Showing papers in "Philosophical Transactions of the Royal Society A in 1927"

Abstract: 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

Abstract: Various problems concerning infinitely many, infinitely small, parts, had been solved before the infinitesimal calculus was invented; for example, Archimedes on the circumference of the circle. The essence of the invention of the calculus appears to be that the passage to the limit was thereby taken at the earliest possible stage, where diverse problems had operations like d / dx in common. Although the infinitesimal calculus has been a splendid success, yet there remain problems in which it is cumbrous or unworkable. When such difficulties are encountered it may be well to return to the manner in which they did things before the calculus was invented, postponing the passage to the limit until after the problem had been solved for a moderate number of moderately small differences. For obtaining the solution of the difference-problem a variety of arithmetical processes are available. This memoir deals with central differences arranged in the simplest possible way, namely, that explained by the writer in the papers cited in the footnote. Advancing differences are ignored, and so are the varieties of central-difference-process in which accuracy is gained by complicating the arithmetic at an early stage.

Abstract: To the pure mathematician of the present day the tensor calculus is a notation of differential geometry, of special utility in connection with multi-dimensional spaces; to the applied mathematician it is the backbone of the general theory of relativity. But when it is recognised that every problem in applied mathematics may be regarded as a geometrical problem (in the widest sense) and that the geometrical forms which many of these problems take are such that the tensor calculus can be directly applied, it is realised that the possibilities of this calculus in the field of applied mathematics can hardly be overestimated. It has a dual importance: first, by its help, known results may be exhibited in the most compact form; secondly, it enables the mathematician to exercise his most potent instrument of discovery, geometrical intuition. In the present paper we are concerned with the development of general dynamical theory with the aid of the tensor calculus. In view of the present close association of the tensor theory with the theory of relativity, it should be clearly understood that this paper only attempts to deal with the classical or Newtonian dynamics of a system of particles or of rigid bodies. The subject is presented in a semi-geometrical aspect, and the reader should visualise the results in order to realise the close analogy between general dynamical theory and the dynamics of a particle. Mathematicians display a strange reluctance in summoning to their assistance the power of visualisation in multidimensional space. They forget that they have studied the geometry of three dimensions largely through the medium of a schematic representation on a two dimensional sheet of paper. The same method is available in the case of any number of dimensions.

Abstract: Since its discovery by W. Swan almost exactly seventy years ago, speculations as to the origin of this familiar band spectrum have been prolific in spectroscopic literature. A good summary of these opinions and the experimental data upon which they were founded is given by Watts (1), writing in 1914. With the exception of one or two writers, the two opposing schools have favoured a Carbon molecule and a Hydrocarbon molecule, respectively. The latter school have for the most part specified in particular an acetylene molecule. Thus among a vast number of experimenters Van der Willigen (1859), Attfield (1862-1875), Dibbits (1864), Morren (1865), Plucker and Hittorf (1865), Huggins (1868), Wullner (1872), Salet (1873), Secchi (1873), Ciamician (1880), Deslandres (1888), and Eder (1890), favoured a carbon molecule (presumably Ca). On the other hand Swan (1856), Angstrom and Thalen (1875), Liveing and Dewar (1880), and many others favoured a hydrocarbon. The latter writers conducted a great deal of careful research (2) on the flames of Carbon compounds, the vacuum tube spectra of Carbon gases, and the Carbon arc in various gases, and they affirm that the emitter is an acetylene molecule, since this gas can be withdrawn from the flames of burning hydrocarbons which, show the bands well. In the light of our present knowledge one or two comments are suggested as we review all this work. It is clear that in the vast quantity of experimental work of the former school the extraordinary difficulty of ensuring the complete absence of Hydrogen—either occluded in the carbons or present as water vapour—was not appreciated. We have, for example, a similar and comparatively recent controversy (3) with reference to the origin of the Nitrogen afterglow, in which the degree of freedom from Oxygen of a sample of Nitrogen was the point at issue. Another very natural idea which prevailed until recently was that the emitters of band spectra were necessarily molecules known to chemistry, the most stable ones being those most likely to radiate under discharge conditions. Our present views are almost the reverse of these, and it seems probable that polar molecules cannot radiate band spectra at all. An excellent paper representative of these views has recently been published by Mulliken (4)—“On a class of one-valence electron emitters.” The old ideas, therefore, which led to the choice of an acetylene molecule as the radiator of the Swan bands were certainly ill-founded, but by a coincidence it happens that a C2H2 molecule must now be quite definitely accepted as their emitter. The present paper contains the evidence for this statement. Hitherto measurements of the fine structure of only three heads have been available, and these from photographs take under arc conditions. It has been found desirable to make a complete re-measurement of the fine structure of these and other heads under new low temperature conditions of production. Some 2,000 lines are tabulated. A considerable number of these have been given series assignment, and the modern quantum theory of band spectra in its various aspects has been applied to the data. The scope of the resulting analysis may be gauged from the table of contents.

Abstract: In an earlier paper on “Optical Rotatory Dispersion” (‘Phil. Trans.,’ 1912, A, vol. 212, pp. 261-297) a description was given of the measurement of the rotatory power of quartz for 24 wave-lengths in the visible region of the spectrum from Li 6708 to Hg 4358. Two important features of this research were:— (1) The discovery, after several years of work on inferior material, of a crystal of quartz of extraordinary optical purity, in which none but mechanical flaws could be detected in a plate 58 mm. in thickness and 150 mm. in diameter. (2) The use of long columns of quartz, made up of cylinders drilled from this crystal, giving even in the visible region rotations of the order of 10,000°, which could be read with an average error amounting only to a few parts per million.

Abstract: During the course of certain experiments conducted at the National Physical Laboratory in 1919, a number of observations were made of the normal pressure at points on the surface of a prolate spheroid of length/diameter ratio 4. The results were compared with the corresponding pressures, calculated by purely theoretical methods, on a similar body in rectilinear motion in an ideal fluid. The comparison showed that agreement between theory and experiment was remarkably good for three-quarters the length of the model, even when the axis of the model was inclined to the direction of motion. The closeness of the agreement suggested that a more extensive investigation would be likely to lead to very interesting results. It was therefore decided to conduct a comprehensive series of experiments with a view to obtaining a more complete comparison. The scope of the investigation was widened to cover not only rectilinear motion which is reproduced in wind-tunnel experiments, but to include purely rotational motion (spin about a minor axis) and also motion in a circle. The whirling arm at the Laboratory provided the necessary means for reproducing this motion in the experimental part of the work.

Abstract: In a previous paper were recorded the results of an investigation into the effects of repetitions of stress on the micro-structure of various metals in the form of crystalline aggregates, the main purpose of the investigation being a study of the causes of fracture under repeated stresses of relatively low magnitude. One important conclusion derived from the experiments was that the action of slipping was not, as had been previously stated, a weakening process in itself. Up to a point the effect of slip was actually to increase the resistance of the metal to further slip. Eventually, however, this strengthening action was exhausted, and failure commenced by the formation of a crack. It was suggested that failure occurred when the amount of strain-hardening by slip exceeded a certain limiting amount. No definite evidence could be obtained on this point, but it was considered that further information might be obtained if attention was directed to a material more simple in structure than a crystalline aggregate. In particular, it was desired to eliminate the effects of the crystal boundaries, whose nature is at present unknown. This could be accomplished if specimens cut entirely from one crystal were employed. Further, it should be possible to verify the assumption, commonly made, that slip bands represent the traces of actual “slip planes” on the surface of the specimen, and to relate these with the atomic structure of the material. Through the kindness of Prof. Carpenter and Miss Elam a number of large single crystals of aluminium were prepared and presented and have been used throughout this work. At that time the necessary experimental facilities for X-ray work were not available to the authors at the National Physical Laboratory. Prof. Carpenter offered to arrange for the X-ray analyses to be undertaken by his assistant, Miss C. F. Elam, at the Royal School of Mines. This offer was gratefully accepted and the authors are greatly indebted to Miss Elam for carrying out this section of the work.

Abstract: In a series of papers the author, partly in collaboration with the late Prof. H. Kamerlingh Onnes, has recently published the results of some investigations on paramagnetism at low temperatures. Included in this work were the measurements of the principal susceptibilities of two crystals (cobalt ammonium sulphate and nickel sulphate) at temperatures ranging from about 300° K. (atmospheric temperature) down to the lowest temperature obtainable with liquid hydrogen, 14° K. These data, with the exception of those of Foex for siderose, are the only ones yet obtained for the principal susceptibilities of paramagnetic crystals at low temperatures over any extended range of temperature. The results showed that, at not too low temperatures, the principal susceptibilities Z 1, Z 2, Z 3 follow the law Z n (T + Δ n = C, ( n = I, 2, 3), in which T = absolute temperature, Δ n is a constant, and C is the Curie constant which has the same value for each of the principal magnetic axes of the crystal. The constants Δ n are intimately connected with the structure of the crystal, being a function of the “spacing” of the paramagnetic atoms in the corresponding directions in the crystal. The precise connection between these quantities could not, however, be deduced, firstly, because of the present scantiness of the data, and, secondly, because the accuracy with which the Δ’s could be determined was small. The susceptibilities themselves were determined with an accuracy of about 1 per cent., but, since Δ is only an additive constant, the error in its determination is greater than that of the determination of the susceptibility. Other interesting points were raised by the results, and a continuation of the research seemed likely to give results of considerable theoretical importance and interest. The present work was undertaken in continuation of that just mentioned. It was decided to carry out the measurements over a range of temperature of from atmospheric temperature down to the lowest obtainable with the aid of liquid air, and to aim at reaching an accuracy of one part in a thousand in the measurements. For this purpose the apparatus described in the following pages was designed and constructed. I he desired accuracy was not attained in the first measurements given later in the present paper, the accuracy of these being about 1 per cent. As experience was gained with the apparatus, it was seen that the desired accuracy was probably attainable when special attention was paid to the working conditions, in particular the constancy of the temperature.