Part II, the continuation was definitely abandoned.

In the second edition many topics are more fully discussed, and the contents include a very valuable account of cycloidal motion (or oscillatory motion, as it is more usually called), and of a revised version of the chapter on Statics which forms the concluding portion of the book, and which discusses some of the great problems of terrestrial and cosmical physics.

Various speculations have been indulged in, from time to time, as to the respective parts contributed to the work by the two authors, but these are generally very wide of the mark. The mode of composition of the sections on cycloidal (oscillatory) motion gives some idea of Thomson's method of working. His proofs (of "T and T-dash" as the authors called the book) were carried with him by rail and steamer, and he worked incessantly (without, however, altogether withdrawing his attention from what was going on around him!) at corrections and additions. He corrected heavily on the proofs, and then overflowed into additional manuscript. Thus, when he came to the short original § 343, he greatly extended that in the first instance, and proceeded from section to section until additions numbered from § 343a to § 343p, amounting in all to some ten pages of small print, had been interpolated. Similarly § 345 was extended by the addition of §§ 345 (i) to 345 (xxviii), mainly on gyrostatic domination. The method had the disadvantage of interrupting the printers and keeping type long standing, but the matter was often all the more inspiring through having been produced under pressure from the printing office. Indeed, much was no doubt written in this way which, to the great loss of dynamical science, would otherwise never have been written at all.

The kinematical discussion begins with the consideration of motion along a continuous line, curved or straight. This naturally suggests the ideas of curvature and tortuosity, which are fully dealt with mathematically, before the notion of velocity is introduced. When that is done, the directional quality of velocity is not so much insisted on as is now the case: for example, a point is spoken of as moving in a curve with a uniform velocity; and of course in the language of the present time, which has been rendered more precise by vector ideas, if not by vector-analysis, the velocity of a point which is continually changing the direction of its motion, cannot be uniform. The same remark may be made regarding the treatment of acceleration: in both cases the reference of the quantity to three Cartesian axes is immediate, and the changes of the components, thus fixed in direction, are alone considered.

There can be no doubt that greater clearness is obtained by the process afterwards insisted on by Tait, of considering by a hodographic diagram the changes of velocity in successive intervals of time, and from these discovering the direction and magnitude of the rate of change at each instant. This method is indeed indicated at § 37, but no diagram is given, and the properties of the hodograph are investigated by means of Cartesians. The subject is, however, treated in the Elements by the method here indicated.

Remarkable features of this chapter are the very complete discussion of simple harmonic or vibratory motion, the sections on rotation, and the geometry of rolling and precessional motion, and on the curvature of surfaces as investigated by kinematical methods. A remark made in § 96 should be borne in mind by all who essay to solve gyrostatic problems. It is that just as acceleration, which is always at right angles to the motion of a point, produces a change in the direction of the motion but none in the speed of the point (it does influence the velocity), so an action, tending always to produce rotation about an axis at right angles to that about which a rigid body is already rotating, will change the direction of the axis about which the body revolves, but will produce no change in the rate of turning.[20]

A very full and clear account of the analysis of strains is given in this chapter, in preparation for the treatment of elasticity which comes later in the book; and a long appendix is added on Spherical Harmonics, which are defined as homogeneous functions of the coordinates which satisfy the differential equation of the distribution of temperature in a medium in which there is steady flow of heat, or of distribution of potential in an electrical field. This appendix is within its scope one of the most masterly discussions of this subject ever written, though, from the point of view of rigidity of proof, required by modern function-theory, it may be open to objection.

In the next chapter, which is entitled "Dynamical Laws and Principles," the authors at the outset declare their intention of following the Principia closely in the discussion of the general foundations of the subject. Accordingly, after some definitions the laws of motion are stated, and the opportunity is taken to adopt and enforce the Gaussian system of absolute units for dynamical quantities. As has been indicated above, the various difficulties more or less metaphysical which must occur to every thoughtful student in considering Newton's laws of motion are not discussed, and probably such a discussion was beyond the scheme which the authors had in view. But metaphysics is not altogether excluded. It is stated that "matter has an innate power of resisting external influences, so that every body, as far as it can, remains at rest, or moves uniformly in a straight line," and it is stated that this property--inertia--is proportional to the quantity of matter in the body. This statement is criticised by Maxwell in his review of the _Natural Philosophy_ in Nature in 1879 (one of the last papers that Maxwell wrote). He asks, "Is it a fact that 'matter has any power, either innate or acquired, of resisting external influences'? Does not every force which acts on a body always produce that change in the motion of the body by which its value, as a force, is reckoned? Is a cup of tea to be accused of resisting the sweetening influence of sugar, because it persistently refuses to turn sweet unless the sugar is put into it?"

