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 refers to a point fixed in space, and if we wish it to refer to a point moving with the system we must write as usual

where $$\tfrac{\partial'w}{\partial t}$$ now means the rate of change measured from the moving point. Likewise, if we wish the velocities which enter into (23) to be expressed in terms of velocities relative to axes moving with the system, we must write

where $$v_{2x}, v_{2y}$$, and $$v_{2z}$$ are the components of these relative velocities.

Substituting (24) and (25) in (23), and remembering the simple proportionality between $$S_{x}, S_{y}$$, and $$S_{z}$$ and the density of momentum $$m_{x}, m_{y}$$, and $$m_{z}$$, we easily obtain

Now ($$\rho\Im_{x}$$) may be expressed in terms of the electric and magnetic force intensities, together with the density of the momentum. This involves only the fundamental equations of electromagnetic theory and has been done in paragraph 7, reference to which will show that with the present notation

Substituting this for the ($$\rho\Im_{x}$$) which occurs on the lefthand side of (26), rearranging the latter, and putting

$w=\frac{1}{8\pi}\{X^{2}+Y^{2}+Z^{2}+\alpha^{2}+\beta^{2}+\gamma^{2}\}$

and

$w_{t}=\frac{1}{8\pi}\{Y^{2}+Z^{2}+\beta^{2}+\gamma^{2}\}$,

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 we get

Now if this expression is integrated through the part of the volume ($$AB$$) which lies on the side ($$A$$) of the partition ($$CD$$), the terms on the left of (28) become equal to

$\int\{(V^{2}+v_{1}^{2})m_{x}-2v_{1}w_{t}\}dS$,

where the integral is taken over the part of the plane ($$C'D'$$) which is included in the surface. This follows from the fact that the rest of the surface of part $$A$$ belongs to the surface ($$AB$$) and outside of ($$AB$$) there is no disturbance whatever. The terms on the right (of 28) give an average value of zero. This last will be evident if they are considered separately. The first two give directly a time average of zero, after great elapse of time, since neither ($$w$$) nor ($$v_{1}m_{x}$$) ever becomes infinite. The terms involving the ($$y$$) and ($$z$$) derivatives when integrated may be written as surface integrals over the bounding surface (of part $$A$$), and they then represent the flux of energy through this surface in a direction perpendicular to ($$v_{1}$$), i. e., in a direction perpendicular to the $$x$$-axis. This flux being everywhere zero over the surface, these terms vanish.

The terms involving $$\Im_{x},\Im_{y},\Im_{z}$$ also give a time average of zero because they represent what might be called the internal activity of the forces which act on the charges in the $$A$$-part of the volume ($$AB$$), and since this part $$A$$ is isolated on all sides except on the side ($$CD),$$ the activity of these forces really means the rate at which the part ($$B$$) by means of them is doing work on $$A$$. In the long run $$B$$'s work on $$A$$ must be equal to $$A$$'s work on $$B$$, if the system is to be conservative and the internal motions are to be stationary.

Thus we learn finally from equation (28) that the average value of

$\int\{(V^{2}+v_{1}^{2})m_{x}-2v_{1}w_{t}\}dS=0$,

where the integral is taken over the enclosed part of the plane ($$CD$$). Since this is true for any position of the plane ($$CD$$) so long as it is perpendicular to ($$v_{1}$$) we evidently have on

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 the average, integrating along ($$x$$) throughout the entire system,

This gives, using the former notation, and remembering that on the average the internal structure is assumed to remain the same,

which, since ($$v_{1}$$) is here along ($$x$$), is precisely the result of equation (16), and becomes (17) on differentiation.

Conclusion.
It has been shown in the foregoing that the electromagnetic mass of an isolated, symmetrical, purely electric system possessing any structure which on the average remains the same, and any internal motions or constraints, is expressible in terms of its velocity as a whole through space together with its "transverse energy" and the derivative of the latter with respect to the velocity. If second-order terms in the velocity be neglected, the mass is a simple constant multiplied by the total included electromagnetic energy.

If the mass of ponderable bodies has an electromagnetic origin, then the inertia of matter is to be considered merely as a manifestation of confined energy. From this point of view, matter and energy are thus very closely related and the laws of the conservation of mass and energy become practically identical.

It has been pointed out that the loss of mass, inevitable on this view, which takes place when energy is lost to the system, is large enough to be detected in the case of radioactive changes. If we assume the disintegration theory of the elements, this loss of mass affords a ready explanation of the general, small irregularities to be found in the list of atomic weights, and thus removes a serious difficulty from the path of the disintegration theory. For this loss of mass to take place however, it is not necessary that the whole of the mass be electromagnetic.

It has been shown that if material mass be electromagnetic and if lighter elements are formed from heavier ones through violent energy changes, it follows that gravity acts between quantities of confined energy and not between masses in any other sense. Several speculations are indulged in as to the results of assuming gravitation between quantities of energy.

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 Finally, the fundamental proposition is dealt with mathematically from an entirely different point of view and the same result obtained.

In conclusion I wish to express my thanks to Prof. J. J. Thomson and to Mr. G. F. C. Searle for several valuable criticisms and suggestions.

Cambridge, England,
 * August 14th, 1907.

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Author talk:Charles Pomeroy Parker

Members of the American Academy of Arts & Sciences: 1780–2010
 * Parker, Charles Pomeroy (1852–1916)
 * Election: 1910,
 * Fellow Affiliation at election: Harvard University
 * Residence at election: Cambridge, MA
 * Career description: Classicist; Educator

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 Byzantines and Seljuk Turks, probably between 1158 and 1174. It had become a fine Moslem city in the 14th century, and was then called Ladik, being famous for the woven and embroidered products of its Greek inhabitants. The delightful gardens of Denizli have obtained for it the name of the “Damascus of Anatolia.”

 DENMAN, THOMAS, (1779–1854), English judge, was born in London, the son of a well-known physician, on the 23rd of July 1779. He was educated at Eton and St John’s College, Cambridge, where he graduated in 1800. Soon after leaving Cambridge he married; and in 1806 he was called to the bar at Lincoln’s Inn, and at once entered upon practice. His success was rapid, and in a few years he attained a position at the bar second only to that of Brougham and Scarlett (Lord Abinger). He distinguished himself by his eloquent defence of the Luddites; but his most brilliant appearance was as one of the counsel for Queen Caroline. His speech before the Lords was very powerful, and some competent judges even considered it not inferior to Brougham’s. It contained one or two daring passages, which made the king his bitter enemy, and retarded his legal promotion. At the general election of 1818 he was returned M.P. for Wareham, and at once took his seat with the Whig opposition. In the following year he was returned for Nottingham, for which place he continued to sit till his elevation to the bench in 1832. His liberal principles had caused his exclusion from office till in 1822 he was appointed common serjeant by the corporation of London. In 1830 he was made attorney-general under Lord Grey’s administration. Two years later he was made lord chief justice of the King’s Bench, and in 1834 he was raised to the peerage. As a judge he is most celebrated for his decision in the important privilege case of Stockdale v. Hansard (9 Ad. & El. I.; 11 Ad. & El. 253), but he was never ranked as a profound lawyer. In 1850 he resigned his chief justiceship and retired into private life. He died on the 26th of September 1854, his title continuing in the direct line.

The (1819–1896), his fourth son, was also a distinguished lawyer, and a judge of the Queen’s Bench from 1872 till his death in 1896.

 DENMARK (Danmark), a small kingdom of Europe, occupying part of a peninsula and a group of islands dividing the Baltic and North Seas, in the middle latitudes of the eastern coast. The kingdom lies between 54° 33′ and 57° 45′ N. and between 8° 4′ 54″ and 12° 47′ 25″ E., exclusive of the island of Bornholm, which, as will be seen, is not to be included in the Danish archipelago. The peninsula is divided between Denmark and Germany (Schleswig-Holstein). The Danish portion is the northern and the greater, and is called Jutland (Dan. Jylland). Its northern part is actually insular, divided from the mainland by the Limfjord or Liimfjord, which communicates with the North Sea to the west and the Cattegat to the east, but this strait, though broad and possessing lacustrine characteristics to the west, has only very narrow entrances. The connexion with the North Sea dates from 1825. The Skagerrack bounds Jutland to the north and north-west. The Cattegat is divided from the Baltic by the Danish islands, between the east coast of the Cimbric peninsula in the neighbourhood of the German frontier and south-western Sweden.

