The loss of energy due to heating of the conducting connection is not

completely understood, though its quantitative laws have been quite

fully ascertained and expressed in terms of magnitudes that are capable

of measurement. It was found by Joule to be proportional to the second

power, or square, of the current, and to a quantity R depending on the

conductor, and called its resistance. The generation of heat in the

conductor seems to be due to some kind of frictional action of particles

of the conductor set up by the penetration of the Faraday tubes into it.

A conductor is unable to bear any tangential action exerted upon it by

Faraday tubes, which, however, when they exist, begin and end at

material particles, except when they are endless, as they may be in the

radiation of energy. When the Faraday tubes are moving with any ordinary

speed they are not at their ends perpendicular to the conducting surface

from which they start or at which they terminate, but are there more or

less inclined to the surface, and consequently there is tangential

action which appears to displace the particles (not merely at the

surface, unless the alternation is very rapid) relatively to one

another and so cause frictional generation of heat.

The time rate of generation of heat is thus Rc², or R(dQ⧸dt)², when the

units in which R and c are expressed are such as to make this quantity a

rate of doing work in the true dynamical sense. This is the rate at

which the sum of energy already found is being diminished, and so the

equation

½d/dt{(Q²⧸C) + L(dQ⧸dt)²} = －R(dQ⧸dt)²

holds, or leaving out the common factor dQ⧸dt, the equation

L(d²Q⧸dt²) + R(dQ⧸dt) + Q⧸C = 0

This last equation was established by Thomson, and is precisely that

which would be obtained for a pendulum bob of mass L, pulled back

towards the position of equilibrium with a force Q⧸C, where Q is the

displacement from the middle position, and having its motion damped out

by resisting force of amount R per unit of the velocity.

It is more instructive perhaps to take the oscillatory motion of a

spiral spring hung vertically with a weight on its lower end, as that

which has a differential equation equivalent to the equation just found.

When the stretch is of a certain amount, there is equilibrium--the

action of the spring just balances the weight,--and if the spring be

stretched further there will be a balance of pull developed tending to

bring the system back towards the equilibrium position. If left to

itself the system gets into motion, which, if the resistance is not too

great, is added to until the equilibrium position is reached; and the

motion, which is continued by the inertia of the mass, only begins to

fall off as that position is passed, and the pull of the spring becomes

insufficient to balance the weight. Thus the mass oscillates about the

position of equilibrium, and the oscillations are successively smaller

and smaller in extent, and die out as their energy is expended finally

in doing work against friction.

If the resisting force for finite motion is very great, as for example

when the vibrating mass of the pendulum or spring is immersed in a very

viscous fluid, like treacle, oscillation will not take place at all.

After displacement the mass will move at first fairly quickly, then more

and more slowly back to the position of equilibrium, which it will,

strictly speaking, only exactly reach after an infinite time. The

resisting force is here indefinitely small for an indefinitely small

speed, but it becomes so great when any motion ensues, that as the

restoring force falls off with the displacement, no work is finally done

by it, except to move the body through the resisting medium.

The differential equation is applicable to the spring if Q is again

taken as displacement from the equilibrium position, L as the inertia of

the vibrating body, 1⧸C as the pull exerted by the spring per unit of

its extension (that is, the stiffness of the spring), and R has the same

meaning as before.

In this case of motion, as well as in that of the pendulum, energy is

carried off by the production of waves in the medium in which the

vibrator is immersed. These are propagated out from the vibrator as

their source, but no account of them is taken in the differential

equation, which in that respect is imperfect. There is no difficulty,

only the addition of a little complication, in supplying the omission.

The formation of such waves by the spiral spring vibrator can be well

shown by immersing the vibrating body in a trough of water, and the much

greater rate of damping out of the motion in that case can then be

compared with the rate of damping in air.

It has been indicated that the differential equation does not represent

oscillatory motion if the value of R is too great. The exact condition

depends on the roots of the quadratic equation Lx² + Rx + 1⧸C = 0,

obtained by writing 1 for Q, and x for d⧸dt, and then treating x as a

quantity. These roots are －R⧸2L ± √(R²⧸4L² － 1⧸CL), and are

therefore real or imaginary according as 4L⧸C is less or greater than

R². If the roots are real, that is, if R² be greater than 4L⧸C, the

discharge will not be oscillatory; the Faraday tubes referred to above

will be absorbed in the wire without any return to the condenser. The

corresponding result happens with the vibrator when R is sufficiently

great, or L⧸C sufficiently small (a weak spring and a small mass, or

both), to enable the condition to be fulfilled.

If, however, the roots of the quadratic are imaginary, that is, if 4L⧸C

be greater than R² (a condition which will be fulfilled in the spring

analogue, by making the spring sufficiently stiff and the mass large

enough to prevent the friction from controlling the motion) the motion

is one in which Q disappears by oscillations about zero, of continually

diminishing amplitude. A complete discussion gives for the period of

oscillation 4πL⧸√(4L⧸C － R²), or if R be comparatively small, 2π√(LC).

The charge Q falls off by the fraction e^{－RT⧸2L} (where e is the

number 2.71828...) in each period T, and so gradually disappears.

Thus electric oscillations are produced, that is to say, the charged

state of the condenser subsides by oscillations, in which the charged

state undergoes successive reversals, with dissipation of energy in the

wire; and both the period and the rate of dissipation can be calculated

if L, C, and R are known, or can be found, for the system. These

quantities can be calculated and adjusted in certain definite cases, and

as the electric oscillations can be experimentally observed, the theory

can be verified. This has been done by various experimenters.

Returning to the pendulum illustration, it will be seen that the

pendulum held deflected is analogous to the charged jar, letting the

pendulum go corresponds to connecting the discharging coil to the

coatings, the motion of the pendulum is the analogue of that motion of

the medium in which consists the magnetic field, the friction of the air

answers to the resistance of the wire which finally damps out the

current. The inertia or mass of the bob is the analogue of what Thomson

called the electromagnetic inertia of the coil and connections; what is

now generally called the self-inductance of the conducting system. The

component of gravity along the path towards the lowest point, answers to

the reciprocal, 1⧸C, of the capacity of the condenser.

It appears from the analogy that just as the oscillations of a pendulum

can be prevented by immersing the bob in a more resisting medium, such

as treacle or oil, so that when released the pendulum slips down to the

vertical without passing it, so by properly proportioning the resistance

in the circuit to the electromagnetic inertia of the coil, oscillatory

discharge of the Leyden jar may also be rendered impossible.