This innate power of resisting is merely the _materiæ vis insita_ of Newton's "Definitio III," given in the Principia, and the statement to which Maxwell objects is only a free translation of that definition. Moreover, when a body is drawn or pushed by other bodies, it reacts on those bodies with an equal force, and this reaction is just as real as the action: its existence is due to the inertia of the body. The definition, from one point of view, is only a statement of the fact that the acceleration produced in a body in certain circumstances depends upon the body itself, as well as on the other bodies concerned, but from another it may be regarded as accounting for the reaction. The mass or inertia of the body is only such a number that, for different bodies in the same circumstances as to the action of other bodies in giving them acceleration, the product of the mass and the acceleration is the same for all. It is, however, a very important property of the body, for it is one factor of the quantum of kinetic energy which the body contributes to the energy of the system, in consequence of its motion relatively to the chosen axes of reference, which are taken as at rest.

The relativity of motion is not emphasised so greatly in the _Natural Philosophy_ as in some more modern treatises, but it is not overlooked; and whatever may be the view taken as to the importance of dwelling on such considerations in a treatise on dynamics, there can be no doubt that the return to Newton was on the whole a salutary change of the manner of teaching the subject.

The treatment of force in the first and second laws of motion is frankly causal. Force is there the cause of rate of change of momentum; and this view Professor Tait in his own writings has always combated, it must be admitted, in a very cogent manner. According to him, force is merely rate of change of momentum. Hence the forces in equations of motion are only expressions, the values of which as rates of change of momentum, are to be made explicit by the solution of such equations in terms of known quantities. And there does not seem to be any logical escape from this conclusion, though, except as a way of speaking, the reference to cause disappears.

The discussion of the third law of motion is particularly valuable, for, as is well known, attention was therein called to the fact that in the last sentences of the Scholium which Newton appended to his remarks on the third law, the rates of working of the acting and reacting forces between the bodies are equal and opposite. Thus the whole work done in any time by the parts of a system on one another is zero, and the doctrine of conservation of energy is virtually contained in Newton's statement. The only point in which the theory was not complete so far as ordinary dynamical actions are concerned, was in regard to work done against friction, for which, when heat was left out of account, there was no visible equivalent. Newton's statement of the equality of what Thomson and Tait called "activity" and "counter-activity" is, however, perfectly absolute. In the completion of the theory of energy on the side of the conversion of heat into work, Thomson, as we have seen, took a very prominent part.

After the introduction of the dynamical laws the most interesting part of this chapter is the elaborate discussion which it contains of the Lagrangian equations of motion, of the principle of Least Action, with the large number of extremely important applications of these theories. The originality and suggestiveness of this part of the book, taken alone, would entitle it to rank with the great classics--the _Mécanique Céleste_, the _Mécanique Analytique_, and the memoirs of Jacobi and Hamilton--all of which were an outcome of the Principia, and from which, with the Principia, the authors of the _Natural Philosophy_ drew their inspiration.

It is perhaps the case, as Professor Tait himself suggested, that no one has yet arisen who can bend to the fullest extent the bow which Hamilton fashioned; but when this Ulysses appears it will be found that his strength and skill have been nurtured by the study of the _Natural Philosophy_. Lagrange's equations are now, thanks to the physical reality which the expositions and examples of Thomson and Tait have given to generalised forces, coordinates, and velocities, applied to all kinds of systems which formerly seemed to be outside the range of dynamical treatment. As Maxwell put it, "The credit of breaking up the monopoly of the great masters of the spell, and making all their charms familiar in our ears as household words, belongs in great measure to Thomson and Tait. The two northern wizards were the first who, without compunction or dread, uttered in their mother tongue the true and proper names of those dynamical concepts, which the magicians of old were wont to invoke only by the aid of muttered symbols and inarticulate equations. And now the feeblest among us can repeat the words of power, and take part in dynamical discussions which a few years ago we should have left to our betters."