There is little variety in the surface of Denmark. It is uniformly low, the highest elevation in the whole country, the Himmelbjerg near Aarhus in eastern Jutland, being little more than 500 ft. above the sea. Denmark, however, is nowhere low in the sense in which Holland is; the country is pleasantly diversified, and rises a little at the coast even though it remains flat inland. The landscape of the islands and the south-eastern part of Jutland is rich in beech-woods, corn fields and meadows, and even the minute islets are green and fertile. In the western and northern districts of Jutland this condition gives place to a wide expanse of moorland, covered with heather, and ending towards the sea in low whitish-grey cliffs. There is a certain charm even about these monotonous tracts, and it cannot be said that Denmark is wanting in natural beauty of a quiet order. Lakes, though small, are numerous; the largest are the Arresö and the Esromsö in Zealand, and the chain of lakes in the Himmelbjerg region, which are drained by the largest river in Denmark, the Gudenaa, which, however, has a course not exceeding 80 m. Many of the meres, overhung with thick beech-woods, are extremely beautiful. The coasts are generally low and sandy; the whole western shore of Jutland is a succession of sand ridges and shallow lagoons, very dangerous to shipping. In many places the sea has encroached; even in the 19th century entire villages were destroyed, but during the last twenty years of the century systematic efforts were made to secure the coast by groynes and embankments. A belt of sand dunes, from 500 yds. to 7 m. wide, stretches along the whole of this coast for about 200 m. Skagen, or the Skaw, a long, low, sandy point, stretches far into the northern sea, dividing the Skagerrack from the Cattegat. On the western side the coast is bolder and less inhospitable; there are several excellent havens, especially on the islands. The coast is nowhere, however, very high, except at one or two points in Jutland, and at the eastern extremity of Möen, where limestone cliffs occur.

Continental Denmark is confined wholly to Jutland, the geographical description of which is given under that heading. Out of the total area of the kingdom, 14,829 sq. m., Jutland, including the small islands adjacent to it, covers 9753 sq. m., and the insular part of the kingdom (including Bornholm), 5076 sq. m. The islands may be divided into two groups, consisting of the two principal islands Fünen and Zealand, and the lesser islands attendant on each. Fünen (Dan. Fyen), in form roughly an oval with an axis from S.E. to N.W. of 53 m., is separated from Jutland by a channel not half a mile wide in the north, but averaging 10 m. between the island and the Schleswig coast, and known as the Little Belt. Fünen, geologically a part of southern Jutland, has similar characteristics, a smiling landscape of fertile meadows, the typical beech-forests clothing the low hills and the presence of numerous erratic blocks, are the superficial signs of likeness. Several islands, none of great extent, lie off the west coast of Fünen in the Little Belt; off the south, however, an archipelago is enclosed by the long narrow islands of Aerö (16 m. in length) and Langeland (32 m.), including in a triangular area of shallow sea the islands of Taasinge, Avernakö, Dreiö, Turö and others. These are generally fertile and well cultivated. Aeröskjöbing and Rudkjöbing, on Aerö and Langeland respectively, are considerable ports. On Langeland is the great castle of Tranekjaer, whose record dates from the 13th century. The chief towns of Fünen itself are all coastal. Odense is the principal town, lying close to a great inlet behind the peninsula of Hindsholm on the north-east, known as Odense Fjord. Nyborg on the east is the port for the steam-ferry to Korsör in Zealand; Svendborg picturesquely overlooks the southern archipelago; Faaborg on the south-west lies on a fjord of the same name; Assens, on the west, a port for the crossing of the Little Belt into Schleswig, still shows traces of the fortifications which were stormed by John of Ranzau in 1535; Middelfart is a seaside resort near the narrowest reach of the Little Belt; Bogense is a small port on the north coast. All these towns are served by railways radiating from Odense. The strait crossed by the Nyborg-Korsör ferry is the Great Belt which divides the Fünen from the Zealand group, and is continued south by the Langelands Belt, which washes the straight eastern shore of that island, and north by the Samso Belt, named from an island 15 m. in length, with several large villages, which lies somewhat apart from the main archipelago.

Zealand, or Sealand (Dan. Sjaelland), measuring 82 m. N. to S. by 68 E. to W. (extremes), with its fantastic coast-line indented by fjords and projecting into long spits or promontories, may be considered as the nucleus of the kingdom, inasmuch as it contains the capital, Copenhagen, and such important towns as Roskilde, Slagelse, Korsör, Naestved and Elsinore (Helsingör). Its topography is described in detail under. Its attendant islands lie mainly to the south and are parts of itself, only separated by geologically recent troughs. The eastern

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ECENT publications of Einstein and Comstock on the relation of mass to energy have emboldened me to publish certain views which I have entertained on this subject and which a few years ago appeared purely speculative, but which have been so far corroborated by recent advances in experimental and theoretical physics that it seems desirable to subject these views to a strict logical development, although in so doing it will be necessary to modify those fundamental principles of the mechanics of ponderable matter which have remained unaltered since the time of Newton.

The recent experiments which indicate a change in the mass of an electron with the speed, together with the phenomenon of radioactivity, have in some minds created a doubt as to the exact validity of some of the most general laws of nature. In the following pages I shall attempt to show that we may construct a simple system of mechanics which is consistent with all known experimental facts, and which rests upon the assumption of the truth of the three great conservation laws, ''Phil. Mag.'' S. 6. Vol. 16. No. 95. Nov. 1908.

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User:Simon Peter Hughes

My name is Simon Peter Hughes and I come from England. I was born in Stratford-upon-Avon, the hometown of William Shakespeare in 1973. I have been teaching English as a foreign language since 1997. I have lived and worked in Turkey, Poland and Japan. I have lived and worked in Taiwan since 2003.

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title: User:Simon Peter Hughes


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706 namely, the law of conservation of energy, the law of conservation of mass, and the law of conservation of momentum. To these we may add, if we will, the law of conservation of electricity.

When a black body is placed in a beam of light it is subject to a pressure or force which tends to move it in the direction in which the light is moving. If $$\frac{d\mathrm{E}}{dt}$$ denotes the time-rate at which the body receives energy, f the force, and V the velocity of light, we have in rational units the formula

This important equation, which was obtained by Maxwell as a consequence of his electromagnetic theory, and by Boltzmann through the direct application of the laws of thermodynamics, has recently been verified with remarkable precision in the beautiful experiments of Nichols and Hull.

A body subjected to the pressure of radiation will acquire momentum, and if we are to accept the law of conservation of momentum, we must conclude that some other system is losing in the same direction an equivalent momentum. We are thus led inevitably, as Poynting has shown, to the idea that the beam of radiation carries not only energy but momentum as well.

The body subject to the constant force of radiation f, will obviously acquire momentum at the rate

Combining equations (1) and (2) gives

The ratio of the acquired energy to the acquired momentum is equal to the velocity of light. The beam of radiation must, therefore, possess energy and momentum in the same ratio. Hence for the beam itself, or any part of it,

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Rh To anyone unfamiliar with the prevailing theories of light, knowing only that light moves with a certain velocity and that in a beam of light momentum and energy are being carried with this same velocity, the natural assumption would be that in such a beam something possessing mass moves with the velocity of light and therefore has momentum and energy. Notwithstanding its apparent divergence from the commonly accepted light theories, I propose to adopt this view and see whither it leads.

Postulating the validity of the fundamental conservation laws mentioned above, we shall need in the following development only this one cardinal assumption, that a beam of radiation possesses not only momentum and energy, but also mass, travelling with the velocity of light, and that a body absorbing radiation is acquiring this mass as it also acquires the momentum and the energy of the radiation. Therefore a body which absorbs radiant energy increases in mass.

The amount of this increase is readily found. If in general we write momentum as the product of mass and velocity, then the momentum of any part of a beam of radiation having the mass m will be given by the equation :

The increase dM in the momentum of the body absorbing the radiation will therefore equal the increase dm in its mass, multiplied by the velocity of light,

Combining this equation with (3) we find

or if we write $$\mathrm{V}=3\times10^{10}$$ centimetres per second,

$$dm=1.111\times10^{-21}d\mathrm{E}$$.

Thus a body receiving or emitting radiant energy gains or loses mass in proportion and by the amount $$1.111\times10^{-21}$$ grams for every erg. This is a small quantity, indeed, but one which is not to be neglected.

Assuming the fundamental conservation law, we must regard mass as a real property of a body which depends upon its state, and not upon its history. Hence it is obvious that if in any other way than by radiation the body gains or loses energy it must still gain or lose mass in just the above proportion. In other words, any change in a body's content of energy is accompanied by a definite change in its mass,

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708 regardless of the nature of the process which the energy change accompanies.

Since therefore when a body loses a given quantity of energy it always loses a definite quantity of mass, we might assume that if it should lose all its energy it would lose all its mass, or, in other words, that the mass of a body is a direct measure of its total energy, according to the equation,

We should then regard mass and energy as different names and different measures of the same quantity, and say that one gram equals $$9 \times10^{20}$$ ergs in the same sense that we say one metre equals 39.37... inches.

