Lord Kelvin Part 11

CHAPTER XI

THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'--GYROSTATIC ACTION--'ELECTROSTATICS AND MAGNETISM'

THE 'NATURAL PHILOSOPHY'

Professor Tait was appointed to the Chair of Natural Philosophy in the University of Edinburgh in 1860, and came almost immediately into frequent contact with Thomson. Both were Peterhouse men, trained by the same private tutor--William Hopkins--both were enthusiastic investigators in mathematical as well as in experimental physics, they taught in the sister universities of Edinburgh and Glasgow, and had much the same kind of cla.s.ses to deal with and the same educational problems to solve. Tait was an Edinburgh man--an old school-fellow of Clerk Maxwell at the Edinburgh Academy--and had therefore been exposed to that contact, in play and in work, with compeers of like age and capabilities, which is one of the best preparations for the larger school and more serious struggles of life. Thomson's early education, under his father's anxious care, had no doubt certain advantages, and his early entrance into college cla.s.ses gave him to a great extent that intercourse with others for which such advantages are never complete compensation. The two men had much community of thought and experience, and the literary partnership into which they entered was hailed as one likely to do much for the progress of science.

In some ways, however, Thomson and Tait were very different personalities. Thomson troubled himself little with metaphysical subtleties, his conceptions were like those of Newton, absolutely clear so far as they went; he never, in his teaching at least, showed any disposition to discuss the "foundations of dynamics," or the conception of motion in a straight line. These were taken for granted like the fundamental ideas in a book on geometry; and the student was left to do what every true dynamical student must do for himself sooner or later--to compare the abstractions of dynamics with the products of his experience in the world of matter and force. Perhaps a little guidance now and then in the difficulties about conceptions, which beset every beginner, might not have been amiss: but Thomson was so intent on the concrete example in hand--pendulum or gyrostat, or what not--that he left each man to form or correct his own ideas by the lessons which such examples afford to every one who carefully examines them.

Tait, on the other hand, though he continually denounced metaphysical discussion, was in reality much more metaphysical than Thomson, and seemed to take pleasure in the somewhat transcendental arguments with regard to matters of a.n.a.lysis which were put forward, especially in the _Elements of Quaternions_, by Sir William Rowan Hamilton, of Dublin, a master whom he much revered. But there is metaphysics and metaphysics!

and the p.r.o.nouncements of professed metaphysicians were often characterised as non-scientific and fruitless, which no doubt they were from the physical point of view.

Then Tait was strongly convinced of the importance for physics of the quaternion a.n.a.lysis: Thomson was not, to say the least; and this was probably the main reason why the vectorial treatment of displacement, velocities, and other directed quant.i.ties, has no place in the joint writings of the two Scottish professors. In controversy Tait was a formidable antagonist: when war was declared he gave no quarter and asked for none, though he never fought an unchivalric battle. He admired foreign investigators--and especially von Helmholtz--but he was always ready to put on his armour and place lance in rest for the cause of British science. Thomson was much less of a combatant, though he also could bravely splinter a spear with an opponent on occasion, as in the memorable discussion with Huxley on the Age of the Earth.

Tait's professorial lectures were always models of clear and logical arrangement. Every statement bore on the business in hand; the experimental ill.u.s.trations, always carefully prepared beforehand, were called for at the proper time and were invariably successful. With Thomson it was otherwise: his digressions, though sometimes inspired and inspiring, were fatal to the success of the utmost efforts of his a.s.sistants to make his lectures successful systematic expositions of the facts and principles of elementary physics.

As has been stated in Chapter IV, two books were announced in 1863 as in course of preparation for the ensuing session of College. These were not published until 1867 and 1873; the first issued was the famous _Treatise on Natural Philosophy_, the second was ent.i.tled _Elements of Natural Philosophy_, and consisted in the main of part of the non-mathematical or large type portions of the Treatise. The scheme of the latter was that of an articulated skeleton of statements of principles and results, printed in ordinary type, with the mathematical deductions and proofs in smaller type. As was to be expected, the Elements, to a student whose mathematical reading was wide enough to tackle the Treatise, was the more difficult book of the two to completely master. But the continued large print narrative, as it may be called, is extremely valuable. It is a memorial of a habit of mind which was characteristic of both authors.

They kept before them always the idea or thing rather than its symbol; and thus the edifice which they built up seemed never obscured by the scaffolding and machinery used in its erection. And as far as possible in processes of deduction the ideas are emphasised throughout; there is no mere putting in at one end and taking out at the other; the result is examined and described at every stage. As in all else of Thomson's work, physical interpretation is kept in view at every step, and made available for correction and avoidance of errors, and the suggestion of new inquiries.

The book as it stands consists of "Division I, Preliminary" and part of "Division II, Abstract Dynamics." Division I includes the chapter on Kinematics already referred to, a chapter on Dynamical Laws and Principles, chapters on Experience and Measures and Instruments.

Division II is represented only by Chapter V, Introductory; Chapter VI, Statics of a Particle and Attractions; and Chapter VII, Statics of Solids and Fluids. Thus Abstract Dynamics is without the more complete treatment of Kinetics to which, as well as to Statics, the discussion of Dynamical Laws and Principles was intended to be an introduction. But to a considerable extent, as we shall see, Kinetics is treated in this introductory chapter: indeed, the discussion of the general theorems of dynamics and their applications to kinetics is remarkably complete.

In Volume II it was intended to include chapters on the kinetics of a particle and of solid and fluid bodies, on the vibrations of solid bodies, and on wave-motion in general. It was expected also to contain a chapter much referred to in Volume I, on "Properties of Matter." That the work was not completed is a matter of keen regret to all physicists, regret, however, now tempered by the fact that many of the subjects of the unfulfilled programme are represented by such works as Lord Rayleigh's _Theory of Sound_, Lamb's Hydrodynamics, and Routh's _Dynamics of a System of Rigid Bodies_. But all deeply lament the loss of the "Properties of Matter." No one can ever write it as Thomson would have written it. His students obtained in his lectures glimpses of the things it might have contained, and it was most eagerly looked for. If that chapter only had been given, the loss caused by the discontinuance of the book would not have been so irreparable.

The first edition of the book was published by the Clarendon Press, Oxford. It was printed by Messrs. Constable, of Edinburgh, and is a beautiful specimen of mathematical typography. In some ways the first edition is exceedingly interesting, for it is not too much to say that its issue had an influence on dynamical science, and its exposition in this country, only second to that due to Newton's Principia. Three other works, perhaps, have had the same degree and kind of influence on mathematical thought--Laplace's _Mecanique Celeste_, Lagrange's _Mecanique a.n.a.lytique_, and Fourier's _Theorie a.n.a.lytique de la Chaleur_.

The second edition was issued by the Cambridge University Press as Parts I and II in 1878 and 1883. Various younger mathematicians now of eminence--Professor Chrystal, of Edinburgh, and Professor Burnside, of Greenwich, may be mentioned--read the proofs, and it is on the whole remarkably free from typographical and other errors. With the issue of 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 ma.n.u.script. 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-a.n.a.lysis, 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 quant.i.ty 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 a.n.a.lysis 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 h.o.m.ogeneous 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 ent.i.tled "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 quant.i.ties. 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 quant.i.ty 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 _materiae 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 circ.u.mstances 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 ma.s.s or inertia of the body is only such a number that, for different bodies in the same circ.u.mstances as to the action of other bodies in giving them acceleration, the product of the ma.s.s 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 quant.i.ties. 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 ent.i.tle it to rank with the great cla.s.sics--the _Mecanique Celeste_, the _Mecanique a.n.a.lytique_, 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, subst.i.tuted 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 quant.i.ty. 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 quant.i.ties), 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.

[Ill.u.s.tration: FIG. 14.]

But the most wonderful of these mechanical arrangements is the machine for a.n.a.lysing 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 a.n.a.lyser 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 bra.s.s 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 bra.s.s, about 2 inches in diameter (called the registering cylinder), and between the disk and this cylinder was laid a metal ball about 2 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 pa.s.sing 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 p.r.o.ngs 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 quant.i.ties 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 const.i.tuent 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 pa.s.sed 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 a.n.a.lyses four harmonic const.i.tuents, is in the Natural Philosophy Department at Glasgow; a much larger machine, to a.n.a.lyse a tidal curve containing five pairs of harmonic terms, or eleven const.i.tuents in all, was made for the British a.s.sociation 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 princ.i.p.al 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 s.p.a.ce 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.