- Alfred Barnard Basset published A treatise on hydrodynamics, with numerous examples in 1888. Here is the Preface of that book:
- Alfred Barnard Basset published An elementary treatise on hydrodynamics and sound (1890). A second revised edition was published in 1900. Here is the Preface of that book:
- Alfred Barnard Basset published A treatise on physical optics (1892). Here is the Preface of that book:
- Alfred Barnard Basset published An elementary treatise on cubic and quartic curves (1901). Here is the Preface of that book:
- Alfred Barnard Basset published A treatise on the geometry of surfaces (1910). Here is the Preface of that book:
In the present Treatise I have endeavoured to lay before the reader in a connected form, the results of the most important investigations in the mathematical theory of Hydrodynamics, which have been made during modem times. The Science of Hydrodynamics may properly be considered to include an enquiry into the motion of all fluids, gaseous as well as liquid; but for reasons which are stated in the introductory paragraph of Chapter I, the present treatise is confined almost entirely to the motion of liquids. The progress of scientific knowledge in all its branches has been the peculiar feature of the present century, and it is therefore not surprising that during the last fifty years a great increase in hydrodynamical knowledge has taken place; but many of the most important results of writers upon this subject have never been inserted in any treatise, and still lie buried in a variety of British and foreign mathematical periodicals and transactions of learned Societies; and it has been my aim to endeavour to collect together those investigations which are of most interest to the mathematician, and to condense them into a form suitable for a treatise.
The present work is divided into two volumes, the first of which deals with the theory of the motion of frictionless liquids, up to and including the theory of the motion of solid bodies in a liquid. In the second volume, a considerable portion of which is already written, it is proposed to discuss the theory of rectilinear and circular vortices; the motion of a liquid ellipsoid under the influence of its own attraction, including Professor G H Darwin's important memoir on dumb-bell figures of equilibrium; the theories of liquid waves and tides; and the theory of the motion of a viscous liquid and of solid bodies therein.
References have been given throughout to the original authorities which have been incorporated or consulted; and a collection of examples has been added, most of which have been taken from University or College Examination Papers, which have been set during recent years.
The valuable report of Mr W M Hicks on Hydrodynamics, to the British Association in 1881-2, has proved of great service in the difficult task of collecting and arranging materials. I have also to express my obligations to the English treatises of Dr Besant and Professor Lamb, from the latter of which I have received considerable assistance in Chapters IV and VI; and also to the German treatise of the late Professor Kirchhoff.
I am greatly indebted to Professor Greenhill for his kindness in having read the proof sheets, and also for having made many valuable suggestions during the progress of the work. In a treatise which contains a large amount of analytical detail, it is probable that there are several undetected errors; and I shall esteem it a favour if those of my readers who discover any errors or obscurities of treatment, or have any suggestions to make, will communicate with me.
The treatise on Hydrodynamics, which I published in 1888, was intended for the use of those who are acquainted with the higher branches of mathematics, and its aim was to present to the reader as comprehensive an account of the whole subject as was possible. But although a somewhat formidable battery of mathematical artillery is indispensable to those who desire to possess an exhaustive knowledge of any branch of mathematical physics, yet there are a variety of interesting and important investigations, not only in Hydrodynamics but also in Electricity and other physical subjects, which are well within the reach of everyone who possesses a knowledge of the elements of the Differential and Integral Calculus and the fundamental principles of Dynamics. I have accordingly, in the present work, abstained from introducing any of the more advanced methods or analysis, such as Spherical Harmonics, Elliptic Functions and the like; and, as regards the dynamical portion or the subject, I have endeavoured to solve the various problems which present themselves, by the aid of the Principles or Energy and Momentum, and have avoided the use or Lagrange's equations. There are a few problems, such as the helicoidal steady motion and stability of a solid or revolution moving in an infinite liquid, which cannot be conveniently treated without having recourse to moving axes; but as the theory of moving axes is not an altogether easy branch of Dynamics, I have as far as possible abstained from introducing them, and the reader who is unacquainted with the use of moving axes is recommended to omit those sections in which they are employed.