Plausible as this view seems, it rests upon an additional hypothesis besides the fundamental postulate which we have chosen. We shall use therefore, not equation (8) but equation (7) as the basis of the following work.

It is to be noted that equation (8) has also been obtained by Einstein (loc. cit.), who derived it from the general equations of the electromagnetic theory, with the aid of the so-called principle of relativity. That a different method of investigation thus leads to the same simple equation we have here deduced, speaks decidedly for the truth of our fundamental postulate.

Comstock (loc. cit.) from electromagnetic theory alone has also concluded that mass is proportional to energy, but his equation is

$$m=\frac{4}{3}\frac{\mathrm{E}}{\mathrm{V}^{2}}$$.

To investigate for the cases studied by Comstock the cause or the justification for this appearance in the equation of the factor 4/3 would lead too far into electromagnetic theory, from which in the present paper I wish to hold entirely aloof.

Before proceeding to develop fully the consequences of equation (7) it may be well to point out an apparent

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Rh inconsistency in the equations for the momentum and the energy of a beam of radiation. The momentum of the beam of mass m we have given in equation (5) as

From our assumption that the energy of the beam is simply the kinetic energy of the moving mass m, we might expect from our knowledge of elementary mechanics to find for the energy the equation

whereas in fact we find from equations (4) and (5) that

We shall see, however, in the next section that this comparison of equations (5) and (9) instead of demolishing our theory actually furnishes a remarkably satisfactory argument in its favour.

One of the interesting branches of modern mathematics has grown out of the study of those geometries which would result from the change of one or more of the axioms of Euclid. These non-Euclidian geometries present the properties of purely imaginary kinds of space and are therefore so far mere exercises in logic, without any physical significance. But their investigation was doubtless prompted in some cases by the belief that experiment itself may sometime show that there are deviations from the ordinary laws of space when these laws are subjected to tests of a different order from those of common mensuration. Indeed it is not unlikely that Euclidian geometry may prove inadequate when we are able to subject to an accurate metric investigation the vast stretches of interstellar space or the minute regions which we believe to be encompassed within an atom or an electron.

The science of mechanics, like geometry, has been built up from a set of simple axioms, which were laid down by Newton. But the conclusions of the previous section lead us to modify one of these axioms and thus lay the foundation of a system of non-Newtonian mechanics.

The axiom which we must surrender is the one which states that the mass of a body is independent of its velocity. We have concluded that mass is proportional to content of energy. When a body is set in motion it gains kinetic energy and therefore its mass must change with its velocity. In place of

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710 the axiom which we have abandoned we must substitute equation (7).

Before investigating the consequences of this step it is necessary to define exactly the principal mechanical quantities which we are to use.

Extension in space (l) and time (t) will be measured in the usual way and the centimetre and the second will be employed as units.

Force (f) will be given its usual significance and the unit, the dyne, will be that force which, acting upon the International standard kilogram, when the latter is at rest, imparts to it an initial acceleration of ⋅001$$\frac{cm.}{sec.^{2}}$$.

The momentum (M) of a moving body will be measured by the time in which it is brought to rest under the influence of a constant opposing force of one dyne acting in the line of its motion.

The mass (m) of a moving body will be defined as the momentum divided by the velocity (v), that is,

The limiting ratio of the momentum of a body to its velocity, when it is brought to rest, will be called its mass when at rest. The unit of mass is the gram.

The kinetic energy (E') of a body will be measured by the distance through which it will move before being brought to rest by a constant opposing force of one dyne, acting in the line or the body's motion. The unit of energy will be the erg.

These definitions, although somewhat unusual in form, are perfectly consistent with the ordinary definitions of Newtonian mechanics. But they have been so chosen as to be consistent also with equation (7) and the fundamental conservation laws. Obviously equation (7) itself is not inconsistent with these conservation laws, for although a body increases in mass as it gains kinetic energy, some other system is losing the same mass as it loses the same energy.

In accordance with the above definitions we may write

Let us consider a body originally moving with a velocity v subjected for the time dt to a force f in the line of its motion.

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Rh Its momentum and kinetic energy will change according to (11) and (12) by the amounts

Hence

So far the equations are those of Newtonian mechanics, but now in substituting for M from equation (10) we must regard m as a variable and write

This will be our fundamental equation connecting the kinetic energy of a body with its mass and velocity.

Introducing now the relation of mass to energy given in equation (7) we may write,

and combining this equation with (14) gives

This equation, containing only two variables, m and v and the constant V, may readily be integrated as follows. By a simple transformation

Writing β=v/V, and noting that

we see that

Hence

where log m0 is the integration constant. Therefore

or

This is the general expression for the mass of a moving body in terms of β, the ratio of its velocity to the velocity of

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712 light. When β is zero, ''m=m0. m0'' represents therefore the mass of the body at rest.

If we substitute in the equation numerical values of β we find that, while the quotient m/m0 becomes infinite when the velocity equals the velocity of light, it remains almost equal to unity until the velocity of light is closely approached. Thus a ton weight given the velocity of the fastest cannon-ball would, according to this equation, gain in mass by less than one millionth of a gram. It is obvious that, except in those unusual cases in which we deal with velocities comparable with that of light, our non-Newtonian equations are identical with those of ordinary mechanics far within the limits of error of the most delicate experiments.

Recently, however, it has been possible to study, in the negative particles emitted by radioactive substances, bodies which sometimes move with a velocity only a little less than that of light. In a series of remarkably skilful experiments Kaufmann was able to measure the ratio of electric charge to mass (e/m) for negative particles moving at different speeds. Assuming that the charge is constant, the fact that e/m varies with the speed of the particle must be attributed to a variation of the mass with the speed. On this assumption it is possible to calculate from Kaufmann's experiments the values of m/m0 at the different velocities.

The mass of a negative particle is usually spoken of as electromagnetic mass, but if we are to hold to our definitions we must recognize only one kind of mass. In general we have defined the mass of a moving body as the quotient of the time during which it will be brought to rest by unit force, divided by the initial velocity. It matters not what the supposed origin of this mass may be. Equation (15) should therefore be directly applicable to the experiments of Kaufmann. In the following table are given the values of

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Rh m/m0 found for the different observed values of β in the second column. The third column shows those values of ft which would correspond with the same values of m/m0 according to equation (15).

It will be seen that the observed values of β follow to a remarkable degree the same trend as those which are calculated by equation (15), but are in every case six to eight per cent higher. I believe that these differences lie within the limits of experimental error of Kaufmann's measurements. It is true that he claims a higher degree of accuracy, but, notwithstanding the extreme care and delicacy with which the observations were made, it seems almost incredible that measurements of this character, which consisted in the determination of the minute displacement of a somewhat hazy spot on a photographic plate, could have been determined with the precision claimed. Moreover, Planck and Stark have pointed out certain corrections which probably should have been made by Kaufmann and which would produce a material change in his results.

That a charged particle must possess mass in virtue of its charge, and that this mass must vary with the velocity of the particle, was shown to be a consequence of the electromagnetic theory by J. J. Thomson and by Heaviside, and numerous attempts have been made to find the exact expression for the change of mass with the velocity. But before this can be done some assumption is necessary as to the shape of the particle and the distribution of its charge. The three theories of the simple negative particle or electron which are now most discussed are due to Abraham, Bucherer, and Lorentz. The first assumes that the electron is and remains a rigid sphere, the second assumes an electron which is spherical when at rest but which in motion contracts in the direction of its translation and expands laterally so as to keep a constant volume. The third assumes an electron similar to the second, which contracts in the direction of translation but which does not change its other dimensions. On the basis of these theories and from known electromagnetic principles, three equations have been obtained for the value

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714 of m/m0 as a function of β, namely,

The extraordinary significance of the similarity of the first two of these equations and the identity of the third with equation (15), which we have derived from strikingly different principles, needs no emphasis. Kaufmann shows that his results agree better with the first two of these equations than with the third, but to regard this as serious evidence as to the validity of equation (15) would, as Planck has pointed out, be laying too great a stress on the accuracy of the experimental observations.

The agreement of Kaufmann's results with the above equations has led him, and all others who have discussed his results, to the conclusion that all of the mass of an electron is electromagnetic.

Their argument is based on the assumption that ordinary mass, the mass of "ponderable matter," is independent of the velocity, while "electromagnetic mass" varies with the velocity according to one of the above equations. But in this paper we have assumed that all mass is one, and that any bodies, whether charged or not, moving at the velocities of Kaufmann's electrons would show the same values of m/m0.

