance between consecutive gold particles or to excessive size of such particles, the latter, however, involving the former. It is also shown that the radiation from radium is capable of producing in gold glass the ruby colour which is generally produced by re-heating. The method adopted enables us to predict from a knowledge of the metal present in metallic form in a glass what colour that glass will be in its “regular” state.

Several reasons have contributed to the prolonged neglect into which the study of statistics, in its theoretical aspects, has fallen. In spite of the immense amount of fruitful labour which has been expended in its practical applications, the basic principles of this organ of science are still in a state of obscurity, and it cannot be denied that, during the recent rapid development of practical methods, fundamental problems have been ignored and fundamental paradoxes left unresolved. This anomalous state of statistical science is strikingly exemplified by a recent paper entitled "The Fundamental Problem of Practical Statistics," in which one of the most eminent of modern statisticians presents what purports to be a general proof of BAYES' postulate, a proof which, in the opinion of a second statistician of equal eminence, "seems to rest upon a very peculiar -- not to say hardly supposable -- relation."

In recent years much information has been accumulated about the flow of fluids past solid boundaries. All experiments so far carried out seem to indicate that in all cases steady motion is possible if the motion be sufficiently slow, but that if the velocity of the fluid exceeds a certain limit, depending on the viscosity of the fluid and the configuration of the boundaries, the steady motion breaks down and eddying flow sets in. A great many attempts have been made to discover some mathematical representation of fluid instability, but so far they have been unsuccessful in every case. The case, for instance, in which the fluid is contained between two infinite parallel planes which move with a uniform relative velocity has been discussed by Kelvin, Rayleigh, Sommerfeld, Orr, Mises, Hope, and others. Each of them cam e to the conclusion that the fundamental small disturbances of this system are stable. Though it is necessarily impossible to carry out experiments with infinite planes, it is generally believed that the motion in this case would be turbulent, provided the relative velocity of the two planes were sufficiently great.

The present memoir is the outcome of an attempt to obtain the conditions under which a given symmetric and continuous function k ( s, t ) is definite, in the sense of Hilbert. At an early stage, however, it was found that the class of definite functions was too restricted to allow the determination of necessary and sufficient conditions in terms of the determinants of § 10. The discovery that this could be done for functions of positive or negative type, and the fact that almost all the theorems which are true of definite functions are, with slight modification, true of these, led finally to the abandonment of the original plan in favour of a discussion of the properties of functions belonging to the wider classes. The first part of the memoir is devoted to the definition of various terms employed, and to the re-statement of the consequences which follow from Hilbert’s theorem.

There are few branches of the Theory of Evolution which appear to the mathematical statistician so much in need of exact treatment as those of Regression, Heredity, and Panmixia. Round the notion of panmixia much obscurity has accumulated, owing to the want of precise definition and quantitative measurement. The problems of regression and heredity have been dealt with by Mr. Francis Galton in his epochmaking work on ‘Natural Inheritance,’ but, although he has shown exact methods of dealing, both experimentally and mathematically, with the problems of inheritance, it does not appear that mathematicians have hitherto developed his treatment, or that biologists and medical men have yet fully appreciated that he has really shown how many of the problems which perplex them may receive at any rate a partial answer. A considerable portion of the present memoir will be devoted to the expansion and fuller development of Mr. Galton’s ideas, particularly their application to the problem of bi-parental inheritance . At the same time I shall endeavour to point out how the results apply to some current biological and medical problems. In the first place, we must definitely free our minds, in the present state of our knowledge of the mechanism of inheritance and reproduction, of any hope of reaching a mathematical relation expressing the degree of correlation between individual parent and individual offspring. The causes in any individual case of inheritance are far too complex to admit of exact treatment; and up to the present the classification of the circumstances under which greater or less degrees of correlation between special groups of parents and offspring may be expected has made but little progress. This is largely owing to a certain prevalence of almost metaphysical speculation as to the causes of heredity, which has usurped the place of that careful collection and elaborate experiment by which alone sufficient data might have been accumulated, with a view to ultimately narrowing and specialising the circumstances under which correlation was measured. We must proceed from inheritance in the mass to inheritance in narrower and narrwoer classes, rather than attempt to build up general rules on the observation of individual instances. Shortly, we must proceed by the method of statistics, rather than by the consideration of typical cases. It may seem discouraging to the medical practitioner, with the problem before him of inheritance in a particular family, to be told that nothing but averages, means, and probabilities with regard to large classes can as yet be scientifically dealt with ; but the very nature of the distribution of variation, whether healthy or morhid, seems to indicate that we are dealing with that sphere of indefinitely numerous small causes, which in so many other instances has shown itself only amenable to the calculus of chance, and not to any analysis of the individual instance. On the other hand, the mathematical theory wall be of assistance to the medical man by answering, inter alia, in its discussion of regression the problem as to the average effect upon the offspring of given degrees of morbid variation in the parents. It may enable the physician, in many cases, to state a belief based on a high degree of probability, if it offers no ground for dogma in individual cases. One of the most noteworthy results of Mr. Francis Galton’s researches is his discovery of the mode in which a population actually reproduces itself by regression and fraternal variation. It is with some expansion and fuller mathematical treatment of these ideas that this memoir commences.