A very remarkable feature in this discussion is the use made of the idea of "ignoration of coordinates." The variables made use of in the Lagrangian equations must be such as to enable the positions of the parts of the system which determine the motion to be expressed for any instant of time. These parts, by their displacements, control those of the other parts, through the connections of the system. They are called the independent coordinates, and sometimes the "degrees of freedom," of the system. Into the expressions of the kinetic and potential energies, from which by a formal process the equations of motion, as many in number as there are degrees of freedom, are derived, the value of these variables and of the corresponding velocities enter in the general case. But in certain cases some of the variables are represented by the corresponding velocities only, and the variables themselves do not appear in the equations of motion. For example, when fly-wheels form part of the system, and are connected with the rest of the system only by their bearings, the angle through which the wheel has turned from any epoch of time is of no consequence, the only thing which affects the energy of the system is the angular velocity or angular momentum of the wheel. The system is said by Thomson and Tait in such a case to be under gyrostatic domination. (See "Gyrostatic Action," p. 214 below.)

Moreover, since the force which is the rate of growth of the momentum corresponding to any coordinate is numerically the rate of variation with that coordinate of the difference of the kinetic and potential energies, every force is zero for which the coordinate does not appear; and therefore the corresponding momentum is constant. But that momentum is expressed by means of the values of other coordinates which do appear and their velocities, with the velocities for the absent coordinates; and as many equations are furnished by the constant values of such momenta as there are coordinates absent. The corresponding velocities can be determined from these equations in terms of the constant momenta and the coordinates which appear and their velocities. The values so found, substituted in the expressions for the kinetic and potential energies, remove from these expressions every reference to the absent coordinates. Then from the new expression for the kinetic energy (in which a function of the constant momenta now appears, and is taken as an addition to the potential energy) the equations of motion are formed for the coordinates actually present, and these are sufficient to determine the motion. The other coordinates are thus in a certain sense ignored, and the method is called that of "ignoration of coordinates."

Theorems of action of great importance for a general theory of optics conclude this chapter; but of these it is impossible to give here any account, without a discussion of technicalities beyond the reading of ordinary students of dynamics.

In an Appendix to Part I an account is given of Continuous Calculating Machines. Ordinary calculating machines, such as the "arithmometer" of Thomas of Colmar, carry out calculations and exhibit the result as a row of figures. But the machines here described are of a different character: they exhibit their results by values of a continuously varying quantity. The first is one for predicting the height of the tides for future time, at any port for which data have been already obtained regarding tidal heights, by means of a self-registering tide-gauge. Two of these were made according to the ideas set forth in this Appendix; one is in the South Kensington Museum, the other is at the National Physical Laboratory at Bushy House, where it is used mainly for drawing on paper curves of future tidal heights, for ports in the Indian Ocean. From these curves tide-tables are compiled, and issued for the use of mariners and others.

Another machine described in this Appendix was designed for the mechanical solution of simultaneous linear equations. It is impossible to explain here the interesting arrangement of six frames, carrying as many pulleys, adjustable along slides (for the solution of equations involving six unknown quantities), which Thomson constructed, and which is now in the Natural Philosophy Department at Glasgow. The idea of arranging the first practical machine for this number of variables, was that it might be used for the calculation of the corrections on values already found for the six elements of a comet or asteroid. The machine was made, but some mechanical difficulties arose in applying it, and the experiments with it were not at the time persevered with. Very possibly, however, it may yet be brought into use.

But the most wonderful of these mechanical arrangements is the machine for analysing the curves drawn by a self-registering tide gauge, so as to exhibit the constants of the harmonic curves, and thus enable the prediction of tidal heights to be carried out either by the tide-predicting machine, or by calculation. One day in 1876, Thomson remarked to his brother, James Thomson, then Professor of Engineering at Glasgow, that all he required for the construction of a tidal analyser was a form of integrating machine more satisfactory for his purpose than the usual type of integrator employed by surveyors and naval architects. James Thomson at once replied that he had invented, a long time before, what he called a disk-globe-cylinder-integrator. This consisted of a brass disk, with its plane inclined to the horizontal, which could be turned about its axis by a wheel gearing in teeth on the edge of the disk, and driven by the operator in a manner which will presently appear. Parallel and close to the disk, but not touching it, was placed a horizontal cylinder of brass, about 2 inches in diameter (called the registering cylinder), and between the disk and this cylinder was laid a metal ball about 2½ inches in diameter. When the disk was kept at rest, and the ball was rolled along between the cylinder and disk, the trace of its rolling on the latter was a straight horizontal line passing through the centre. Supposing then that the point of contact of the ball with the disk was on one side, at a distance from the centre, and that the disk was then turned, the ball was by the friction between it and the disk made to roll, and so to turn the cylinder. The angular velocity of rolling, and therefore the angular velocity of the cylinder, was proportional to the speed of the part of the disk in contact with it, that is, to y. It was also proportional to the speed of turning of the disk.