There is some hope that the correctness of this view may be decided by an experimental study of the mass of a positive or α particle at different speeds. According to the ordinary view, the mass of such a positive particle as issues from a radioactive source is chiefly that of its "ponderable" matter and only to a very small extent "electromagnetic mass." It would therefore be generally assumed that at the highest velocity of the particle, about one-tenth of the velocity of light, it would have substantially the same mass as at rest. According to our view, on the other hand, the mass of this or any other particle would change with the velocity in the same way as the mass of an electron. From equation (15) we should therefore expect the particle moving with one- tenth of the velocity of light to have a mass two per cent. greater than when at rest. The experimental difficulties in

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Rh testing this point would be very great, of course, but perhaps not insurmountable.

The plausibility of our fundamental assumption which led directly to equation (15) has been greatly increased by the agreement, between this equation and Kaufmann's results, and also perhaps by the similarity between this equation and those deduced from electromagnetic theory. But the simplest as well as the strongest evidence of the correctness of our point of view comes from a consideration of the non-Newtonian equation for kinetic energy.

The integration of equation (14) obviously does not yield the simple Newtonian equation,

$$\mathrm{E}'=1/2mv^2$$.

This equation must be replaced by one that is obtained as follows : —

Let a body, which at rest has the mass m0, be given the velocity v. As its internal energy changes, its mass will change according to equation (7), and

$$m-m_{0}=\frac{\mathrm{E}'}{\mathrm{V}^{2}}$$

where E' is the acquired kinetic energy and m—m/m0 is the increase in mass.

Eliminating m0 between this equation and (15) gives

This is the general formula for the kinetic energy of a moving body. As usual β represents v/V, the ratio of this velocity to the velocity of light.

From equations (10), (15), and (16) may be constructed the whole science of non-Newtonian dynamics, of which Newtonian dynamics furnishes a limiting case, namely, for velocities which are negligible in comparison with the velocity of light.

For example, expanding (16) by the binomial theorem gives

For low values of β the higher terms may be neglected and

$$\mathrm{E}'=\frac{1}{2}mv^{2}$$

That is, the limit approached by the kinetic energy of a body as the velocity approaches zero is, as in ordinary mechanics, one half the product of the mass and the square of the velocity. At the other limit of velocity when β=1, it follows from (16) that

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716 Between these two limits it is obvious that

The momentum and the kinetic energy of any mass moving with the velocity of light are, therefore,

but these equations are identical with (5) and (9) which we obtained for the momentum and the energy of a beam of light.

The view here proposed, which appears at first sight a reversion to the old corpuscular theory of light, must seem to many incompatible with the electromagnetic theory. If it were really so I should not have ventured to advance it, for the ideas announced by Maxwell constitute what may no longer be regarded as a theory, but rather a body of experimental fact. The new theory is offered, not in any sense to replace, but to supplement the accepted theories of light. I hope in another paper to show that it is entirely consistent with those theories. Such a proof may constitute a step towards one of the obvious goals of present day science, the complete mechanical explanation of electromagnetic phenomena, or, what is very nearly the same thing, an electromagnetic explanation of the phenomena of ordinary mechanics. In the meantime a few of the more salient conclusions of our theory may be cursorily examined.

In the first place it should be noticed that, while the theory is consistent with a modified corpuscular theory of light, it does not necessarily imply that light is corpuscular. The stream of mass issuing from a radiating body may be made up of discrete particles or it may be continuous. Whatever it may be that is emitted it is not matter in the ordinary sense, as is to be seen from the following considerations : —

According to equation (15) any body of finite mass increases in mass as it increases in velocity and would possess infinite mass if it could be given the velocity of light. Therefore that which in a beam of light has mass, momentum, and energy, and is travelling with the velocity of light, would have no energy, momentum, or mass if it were at rest, or indeed if it were moving with a velocity even by the smallest fraction less than that of light. After this extraordinary conclusion it would be at present idle to discuss whether the same substance or thing which carries the radiation from the emitting body continues to carry it through space, or, indeed, whether there is any substance or thing connected with the process.

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Rh If we assume an aether pervading space, and assume that this æther possesses no mass except when it moves with the velocity of light, it is obvious that an æther drift could in no way affect a beam of radiation nor could it be detected by any mechanical means. If we are to assume such an æther we may as well assume it to be at rest.

The question whether a method is conceivable by which absolute motion in space may be distinguished from relative motion must be answered definitely in the affirmative by one who accepts the above equations of non-Newtonian mechanics. A body is absolutely at rest when any motion imparted to it increases its mass, or when a certain force will give it the same acceleration in any direction. It is true that metaphysicians hold that in the strictest sense absolute motion is not only unknowable but unthinkable, but we may say at least that the above method permits theoretically the detection of absolute translational motion in the same sense that a study of centrifugal forces enables us to detect absolute rotational motion.

It is postulated that the energy and momentum of a beam of radiation are due to a mass moving with the velocity of light.

From this postulate alone it is shown that the mass of a body depends upon its energy content. It is therefore necessary to replace that axiom of Newtonian mechanics according to which the mass of a body is independent of its velocity, by one which makes the mass increase with the kinetic energy.

Retaining all the other axioms of Newtonian mechanics and assuming the conservation laws of mass, energy, and momentum, a new system of mechanics is constructed.

In this system momentum is mv, kinetic energy varies between 1/2 mv² at low velocity and mv² at the velocity of light, while the mass of a body is a function of the velocity and becomes infinite at the velocity of light. The equation obtained agrees with the experiments of Kaufmann on the relation between the mass of an electron and its velocity. It is, moreover, strikingly similar to the equations that have been obtained for electromagnetic mass.

The new view leads to an unusual conception of the nature of light. It offers theoretically a method of distinguishing between absolute and relative motion.

 * Research Laboratory of Physical Chemistry,
 * Massachusetts Institute of Technology,
 * May 14, 1908.

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 where rose the priory of St-Denis-de-l’Estrée, which existed until the 18th century. In the 5th century the clergy of the diocese of Paris built a basilica over the tomb. About 625 Dagobert, son of Lothair II., founded in honour of St Denis, at some distance from the basilica, the monastery where the greater number of the kings of France have been buried. The festival of St Denis is celebrated on the 9th of October. With his name are already associated in the Martyrologium Hieronymianum the priest Rusticus and the deacon Eleutherius. Other traditions—of no value—are connected with the name of St Denis. A false interpretation of Gregory of Tours, apparently dating from 724, represented St Denis as having received his mission from Pope Clement, and as having suffered martyrdom under Domitian (81–96). Hilduin, abbot of St-Denis in the first half of the 9th century, identified Denis of Paris with Denis (Dionysius) the Areopagite (mentioned in Acts xviii. 34), bishop of Athens (Eusebius, Hist. Eccl. iii. 4. 10, iv. 23. 3), and naturally attributed to him the celebrated writings of the pseudo-Areopagite. St Denis is generally represented carrying his head in his hands.

 DENIS, JOHANN NEPOMUK COSMAS MICHAEL (1729–1800), Austrian poet, was born at Schärding on the Inn, on the 27th of September 1729. He was brought up by the Jesuits, entered their order, and in 1759 was appointed professor in the Theresianum in Vienna, a Jesuit college. In 1784, after the suppression of the college, he was made second custodian of the court library, and seven years later became chief librarian. He died on the 29th of September 1800. A warm admirer of Klopstock, he was one of the leading members of the group of so-called “bards”; and his original poetry, published under the title Die Lieder Sineds des Barden (1772), shows all the extravagances of the “bardic” movement. He is best remembered as the translator of Ossian (1768–1769; also published together with his own poems in 5 vols. as Ossians und Sineds Lieder, 1784). More important than either his original poetry or his translations were his efforts to familiarize the Austrians with the literature of North Germany; his Sammlung kürzerer Gedichte aus den neuern Dichtern Deutschlands, 3 vols. (1762–1766), was in this respect invaluable. He has also left a number of bibliographical compilations, Grundriss der Bibliographie und Bücherkunde (1774), Grundriss der Literaturgeschichte (1776), Einleitung in die Bücherkunde (1777) and Wiens Buchdruckergeschichte bis 1560 (1782).