The present work is principally designed for those who are reading for the Mathematical Tripos and for other examinations in which an elementary knowledge of Hydrodynamics and Sound is required; but I also trust that it will not only be of service to those who have neither the time nor the inclination to become conversant with the intricacies of the higher mathematics, but that it will also prepare the way for the acquisition of more elaborate knowledge, on the part of those who have an opportunity of devoting their attention to the more recondite portions of these subjects.
The first part, which relates to Hydrodynamics, has been taken with certain alterations and additions from my larger treatise, and the analytical treatment has been simplified as much as possible. I have thought it advisable to devote a chapter to the discussion of the motion of circular cylinders and spheres, in which the equations of motion are obtained by the direct method of calculating the resultant pressure exerted by the liquid upon the solid; inasmuch as this method is far more elementary, and does not necessitate the use of Green's Theorem, nor involve any further knowledge of Dynamics on the part of the reader than the ordinary equations of motion of a rigid body. The methods of this chapter can also be employed to solve the analogous problem of determining the electrostatic potential of cylindrical and spherical conductors and accumulators. and the distribution of electricity upon such surfaces. The theory of the motion of a solid body and the surrounding liquid, regarded as a single dynamical system, is explained in Chapter III, and the motion of an elliptic cylinder in an infinite liquid, and the motion of a circular cylinder in a liquid bounded by a rigid plane are discussed at length.
The Chapters on Waves and on Rectilinear Vortex Motion comprise the principal problems which admit of treatment by elementary methods, and I have also included an investigation due to Lord Rayleigh, respecting one of the simpler cases of the instability of fluid motion.
In the second part, which deals with the Theory of Sound, I have to acknowledge the great assistance which I have received from Lord Rayleigh's classical treatise. This part contains the solution of the simpler problems respecting the vibrations of strings, membranes, wires and gases. A few sections are also devoted to the Thermodynamics of perfect gases, principally for the sake of supplementing Maxwell's treatise on Heat, by giving a proof of some results which require the use of the Differential Calculus.
The present edition has been carefully revised throughout, and a certain amount of new matter has been added. I have devoted Chapter IX to the flexion and vibrations of naturally straight wires and rods; whilst an entirely new chapter has been added on the finite deformation of naturally straight and curved wires, in which I have discussed a variety of questions which admit of fairly simple mathematical treatment.
The Science of Optics embraces so large a class of phenomena, that any treatise which attempted to give a comprehensive account of all the various practical applications of Optics, in addition to the experimental and theoretical portions of the subject, would necessarily be of an exceedingly voluminous character. I have accordingly limited the present work to one special branch of Optics, and have endeavoured to place before the reader a concise a treatise upon the Mathematical Theory of Light, and such experimental phenomena as are immediately connected therewith, as the nature of the case will admit.
Those who are acquainted with the Mathematical Theories of Hydrodynamics, Sound and Elasticity on the one hand, and of Electricity and Light on the other, cannot fail to have been struck with the difference, which exists between the two classes of subjects. In the former class, certain equations are obtained, which approximately, though not quite accurately, specify in a mathematical form, the physical state of fluids and solids as they exist in Nature; and the subsequent, investigation of these branches of Science, is thereby in great measure reduced to a question of mathematics. As soon as the fundamental equations are established, the subject is brought within the dominion of mathematical analysis and mathematicians are enabled to exercise their ingenuity and analytical skill, in elaborating and developing the results which flow from them.
But in the Theory of Light, we are confronted with a totally different state of things. Although the existence of the luminiferous ether may, at the present day, be regarded as a scientific axiom, which is as firmly established as any other scientific law, yet the properties of the ether are almost entirely unknown to us. We are therefore unable to start with certain definite equations, of whose approximate correctness we can feel assured; but are compelled to formulate certain hypotheses concerning the physical constitution of the ether, which are capable of being expressed in mathematical form, and then to trace the consequences to which they lead us. The only means at our disposal for testing the correctness of any hypothesis, which forms the foundation of any dynamical theory of light, is to compare the results furnished by theory, with known experimental facts; accordingly a knowledge of the facts, which it is the object of the theory to explain, is of the utmost importance.
Under these circumstances, it has been necessary to describe at length, a variety of phenomena of an experimental character; but as the object of this work is to investigate the dynamical theory of light in relation to experimental phenomena, I have abstained from entering into many details respecting the methods of performing optical experiments, or the description of the necessary instrumental appliances. Those who desire a fuller acquaintance with the experimental portions of the subject, are recommended to consult Verdet's Leçons d'Optique Physique, Mascart's Traité d'Optique, and Preston's Theory of Light, where ample information concerning these matters will be found.
The first ten chapters are devoted to the consideration of Interference, Colours of Thick and Thin Plates, Diffraction, Double Refraction, Rotary Polarization, and Reflection and Refraction of Polarized Light; and in these chapters, dynamical theories are as far as possible dispensed with. The remaining ten chapters are of a more speculative character, and contain an account of some of the dynamical theories, which have been proposed to explain optical phenomena. The investigations of numerous physicists upon the theories of Reflection and refraction at the Surfaces of isotopic and Crystalline Media, upon Double Refraction, Absorption, Anomalous Dispersion and Metallic reflection are considered; and a description of a variety of experimental results, which require a dynamical theory to account for them, is given. The last two chapters are devoted to Maxwell's Electromagnetic Theory, together with the additions made to it since the death of the author. The last chapter of all, contains a description of the experimental results of Faraday, Kerr and Kundt on the action of electromagnetism on light; together with a development of Maxwell's theory, which is believed to be capable of accounting for the action of a magnetic field, when light is propagated through a transparent medium.
It must be admitted, that some of the theories discussed in the later Chapters are of a somewhat speculative character, and will require reconsideration as our knowledge of the properties of matter increases; but at the same time, it is a great assistance to the imagination to be able to construct a mechanical model of a medium, which represents, even imperfectly, the action of ponderable matter upon ethereal waves. A full and complete investigation, by the aid of rigorous mathematical analysis, of the peculiar action of a medium, which is assumed to possess certain definite properties, enables us to understand the reason why certain effects are produced, and cannot fail to impress upon the mind the conviction, that the dynamical theory of light is a reality, rather than a scientific speculation. I have a profound distrust of vague and obscure arguments, based upon general reasoning instead of rigorous mathematical analysis. Investigations founded upon such considerations are always difficult to follow, are frequently misleading, and are sometimes erroneous. When the conditions of a dynamical problem are completely specified, all the circumstances connected with the motion can be expressed in mathematical language, and equations obtained, which are sufficient for the solution of every conceivable problem; and if the number of equations obtained is insufficient, the specification is incomplete, or some necessary condition has been overlooked. I consider it most important to the interests of mathematical physics, that the solutions of the various problems which present themselves, should be properly worked out by rigorous analysis, whenever it is possible to do so, and that definite mathematical results should be obtained and interpreted.
At the end of some of the earlier Chapters, examples and problems have been inserted, which have been derived from the examination papers set in the University of Cambridge. During recent years, there has been a disposition in certain quarters to question the educational value of examples; and a little sarcasm has occasionally been indulged in, with reference to so-called mathematical conundrums. One of the difficulties, which the Examiners for the Mathematical Tripos have to contend against, is the tendency on the part of the Candidates to devote their time to learning certain pieces of book-work, which are likely to be set, instead of endeavouring to acquire an accurate and thorough knowledge of the fundamental facts and principles of the subjects, which they take up; and well-selected examples and problems illustrating the book-work are of great assistance to an examiner, in enabling him to discriminate between candidates, who have acquired a perfunctory knowledge of a subject, and those who have endeavoured to master it. But it would be a fallacy to imagine, that the utility of examples and problems is exclusively confined to the particular examination in which they are set; or that the practical value of a problem is to be estimated solely by the scientific value of the result, which it embodies. The existence of a large collection of examples and problems, is of great assistance to future generations of students, in enabling them to grasp the fundamental principles of a subject, and to acquire facility in the application of mathematical analysis to physics; and the severe course of training, which the University of Cambridge extracts from students of the higher branches of Mathematics, is of inestimable benefit to Science, in producing a body of men, who are thoroughly conversant with dynamical principles, and are able to employ with ease the more recondite processes of mathematical analysis.