All this was worked out in an exceedingly instructive manner in

Thomson's paper; the account of the matter by the motion of Faraday

tubes is more recent, and is valuable as suggesting how the inertia

effect of the coil arises. The analogy of the pendulum is a true one,

and enables the facts to be described; but it is to be remembered that

it becomes evident only as a consequence of the mathematical treatment

of the electrical problem. The paper was of great importance for the

investigation of the electric waves used in wireless telegraphy in our

own time. It enabled the period of oscillation of different systems to

be calculated, and so the rates of exciters and receivers of electric

waves to be found. For such vibrators are really Leyden jars, or

condensers, caused to discharge in an oscillatory manner.

This application was not foreseen by Thomson, and, indeed, could hardly

be, as the idea of electric waves in an insulating medium came a good

deal later in the work of Maxwell. Yet the analogy of the pendulum, if

it had then been examined, might have suggested such waves. As the bob

oscillates backwards and forwards the air in which it is immersed is

periodically disturbed, and waves radiate outwards from it through the

surrounding atmosphere. The energy of these waves is exceedingly small,

otherwise, as pointed out above, a term would have to be included in the

theory of the resisted motion of the pendulum to account for this energy

of radiation. So likewise when the electric vibrations proceed, and the

insulating medium is the seat of a periodically varying magnetic field,

electromagnetic waves are propagated outwards through the surrounding

medium--the ether--and the energy carried away by the waves is derived

from the initial energy of the charged condenser. In strictness also

Thomson's theory of electric oscillations requires an addition to

account for the energy lost by radiation. This is wanting, and the whole

decay of the amount of energy present at the oscillator is put down to

the action of resistance--that is, to something of the nature of

frictional retardation. Notwithstanding this defect of the theory, which

is after all not so serious as certain difficulties of exact calculation

of the self-inductance of the discharging conductor, the periods of

vibrators can be very accurately found. When these are known it is only

necessary to measure the length of an electrical wave to find its

velocity of propagation. When electromagnetic waves were discovered

experimentally in 1888 by Heinrich Hertz, it was thus that he was able

to demonstrate that they travelled with the velocity of light.

Thomson suggested that double, triple and quadruple flashes of lightning

might be successive flashes of an oscillatory discharge. He also pointed

out that if a spark-gap were included in a properly arranged condenser

and discharging wire, it might be possible, by means of Wheatstone's

revolving mirror, to see the sparks produced in the successive

oscillations, as "points or short lines of light separated by dark

intervals, instead of a single point of light, or of an unbroken line of

light, as it would be if the discharge were instantaneous, or were

continuous, or of appreciable duration."

This anticipation was verified by experiments made by Feddersen, and

published in 1859 (_Pogg. Ann._, 108, 1859). The subject was also

investigated in Helmholtz's laboratory at Berlin, by N. Schiller, who,

determining the period for condensers with different substances between

the plates, was able to deduce the inductive capacities of these

substances (_Pogg. Ann._, 152, 1874). [The specific inductive capacity

of an insulator is the ratio of the capacity of a condenser with the

substance between the plates to the capacity of an exactly similar

condenser with air between the plates.]

The particular case of non-oscillatory discharge obtained by supposing C

and Q both infinitely great and to have a finite ratio V (which will be

the potential, p. 34, of the charged plate), is considered in the paper.

The discharging conductor is thus subjected to a difference of potential

suddenly applied and maintained at one end, while the other end is kept

at potential zero. The solution of the differential equation for this

case will show how the current rises from zero in the wire to its final

steady value. If c be put as before for the current －dQ⧸dt, and the

constant value V for Q⧸C, the equation is

L(dc⧸dt) + Rc = V

which gives, since c = 0 when t = 0,

c = (V⧸R)[1 － e^{－(R⧸L)t}].

Thus, when an infinite time has elapsed the current has become V⧸R, the

steady value.

Thomson concludes by showing how, by measuring the non-oscillatory

discharge of a condenser (the capacity of which can be calculated) by

means of an electrodynamometer and an ordinary galvanometer arranged in

series, what W. Weber called the duration of the discharging current may

be determined. From this Thomson deduced a value for the ratio of the

electromagnetic unit of electricity to the electrostatic unit, and

indicated methods of determining this ratio experimentally. This ratio

is of fundamental importance in electromagnetic theory, and is

essentially of the nature of a speed. According to Maxwell it is the

speed of propagation of electromagnetic waves in an insulating medium

for which the units are defined. It was first determined in the Glasgow

laboratory by Mr. Dugald McKichan, and has been determined many times

since. It is practically identical with the speed of light as

ascertained by the best experiments.

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 classes 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 classes 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 analysis 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 pronouncements 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 analysis: Thomson was not, to say the least; and this was

probably the main reason why the vectorial treatment of displacement,

velocities, and other directed quantities, 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 illustrations, 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

assistants 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 entitled _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 _Mécanique Céleste_, Lagrange's

_Mécanique Analytique_, and Fourier's _Théorie Analytique 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

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.

[Illustration: FIG. 14.]

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.

[Illustration: FIG. 15.]

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.

[Illustration: FIG. 16.]

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.

[Illustration: FIG. 17.]

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.

CHAPTER XII

THE AGE OF THE EARTH

From his student days throughout his life, Lord Kelvin took a keen

interest in geological questions. He was always an active member of the

Geological Society of Glasgow, and was its president for twenty-one

years (1872-1893). The distribution of heat in the substance of the

earth was the subject of his inaugural dissertation as Professor of

Natural Philosophy; and previously, as a student, he had written an

essay on "The Figure of the Earth," for which he had been awarded a

University Gold Medal. He never ceased to ponder over the problems of

terrestrial physics, and he wrote much on the subject. His papers are to

be found as Appendices to Thomson and Tait's _Natural Philosophy_, and

in vol. ii of his _Popular Lectures and Addresses_, which is devoted to

geology and general physics.

His conclusions regarding the age of the earth have been referred to in

the last chapter. The first allusion to the subject was contained (see

p. 65 above) in his inaugural dissertation "_De Caloris distributione in

Terræ Corpus_"; but he returned to it again in a communication made to

the Royal Society of Edinburgh in December, 1865, and entitled "The

Doctrine of Uniformity in Geology briefly refuted." On February 27,

1868, he delivered to the Geological Society of Glasgow an address

entitled "On Geological Time," in which the necessity for limiting

geological and other changes to an almost infinitesimal fraction of the

vast periods at that time demanded was insisted on, and which gave rise

to much discussion.