 DENISON, GEORGE ANTHONY (1805–1896), English churchman, brother of John Evelyn Denison (1800–1873; speaker of the House of Commons 1857–1872; Viscount Ossington), was born at Ossington, Notts, on the 11th of December 1805, and educated at Eton and Christ Church, Oxford. In 1828 he was elected fellow of Oriel; and after a few years there as a tutor, during which he was ordained and acted as curate at Cuddesdon, he became rector of Broadwindsor, Dorset (1838). He became a prebendary of Sarum in 1841 and of Wells in 1849. In 1851 he was preferred to the valuable living of East Brent, Somerset, and in the same year was made archdeacon of Taunton. For many years Archdeacon Denison represented the extreme High Tory party not only in politics but in the Church, regarding all “progressive” movements in education or theology as abomination, and vehemently repudiating the “higher criticism” from the days of Essays and Reviews (1860) to those of Lux Mundi (1890). In 1853 he resigned his position as examining chaplain to the bishop of Bath and Wells owing to his pronounced eucharistic views. A suit on the complaint of a neighbouring clergyman ensued and after various complications Denison was condemned by the archbishops’ court at Bath (1856); but on appeal the court of Arches and the privy council quashed this judgment on a technical plea. The result was to make Denison a keen champion of the ritualistic school. He edited The Church and State Review (1862–1865). Secular state education and the “conscience clause” were anathema to him. Until the end of his life he remained a protagonist in theological controversy and a keen fighter against latitudinarianism and liberalism; but the sharpest religious or political differences never broke his personal friendships and his Christian charity. Among other things for which he will be remembered was his origination of harvest festivals. He died on the 21st of March 1896.

 DENISON, GEORGE TAYLOR (1839–), Canadian soldier and publicist, was born in Toronto on the 31st of August 1839. In 1861 he was called to the bar, and was from 1865–1867 a member of the city council. From the first he took a prominent part in the organization of the military forces of Canada, becoming a lieutenant-colonel in the active militia in 1866. He saw active service during the Fenian raid of 1866, and during the rebellion of 1885. Owing to his dissatisfaction with the conduct of the Conservative ministry during the Red River Rebellion in 1869–70, he abandoned that party, and in 1872 unsuccessfully contested Algoma in the Liberal interest. Thereafter he remained free from party ties. In 1877 he was appointed police magistrate of Toronto. Colonel Denison was one of the founders of the “Canada First” party, which did much to shape the national aspirations from 1870 to 1878, and was a consistent supporter of imperial federation and of preferential trade between Great Britain and her colonies. He became a member of the Royal Society of Canada, and was president of the section dealing with English history and literature. The best known of his military works is his History of Modern Cavalry (London, 1877), which was awarded first prize by the Russian government in an open competition and has been translated into German, Russian and Japanese. In 1900 he published his reminiscences under the title of Soldiering in Canada.

 DENISON, a city of Grayson county, Texas, U.S.A., about 2 m. from the S. bank of the Red river, about 70 m. N. of Dallas. Pop. (1890) 10,958; (1900) 11,807, of whom 2251 were negroes; (1910 census) 13,632. It is served by the Houston & Texas Central, the Missouri, Kansas & Texas, the Texas & Pacific, and the St Louis & San Francisco (’Frisco System) railways, and is connected with Sherman, Texas, by an electric line. Denison is the seat of the Gate City business college (generally known as Harshaw Academy), and of St Xavier’s academy (Roman Catholic). It is chiefly important as a railway centre, as a collecting and distributing point for the fruit, vegetables, hogs and poultry, and general farming products of the surrounding region, and as a wholesale and jobbing market for the upper Red river valley. It has railway repair shops, and among its manufactures are cotton-seed oil, cotton, machinery and foundry products, flour, wooden-ware, and dairy products. In 1905 its factory products were valued at $1,234,956, 47.0% more than in 1900. Denison was settled by Northerners at the time of the construction of the Missouri, Kansas & Texas railway to this point in 1872, and was named in honour of George Denison (1822–1876), a director of the railway; it became a city in 1891, and in 1907 adopted the commission form of government.

 DENIZEN (derived through the Fr. from Lat. de intus, “from within,” i.e. as opposed to “foreign”), an alien who obtains by letters patent (ex donatione regis) certain of the privileges of a British subject. He cannot be a member of the privy council or of parliament, or hold any civil or military office of trust, or take a grant of land from the crown. The Naturalization Act 1870 provides that nothing therein contained shall affect the grant of any letters of denization by the sovereign.

 DENIZLI (anc.  (q.v.) ad Lycum), chief town of a sanjak of the Aidin vilayet of Asia Minor, altitude 1167 ft. Pop. about 17,000. It is beautifully situated at the foot of Baba Dagh (Mt. Salbacus), on a tributary of the Churuk Su (Lycus), and is connected by a branch line with the station of Gonjeli on the Smyrna-Dineir railway. It took the place of Laodicea when that town was deserted during the wars between the

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text: 1911 Encyclopædia Britannica/Denis, Johann Nepomuk Cosmas Michael


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text: 1911 Encyclopædia Britannica/Denison, George Anthony


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Rh XVIII. On the Second Postulate of the Theory of Relativity: Experimental Demonstration of the Constancy of Velocity of the Light reflected from a Moving Mirror.

By , Professor of Physics at the Polytechnic School of Turin.

HE Theory of Relativity is based upon two well-known fundamental postulates. The first affirms the impossibility of discovering the movement of a system without referring this system to other systems; that is to say, it denies the physical reality of absolute motion. The second postulate affirms that the velocity c of propagation of light in vacuo is a universal constant. Both these postulates are generalizations of facts or principles already admitted by physicists.

In fact, we may regard the first as the extension to optical or electrical phenomena of a classical principle of mechanics,

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164 an extension justified by the negative results of certain experiments (Michelson and Morley, Trouton and Noble) by which it was sought to discover the absolute motion of the earth, or the æthereal wind which must traverse all terrestrial objects. The second postulate is the generalization of a fundamental principle in the theory of æthereal or electromagnetic undulations.

But if these two principles, derived from quite different chapters of physics, have been fully accepted severally by modern physicists, their origin has been almost forgotten; an ingenious structure arose upon their union: the theory of relativity. This theory, while repudiating according to Einstein and others a theoretical conception which had given occasion for the formulation of the second postulate (the æther), serves well to explain the insuccess of the above-cited experiments.

Now our imagination, accustomed, as W. Ritz has said, to "substantialize" physical phenomena, if it easily grasps the essence of the first postulate, does not do so in the case of the second; and the more so since, as has already been said, some adherents to relativistic theories do not retain as necessary the existence of a medium of transmission (the æther) in order to explain the constancy of c. Moreover on the second postulate, or, more precisely, on a certain portion of this, depend the conclusions which appear artificial or extraordinary in the whole relativistic theory. The second postulate must be understood in the sense that an observer who measures the velocity of light finds always the same value if both he and the source be at rest, relative or (if the possibility be admitted) absolute, or if the source or the observer, or both, have a uniform motion of translation. In short, the second postulate affirms the absolute independence of c of any contingency whatever of uniform motion of translation of the source or of the observer.

It is known that an hypothesis of a mechanical character (emissive or ballistic), according to which to the ordinary velocity of light must be added that of the source, can explain, like the theory of relativity, the failure of the above-quoted experiments. But this hypothesis is radically in contrast with the electromagnetic theory, and consequently is not much favoured. But in any case laboratory experiments can be conceived which should decide between the

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Rh above-mentioned hypothesis or mechanical theory and the relativistic one. It is indeed possible to see that some method, one moreover already in use, adopted for the verification of Doppler's principle may serve for the solution of the above-quoted problem.

In order to see this, let us consider a luminous source S which emits waves of length λ and of frequency n moving towards the observer fixed at O (fig. I). If we suppose

that the waves are transmitted through a stationary æther, the n waves emitted in a second by S will be distributed over the segment S'A = c — v. In the same time all the n waves distributed in the segment OB = c will have passed through O; we have therefore

$\frac{c-v}{n}=\frac{c}{n}$, or $n'=n\frac{c}{c-v}$.

If we put v/c = β and neglect terms of higher order than the first in β we have

n' = n(1 + β).

The new wave-length is obtained by the relations

c = nλ = n'λ': λ' = λ(1 - β).

If now instead of the hypothesis of a stationary medium we adopt the ballistic or emissive hypothesis of which we have spoken above, we shall find that in one second the n waves emitted by S will be distributed over the segment S'A' = c. In the same time there will pass through O, n' waves which will be distributed over the segment OB' = c + v. We have, therefore,

$\frac{c}{n}=\frac{c+v}{n'}$, or n' = n(1 + β).

And since c = nλ and c + v = n'λ' we see that, in this case, λ' = λ.

As regards the frequency we arrive, therefore, at the same conclusions (with the exception of the terms in β²) whether we adopt the æthereal or the ballistic hypothesis; but for the wave-length we obtain different values from the two

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166 hypotheses, and these values differ by a term of the first order in β. If then the Doppler effect is measured, by observation of the wave-length, different results should lie obtained according as the one or the other hypothesis is adopted. Now observations of the Doppler effect have already been made by measuring the displacement of the spectral lines, employing either prisms or diffraction-gratings. In the cage of prisms it may be observed that all the theories of dispersion hitherto admitted lead to the supposition that this phenomenon can only depend on the frequency of the incident luminous vibrations. Consequently the displacement of the spectral lines may be caused by the simple variation in frequency due to the Doppler effect, and this whether, for the light, the hypothesis of a stationary æther is adopted, or a ballistic or emissive theory. From this point of view, therefore, the question whether the velocity of propagation of the light emitted by a source does or does not change with the velocity of the latter remains unanswered.