The mode by which this machine effects an integration will now be evident. Imagine the area to be found to lie between a curve and a straight datum line, drawn on a band of paper. This is stretched on a large cylinder, with the datum line round the cylinder. We call this the paper-cylinder. The distances of the different points of the curve from the datum line are values of y. A horizontal bar parallel to the cylinder carries a fork at one end and a projecting style at the other. The globe just fits between the prongs of the fork, and when the bar is moved in the direction of its length carries the ball along the disk and cylinder. When the style at the other end is on the datum line, the centre of the ball is at the centre of the disk, and the turning of the disk does not turn the cylinder. When the bar is displaced in the line of its own length to bring the style from the datum line to a point on the curve, the ball is displaced a distance y, and there is a corresponding turning of the cylinder by the action of the ball. In the use of the instrument the paper-cylinder is turned by the operator while the style is kept on the curve, and the disk is turned by the gearing already referred to, which is driven by a shaft geared with that of the paper-cylinder. Thus the displacement of the ball is always y, the ordinate of the curve, and for any displacement dx along the datum line, the registering cylinder is turned through an angle proportional to ydx. Thus any finite angle turned through is proportional to the integral of ydx for the corresponding part of the curve: a scale round one end of the registering cylinder gives that angle. Thomson immediately perceived that this extremely ingenious integrating machine was just what he required for his purpose. The curve of tidal heights drawn (on a reduced scale, of course) by a tide-gauge, is really the resultant of a large number of simple curves, represented by a series of harmonic terms, the coefficients of which are certain integrals. The problem is the evaluation of these integrals; and the method usually employed is to obtain them by measurement of ordinates of the curve and an elaborate process of calculation. But one of them is simply the integral area between the curve and the datum line corresponding to the mean water level, and the others are the integrals of quantities of the type y sin nx.dx, where y is the ordinate of the curve, and n a number inversely proportional to the period of the tidal constituent represented by the term.

All that was necessary, in order to give the integral of a term y sin nx.dx, was to make the disk oscillate about its axis as the paper-cylinder was turned through an angle proportional to x. Thus one disk, globe, and cylinder was arranged exactly as has been described for the integral of ydx, and with this as many others as there were harmonic terms to be evaluated from the curve were combined as follows. The disks were placed all in one plane with their centres all on one horizontal line, and the cylinders with their axes also in line, and a single sliding bar, with a fork for each globe, gave in each case the displacement y from the centre of the disk.

The requisite different speeds of oscillation were given to the disks by shafts geared with the paper-cylinder, by trains of wheels cut with the proper number of teeth for the speed required.

Thus the angles turned through by the registering cylinders when a curve on the paper-cylinder was passed under the style were proportional to the integrals required, and it was only necessary to calibrate the graduation of the scales of these cylinders by means of known curves to obtain the integrals in proper units.

One of these machines, which analyses four harmonic constituents, is in the Natural Philosophy Department at Glasgow; a much larger machine, to analyse a tidal curve containing five pairs of harmonic terms, or eleven constituents in all, was made for the British Association Committee on Tidal Observations, and is probably now in the South Kensington Museum.

But still more remarkable applications which Thomson made of his brother's integrating machine were to the mechanical integration of linear differential equations, with variable coefficients, to the integration of the general linear differential equation of any order, and, finally, to the integration of any differential equation of any order.

These applications were all made in a few days, almost in a few hours, after James Thomson first described the elementary machine, and papers containing descriptions of the combinations required were at once dictated by Thomson to his secretary, and despatched for publication. Very possibly he had thought out the applications to some extent before; but it is unlikely that he had done so in detail. But, even if it were so, the connection of a series of machines by the single controlling bar, and the production of the oscillations of the disks, all controlled, as they were, by the motion of a simple point along the curve, so as to give the required Fourier coefficients, were almost instantaneous, and afford an example of invention amounting to inspiration.

There should be noticed here also the geometrical slide for use in safety-valves, cathetometers and other instruments, and the hole-slot-and-plane mode of so supporting an instrument now used in all laboratories. These were Thomson's inventions, and their importance is insisted on in the _Natural Philosophy_.

In Part II, the principal subjects treated are attractions, elasticity, such great hydrostatical examples as the equilibrium theory of the tides and the equilibrium of rotating liquid spheroids, and such problems of astronomical and terrestrial dynamics as the distribution of matter in the earth, with the bearing on this subject of the precession of the equinoxes, tidal friction, the earth's rigidity, the effects of elastic tides, the secular cooling of the earth, the age of the earth, and the "age of the sun's heat." Of these, with the exception of the age of the earth, we shall not attempt to give any account. The importance of the original contributions to elasticity contained in the book is indicated by the large space devoted to the _Natural Philosophy_ in Professor Karl Pearson's continuation of Todhunter's _History of Elasticity_. The heavy task of editing Part II was performed mainly by Sir George Darwin, who made many notable additions from his own researches to the matter contained in the first edition.

In the next chapter an attempt will be made to present Thomson's views on the subject of the age of the earth. These, when they were published, attracted much attention, and received a good deal of hostile criticism from geologists and biologists, whose processes they were deemed to restrict to an entirely inadequate period of time.

GYROSTATIC ACTION

Thomson in his lectures and otherwise gave a great deal of attention to the motion of gyrostats, and to the effect of the inclusion of gyrostats in a system on its properties. Reference has been made to the treatment of "gyrostatic domination" in "Thomson and Tait." A gyrostat consists of a disk or wheel with a massive rim, which revolves within a case or framework, by which the whole arrangement can be moved about, or supported, without interfering with the wheel. The ordinary toy consisting of wheel with a massive rim, and a light frame, is an example. But much larger and more carefully made instruments, in which the wheel is entirely enclosed, give the most interesting experiments. The body seems to have its properties entirely altered by the rotation of the wheel, and of course the case prevents any outward change from being visible.

Figure 15 shows one form of gyrostat mounted on a horizontal frame, held in the hands of an experimenter. The axis of the fly-wheel is vertical within the tubular part of the case; the fly-wheel is within the part on which is engraved an arrow-head to show the direction of rotation. Round the case in the plane of the wheel is a projecting rim sharpened to an edge, on which the gyrostat can be supported in other experiments. To the rim are screwed two projecting pivots, which can turn in bearings on the two sides of the frame as shown. The centre of mass of the wheel is on the level of these pivots, so that the instrument will remain with either end of the axis up.

If the fly-wheel be not in rotation, the experimenter can carry the arrangement about, and the fly-wheel and case move with it as if the gyrostat were merely an ordinary rigid body. But now remove the gyrostat from the frame, and set the wheel in rotation. This is done by an endless cord wrapped round a small pulley fast on the axle (to which access is obtained by a hole just opposite in the case) and passed also round a larger pulley on the shaft of a motor. When the motor is started the cord must be tightened only very gently at first, so that it slips on the pulley, otherwise the motor would be retarded, and possibly burned by the current. The fly-wheel gradually gets up speed, and then the cord can be brought quite tight so that no slipping occurs. When the speed is great enough the cord is cut with a stroke from a sharp knife and runs out.

The gyrostat is now replaced on its pivots in the frame, with its axis vertical, and moved about as it was before. If the experimenter, holding the frame as shown, turns round in the direction of the arrow, which is that of rotation, nothing happens. If, however, he turns round the other way, the gyrostat immediately turns on its pivots so as to point the other end of the axis up. If the experimenter continues his turning motion, the gyrostat is now quiescent: for it is being carried round now in the direction of rotation. Thus, with no gravitational stability at all (since the centre is on a level with the pivots) the gyrostat is in stable equilibrium when carried round in the direction of rotation, but is in unstable equilibrium when carried round the opposite way.

Thus, if the observer knew nothing of the rotation of the fly-wheel, and could see and feel only the outside of the case, the behaviour of the instrument might well appear very astonishing.

This is a case of what Thomson and Tait call "gyrostatic domination," which is treated very fully in their Sections 345 (vi) to 345 (xxviii) of Part I. It may be remarked here that this case of motion may be easily treated mathematically in an exceedingly elementary manner, and the instability of the one case, and the stability of the other, made clear to the beginner who has only a notion of the composition of angular momenta about different axes.

A year or two ago it was suggested by Professor Pickering, of Harvard, that the fact that the outermost satellite of Saturn revolves in the direction opposite to the planet's rotation, may be due to the fact that originally Saturn rotated in the direction of the motion of this moon, but inasmuch as his motion round the sun was opposite in direction to his rotation, he was turned, so to speak, upside down, like the gyrostat! The other satellites, it is suggested, were thrown off later, as their revolution is direct. Professor Pickering refers to an experiment (similar to that described above) which he gives as new. Thomson had shown this experiment for many years, as an example of the general discussion in "Thomson and Tait," and its theory had already been explicitly published.[21]

Many other experiments with gyrostats used to be shown by Thomson to visitors. Many of these are indicated in "Thomson and Tait." The earth's precessional motion is a gyrostatic effect due to the differential attraction of the sun, which tends to bring the plane of the equator into coincidence with the ecliptic, and so alters the direction of the axis of rotation. Old students will remember the balanced globe--with inclined material axis rolling round a horizontal ring--by which the kinematics of the motion could be studied, and the displacement of the equinoxes on the ecliptic traced.

Another example of the gyrostatic domination discussed in "Thomson and Tait" is given in the very remarkable address entitled "A Kinetic Theory of Matter," which Sir William Thomson delivered to Section A of the British Association at Montreal, in 1884. Figure 16 shows an ordinary double "coach spring," the upper and lower members of which carry two hooked rods as shown. If the upper hook is attached to a fixed support, and a weight is hung on the lower, the spring will be drawn out, and the arrangement will be in equilibrium under a certain elongation. If the weight be pulled down further and then left to itself, it will vibrate up and down in a period depending upon the equilibrium elongation produced by the weight. The same thing will happen if a spiral spring be substituted for the coach spring. A spherical case, through which the hooked rods pass freely, hides the internal parts from view.

Figure 17 shows two hooked rods, as in the former case, attached by swivels to two opposite corners of a frame formed of four rods jointed together at their ends. Each of these is divided in the middle for the insertion of a gyrostat, the axis of which is pivoted on the adjacent ends of the two halves of the rod. A spherical case, indicated by the circle, again hides the internal arrangement from inspection, but permits the hooked rods to move freely up and down. The swivels allow the frame, gyrostats and all, to be turned about the line of the hooks.

If now the gyrostats be not in rotation, the frame will be perfectly limp, and will not in the least resist pull applied by a weight. But if the gyrostats be rotated in the directions shown by the circles, with arrow-heads drawn round the rods, there will be angular momentum of the whole system about the line joining the hooks, and if a weight or a force be applied to pull out the frame along that line, the pull will be resisted just as it was in the other case by the spring. Moreover, equilibrium will be obtained with an elongation proportional to the weight hung on, and small oscillations will be performed just as if there were a spring in the interior instead of the gyrostats.

According as the frame is pulled out, or shortened, the angular momentum of the gyrostats about the line joining the hooks is increased or diminished, and the frame, carrying the gyrostats with it, turns about the swivels in one direction or the other, at the rate necessary to maintain the angular momentum at a constant value. But this will not be perceived from without.

The rotation of the fly-wheels thus gives to the otherwise limp frame the elasticity which the spring possesses; without dissection of the model the difference cannot be perceived. This illustrates Thomson's idea that the elasticity of matter may be due to motion of molecules or groups of molecules of the body, imbedded in a connecting framework, deformed by applied forces as in this model, and producing displacements which are resisted in consequence of the motion.

And here may be mentioned also Thomson's explanation of the phenomenon, discovered by Faraday, of the rotation of the plane of a beam of polarised light which is passed along the lines of force of a magnetic field. This rotation is distinct altogether from that which is produced when polarised light is passed along a tube filled with a solution of sugar or tartaric acid. If the ray be reflected after passage, and made to retraverse the medium, the rotation is annulled in the latter case, it is doubled in the former. This led Thomson to the view that in sugar, tartaric acid, quartz, etc., the turning is due to the structure of the substance, and in the magnetic field to rotation already existing in the medium. He used to say that a very large number of minute spiral cavities all in the same direction, and all right-handed or all left-handed, in the sugar or quartz, would give the effect; on the other hand, the magnetic phenomenon could only be produced by some arrangement analogous to a very large number of tops, or gyrostats, imbedded in the medium with their axes all in one direction (or preponderatingly so) and all turning the same way. The rotation of these tops or gyrostats Thomson supposed to be caused by the magnetic field, and to be essentially that which constitutes the magnetisation of the medium.

Let the frame of the gyrostatic spring-balance described above, turn round the line joining the hooks so as to exactly compensate, by turning in the opposite direction, the angular momentum about that line given by the fly-wheels; then the arrangement will have no angular momentum on the whole; and a large number of such balances, all very minute and hooked together, will form a substance without angular momentum in any part. But now by the equivalent of a magnetic force along the lines of the hooks, let a different angular turning of the frames be produced; the medium will possess a specific angular momentum in every part. If a wave of transverse vibrations which are parallel to one direction (that is, if the wave be plane-polarised) enter the medium in the direction of the axes of the frames, the direction of vibration will be turned as the wave proceeds, that is, the plane of polarisation will be turned round.

More recent research has shown an effect of a magnetic field on the spectrum of light produced in the field, and viewed with a spectroscope in a direction at right angles to the field--the Zeeman effect, as it is called--and the explanation of this effect by equations of moving electric charges, which are essentially gyrostatic equations, is suggestive of an analogy or correspondence between the systems of moving electrons which constitute these charges, and some such gyrostatic molecules as Thomson imagined. It has been pointed out that the Zeeman effect, in its simple forms at least, can be exactly imitated by the motion of an ordinary pendulum having a gyrostat in its bob, with its axis directed along the suspension rod.[22]

ELECTROSTATICS AND MAGNETISM

In the ten years from 1863 to 1873 Thomson was extremely busy with literary work. In 1872, five years after the publication of the treatise on _Natural Philosophy_, and just before the appearance of the Elements, Messrs. Macmillan & Co. published for him a collection of memoirs entitled _Reprint of Papers on Electrostatics and Magnetism_. The volume contains 596 pages, and the subjects dealt with range from the "Uniform Motion of Heat and its Connection with the Mathematical Theory of Electricity" (the paper already described in Chapter II above) and the discussion of Electrometers and Electrostatic Measuring Instruments, to a complete mathematical theory of magnetism. The subject of electrostatics led naturally to the consideration of electrical measuring instruments as they existed forty years ago (about 1867), and their replacement by others, the indications of which from day to day should be directly comparable, and capable of being interpreted in absolute units. Down to that time people had been obliged to content themselves with gold-leaf electroscopes, and indeed it was impossible for accurate measuring instruments to be invented until a system of absolute units had been completely worked out. The task of fixing upon definitions of units and of realising them in suitable standards had been begun by the British Association, and it was as part of the Report of that Committee to the Dundee Meeting in 1867 that Thomson's paper on Electrometers first appeared.

It was there pointed out that an electrometer is essentially an instrument for measuring differences of electric potential between conductors, by means of effects of electrostatic force. Such a difference is what a gold-leaf electroscope indicates for its gold leaves and the walls surrounding the air-space in which they are suspended. As electroscopes used to be constructed, these walls were made of glass imperfectly covered, if at all, by conducting material, and the electroscope was quite indefinite and uncertain in its action. The instrument was also, as made, quite insensitive. Recently, however, it has been rehabilitated in reputation, and brought into use as a very sensitive indicator of effects of radio-activity.

Thomson described in this paper six species of electrometers of his own devising. The best known of these are his quadrant electrometer and his attracted-disk electrometers. The former is to be found in some form or other in every laboratory nowadays, and need not be described in detail. The action is of two conductors--the two pairs of opposite quadrants of a shallow, horizontal, cylindrical box, made by dividing the box into four by two slits at right angles--upon an electrified slip of aluminium suspended by a two-thread suspension within the box, with its length along one of the slits. The two pairs of opposite quadrants are at the potential difference to be measured, and the slip of aluminium, or "needle," has each end urged round from a quadrant at higher potential towards one at a lower, and these actions conspire to turn the slip against its tendency to return to the position in which the two threads are in one plane. Thus the deflection (measured by the displacement of a reflected ray of light used as index) gives an indication of the amount of the potential difference.