The address began with a protest against the old uniformitarian view of

geological changes as expressed by Playfair in his _Illustrations of the

Huttonian Theory_. The first objection taken to the idea that "in the

continuation of the different species of animals and vegetables that

inhabit the earth, we discern neither a beginning nor an end; in the

planetary motions where geometry has carried the eye so far, both into

the future and the past, we discover no mark either of the commencement

or the termination of the present order" is, that the stability of the

motions of the heavenly bodies, to which reference is made in this

statement, is founded upon what is essentially an approximate

calculation, which leaves out, by intention, the consideration of

frictional resistance.

He points out, for example, that the friction which accompanies the

relative motion of the waters of the earth and the land is attended by

the production of heat, and that, by the doctrine of the conservation of

energy, heat cannot be produced without a disappearance of an equivalent

quantity of energy, either of motion or of position. The chief source of

this energy is the earth's rotation. Since the earth turns under the

moon and the tidal spheroid--that is, the earth's shape as distorted by

the heaping up of the waters in the tides--remains on the whole

stationary with respect to the moon, the solid matter of the earth turns

under the distribution of the water, held more or less fixed by the

moon, as does a fly-wheel under a stationary friction band round its

rim. Then just as the band held fixed retards the fly-wheel, so the

earth must be retarded in its rotation by this water-brake. In the

earth's rotation there is a store of kinetic energy which, roughly

estimated, would not be exhausted in less than ten million million

years, although drawn upon continuously by friction, or other actions,

at the rate of one million horse-power; so that, no immediate

catastrophe, such as that we should be involved in by the stoppage or

considerable retardation of the spinning motion of the earth, is

possible. But it was pointed out by Thomson that the best results of

astronomical observation show that the earth would in one hundred years

fall behind a perfect time-keeper, with which its rotation kept pace at

the beginning of the time, by about twenty seconds. The tendency is to

make the earth turn slower, and the moon to increase its distance and

move more slowly in its orbit, but with a resultant effect towards

coincidence of the period of the earth's rotation with that of

revolution of the moon round the earth. After this coincidence has been

attained, however, the solar tides will tend to make the moon fall in

towards the earth.

If then the earth be rotating more and more slowly, as time goes on, at

present, it must have been rotating more rapidly in past time. A

thousand million years ago, at the present rate of retardation, the

earth must have been rotating one seventh part of its speed faster than

it is rotating at present, and this would give for centrifugal force at

the surface one thousand million years ago, greater than the centrifugal

force at present, in the ratio of 64 to 49. Apparently therefore the

earth must have solidified at a much later date than that epoch, a date

when it was rotating much more nearly with the angular speed which it

has now; otherwise the figure of the earth would have deviated much more

from the spherical form than it actually does. On the other hand, one

hundred million years ago centrifugal force would be only three per

cent. greater than it is at present, and consolidation of the earth at

that less remote period would give a shape to the earth not very

different from that which it now possesses. The argument therefore from

tidal retardation would cut down the time available for geological and

biological changes to something not much more than one hundred million

years, perhaps to less.

A second argument for limitation of the time available for such

processes is derived from the sun's heat. The sun cannot be regarded as

a miraculous body producing its light and heat from nothing. Changes of

the constitution of the sun must be continually proceeding, to account

for its enormous radiation of energy into space, a radiation of which

only an infinitesimal part is received by the bodies of the solar

system, and a still more minute portion by the earth. The effects of the

sun's light and heat on the earth show how enormous must be the quantity

of energy lost from the sun in a year. How is this loss of energy to be

accounted for? What is the physical change which gives rise to it? In

1854 Thomson put forward the theory that the sun's heat is kept up by

the falling in of meteors on the sun's surface, but he afterwards saw

reason to abandon that view. Helmholtz had advocated the theory that the

sun was a body heated by the coming together of the matter composing it

by its mutual attraction, a process which, although the sun is now a

continuous mass, is to be regarded as still going on. It is easy to

calculate the exhaustion of potential energy caused by the coming

together of the matter of the sun from universal dispersion through

infinite space to a sphere of uniform density of the present size of the

sun. The result is about as much energy as would be generated by burning

seven million million million million million tons of coal. The amount

radiated in each hour is about as much as would be generated by burning

something like nine tons of coal every hour on every square yard of the

sun's surface. It is certain that the sun must be still contracting, and

if it contracts sufficiently to just make good this expenditure by the

further exhaustion of potential energy involved in the closer

aggregation of the matter, it must diminish in radius in each year by as

much as 130 feet.

The amount of energy generated by the falling together of the matter of

the sun from universal diffusion to the dimensions which the sun has at

present, is only about 13,000,000 times the amount now radiated per

annum. In Thomson's paper Pouillet's estimate of the energy radiated per

second is used, and this number is raised to 20,000,000. Taking the

latter estimate, the whole potential energy exhausted by the

condensation of the sun's mass to uniform density would suffice for only

20,000,000 years' supply. But the sun is undoubtedly of much greater

density in the central parts than near the surface, and so the energy

exhausted must be much greater than that stated above. This will raise

the number of years provided for. On the other hand, a considerable

amount of energy would be dissipated during the process of

condensation, and this would reduce the period of radiation estimated.

Thomson suggests that 50,000,000, or 100,000,000, years is a possible

estimate.

It is not unlikely that the rate of radiation in past time, when the sun

had not nearly condensed to its present size, was so much less than it

is at present that the period suggested above may have to be

considerably augmented. Another source of radiation, which seems to be

regarded by some authorities as a probable, if not a certain, one, has

been suggested in recent years--the presence of radio-active substances

in the sun. So far as we know, Lord Kelvin did not admit that this

source of radiation was worthy of consideration; but of course, granted

its existence to an extent comparable with the energy derivable from

condensation of the sun's mass, the "age of the sun's heat" would have

to be very greatly extended. These are matters, however, on which

further light may be thrown as research in radio-activity progresses.

Lord Kelvin was engaged when seized with his last illness in discussing

the changes of energy in a gaseous, or partially gaseous, globe, slowly

cooling and shrinking in doing so; and a posthumous paper on the subject

will shortly be published which may possibly contain further information

on this question of solar physics.

But Thomson put forward a third argument in the paper on Geological

Time, which has always been regarded as the most important. It is

derived from the fact, established by abundant observations, that the

temperature in the earth's crust increases from the surface inwards; and

that therefore the earth must be continually losing heat by conduction

from within. If the earth be supposed to have been of uniform

temperature at some period of past time and in a molten state, and

certain assumptions as to the conductive power and melting point of its

material be made, the time of cooling until the gradient of temperature

at the surface acquired its present value can be calculated. This was

done by Thomson in a paper published in the _Transactions, R.S.E._, in

1862. We propose to give here a short sketch of his argument, which has

excited much interest, and been the cause of some controversy.