But the Doppler effect has been established with diffraction-gratings as well as with prisms, and for astronomical as well as terrestrial sources. Now the function of a grating, from the geometrical point of view, may be regarded as depending exclusively on the values of the incident wave-lengths; the positions of the successive spectral lines remain exactly determined by those values. But since, according to the ballistic or emissive hypothesis, the value of λ does not vary with the velocity of the source, we see that the grating should not give an appreciable result in the study of the Doppler effect, and this, as is known, is not in agreement with experience. We may then conclude from observations of the Doppler phenomenon in the stars and the limb of the sun with moving mirrors (Galitzin & Wilip), or again in the canal rays (Stark, Paschen), that the velocity of light is absolutely constant and independent of the movement of the source; this is equivalent to the rejection of the ballistic or emissive theory. This is Tolman's opinion, in contradiction to that of Stewart. Indeed, it should be borne in mind that the ordinary grating theory may not apply exactly in the case of a mechanical (ballistic or emissive) theory of

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Rh light. In any case it should be remarked that astronomical observations of the Doppler effect are not always made with an à priori knowledge of the relative velocities of source and observer. In the case of the solar limb it is necessary, moreover, to be cautious in establishing a relation between the measure of displacement of the lines and the velocity of the limb established by observation of the solar spots; in fact, the light from the limb may be strongly refracted by the perispherical incandescent gases, and consequently the value of the Doppler effect may vary considerably. So far as terrestrial observations are concerned, and those on the canal rays (Stark, Paschen), they give measures of the phenomenon of only small precision, and it is impossible to foresee by another method the exact velocity of the luminous particles; finally, observations made with moving mirrors bear no relation to those with moving sources, and these may produce different consequences.

From all this we may conclude that up to the present time we do not possess any quite certain evidence of the immutability of c with variable velocity of the source if, be it understood, we are not willing to admit as conclusive the simple electromagnetic theory or that of bodies in motion according to Lorentz or else Einstein's theory of relativity. This conclusion is confirmed by the study of the works of the chief supporters of the last theory, and, implicitly, of the second postulate. In these works we frequently find expressed the desire to discover further facts which will definitely confirm the said theory: this desire corresponds with the crisis of the latter years of the said theory.

But on the other hand, as Levi-Civita observes, after the latest researches of Einstein, which collect in an admirably comprehensive synthesis all the physical phenomena (gravitation included), it is difficult to avoid the impression that we are, as regards the theory of relativity, face to face with some definite acquisition. But, while taking account of this, it is not expedient to neglect any attempt at a definite confirmation, from an experimental point of view, of a theory which has subverted to so large an extent our simplest physical notions. This confirmation may follow from a precise study of the velocity of propagation of light emitted from a moving source, or, which is equivalent, of the wavelength of this light.

To realize this study we must devise an arrangement

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168 which will permit us to identify the structure of the luminous wave, freed from all external action, in its free propagation (or transmission) when the velocity of translation of the source can be varied at will. But, apart from the fact that we must inevitably experiment under the eventual action of our earth, two serious and almost insurmountable difficulties oppose themselves to the realization of such a programme. In the first place, it is not easy artificially to endow a luminous source with rapid movement, especially if this source (as is necessary in some interference methods) has to be very rigorously monochromatic; moreover, I shall publish an account in a forthcoming paper of a disposition of this nature with which I am about to experiment. Secondly, in order to be able to examine the structure of the light emitted by a moving source, with whatever disposition, the light itself has to be subjected to reflexion's, refractions, &c., sometimes fairly numerous; that is to say, the luminous ray must encounter ponderable matter after leaving the source. It does not follow, therefore, that even if c in a vacuum varies with the particular velocity of the source, this quantity does not return to the same fixed value after the said phenomena of reflexion, refraction, &c. It will be well, therefore, to endeavour to eliminate as far as possible, in an experiment of this nature, all causes tending to complicate the phenomenon, and in every case to consider its results carefully.

Meanwhile, to begin with a relatively simple experiment, we may undertake the study of the wave-length of a ray of light reflected by a moving mirror. This may correspond with the experiment already realized, some years ago, by Belopolski, and afterwards repeated by Galitzin & Wilip; but if the first of these authors employed prisms for the observation of the Doppler effect (and consequently the question of the eventual variation of λ remained unsolved), the other two made use of a diffraction-grating, by which the controversy spoken of above arises. It would be better

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Rh therefore to examine the ray reflected from a moving mirror by an interference method simpler than that on which the action of the diffraction depends, as has been said above.

Before expounding this method it is well to recall that many theoretical researches have been made on the influence of the motion of the mirror upon the reflected luminous wave, amongst them those of Abraham, Brown, Edser, Harnack, Larmor, Planck. These researches make of the problem either a simply geometrical investigation, or an application of the electromagnetic theory of light. But without discussing the result of these researches we may accept the conclusion of Harnack respecting the frequency of the vibrations reflected by a mirror in uniform motion. Let v be the velocity of the latter, normal to its plane, reckoned as positive towards the source; c the velocity of the luminous ray in vacua which makes the angle of incidence θ with the mirror; n, n' the frequencies of the ray before and after reflexion, the source and observer being at rest. If we put β = v/c we shall have

$$n'=n\frac{1+2\beta\ \cos\theta+\beta^{2}}{1-\beta^{2}}$$

which, neglecting the terms in β², reduces to

This relation is the same as that of Ketteler, which was employed by Belopolski in his investigation of the Doppler effect, and follows simply from the consideration that the image of the source moves with the velocity 2v in the direction of the normal to the mirror and, consequently, the component of this velocity in the direction of the reflected ray is 2v cos θ.

If now we suppose that the ray is, by suitable arrangements, reflected with the incidence θ, k times from several mirrors in motion with the velocity v, we shall have

Therefore, according to the hypothesis of constant velocity of light, neglecting the terms in β² we shall have

If, on the contrary, we suppose that the velocity of the

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170 reflected light is variable (and equal to the sum of c = 3.1010cm. and the component of the velocity of the image in the direction of the ray) we shall have

c' = c + 2kv cos θ.

And since c' = n'λ' and c = nλ we have λ' = λ. It remains then to see by experiment whether or not we can observe, in addition to the Doppler effect, a variation in the value of λ; from this we can ascertain whether c remains constant or not on reflexion from a moving mirror. I have not proceeded to the observation of the Doppler effect in these researches since there is no doubt about its existence, already proved experimentally by the authors quoted; I have rather sought to find out whether and in what way λ varies when the velocity of the moving mirrors changes.

Belopolski's arrangement for the study of the Doppler effect was inconvenient on account of the excessive subtility of the luminous ray necessary to obtain multiple reflexions from the same mirrors; for this reason the author mentioned was unable to observe the displacement of the rays except by photography. I prefer to adopt the arrangement shown diagrammatically in fig. 2. A horizontal brass wheel R, 35 cm.

in diameter (6 mm. thick), which can be made to revolve with a maximum velocity of 80 turns per second, bears on its periphery ten mirrors similar to M, rectangular, plane, vertical, of glass silvered at the back. The velocity of the centres of the mirrors, corresponding to the greatest velocity of rotation, amounts therefore to more than 100 metres per

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Rh second. The number of revolutions of the wheel was determined acoustically in each experiment. The mirrors, at equal intervals on the periphery of the wheel, are inclined to the radius from R passing through the centre of each of them at an angle α = 29°. They are fixed solidly to R by screw movements capable of permitting a rigorous adjustment. The support for the bearings of the axle of R carries also fixed mirrors F, vertical like M, of which the number in the figure is three; but this number may, at will, be reduced, or increased up to nine. The position of the F's and M's is such that a parallel beam of light L, after a certain number of reflexions from the F's and M's (seven in the figure), may be received at L' when R has determinate angular positions. Naturally the intensity of L' is much weaker than that of L, and this enfeeblement is much more marked if R is in rotation, because in this case the light arrives at L' only during certain very short instants (ten times per revolution). I have observed in practice, however, that the four moving and three fixed reflexions of the figure allow of experimenting with light sufficiently intense at L' even if R is in motion: that is to say, that direct observation (without photography) suffices to establish the luminous phenomenon of which we have spoken above.

To study the value of λ the light L' was examined with the well-known interferometer of Michelson, shown diagrammatically in the figure. It is known that if the distances S1S3 and S2S3 are exactly equal fringes are observed with the telescope C even if the light is not monochromatic; these fringes then have the coloration of Newton's rings. As soon as a difference of path occurs (even if only of a few microns) observation with white light is no longer possible. Monochromatic light must then be used, and the order of the interference fringes increases with this difference. Their visibility is greater, the simpler the luminous vibrations. From the researches of Michelson it is known that from this point of view the line that gives the greatest visibility of the fringes with the greatest difference in path is the green one of mercury (λ = 546μμ). In this case numberless circular fringes are visible even for a difference of path l = 2(S1S3 - S2S3) = 40 cm. I have therefore employed as source L a mercury arc in vacuo the light of which is conveniently filtered by solutions of chromate of potassium and chloride of nickel to absorb the violet and yellow rays;

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172 in this manner I have been able to observe with the telescope C, with sufficient clearness, countless circular fringes, even for l = 32 cm. But for these researches I have limited the difference of path to l = 13 cm., or still less.

The disposition described above is particularly suitable for detecting very small differences in the value of the incident wave-length; in fact, the value of l being large a very great number of wave-lengths is comprised in this length (e.g., 200,000 if λ = 0.5μ, and l = 10 cm.), and correspondingly for the same variations very sensible displacements can be observed in the position of a fringe.

With the apparatus disposed as above, let us note with the micrometer wire of the telescope the position of a fringe, for instance the first central bright one, when R is in the position shown in the figure, or, still better, when it revolves with a negligible velocity (one turn per second). If, now, this velocity be increased to sixty turns per second a displacement of the fringe under observation is distinctly visible; if the mirrors are moving against the incident ray this displacement indicates a diminution of λ, and it changes sign when the direction of rotation of the wheel is reversed, and this indicates an increase of λ. In order to define the sense of the displacement, I will say that on examining the system of circular fringes with the telescope focussed for infinite distance the diameter of each of these increases when the mirrors move against the incident ray, and the fringes themselves crowd together as those of large diameter are very little displaced; at the same time some new fringes come out from the centre of the system. On the other hand, when the mirrors are moving in the sense of propagation of the incident light the diameter of each fringe diminishes; they become more widely separated, and some of them are as it were swallowed up by the centre.

Before stating the measure of the displacement observed we will see what it should amount to, making the hypothesis that the velocity of the light reflected from a mirror is the same as that of the incident light. Let g be the number of revolutions of R per second and d its diameter, reckoned between the centres of two opposite mirrors M, then πdg will be the instantaneous velocity of translation of the latter. Since the mirrors are inclined at an angle α to the radius of the wheel passing through each of them, the component of the given velocity in the direction normal to the plane of each mirror will be

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Rh We have, therefore,

$n'=n\left(1+\frac{2k\pi dg\ \cos\alpha\ \cos\theta}{c}\right)$

and, by the hypothesis of the invariability of c,

$\lambda'=\lambda\left(1-\frac{2k\pi dg\ \cos\alpha\ \cos\theta}{c}\right)$

If l is the difference of path of two interfering rays in Michelson's apparatus, the number of fringes which are seen to cross the micrometer thread of the telescope when λ becomes λ' (that is to say when the velocity of rotation varies between zero and g turns per second) is

$f=\frac{l}{\lambda}\frac{2k\pi dg\ \cos\alpha\ \cos\theta}{c}$.

If the observation is made by noting the position of the fringes when the wheel turns in one sense with the velocity g, and that corresponding to an equal and contrary velocity, the number of fringes crossing the micrometer thread will be 2f.

Now, in my apparatus d = 38 cm., α = 29°, θ = 27°, k = 4 (as in the figure); if λ is put equal to 0.546μ (green mercury line), l = 13 cm., c = 3·1010 cm., and g = 60 (turns of R per second in one sense and afterwards in the other), we may expect, according to the preceding formula, a fringe displacement 2f = 0.71.

Experiment gives, for the case mentioned, a displacement of between 0.7 and 0.8 fringes; and it is not possible, for reasons of visibility, to carry the precision of the observations further. But, as is seen, the agreement between the predicted result and observation is sufficient; this agreement is confirmed by observations made by choosing other convenient values of l and g, of which for brevity's sake I shall not speak here.

Experiment, therefore, authorizes the conclusion that reflexion of light by a moving metallic mirror does not alter the velocity of propagation of the light itself, in air, and consequently, with great probability, also in vacuo; at least, in the conditions of the experiment above described. This experimental result, as to which no doubt can be entertained, is contrary to the hypothesis of some physicists who, like Stewart, basing themselves upon the electromagnetic emission theory of Thomson, maintain the possibility that

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174 light, after reflexion, is propagated with the velocity c + v, where v is the component of the velocity of the image in the direction of the reflected ray.

To complete these researches I intend, as I have said above, to investigate further with the same interferential arrangements, the velocity of propagation of light from a source set in motion artificially; but of these, and of the general conclusions to be drawn from these investigations, I reserve mention for a future occasion.

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text:
Author:John Hunter

 * John Hunter (1728–1793), Scottish surgeon
 * John Hunter (1738–1821), British naval officer and governor of New South Wales

title: Author:John Hunter


text: 1911 Encyclopædia Britannica/Denison

title: 1911 Encyclopædia Britannica/Denison


title: File:MajoranaEmission2.djvu


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a preceding paper on the second postulate of the theory of relativity I described an experimental arrangement of mine by means of which I was able to demonstrate that light propagates itself with constant velocity, independently of the conditions of movement, or rest of the mirror by which the light is reflected. At the end of the above-mentioned paper I hinted at my intention of studying experimentally the eventual influence of the movement of the source on the velocity of the propagation of the light; the object of the present note is to communicate the result of these researches.

As is known, the only studies made with luminous sources in motion are astronomical ones, and those with the canal rays. Particularly with the former it has been possible to deduce the measure of the Doppler effect (and therefore the value of the velocity of displacement) for different sources, such as the fixed stars or planets and the limb of the sun. I am not aware of any attempt to prove the Doppler effect with the artificial movement of a common luminous source; the difficulty of this research consists principally in the necessity for giving a specially high rate of velocity of displacement to the source.

But even if an arrangement of this kind could be realized, its interest does not lie in the verification of the Doppler effect (change of frequency), upon which no doubt any longer exists, so much as in the control of the value of the velocity of the propagation of the light, also in the case of a moving source. This is the reason why the examination of the latter must not be made either with prisms, as in the arrangement of Belopolski, or with diffraction-gratings, as I have before explained. In making my preparations to set up an apparatus with moving source, I resolved, from the first, to examine the latter with the interference method already described by me, which is founded on the use of the

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text: 1911 Encyclopædia Britannica/Denizen

title: 1911 Encyclopædia Britannica/Denizen


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146 Michelson interferometer, with a great difference in path. Admitting the second postulate of the theory of relativity, or if, in any way, the velocity of propagation of light by earthly sources seems to us unchangeable, supposing such an apparatus were realized, a certain number f of fringes would be seen to cross the wire of the eyepiece, when the source passes from rest to velocity v,

$$f=\frac{lv}{\lambda c},$$

where l is the difference of path of the interfering rays, λ the length of the wave, and c the velocity of light. This is analogous to my affirmation previously made with regard to moving mirrors. Now, the values which it is hoped may be attained in a laboratory for v are rather modest, even with regard to those of astronomical luminous sources; it is therefore necessary to give to l the highest possible value, which is only to be obtained by using a source of light with a very long visibility curve.

This can only be the green line of mercury, which, however, being more complex in its structure than the lines of cadmium, allows the observation of the fringes up to the value of l = 32 cm., using excitation of vapours by means of a voltaic arc in vacuum. Besides, this source is particularly suited for the present researches because of its exceptional intensity. I believe, therefore, that they would hardly be repeated, using a different source.

I established a new plan of experiments, intending to endow with swift rotatory movement some mercurial arcs held by airless glass tubes, and to examine by means of the Michelson interferometer the light emitted by them tangentially to the trajectory line. Now in the attaining of a peripheral velocity of nearly 100 m. per second, this being the necessary velocity for a sure appreciation of a displacement in the fringes, two principal mechanical difficulties are found: the enormous centrifugal force, and the very great resistance of air. To diminish the first, it is convenient to enlarge as much as possible the diameter of the trajectory and lessen the number

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 per second of the revolutions. It is known that the centrifugal velocity increases as the square of that number, and the velocity of displacement grows as the first power. After different attempts I fixed the diameter at 2 m.: because since the small tubes with the mercury weigh 35 gr., the centrifugal force that excites them at a velocity of only 14 revolutions per second (corresponding to nearly 90 m. of peripheral velocity) amounts to 30 kg.

That is nearly the highest point of capacity of charge to which the glass material can be brought. In my experiments the glass tubes still broke very often, although allowing a sufficient interval of time for the observation and measures.

In respect to the resistance of air, it has been reduced to the minimum by using fine steel wires of high mechanical resistance as connexions between the tubes and the rotating axle. Notwithstanding the aforesaid conditions of velocity, the apparatus being provided with only two tubes in diametral position, n power of about 5 kw. was necessary. I give now a short description of it. The figure illustrates schematically

the details, not, presenting them on a uniform scale. O is the rotating axle connected with a pulley and strap to a motor with velocity of rotation capable of regulation and inversion, and of the maximum power 10 H.P. Two airless tubes of a

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 particular form (maximum diameter 4 cm.) bear each three electrodes A, B, E, and are joined by means of strong steel wires with the axle O. The latter is encircled by two collector-rings K1, K2 communicating with the electrodes A, B. Two brushes P1, P2 bring a continuous current at 70v. A certain quantity of mercury is contained in each tube (28 gr.) and is distributed on A and B when the apparatus is rotating. The electrodes E are provided with light segments of aluminium L, which, when the tubes revolve, pass near a metallic arc M, which has the centrum O. An induction-coil I with its terminals connected as in the figure, placed in action for few seconds, gives discharges, which excite the tubes passing near M. In these the current can be regulated, by interposing the necessary resistance, to between 2 or 3 amperes. I must observe that it is often useless to employ the coil I after having started the rotation. Although in tact the quantity of mercury contained in each tube is absolutely insufficient to establish a continued metallic connexion between A and B, the excitement frequently happens spontaneously by reason of ionization of the remaining gas, caused by the mechanical shaking during rotation. Besides K1 and K2, the axle O is surrounded by a series of small metallic blocks (not visible in the figure) across which wipes a third brush. This apparatus, like an electric siren, allows a sound to be heard by means of the necessary connexion with a battery and a telephone, by which it is possible to deduce the value of the velocity of the rotation. The light emitted by the tubes is the highest, for constructive reasons, in the tangential direction of the movement. The Michelson interferometer is disposed as in the figure, and on it the light arrives parallel on the mirror S2 by means of a lens not shown. With the telescope C it is possible to receive a luminous sensation, sufficiently intense in spite of its discontinuity. (20-30 spots per second.)

Thanks to a sufficient intensity of light, I can use (instead of the above-mentioned experiments with mirrors) a value of 1 = 232 mm., by which I have observed a maximum of visibility of fringes. In these conditions, and giving to the apparatus a velocity of from 10 to 14 rotations a second, one perceives a displacement of the fringes when the velocity passes from one part to the other. This displacement observed with an eyepiece with micrometer has really the direction that is demanded by the principle of constancy of velocity of propagation of light. Let us foresee its value on this basis. During a long series of observations there is an average:

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 v = 79.77 m./sec.; l = 232 mm., λ = 0.546 μ; so that a displacement is foreseen of

Experimentally, I have been able to augment considerably the exactitude of observation of the fringes; and this because of the higher luminosity of the phenomena. The circular fringes follow each other as is known, with increasing diameter according to the law:

in which l and λ have the usual meaning, n is the increasing ordinal number of the fringes, beginning at the centre, and α is their radius measured in visual angle from the eye of the observer. So that when a displacement of a fringe is observed, for obtaining a higher precision it is necessary to keep in mind that parabolical law referring to the value of the length of the wave. This is what I have done, studying previously the distribution of the above-mentioned fringes in the field of the telescope. Want of space does not allow me to explain this more at length. I may only say that in the series of observations quoted, I obtained an average of displacement

As we see, this value is somewhat superior to that expected, about 5 per cent. Until now, although I took the greatest care to keep in mind the precision of the different measures which are necessary to arrive at this result, I do not know if any systematic error is the reason for this small difference. Certainly it appears superior to the probable error of the result; and that is why I have mentioned it. But in consideration of the delicacy of the measures I do not register the value of the displacement of the fringes before admitting the above-mentioned discordancy (however slight). For now we may conclude that, under the conditions of the experiment, and within the limits of exactitude of the observations made, the velocity of light does not change by the movement of the source along the direction of propagation.

From the researches made by Michelson, Fabry and Buisson, and by myself, it results that the velocity of light

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 is not influenced by reflexion on mirrors, or reflecting surfaces; from those now described by me, it results that the said velocity does not change by the movement of the source. These facts are surely in harmony with the theory of relativity; but really, in spite of their evident interest, they cannot logically be cited as sure experimental proof of this theory. In fact, two experimental circumstances must not be forgotten: first, the presence of materials which are traversed by the interfering rays (air, glass, metals); and second, the gravitation field of our earth. While it is possible to imagine experiments entirely apart from the former, it cannot be foreseen if later experimental results will bring into evidence the eventual influence of the second.

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text: 1911 Encyclopædia Britannica/Denizli

title: 1911 Encyclopædia Britannica/Denizli


text: 1911 Encyclopædia Britannica/Denman, Thomas


text: Index:BumsteadContraction.djvu

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Rh but, granting that, it is evident we must accept the situation, and it is for us to do our duty in it. Wishing you success in your efforts—I am yours, &c., —Queenslander, June 19, 1880.

,—Allow me one word of comment upon "Never Never's" letter in your last issue. Callousness, to use a very mild word, seems to have outgrown any finer feelings this writer may have once possessed.

I have known of blacks when being dispersed seek refuge in trees, where, shrieking for mercy, they were shot down by their murderers beneath. "Never Never" would fain as easily dispose of the Queenslander's good taste and honesty of purpose. That blackfellows take a deal of killing sometimes and "die hard" he seems to know, and I trust that he may yet learn that naked truth is not to be dispersed by the pen, however facile, of any unhesitating utilitarian such as he undoubtedly is. Let him not seek to strengthen his position by dragging in the outside men. They are, I am happy to say, the great majority, strangers to his line of thought.—Yours, &c., June 2. —Queenslander, June 19, 1880.

,—In reply to "Outis'" correction of my communication relative to the "gentlemen's" reception of officers of the Native Police force, I beg the favor of stating that undoubtedly "gentlewomen" should have been the word used; and if I omitted doing so it is simply a mistake—one, however, that does not affect the question at issue; for I include "gentlemen" as well as "gentlewomen" as those from whom I and others of the force invariably received the greatest kindness, not only when in their own homes, but we were also honored by their visiting us in our barracks.

As regards the mode by which "Outis" obtained his information in re the atrocities, I am quite prepared to admit the reliability of his informants, namely, the relatives or friends of the blacks dispersed. Would the honest womanly feelings of those two "gentlewomen," think you, have undergone a change towards the Native Police officers had their homes been surrounded by hostile myalls, or would "Outis" consider such wholesale murderers and refuse their assistance had they come to their relief? I have known of two similar occurrences.

In all services there are without doubt black sheep, but I assert emphatically that as a body the Native Police officers are as honorable and, I maintain, as humane as any body of officers in the service of the Crown, and would regard with as great horror as "Outis" himself the atrocities with which he charges them.—Yours, &c., —Queenslander, June 19, 1880.

,—I have read with painful interest and deep sympathy your article No. III., "How we civilise the blacks," and I desire to support the philanthropic view which you justly take of the cause of what are usually termed "atrocities of the aborigines" being really "white aggression." I need only point to the first white sufferer who fell from black spears, in what is now known as Queensland—my very esteemed and noble friend Kennedy, the last to provoke the enmity of the savage. Then, why was he killed? is the natural enquiry. If the Queensland Government have preserved the book which was kept at the Booby Island post office, you may find there the record in the handwriting of Kennedy's real murderer, though indirectly, about seven years previously—namely, about June, 1841—in which Grayburne, the master of the Brothers, records the fact of while sailing past the York Islands he "shot a blackfellow." Here we have a picture of a British vessel sailing along the coast while natives are peacefully gathering shellfish for their meal, when this unblushing British tar (no, not tar, but brute) tries the range of his musket, and thinks himself a clever shot. I read this a few days after its entry, and remarked that I pitied the first white man that came within reach of that tribe. The opportunities were rare, but Kennedy came at last. Who, again, carried off two young children from the tribe on poor Wills' run and brought them to Sydney never to return? But Wills, unconscious

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Rh —Applications of the Lorentz-FitzGerald Hypothesis to Dynamical and Gravitational Problems; by  is at the present time a general consensus of opinion, among those best qualified to judge, that the fundamental facts of optics and electro-dynamics require us to assume that the ether does not partake (to any sensible extent) in the motion of material bodies which pass through it. The aberration of light is perhaps the most conspicuous of those phenomena which it has hitherto been found impossible to account for on any other hypothesis without becoming involved in serious difficulties. All the phenomena in which there is relative motion of the source of light with respect to the observer, or of a material medium (through which light is propagated) with respect to source and observer, appear to require the above assumption, even those which at first seemed to lead to a conclusion somewhat different in form. