- Alfred B Basset, On the Motion of a Sphere in a Viscous Liquid [Abstract], Proceedings of the Royal Society of London 43 (1887-1888), 174-175.
The determination of the small oscillations and steady motion of a sphere which is immersed in a viscous liquid, and which is moving in a straight line, was first effected by Professor Stokes in his well-known memoir "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums;" and in the appendix he also determines the steady motion of a sphere which is rotating about a fixed diameter. The same subject has also been subsequently considered by Helmholtz and other German writers; but, so far as I have been able to discover, very little appears to have been effected with respect to the solution of problems in which a solid body is set in motion in a viscous liquid in any given manner, and then left to itself. In the present paper I have endeavoured to determine the motion of a sphere which is projected vertically upwards or downwards with given velocity, and allowed to ascend or descend under the action of gravity (or any constant force), and which is surrounded by a viscous liquid of unlimited extent, which is initially at rest excepting so far as it is disturbed by the initial motion of the sphere. In solving this problem, mathematical difficulties have compelled me to neglect the squares and products of velocities, and quantities depending thereon, which involves the assumption that the velocity of the sphere is always small throughout the motion; and I have also assumed that no slipping takes place at the surface of the sphere.
- Alfred B Basset, On the Motion of a Sphere in a Viscous Liquid, Philosophical Transactions of the Royal Society of London A 179 (1888), 43-63.
The first problem relating to the motion of a solid body in a viscous liquid which was successfully attacked was that of a sphere, the solution of which was given by Professor Stokes in 1850, in his memoir "On the Effect of the Internal Friction of Fluids on Pendulums," Cambridge Phil. Soc. Trans. 9, in the following cases: (i) when the sphere is performing small oscillations along a straight line; (ii) when the sphere is constrained to move with uniform velocity in a straight line; (iii) when the sphere is surrounded by an infinite liquid and constrained to rotate with uniform angular velocity about a fixed diameter: it being supposed, in the last two cases, that sufficient time has elapsed for the motion to have become steady. In the same memoir he also discusses the motion of a cylinder and a disc. The same class of problems has also been considered by Meyer and Oberbeck, the latter of whom has obtained the solution in the case of the steady motion of an ellipsoid, which moves parallel to any one of its principal axes with uniform velocity. The torsional oscillations about a fixed diameter, of a sphere which is either filled with liquid or is surrounded by an infinite liquid when slipping takes place at the surface of the sphere, forms the subject of a joint memoir by Helmholtz and Piotrowski. Very little appears to have been effected with regard to the solution of problems in which a viscous liquid is set in motion in any given manner and then left to itself. The solution, when the liquid is bounded by a plane which moves parallel to itself, is given by Professor Stokes at the end of his memoir referred to above; and the solutions of certain problems of two-dimensional motion have been given by Stearn. In the present paper I propose to obtain the solution for a sphere moving in a viscous liquid in the following cases:- (i) when the sphere is moving in a straight line under the action of a constant force, such as gravity; (ii) when the sphere is surrounded by viscous liquid and is set in rotation about a fixed diameter and then left to itself.
- Alfred B Basset, On the Steady Motion of an Annular Mass of Rotating Liquid, American Journal of Mathematics 11 (2) (1889), 172-181.
The recent investigations of Poincare and Professor G H Darwin have drawn attention to the problem of the figures of equilibrium of rotating masses of liquid; and in the present paper it is proposed to consider the steady motion of an annular mass of liquid whose cross-section is approximately circular, and which is rotating as a rigid body under the influence of its own attraction, about an axis through its centre of inertia which is perpendicular to the plane of its central line. This problem has to some extent been dealt with by Poincaré, who has proved that such figures are possible forms of surfaces of equilibrium; but the subject is capable of further development, and the object of this paper is to show how a solution may be obtained to any degree of approximation by the aid of the Toroidal Function analysis which has been so successfully employed by Mr W M Hicks in his investigations on circular vortex rings.
- Alfred B Basset, On the Extension and Flexure of Cylindrical and Spherical Thin Elastic Shells, Philosophical Transactions of the Royal Society of London A 181 (1890), 433-480.