The electrification of the "needle" was kept up by enclosing the quadrantal box within an electrified Leyden jar, to the interior coating of which contact is made by a platinum wire, depending from the needle to sulphuric acid contained in the jar. The whole apparatus was enclosed in a conducting case connected to earth. This made its action perfectly definite. Variations of this electrification of the jar were shown by an attached attracted-disk electrometer, the principle of which we shall merely indicate.

The quadrant electrometer has now been vastly increased in sensibility by the use of a single quartz fibre as suspension. By the invention of this fibre, which is exceedingly strong and is, moreover, so definite in its elastic properties that it comes back at once exactly to its former zero state after twist, Mr. C. V. Boys has increased the delicacy of all kinds of suspended indicators many fold. But it ought to be remembered that a Dolezalek electrometer, with some hundred or more times the sensibility of the bifilar instrument, was only made possible by its predecessor.

Attracted-disk-electrometers simply measure, either by weighing or by the deflection of a spring, the attractive force between two parallel disks at different potentials. From the determination of this force, and the measurement of the distance between the disks (or better, of an alteration of the distance) a difference of potentials can be determined, and a unit for it obtained, which is in direct and known relation to ordinary dynamical units. Thomson's "Absolute Electrometer" was designed specially for accurate determinations of this kind. Another form, called the Long Range Electrometer, was devised for the measurement of the potentials of the charged conductors in electric machines and Leyden jars.

Accurate determinations of the sparking resistance between parallel plates charged to different potentials in air were made by means of attracted-disk-electrometers in the course of some important experiments described in the _Electrostatics and Magnetism_. These results have been much referred to in later researches.

A small attracted-disk-electrometer was used as indicated above to keep a watch on the electrification of the Leyden jar of the quadrant instrument, and a small induction machine was added, by turning which the operator could make good any loss of charge of the jar.

This electrical machine was an example of an apparatus on precisely the same principle as the Voss or Wimshurst machines of the present day. In it by a set of moving carriers, influenced by conductors, the charges of the latter were increased according to a compound interest principle only interfered with by leakage to the air or by the supports. Several forms of this machine, on the same principle, were constructed by Thomson, and described in 1868; but he afterwards found that he had been anticipated by C. F. Varley in 1860. Still later it was discovered that a similar instrument had been made a century before by Nicholson, and called by him the "Revolving Doubler."

The experiments which Thomson made on atmospheric electricity at the old College tower, and by means of portable electrometers in Arran and elsewhere, can only be mentioned. They led no doubt to some improvements on electrometers which he made, the method of bringing the nozzle of a water-dropper, or a point on a portable electrometer to the potential of the air, by the inductive action on a stream of water-drops in the one case, or the particles of smoke from a burning match in the other. He invented a self-acting machine, worked by a stream of water-drops, for accumulating electric charges, on the principle of the revolving doubler. It was this apparently that led to the machines with revolving carriers, to which reference has been made above.

The mathematical theory of magnetism which Thomson gave in 1849, in the _Phil. Trans. R.S._, was, when completed by various later papers, a systematic discussion of the whole subject, including electromagnetism and diamagnetism. To a large extent the ground covered by the 1849 paper had been traversed before by Poisson, and partially by Murphy and Green; but Thomson stated that one chief object of his memoir was to formally construct the theory without reference to the two magnetic fluids, by means of which the facts of experiment and conclusions of theory had so far been expressed. He found it, however, convenient to introduce the idea of positive and negative magnetic matter (attracting and repelling as do charges of positive and negative electricity), which are to be regarded as always present in equal amounts, not only in a magnet as a whole, but in every portion of a magnet; and at first sight this might appear like a return to the magnetic fluids. But it amounts on the whole rather to a conception of a magnet as a conglomeration of doublets of magnetic matter (that is, very close, equal and inseparable charges of the two kinds of matter), the arrangement of which can be changed by the action of magnetic force. This idea is set forth now in all the books on magnetism and electricity. There can be no doubt that the systematic presentment of the subject by Thomson, and the theorems and ideas of magnetic force and magnetic permeability by which he rendered the clear, and therefore mathematical, notions of Faraday explicitly quantitative, had much influence in furthering the progress of electrical science, and so leading on the one hand to the electromagnetic theories of Maxwell, and on the other to modern research on the magnetic properties of iron, and to the correct ideas which now prevail as to construction of dynamo-electric machines and motors.