In order to understand this argument, the reader must bear in mind some

fundamental facts of the flow of heat in a solid. Let him imagine a slab

of any uniform material, say sandstone or marble, the two parallel faces

of which are continually maintained at two different temperatures,

uniform over each face. For example, steam may be continually blown

against one face, while ice-cold water is made to flow over the other.

Heat will flow across the slab from the hotter face to the colder. It

will be found that the rate of flow of heat per unit area of face, that

is per square centimetre, or per square inch, is proportional to the

difference of the temperatures in the slab at the two faces, and

inversely proportional to the thickness of the slab. In other words, it

is proportional to the fall of temperature from one face to the other

taken per unit of the thickness, that is, to the "gradient of

temperature" from one face to the other. Moreover, comparing the flow in

one substance with the flow in another, we find it different in

different substances for the same gradient of temperature. Thus we get

finally a flow of heat across unit area of the slab which is equal to

the gradient of temperature multiplied by a number which depends on the

material: that number is called the "conductivity" of the substance.

Now, borings made in the earth show that the temperature increases

inwards, and the same thing is shown by the higher temperatures found in

deeper coal mines. By means of thermometers sunk to different depths,

the rate of increase of temperature with depth has been determined.

Similar observations show that the daily and annual variations of

temperature caused by the succession of day and night, and summer and

winter, penetrate to only a comparatively small depth below the

surface--three or four feet in the former case, sixty or seventy in the

latter. Leaving these variations out of account, since the average of

their effects over a considerable interval of time must be nothing, we

have in the earth a body at every point of the crust of which there is a

gradient of increasing temperature inwards. The amount of this may be

taken as one degree of Fahrenheit's scale for every 50 feet of descent.

This gradient is not uniform, but diminishes at greater depths.

Supposing the material of uniform quality as regards heat-conducting

power, the mathematical theory of a cooling globe of solid material (or

of a straight bar which does not lose heat from its sides) gives on

certain suppositions the gradients at different depths. The surface

gradient of 1° F. in 50 feet may be taken as holding for 5000 feet or

6000 feet or more.

This gradient of diminution of temperature outwards leads inevitably to

the conclusion that heat must be constantly flowing from the interior of

the earth towards the surface. This is as certain as that heat flows

along a poker, one end of which is in the fire, from the heated end to

the other. The heat which arrives at the surface of the earth is

radiated to the atmosphere or carried off by convection currents; there

is no doubt that it is lost from the earth. Thus the earth must be

cooling at a rate which can be calculated on certain assumptions, and it

is possible on these assumptions to calculate backwards, and determine

the interval of time which must have elapsed since the earth was just

beginning to cool from a molten condition, when of course life cannot

have existed on its surface, and those geological changes which have

effected so much can hardly have began.

Considering a globe of uniform material, and of great radius, which was

initially at one temperature, and at a certain instant had its surface

suddenly brought to, let us say, the temperature of melting ice, at

which the surface was kept ever after, we can find, by Fourier's

mathematical theory of the flow of heat, the gradient of temperature at

any subsequent time for a point on the surface, or at any specified

distance within it. For a point on the surface this gradient is simply

proportional to the initial uniform temperature, and inversely

proportional to the square root of the product of the "diffusivity" of

the material (the ratio of the conductivity to the specific heat) by the

interval of time which has elapsed since the cooling was started. Taking

a foot as the unit of length, and a year as the unit of time, we find

the diffusivity of the surface strata to be 400. If we take the initial

temperature as 7000 degrees F.--which is high enough for melting

rock--and take the interval of time which has elapsed as 100,000,000

years, we obtain at the surface a gradient approximately equal to that

which now exists. A greater interval of time would give a lower

gradient, a smaller interval would give a higher gradient than that

which exists at present. A lower initial temperature would require a

smaller interval of time, a higher initial temperature a longer interval

for the present gradient.

With the initial temperature of 7,000 degrees F., an interval of

4,000,000 years would give a surface gradient of 1° F. in 10 ft. Thus,

on the assumption made, the surface gradient of temperature has

diminished from 1⧸10 to 1⧸50 in about 96,000,000 years. After 10,000

years from the beginning of the cooling the gradient of temperature

would be 2° F. per foot. But, as Thomson showed, such a large gradient

would not lead to any sensible augmentation of the surface temperature,

for "the radiation from earth and atmosphere into space would almost

certainly be so rapid" as to prevent this. Hence he inferred that

conducted heat, even at that early period, could not sensibly affect the

general climate.

Two objections (apart from the assumptions already indicated) will

readily occur to any one considering this theory, and these Thomson

answered by anticipation. The first is, that no natural action could

possibly bring the surface of a uniformly heated globe instantaneously

to a temperature 7000° lower, and keep it so ever after. In reply to

this Thomson urged "that a large mass of melted rock, exposed freely to

our earth and sky, will, after it once becomes crusted over, present in

a few hours, or a few days, or at most a few weeks, a surface so cool

that it can be walked over with impunity. Hence, after 10,000 years, or

indeed, I may say, after a single year, its condition will be sensibly

the same as if the actual lowering of temperature experienced by the

surface had been produced in an instant, and maintained constant ever

after." The other objection was, that the earth was probably never a

uniformly heated solid 7000° F. above the present surface temperature as

assumed for the purpose of calculation. This Thomson answers by giving

reasons for believing that "the earth, although once all melted, or

melted all round its surface, did, in all probability, really become a

solid at its melting temperature all through, or all through the outer

layer which has been melted; and not until the solidification was thus

complete, or nearly so, did the surface begin to cool."

Thomson was inclined to believe that a temperature of 7000° F. was

probably too high, and results of experiments on the melting of basalt

and other rocks led him to prefer a much reduced temperature. This, as

has already been pointed out, would give a smaller value for the age of

the earth. In a letter on the subject published in Nature (vol. 51,

1895) he states that he "is not led to differ much" from an estimate of

24,000,000 years founded by Mr. Clarence King (_American Journal of

Science_, January 1893) on experiments on the physical properties of

rocks at high temperatures.

It is to be observed that the assumptions made above that the physical

constants of the material are constant throughout the earth, and at all

temperatures, are confessedly far from the truth. Nevertheless Thomson

strongly held that the uncertainty of the data can at most extend the

earth's age to some value between 20,000,000 and 200,000,000 of years,

and that the enormously long periods which were wont to be asked for by

geologists and biologists for the changes of the earth's surface and the

development of its flora and fauna, cannot possibly be conceded.