That the scientific discoveries of the present century have been of incalculable benefit to mankind, will be admitted by all; but it is also a distinct advantage to Science, when any discovery in abstract Science turns out to be of practical utility. The optical properties of chemical compounds have already been applied as a test of their purity; and it is probable, that further investigations of this character will be found to place to place a powerful weapon in the hands of Chemists.
I have to acknowledge the great assistance, which I have received from Verdet's Leçons d'Optique Physique, as well as from the original papers of the eminent mathematicians and experimentalists, which are referred to in the body of this treatise. I am also much indebted to Mr J Larmor for having read the proof sheets, and for having made numerous valuable suggestions during the progress of the work.
The present work originated in certain notes, made about twenty-five years ago, upon the properties of some of the best-known higher plane curves; but upon attempting to revise them for the press, it appeared to me impossible to discuss the subject adequately without investigating the theory of the singularities of algebraic curves. I have accordingly included Plücker's equations, which determine the number and the species of the simple singularities of any algebraic curve; and have also considered all the compound singularities which a quartic curve can possess.
This treatise is intended to be an elementary one on the subject. I have therefore avoided investigations which would require a knowledge of Modern Algebra, such as the theory of the invariants, covariants and other concomitants of a ternary quantic: and have assumed scarcely any further knowledge of analysis on the part of the reader, than is to be found in most of the ordinary text-books on the Differential Calculus and on Analytical Geometry. I have also endeavoured to give special prominence to geometrical methods, since the experience of many years has convinced me that a judicious combination of geometry and analysis is frequently capable of shortening and simplifying, what would otherwise be a tedious and lengthy investigation.
The introductory Chapter contains a few algebraic definitions and propositions which are required in subsequent portions of the work. The second one deals with the elementary theory of the singularities of algebraic curves and the theory of polar curves, The third Chapter commences with an explanation of tangential coordinates and their uses, and then proceeds to discuss a variety of miscellaneous propositions connected with reciprocal polars, the circular points at infinity and the foci of curves. Chapter IV is devoted to Plücker's equations; whilst Chapter V contains an account of the general theory of cubic curves, including the formal proof of the principal properties which are common to all curves of this degree. In this Chapter I have almost exclusively employed trilinear coordinates, since the introduction of a triangle of reference, whose elements can be chosen at pleasure, constitutes a vast improvement on the antiquated methods of homogeneous coordinates and abridged notation. Chapter VI is devoted to the consideration of a variety of special cubics, including the particular class of circular cubics which are the inverses of conics with respect to their vertices; and in this Chapter the method of Cartesian coordinates is the most appropriate. A short Chapter then follows on curves of the third class, after which the discussion of quartic curves commences.
To adequately consider such an extensive subject as quartic curves would require a separate treatise. I have therefore confined the discussion to the simple and compound singularities of curves of this degree, together with a few miscellaneous propositions; and in Chapter IX, I proceed to investigate the theory of bicircular quartics and cartesians, concluding with the general theory of circular cubics, which is better treated as a particular case of bicircular quartics than as a special case of cubic curves. Chapter X is devoted to the consideration of various well known quartic curves, most of which are bicircular or are cartesians; whilst Chapter XI deals with cycloidal curves, together with a few miscellaneous curves which frequently occur in mathematical investigations. The theory of projection, which forms the subject of the last Chapter, is explained in most treatises on Conics; but except in the case of conics, due weight has not always been given to the important fact that the projective properties of any special class of curves can be deduced from those of the simplest curve of the species. Thus all the projective properties of tricuspidal quartics can be obtained from those of the three-cusped hypocycloid or the cardioid; those of quartics with a node and a pair of cusps from the limaçon; those of quartics with three biflecnodes from the lemniscate of Bernoulli or the reciprocal polar of the four-cusped hypocycloid; whilst the properties of binodal and bicuspidal quartics can be obtained from those of bicircular quartics and cartesians.