The various theories of thin elastic shells which have hitherto been proposed have been discussed by Mr Love in a recent memoir, and it appears that most, if not all of them, depend upon the assumption that the three stresses which are usually denoted by R, S, T are zero; but, as I have recently pointed out, a very cursory examination of the subject is sufficient to show that this assumption cannot be rigorously true. It can, however, be proved that, when the external surfaces of a plane plate are not subjected to pressure or tangential stress, these stresses depend upon quantities proportional to the square of the thickness, and whenever this is the case they may be treated as zero in calculating the expression for the potential energy due to strain, because they give rise to terms proportional to the fifth power of the thickness, which may neglected, since it is usually unnecessary to retain powers of the thickness higher than the cube. It will also, in the present paper, be shown by an indirect method that a similar proposition is true in the case of cylindrical and spherical shells, and, therefore, the fundamental hypothesis upon which Mr Love has based his theory, although unsatisfactory as an assumption, leads to correct results. A general expression for the potential energy due to strain in curvilinear coordinates has also been obtained by Mr Love, and the equations of motion and the boundary conditions have been deduced therefrom by means of the Principle of Virtual Work, and if this expression and the equations to which it leads were correct, it would be unnecessary to propose a fresh theory of thin shells; but although those portions of Mr Love's results which depend upon the thickness of the shell are undoubtedly correct, yet, for reasons which will be more fully stated hereafter, I am of opinion that the terms which depend upon the cube of thickness are not strictly accurate, inasmuch as he has omitted to take into account several terms of this order, both in the expression for the potential energy and elsewhere. His preliminary analysis is also of an exceedingly complicated character.
- Alfred B Basset, On the Reflection and Refraction of Light at the Surface of a Magnetized Medium, Philosophical Transactions of the Royal Society of London A 182 (1891), 371-396.
The object of this investigation is to endeavour to ascertain how far the electromagnetic theory of light, as at present developed, is capable of giving a theoretical explanation of Dr Kerr's experiments on the effect of magnetism on light. In the first series of experiments polarized light was reflected from the polished pole of an electromagnet, and it was found that when the circuit was closed, so that the reflecting surface became magnetized perpendicularly to itself, the reflected light exhibited certain peculiarities, which disappeared when the circuit was broken. In the second series of experiments the reflector was a polished plate of soft iron laid upon the poles of a horse-shoe electromagnet, so that the direction of magnetization was parallel, or approximately so, to the reflecting surface; and it was found that the effect of the current was analogous to, though by no means identical with, the effect produced in the first series of experiments. It was also found that when the incidence was normal, or when the plane of incidence was perpendicular to the direction of magnetization, no effect was produced.
In both series of experiments it was found that the effects produced by magnetization materially varied with the angle of incidence. It was also found that these effects were in most cases reversed, when the direction of the magnetizing current was reversed; that is to say, if the intensity of the reflected light was strengthened by a right-handed current, it was weakened by a left-handed one. Since the effects produced by the current were in most cases reversed when the direction of the current was reversed, it follows that the first power of the magnetic force must enter into the expression for the intensity of the reflected light. It will be noticed that in all these experiments a metallic reflector was employed, and consequently the results were complicated by the influence of metallic reflection. It therefore seems hopeless to attempt to construct a theory which shall furnish a complete explanation of these phenomena, until a perfectly satisfactory electromagnetic theory of metallic reflection has been obtained, and, so far as I am aware, no such theory has been discovered. Lord Rayleigh has shown that Cauchy's formulae may be derived from Green's theory of elastic media, by assuming that effect of metallic reflection is represented by a term proportional to the velocity; and in a subsequent paper he has shown that the same formulae may be obtained from the electromagnetic theory by taking into account the conductivity of the metallic reflector. The results obtained by either of these theories are open to various objections, which Lord Rayleigh has discussed in the papers referred to; and he considers "that much remains to be done before the electrical theory of metallic reflection can be accepted as complete."
There are, however, several non-metallic reflecting media (such as strong solutions of certain chemical compounds of iron), which are capable, when magnetized, of producing an effect upon light; and the theoretical explanation of the magnetic of such media is accordingly free from the difficulties surrounding metallic reflection. It would be exceedingly desirable that experiments upon magnetic solutions should be made; and in view of the possibility of such experiments I have thought it worth white to develop a theory applicable to them. Whether the results of the present paper will stand the test of experiment cannot be finally decided until the experiments alluded to have been made; I but it must not excite surprise if the results of the theory do not agree very well with Kerr's experiments, since the conditions of the problem are materially different.