In Nature for January 3, 1895, Professor John Perry suggested that very

possibly the conductivity of the material composing the interior of the

earth was considerably higher than that of the surface strata. If this

were so, then, as can be shown without difficulty, the attainment of the

present gradient would be very greatly retarded, and therefore the age

of the earth correspondingly increased. The question then arose, and was

discussed, as to whether the rocks and other materials at high

temperatures were more or less conducting than at low temperatures, and

experiments on the subject were instituted and carried out. On the

whole, the evidence seemed to show that the conductivity of most

substances is diminished, not increased, by the rise of temperature, and

so far as it went, therefore, the evidence was against Professor Perry's

suggestion. On the other hand, he contended that the inside of the earth

may be a mass of great rigidity, partly solid and partly fluid,

possessing a "quasi-conductivity" which might greatly increase the

period of cooling. The subject is a difficult one both from a

mathematical and from the physical point of view, and further

investigation is necessary, especially of the behaviour of materials

under the enormous stresses which they undoubtedly sustain in the

interior of the earth.

After the publication of the paper on Geological Time a reply to it was

made by Professor Huxley, in an address to the Geological Society of

London, delivered on February 19, 1869. He adopted the rôle of an

advocate retained for the defence of geology against what seems to have

been regarded as an unwarranted attack, made by one who had no right to

offer an opinion on a geological question. For, after a long and

eloquent "pleading," he concludes his address with the words: "My

functions, as your advocate, are at an end. I speak with more than the

sincerity of a mere advocate when I express the belief that the case

against us has entirely broken down. The cry for reform which has been

raised from without is superfluous, inasmuch as we have long been

reforming from within with all needful speed; and the critical

examination of the grounds upon which the very grave charge of

opposition to the principles of Natural Philosophy has been brought

against us, rather shows that we have exercised a wise discrimination in

declining to meddle with our foundations at the bidding of the first

passer-by who fancies our house is not so well built as it might be." To

this Thomson rejoined in an address entitled "Of Geological Dynamics,"

also delivered to the Geological Society of Glasgow on April 5, 1869;

and to this, with Professor Huxley's address, the reader must be

referred for the objection, brought against Thomson's arguments, and the

replies which were immediately forthcoming. This is not the place to

discuss the question, but reference may be made to an interesting paper

on the subject in the _Glasgow Herald_ for February 22, 1908, by

Professor J. W. Gregory, in which the suggestion of Professor Perry, of

a nearer approach to uniformity of temperature in the interior of the

earth than Thomson had thought possible, is welcomed as possibly

extending the interval of time available to a period sufficient for all

purposes. In Professor Gregory's opinion, "Lord Kelvin in one respect

showed a keener insight than Huxley, who, referring to possible changes

in the rate of rotation of the earth, or in the heat given forth from

the sun or in the cooling of the earth, declared that geologists are

Gallios, 'who care for none of these things.' An ever-increasing school

of geologists now cares greatly for these questions, and reveres Lord

Kelvin as one of the founders of the geology of the inner earth."

After all, the problem is not one to be dealt with by the geologist or

biologist alone, but to be solved, so far as it can be solved at all, by

a consideration of all relevant evidence, from whatsoever quarter it may

come. It will not do in these days for scientific men to shut themselves

up within their special departments and to say, with regard to branches

of science which deal with other aspects of nature and other problems of

the past, present and future of that same earth on which all dwell and

work, that they "care for none of these things." This is an echo of an

old spirit, not yet dead, that has done much harm to the progress of

science. The division of science into departments is unavoidable, for

specialisation is imperative; but it is all the more necessary to

remember that the divisions set up are more or less arbitrary, and that

there are absolutely no frontiers to be guarded and enforced. Chemistry,

physiology, and physics cannot be walled off from one another without

loss to all; and geology has suffered immensely through its having been

regarded as essentially a branch of natural history, the devotees of

which have no concern with considerations of natural philosophy. Lord

Kelvin's dignified questions were unanswerable. "Who are the occupants

of 'our house,' and who is the 'passer-by'? Is geology not a branch of

physical science? Are investigations, experimental and mathematical, of

underground temperature not to be regarded as an integral part of

geology?... For myself, I am anxious to be regarded by geologists not as

a mere passer-by, but as one constantly interested in their grand

subject, and anxious in any way, however slight, to assist them in their

search for truth."

CHAPTER XIII

BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL STANDARDS

When Professor Thomson began his work as a teacher in the University of

Glasgow, there was, as has already been noticed, great vagueness of

specification of physical quantities. Few of the formal definitions of

units of measurement, now to be found in the pages of every elementary

text book, had been framed, and there was much confusion of quantities

essentially distinct, a confusion which is now, to some extent at least,

guarded against by the adoption of a definite unit, with a distinctive

name for each magnitude to be measured. Thus rate of working, or

activity, was confused with work done; the condition for maximum

activity in the circuit of a battery or dynamo was often quoted as the

condition of greatest efficiency, that is of greatest economy of energy,

although it was exactly that in which half the available energy was

wasted.

Partly as a consequence of this vagueness of specification, there was a

great want of knowledge of the values of physical constants; for without

exact definitions of quantities to be determined, such definitions as

would indicate units for their measurement, related to ordinary

dynamical units according to a consistent scheme, it was impossible to

devise satisfactory experimental methods to do for electricity and

magnetism what had been done by Regnault and others for heat.

The first steps towards the construction of a complete system of units

for the quantitative measurement of magnetic and electric quantities

were taken by Gauss, in his celebrated paper entitled _Intensitas vis

magneticæ terrestris ad mensuram absolutam revocata_, published in 1832.

In this he showed how magnetic forces could be expressed in absolute

units, and thus be connected with the absolute dynamical units which

Gauss, in the same paper, based on chosen fundamental units of length,

mass, and time. Thus the modern system of absolute units of dynamical

quantities, and its extension to magnetism, are due to the practical

insight of a great mathematician, not to the experimentalists or

"practicians" of the time.

Methods of measuring electric quantities in absolute units were

described by W. Weber, in Parts II and III of his _Elecktrodynamische

Maassbestimmungen_, published in 1852. These were great steps in

advance, and rendered further progress in the science of absolute

measurement comparatively easy. But they remained the only steps taken

until the British Association Committee began their work. We have

already (pp. 74-76) referred to the great importance of that work, not

only for practical applications but also for the advancement of science.