Whenever the medical profession require a new word they usually have recourse to the Greek language, and mathematicians would do well to follow their example; since the choice of a suitable Greek word supplies a concise and pointed mode of expression which saves a great deal of circumlocution and verbosity. The old-fashioned phrase "a non-singular cubic or quartic curve" involves a contradiction of terms, since Plücker has shown that all algebraic curves except conics possess singularities; and I have therefore introduced the words 'autotomic' and 'anautotomic' to designate curves which respectively do and do not possess multiple points. The words 'perigraphic', 'endodromic' and 'exodromic', which are defined on page 14, are also useful; in fact a word such as 'aperigraphic' is indispensable in order to avoid the verbose phrase "a curve which has branches extending to infinity."
At the present day the subject of Analytical Geometry covers so vast a field that it is by no means easy to decide what to insert and what to leave out. I trust, however, that the present work will form a useful introduction to the higher branches of the subject; and will facilitate the study of a variety of curves whose properties, by reason of their beauty and elegance, deserve at least as much attention as the well-worn properties of conics.
The last edition of Salmon's Analytic Geometry of Three Dimensions, which was published in 1884, has been out of print for some years; and although there are several excellent works on Quadric Surfaces and other special branches of the subject, such as those of Mr Blythe on Cubic Surfaces and of the late Mr Hudson on Kummer's Quartic Surface, yet there is no British treatise exclusively devoted to the theory of surfaces of higher degree than the second. I have therefore endeavoured to supply this want in the present work.
The Theory of Surfaces is an extensive one, and a thoroughly comprehensive treatise would necessarily be voluminous. I have therefore decided to limit this work to the more elementary portions of the subject, and have abstained front introducing investigations which require a knowledge of the Theory of Functions and of the higher branches of Modern Algebra. The ordinary methods of Analytical Geometry arc quite sufficient to enable the properties of cubic and quartic surfaces and twisted curves, and also the point and plane singularities of surfaces, to be discussed with tolerable completeness, and to demonstrate a number of interesting and important theorems connected with them; but for the purpose of confining this treatise within a moderate compass, I have abstained from any general discussion of surfaces of higher degree than the fourth.
The properties of a point singularity may usually be examined by means of a surface of low degree just all well as by one of the nth degree; but if the degree is less than a certain limit, which depends on the character of the singularity, the latter appears in an incomplete form on the surface. Thus the properties of a triple line cannot be fully investigated without employing a surface of the seventh degree, and this fact has rendered it necessary to partially discuss surfaces of higher degree than a quartic.
The resolution of a multiple point into its constituents has been discussed by Professor Segre of Turin, and other Italian mathematicians, in various papers published in the Annali di Matematica; and these researches have shown that an important analogy exists between the theories of plane curves and of surfaces. The class of an anautotomic plane curve of degree n, and also the reduction of class produced by a multiple point of order n, the tangents at which are distinct, are both equal to n(n - 1); whilst the constituents of the multiple point are n(n - 1)/2 nodes. The class of an anautotomic surface of degree n, and also the reduction of class produced by a multiple point of order n, the tangent cone at which is anautotomic, are both equal to n(n - 1)2; and from analogy I concluded that the constituents of the multiple point were f conic nodes. In 1908 I succeeded in obtaining a formal proof of the last theorem, which enables a large number of singular points to be resolved into their constituent conic nodes and binodes.
In the present treatise I have incorporated a variety of results, originally due to Italian and German mathematicians, many of which have been published since the last edition of Salmon's work; and I have endeavoured to modernize the analysis and the terminology by discarding antiquated methods and inappropriate symbols and phrases. I have also to express my obligations to the late Professor Cayley's papers, references to which are denoted by the letters C. M. P.; as well as to the Repertorio di Matematiche Superiori by Professor E Pascal, which contains a valuable epitome of the subject, together with an exhaustive collection of references to the original papers of British and foreign mathematicians, who have studied the subject.