- Alfred B Basset, Stability and Instability of Viscous Liquids [Abstract], Proceedings of the Royal Society of London 52 (1892-1893), 273-276.
The principal object of this paper is to endeavour to obtain a theoretical explanation of the instability of viscous liquids, which was experimentally studied by Professor Osborne Reynolds. The experiment, which perhaps most strikingly illustrates this branch of hydrodynamics, consisted in causing water to flow from a cistern through a long circular tube, and by means of suitable appliances a fine stream of coloured liquid was made to flow down the centre of the tube along with the water. When the velocity was sufficiently small, the coloured stream showed no tendency to mix with the water; but when the velocity was increased, it was found that as soon as it had attained a certain critical value, the coloured stream broke off at a certain point of the tube and began to mix with the water, thus showing that the motion was unstable. It was also found that as the velocity was still further increased the point at which instability commenced gradually moved up the tube towards the end at which the water was flowing in. ...
The results of the investigation may be summed up as follows: (i) The tendency to instability increases as the velocity of the liquid, the radius of the tube, and the coefficient of sliding friction increases but diminishes as the viscosity increases; (ii) The tendency to instability increases as the wave-length (2π/m) of the disturbance increases.
The remainder of the paper is occupied with the discussion of a variety of problems relating to jets and wave motion. I find that when a cylindrical jet is moving through the atmosphere, the tendency of the viscosity of the jet is always in the direction of stability. The velocity of the jet does not affect the stability unless the influence of the surrounding air is taken into account; if, however, this is done, it will be found that it gives rise to a term proportional to the product of the density of the air and the square of the velocity of the jet, whose tendency is to render the motion unstable. The tendency of surface tension (as has been previously shown by Lord Rayleigh) is in the direction of stability or instability according as the wave-length of the disturbance is less or greater than the circumference of the jet.
- Alfred B Basset, Publication of scientific papers, The Journal of the Society of Arts 41 (2135) (1893), 990-992.
Two suggestions have been made with regard to the publication of scientific papers - first, that all papers of importance should be published in a central organ; secondly, that a digest containing an abstract of such papers should from time to time be published. I do not think the first scheme could be carried out so as to cover any useful purpose; for, although it might suit the requirements of a few juvenile societies, it is unlikely that societies of position and standing, which have ample funds at their command for the publication of their proceedings and transactions, would consent to sink their individuality by giving up the publication of papers communicated to them. Moreover, as many societies derive a considerable portion of their income from the sale of their proceedings, it would be impossible for them to allow the concurrent publication of papers in the central organ, as this might seriously diminish their revenue. ... The only feasible scheme seems to be the publication of a digest of papers by the co-operation of the various scientific societies; and, if thought desirable, papers published in foreign countries might also be included.
- Alfred B Basset, On the Deformation of Thin Elastic Plates and Shells, American Journal of Mathematics 16 (3) (1894), 254-290.
The mathematical theory of the deformation of a thin elastic plate or shell involves difficulties of a formidable nature. This is partly owing to the fact that an approximate solution, which is correct as far as terms involving the cube of the thickness of the plate or shell, cannot be obtained without having recourse to an extremely lengthy and troublesome process which requires the Calculus of Variations, and partly because the older writers upon this subject based their investigations upon hypotheses which were in mos.t cases inadequate and erroneous. When a mathematician of standing and reputation gravely propounds a hypothesis which turns out to be incorrect, or condemns as unsound some method which is perfectly legitimate, and in addition obtains by means of his erroneous hypothesis many results Which are substantially correct, the mischief done to science can hardly be exaggerated; for subsequent investigators not only are led astray from the path which must be followed in order to obtain a satisfactory theory, but are hampered by the difficulty of convincing mankind of the errors of their predecessors. The development of the theory of thin elastic plates and shells has been retarded by-two errors for which Clebsch and Saint-Venant are mainly responsible. The first error is that the three stresses R, S, T are accurately zero throughout the substance of the plate or shell; the second error is that it is not permissible to expand the various quantities involved, in a series of ascending powers of the distance of a point from the middle surface. In England the first error may, I think, now be regarded as exploded; but the second one has not yet been completely driven from the field, for a perusal of Mr Love's recent 'Treatise on Elasticity' shows that he entertains some bias against the method of expansion, though on what grounds I am at a loss to conceive. Owing to the unsatisfactory manner in which Mr Love has dealt with the theories of thin plates, shells and wires in his book, I propose in the present paper to give as concise an account of the first two theories as the nature of the case will admit. I shall commence by expounding the fundamental principles of these theories; I shall then proceed to develop them in a form suitable for mathematical calculation; I shall endeavour to avoid all unnecessary mathematical complications; and I shall omit or pass over very lightly those parts which are not of much physical interest, or which require lengthy and difficult analysis for their elucidation.
- Alfred B Basset, Waves and Jets in a Viscous Liquid, American Journal of Mathematics 16 (1) (1894), 93-110.
In an able article which appeared in the ninth volume of this Journal, Prof Greenhill discussed at considerable length the principal cases of wave-motion in a frictionless liquid which have hitherto been solved. In the present paper, I propose to consider certain problems of a similar character when the viscosity of the liquid is taken into account. ...
The motion of waves at the surface of separation of two frictionless liquids which are moving with independent velocities was investigated by Greenhill in 1878, and his solution has been discussed by Lord Rayleigh with reference to the question of stability, but the corresponding solution for two viscous liquids would entail difficulties of a rather formidable character, owing to the fact that it would be necessary to retain all quadratic terms upon which the undisturbed motion depends. If, however, one of the liquids is frictionless whilst the other is viscous, the solution can be arrived at without difficulty. It is always a great advantage when the conditions of a mathematical problem can be expressed in terms of a single function, which satisfies a certain partial differential equation together with given boundary conditions. In most problems of interest relating to wave-motion in frictionless liquids this can be affected by means of the velocity potential; but when the liquid is viscous, a velocity potential cannot as a rule exist, owing to the fact that the motion almost always involves molecular rotation. Of course any solution giving a possible irrotational motion must necessarily satisfy the equations of motion of a viscous liquid; but the solution will not satisfy the boundary conditions except in the single case in which the liquid moves like a rigid body having a motion of translation alone. Moreover, viscous motion involves a conversion of energy into heat. If, however, the motion of a viscous liquid is in two-dimensions, or is symmetrical with respect to an axis, the former can always be expressed by means of the current function; and this function will be used throughout the present paper.
- Alfred B Basset, On the Deformation of Thin Elastic Wires, American Journal of Mathematics 17 (4) (1895), 281-317.
An account of the various attempts which have been made to construct a theory of the deformation of a thin elastic wire, together with the solution of various problems of interest, will be found in the second volume of Mr Love's recent 'Treatise on Elasticity'. The above work also contains a variety of geometrical investigations connected with this subject, and the methods employed are of considerable novelty, power and elegance. But Mr Love's treatment of the physical portion of the subject is not at all so satisfactory; and this is in great measure due to the fact, which I have commented upon in my recent paper on the 'Deformation of Thin Elastic Plates and Shells', that he appears to entertain some objection against the method of expansion, and has also been unable to emancipate himself from the imperfect methods of the German and French schools.
I am of opinion that the most satisfactory way of constructing a complete theory of the small deformations of thin wires is to employ the method explained in my paper on the Theory of Elastic Wires; but unfortunately that investigation contains a slight slip in the work, which arose from my having copied an equation wrongly and used the wrong equation in a subsequent portion of the paper. In consequence of this, the values of the two flexural couples are not proportional to the changes of curvature, as ought to be the case. This circumstance may possibly have led Mr Love to entertain doubts as to the soundness of the principles upon which the theory was based; whereas the real fact is that the theory is a perfectly sound and unimpeachable one, and when the error is corrected it leads to results which have been established by methods of a more or less imperfect character, which agree with those obtained by Mr Love, and are now generally admitted to be correct. Under these circumstances I think that a further exposition of the theory of wires is needed, and this is what I propose to give in the present paper. I shall commence with the theory of the small deformations of a naturally curved wire; I shall then discuss the theory of finite deformations, in which finite changes of curvature and twist occur; and I shall lastly work out the solutions of various problems of interest.
In most problems of practical interest, the wire is made of flexible and well-tempered metal such as steel; also its cross-section is uniform and circular, and the radius of the latter is small in comparison with the radius of principal curvature of the wire at any point of its length. Wires of this description are called thin wires, and to such wires the following investigation will be exclusively confined. It is also obvious that the central axis of a metal wire may usually be regarded as inextensible, since any extension of the axis which might possibly be produced by any given forces is extremely small in comparison with the flexion and torsion actually produced. We shall therefore suppose that the extension of the central axis may be neglected.
- Alfred B Basset, Theories of the Action of Magnetism on Light, American Journal of Mathematics 19 (1) (1897), 60-74.
About five years ago I worked out a theory of the reflection and refraction of light at the surface of a magnetized transparent medium by means of a suggestion of Professor Rowland, which consisted in introducing Hall's effect into the general equations of the electro-magnetic field. I did not, however, at the time attempt to apply the results to the phenomenon of reflection from a magnetized metal, for being aware that metallic reflection could not be explained upon the electro-magnetic theory by taking into consideration the conductivity of the metal, and that no satisfactory theories of metallic reflection existed except those in which the ether is regarded as an elastic medium, I was uncertain how to proceed. I subsequently determined to try what could be done by transforming the expressions for the intensities of the reflected waves in the same manner as Cauchy and Eisenlohr transformed Fresnel's sine and tangent formulae; and I found that the agreement between the theoretical results thus obtained and the experiments of Kerr was perfect as far as qualitative results were concerned, and in one case the theory showed the existence of a phenomenon which Kerr had failed to detect, but which was shortly afterwards discovered by Kundt upon repeating Kerr's experiments. The agreement as regards quantitative results was not, however, in all cases, so close as might be desired. But irrespective of the question of metallic reflection, the theory was not entirely satisfactory, since it required the tangential component of the electro-motive force to be discontinuous at the surface of separation of a magnetised and an unmagnetised medium. I was fully aware of this objection at the time, and endeavoured to explain the difficulty by suggesting that possibly the transition from one medium to the other was not abrupt, but that a thin interfacial layer might exist through which there was a rapid but continuous change in the value of the tangential component of the electro-motive force. 1I must however confess that this explanation savours of a device for evading rather than accounting for a difficulty.
Mr Larmor has recently attempted to resuscitate a modification of Maxwell's theory, which was proposed in 1879 by Professor FitzGerald. FitzGerald's theory contains a serious defect owing to the boundary conditions being too numerous; and I accordingly awaited the publication of Mr Larmor's papers with much interest, in the hope that he had succeeded in overcoming the difficulties to which I have alluded. But in this I was disappointed, for the papers in question, instead of containing a careful mathematical investigation of a phenomenon whose interest is equal to the difficulty of satisfactorily accounting for it, consisted for the most part of vague and obscure generalities, which are calculated to envelop the subject in a cloud of mystery rather than to enlighten the understanding. The superficial reader may possibly be impressed with their apparent profundity, but when examined they turn out to be a dry husk without a kernel.
The object of the present communication is two-fold. In the first place I shall subject Mr Larmor's theory to a searching examination for the purpose of exposing its imperfections, and shall show that instead of being an improvement on its predecessors it is open to a variety of additional objections and defects. In the second place I shall show that by means of a slight modification of the fundamental hypothesis, the theory of Rowland and myself may be placed on a perfectly satisfactory basis, and that the difficulty with regard to the discontinuity of the tangential component of the electro-motive force at an interface may be removed.
- Alfred B Basset, On Certain Conics Connected with Trinodal Quartics, American Journal of Mathematics 26 (2) (1904), 169-176.
It is well known that three conics can be described which respectively pass, (i) through the six points in which the nodal tangents intersect the quartic, (ii) through the six points of contact of the tangents drawn from the nodes, (iii) through the six points of inflexion; also, that each of the three conics intersect the quartic at two points S, S', which will be called the S points. A somewhat lengthy proof of the first theorem is given in ¤194 of my book on "Cubic and Quartic Curves;" and concise proofs of the three theorems, together with the equations of the three conics, appear to be a desideratum. There is also a fourth conic which passes through the six Q points of a trinodal quartic. The equation of this conic will be found, and it will be shown to pass through two points T, T', which will be called the T points, which are the two remaining points in which the line SS' cuts the quartic. The T points are also points of importance in the theory of trinodal quartics.