1. Introduction.— 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a “Freehand Graphic Way of Determining Stream Lines and Equipotentials” (‘Phil. Mag.,’February, 1908; also ‘Proc. Physical Soc.,’ London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited.

1. This paper treats of the propagation of vibrations over the surface of a “semiinfinite” isotropic elastic solid, i. e. ,a solid bounded only by a plane. For purposes of description this plane may be conceived as horizontal, and the solid as lying below it, although gravity is not specially taken into account. The vibrations are supposed due to an arbitrary application of force at a point. In the problem most fully discussed this force consists of an impulse applied vertically to the surface; but some other cases, including that of an internal source of disturbance, are also (more briefly) considered. Owing to the complexity of the problem, it has been thought best to concentrate attention on the vibrations as they manifest themselves at the free surface. The modifications which the latter introduces into the character of the waves propagated into the interior of the solid are accordingly not examined minutely.

If we take a curve representing a simple harmonic function of the time, and superpose on the ordinates small random errors, the only effect is to make the graph somewhat irregular, leaving the suggestion of periodicity still quite clear to the eye. Fig. 1 ( a ) shows such a curve, the random errors having been determined by the throws of dice. If the errors are increased in magnitude, as in fig. 1 ( b ), the graph becomes more irregular, the suggestion of periodicity more obscure, and we have only sufficiently to increase the “errors” to mask completely any appearance of periodicity. But, however large the errors, periodogram analysis is applicable to such a curve, and, given a sufficient number of periods, should yield a close approximation to the period and amplitude of the underlying harmonic function. When periodogram analysis is applied to data respecting any physical phenomenon in the expectation of eliciting one or more true periodicities, there is usually, as it seems to me, a tendency to start from the initial hypothesis that the periodicity or periodicities are masked solely by such more or less random superposed fluctuations — fluctuations which do not in any way disturb the steady course of the underlying periodic function or functions. It is true that the periodogram itself will indicate the truth or otherwise of the hypothesis made, but there seems no reason for assuming it to be the hypothesis most likely a priori .

1. In the ordinary theory of statistical correlation, normal or otherwise, we are always supposed to be dealing with material susceptible of continuous variation, or at least of variation by a considerable number of discontinuous steps. The correlations of lengths or measurements on portions of the body form examples of the first kind; of numbers of children in families, petals or other parts of flowers, are examples of the second. Certain practical cases arise, however, where either no variation is thinkable at all, or else is not measured or possibly measurable. We may class a number of individuals into deaf and not deaf, blind and not blind, imbecile and not imbecile, without attempting to go further (although gradations of deafness, blindness, and imbecility occur), and demand on the basis of the enumeration a discussion of the association of the three infirmities. Or again the data may be the mortality from some disease with and without the administration of, say, a new antitoxin, the statistics giving number who died to whom antitoxin was administered, number who died to whom antitoxin was not administered; number who did not die to whom antitoxin was not administered, number who did not die to whom antitoxin was administered; and from these data a discussion of the value of the cure is required. Here there is no scale of “death”; there may be a scale of “antitoxin” if the dose varied, but not otherwise.