But it was not a task which struck the imagination or excited the wonder

of the multitude. For the realisation of standards of resistance, for

example, involved long and tedious investigations of the effects of

impurities on the resistance of metals, and the variation of resistance

caused by change of temperature and lapse of time. Then alloys had to

be sought which would have a temperature effect of small amount, and

which were stable and durable in all their properties.

The discoveries of the experimentalist who finds a new element of

hitherto undreamed-of properties attract world-wide attention, and the

glory of the achievement is deservedly great. But the patient, plodding

work which gives a universal system of units and related standards, and

which enables a great physical subject like electricity and magnetism to

rise from a mere enumeration of qualitative results to a science of the

most delicate and exact measurement, and to find its practical

applications in all the affairs of daily life and commerce, is equally

deserving of the admiration and gratitude of mankind. Yet it receives

little or no recognition.

The construction of a standard of resistance was the first task

undertaken by the committee; but other units, for example of quantity of

electricity, intensity of electric field and difference of potential,

had also to be defined, and methods of employing them in experimental

work devised. It would be out of place to endeavour to discuss these

units here, but some idea of the manner in which their definitions are

founded on dynamical conceptions may be obtained from one or two

examples. Therefore we shall describe two simple experiments, which will

illustrate this dynamical foundation. An account has been given in

Chapter XI of the series of electrometers which Thomson invented for the

measurement of differences of electric potential. These all act by the

evaluation in terms of ordinary dynamical units of the force urging an

electrified body from a place of higher towards a place of lower

potential.

Some indication of the meaning of electrical quantities has been given

in Chapter IV. Difference of electric potential between two points in an

electric field was there defined as the dynamical work done in carrying

a unit of positive electricity against the forces of the field from the

point of lower to the point of higher potential. Now by the definition

of unit quantity of electricity given in electrical theory--that

quantity which, concentrated at a point at unit distance from an equal

quantity also concentrated at a point, is repelled with unit force--we

can find, by the simple experiment of hanging two pith balls (or,

better, two hollow, gilded beads of equal size) by two fine fibres of

quartz, a metre long, say, electrifying the two balls as they hang in

contact, and observing the distance at which they then hang, the

numerical magnitude in absolute units of a charge of electricity, and

apply that to finding the charge on a large spherical conductor and the

potential at points in its field also in absolute units. If m be

the mass of a ball, g gravity in cm. sec. units, d the distance in

cms. of the centres of the balls apart, and l the length in cms. of

a thread, the charge q, say, on each ball is easily found to be

√[mgd³⧸√{4^(l² － d²)}]. Thus the charge is got in absolute

centimetre-gramme-second units in terms of the mass m obtained by

ordinary weighing, and l and d obtained by easy and exact measurements.

If one of the balls be now taken away without discharging the other, and

the latter be placed in the field of a large electrified spherical

conductor, the fibre will be deflected from the vertical by the force on

the ball. Let the two centres be now on the same level. That force is

got at once from the angle of deflection (which is easily observed),

the charge on the ball, and the value of m. The electric field-intensity

is obtained by dividing the value of the force by q. The field intensity

multiplied by D, the distance apart in cms. of the centres of the ball

and the conductor, gives the potential at the centre of the ball in

C.G.S. units. Multiplication again by D gives the charge on the

conductor.

When it made its first Report in 1862 (to the meeting at Cambridge) the

committee consisted of Professors A. Williamson, C. Wheatstone, W.

Thomson, W. H. Miller, Dr. A. Matthiessen, and Mr. F. Jenkin. At the

next meeting, at Newcastle, it had been augmented by the addition of

Messrs. Balfour Stewart, C. W. Siemens, Professor Clerk Maxwell, Dr.

Joule, Dr. Esselbach, and Sir Charles Bright. The duty with which the

committee had been charged was that of constructing a suitable standard

of resistance. A reference to the account given in Chapter X above, of

the derivation of what came to be called the electromagnetic unit of

difference of potential, or electromotive force, by means of a simple

magneto-electric machine--a disk turning on a uniform magnetic field, or

the simple rails and slider and magnetic field arrangement there

described--will show how from this unit and the electromagnetic unit of

current (there also defined) the unit of resistance is defined. It is

the resistance of the circuit of slider, rails, and connecting wire,

when with this electromagnetic unit of electromotive force the unit of

current is made to flow.

This was one clear and definite way of defining the unit of current, and

of attaining the important object of connecting the units in such a way

that the rate of working in a circuit, or the energy expended in any

time, should be expressed at once in ordinary dynamical units of

activity or energy. A considerable number of proposals were discussed by

the committee; but it was finally determined to take the basis here

indicated, and to realise a standard of resistance in material of

constant and durable properties, which should have some simple multiple

of the unit of resistance, in the system of dynamical units based on the

centimetre as unit of length, the gramme as unit of mass, and the second

as unit of time--the so-called C.G.S. system. The comparison of the

different metals and alloys available was a most important but

exceedingly laborious series of investigations, carried out mainly by

Dr. Matthiessen and Professor Williamson.

Professor Thomson suggested to the committee the celebrated method of

determining the resistance of a circuit by revolving a coil, which

formed the main part of the circuit about a vertical axis in the earth's

magnetic field. An account of the experiments made with this method is

contained in the Report of 1863. They were carried out at King's

College, London, where Maxwell was then Professor of Experimental

Physics, by Maxwell, Balfour Stewart, and Fleeming Jenkin. The

theoretical discussion and the description of the experiments was

written by Maxwell, the details of the apparatus were described by

Jenkin.

The principle of the method is essentially the same as that of the

simple magneto-electric machine, to which reference has just been made.

Two parallel coils of wire were wound in channels cut round rings of

brass, which, however, were cut across by slots filled with vulcanite,

to prevent induced currents from circulating in the brass. These coils

were mounted in a vertical position and could be driven as a rigid

system, at a constant measured speed, about a vertical axis passing

through the centre of the system. Between the coils at this centre was

hung, from a steady support, a small magnetic needle by a single fibre

of silk; and a surrounding screen prevented the needle and suspension

from being affected by currents of air.

The ends of the coil were connected together so that the whole revolved

as a closed circuit about the vertical axis. When the coil system was at

right angles to the magnetic meridian there was a magnetic induction

through it of amount AH, where A denotes the effective area of the

coils, and H the horizontal component of the earth's magnetic field. By

one half-turn the coil was reversed with reference to this magnetic

induction, and as the coil turned an induced current was generated,

which depended at any instant on the rate at which the magnetic

induction was varying at the instant, on the inductive electromotive

force due to the varying of the current in the coil itself, and on the

resistance of the circuit. A periodic current thus flowed in one

direction _relatively to the coil_ in one half-turn from a position

perpendicular to the magnetic meridian, and in the opposite direction in

the next half-turn. But as the position of the coil was reversed in

every half-turn as well as the current in it, the current flowed on the

whole in the same average direction relatively to the needle, and but

for self-induction would have had its maximum value always when the

plane of the coil was in the magnetic meridian.

The needle was deflected as it would have been by a certain average

current, and the deflection was opposed by the action of the earth's

horizontal magnetic field H. But this was the field cut by the coil as

it turned, and therefore (except for a small term depending on the

turning of the coil in the field of the needle) the value of H did not

appear in the result, and did not require to be known.

Full details of the theory of this method and of the experiments carried

out to test it will be found in various memoirs and treatises[23]; but

it must suffice here to state that the resistance of the coil was

determined in this way, by a large series of experiments, before and

after every one of which the resistance was compared with that of a

German-silver standard. The resistance of this standard therefore became

known in absolute units, and copies of it, or multiples or sub-multiples

of it, could be made.

A unit called the B.A. unit, which was intended to contain 10^9 C.G.S.

electromagnetic units of resistance, was constructed from these

experiments, and copies of it were soon after to be found in nearly all

the physical laboratories of the world. Resistance boxes were

constructed by various makers, in which the coils were various multiples

of the B.A. unit, so that any resistance within a certain range could be

obtained by connecting these coils in series (which was easily done by

removing short circuiting plugs), and thus the absolute units of current

electromotive force and resistance came into general use.

In 1881 Lord Rayleigh and Professor Schuster carried out a very careful

repetition of the British Association experiments with the same

apparatus at the Cavendish Laboratory, and obtained a somewhat different

result. They found that the former result was about 1.17 per cent. too

small. Lord Rayleigh next carried out an independent set of experiments

by the same method with improved apparatus, and found that this

percentage error must be increased to about 1.35.

It may be noticed here that the simple disk machine, of Thomson's

illustration of the absolute unit of electromotive force, has been used

by Lorenz to give a method of determining resistance which is now

recognised as the best of all. It is sketched here that the reader may

obtain some idea of later work on this very important subject; work

which is a continuation of that of the original British Association

Committee by their successors. A circuit is made up of a standard coil

of wire, the ends of which are made to touch at the circumference and

near the centre of the disk, which is placed symmetrically with respect

to a cylindrical coil, and within it. A current is sent round this coil

from a battery, and produces a magnetic field within the coil, the lines

of magnetic force of which pass across the plane of the disk. This

current, or a measured fraction of it, is also made to flow through the

standard coil. The disk is now turned at a measured speed about its

axis, so that the electromotive force due to the cutting of the field

tends to produce a current in the standard coil of wire. The

electromotive force of the disk is made to oppose the potential

difference between the ends of this coil due to the current, so that no

current flows along the disk or the wires connecting it with the

standard coil. The magnetic field within the coil can be calculated from

the form and dimensions of the coil and the current in it (supposed for

the moment to be known), and the electromotive force of the disk is

obtained in terms of its dimensions and its speed and the field

intensity. But this electromotive force, which is proportional to the

current in the coil, is equal to the product of the resistance of the

wire and the same current, or a known fraction of it. Thus the current

appears on both sides of the equation and goes out, and the value of the

resistance is found in absolute units.

Lord Rayleigh obtained, by this method, a result which showed that the

B.A. unit was 1.323 per cent. too small; and exact experiments have been

made by others with concordant results. Values of the units have been

agreed on by International Congresses as exact enough for general work,

and with these units all electrical researches, wherever made, are

available for use by other experimenters.

A vast amount of work has been done on this subject during the last

forty years, and though the value of the practical unit of

resistance--10^9 C.G.S. units, now called the "ohm"--is taken as

settled, and copies can now be had in resistance boxes, or separately,

adjusted with all needful accuracy, at the National Physical Laboratory

and at the Bureau of Standards at Washington, and elsewhere, experiments

are being made on the exact measurement of currents; while a careful

watch is kept on the standards laid up at these places to see whether

any perceptible variation of their resistance takes place with lapse of

time.

The British Association Committee also worked out a complete system of

units for all electrical and magnetic quantities, and gave the first

systematic statement of their relations, that is, of the so-called

dimensional equations of the quantities. This will be found in the works

to which reference has already been made (p. 251).

CHAPTER XIV

THE BALTIMORE LECTURES

The Baltimore Lectures were delivered in 1884 at Johns Hopkins

University, soon after the Montreal meeting of the British Association.

The subject chosen was the Wave Theory of Light; and the idea underlying

the course was to discuss the difficulties of this theory to

"Professorial fellow-students in physical science." A stenographic

report of the course was taken by Mr. A. S. Hathaway, and was published

soon after. The lectures were revised by Lord Kelvin, and the book now

known as _The Baltimore Lectures_ was published just twenty years later

(in 1904) at the Cambridge University Press. It is absolutely impossible

in such a memoir as the present to give any account of the discussions

contained in the lectures as now published. The difficulties dealt with

can for the most part only be understood by those who are acquainted

with the wave theory of light in its details, and such readers will

naturally go direct to the book itself.

Some of the difficulties, however, were frequently alluded to in Lord

Kelvin's ordinary lectures, and all his old students will remember the

animation with which he discussed the apparent anomaly of a medium like

the luminiferous ether, which is of such enormous rigidity that (on the

elastic solid theory) a wave of transverse oscillation is propagated

through it with a speed of 3 × 10^10 centimetres (186,000 miles) per

second, and yet appears to offer no impediment to the slow motion of the

heavenly bodies. For Lord Kelvin adopted the elastic solid theory of

propagation of light as "the only tenable foundation for the wave theory

of light in the present state of our knowledge," and dismissed the

electromagnetic theory (his words were spoken in 1884, it is to be

remembered) with the statement of his strong view that an electric

displacement perpendicular to the line of propagation, accompanied by a

magnetic disturbance at right angles to both, is inadmissible.

And he goes on to say that "when we have an electromagnetic theory of

light," electric displacement will be seen as in the direction of

propagation, with Fresnelian vibrations perpendicular to that direction.

In the preface, of date January 1904, the insufficiency of the elastic

solid theory is admitted, and the question of the electromagnetic theory

again referred to. He says there that the object of the Baltimore

Lectures was to ascertain how far the phenomena of light could be

explained within the limits of the elastic solid theory. And the answer

is "everything _non-magnetic; nothing magnetic_." But he adds, "The

so-called electromagnetic theory of light has not helped us hitherto,"

and that the problem is now fully before physicists of constructing a

"comprehensive dynamics of ether, electricity, and ponderable matter

which shall include electrostatic force, magnetostatic force,

electromagnetism, electrochemistry, and the wave theory of light."

All this is exceedingly interesting, for it seems to make clear Lord

Kelvin's attitude with respect to the electromagnetic theory of Maxwell,

which is now regarded by most physicists as affording on the whole a

satisfactory account, if not a dynamical theory in the sense understood

by Lord Kelvin, of light-propagation. That there is an electric

displacement perpendicular to the direction of propagation and a

magnetic displacement (or motion) perpendicular to both seems proved by

the experiments of Hertz, and the velocity of propagation of these

disturbances has been found to be that of light. Of course it remains to

be found out in what the electric and magnetic changes consist, and

whether the ether has or has not an atomic structure. Towards the answer

to this question on electromagnetic presuppositions some progress has

already been made, principally by Larmor. And, after all, while we may

imagine that we know something more definite of dynamical actions on

ponderable matter, it is not quite certain that we do: we are more

familiar with them, that is almost all. We know, for example, that at

every point in the gravitational field of the earth we may set up a

gravitation vector, or field-intensity; for a particle of matter there

is subjected to acceleration along that direction. But of the rationale

of the action we know nothing, or next to nothing. So we set up electric

and magnetic vectors in an insulating medium, corresponding to electric

and magnetic effects which we can observe; and it is not too much to say

that we know hardly less in this case than we do in the other, of the

inner mechanism of the action of which we see the effects.

Returning to the difficulty of the elastic solid theory, that while its

rigidity is enormous, it offers no obstacle to the planets and other

heavenly bodies which move through it, it may be interesting to recall

how Lord Kelvin used to deal with it in his elementary lectures. The

same discussion was given in the Introductory Lecture at Baltimore. The

difficulty is not got over by an explanation of what takes place: it is

turned by showing that a similar difficulty exists in reconciling

phenomena which can be observed every day with such ordinary materials

as pitch or shoemakers' wax. A piece of such wax can be moulded into a

tuning-fork or a bell, and will then, if struck, sound a musical note of

definite pitch. This indicates, for rapidly alternating deformations

started by a force of short duration, the existence of internal forces

of the kind called elastic, that is, depending on the amount of

deformation caused, not on the rate at which the deformation is

increasing or diminishing, as is the case for the so-called "viscous

forces" which are usually displayed by such material. But the

tuning-fork or bell, if left lying on the table, will gradually flatten

down into a thin sheet under only its own weight. Here the deformation

is opposed only by viscous forces, which, as the change is very slow,

are exceedingly small.

But let a large slab of it, three or four inches thick, be placed in a

glass jar ten or twelve inches in diameter, already partly filled with

water, and let some ordinary corks be imprisoned beneath, while some

lead bullets are laid on the upper surface. After a month or two it will

be found that the corks have disappeared from the water into the wax,

and that the orifices which they made in entering it have healed up

completely; similarly the bullets have sunk down into the slab, leaving

no trace behind. After two or three months more, the corks will be seen

to be bursting their way out through the upper surface of the slab, and

the bullets will be found in the water below. The very thing has taken

place that would have happened if water had been used instead of pitch,

only it has taken a very much longer time to bring it about. The corks

have floated up through the wax in consequence of hydrostatic upward

force exerted by the wax acting as a fluid; and the bullets have sunk

down in consequence of the excess of their weights above the upward

hydrostatic force exerted on them as on the corks. The motion in both

cases has been opposed by the viscous forces called into play.

The application of this to the luminiferous ether is immediate. Let the

ether be regarded as a substance which can perform vibrations only "when

times and forces are suitable," that is, when the forces producing

distortion act for only an infinitesimal time (as in the starting of the

tuning-fork by a small blow), and are not too great. Vibrations may be

set up locally, and the medium may have a true rigidity by which they

are propagated to more remote parts; that is to say, waves travel out

from the centre of disturbance. On the other hand, if the forces are

long continued, even if they be small, they produce continuously

increasing change of shape. Thus the planets move seemingly without

resistance.

The conclusion is that the apparently contradictory properties of the

ether are no more mysterious than the properties of pitch or shoemakers'

wax. And, after all, matter is still a profound mystery.

Dynamical illustrations, which old Glasgow students will recognise,

appear continually in the lectures. They will remember, almost with

affection, the system of three particles (7 lb. or 14 lb. weights!)

joined together in a vertical row by stout spiral springs of steel,

which were always to be taken as massless, and will recall Lord Kelvin's

experiments with them, demonstrating the three modes of vibration of a

system of three masses, each of which influenced those next it on the

two sides. Here they will find the problem solved for any number of

particles and intervening springs, and the solution applied to an

extension of the massive molecule which von Helmholtz imbedded in the

elastic ether, and used to explain anomalous dispersion. A highly

complex molecule is suggested, consisting of an outer shell embedded in

the ether as in the simpler case, a second shell within that connected

to the outer by a sufficient number of equal radial springs, a third

within and similarly connected to the second by radial springs, and so

on. This molecule will have as many modes of vibration as there are sets

of springs, and can therefore impart, if it is set into motion, a

complex disturbance to the ether in which it is imbedded.

The modification of this arrangement by which Lord Kelvin explained the

phosphorescence of such substances as luminous paint is also described,

and will be recognised by some as an old friend. A number, two dozen or

so, of straight rods of wood eighteen inches long are attached to a

steel wire four or five inches apart, like steps on a ladder made with a

single rope along the centres of the steps. The wire is so attached to

each rod that the rod must turn with the wire if the latter is twisted

round. Each rod is loaded with a piece of lead at each end to give it

more moment of inertia about the wire. The wire, with this "ladder"

attached to it, is rigidly attached to the centre of a cross-bar at the

top, which can be made to swing about the wire as an axis and so impart

twisting vibrations to the wire in a period depending on this driver.

Sliding weights attached to the bar enable its moment of inertia to be

changed at pleasure. The lower end of the wire carries a cross-bar with

two vanes, immersed in treacle in a vessel below. When the period of the

exciter was very long the waves of torsion did not travel down the

"ladder," but when the period was made sufficiently short the waves

travelled down and were absorbed in the treacle below. In the former

case the vibrations persisted; the case was analogous to that of

phosphorescence.

[Illustration: FIG. 18.]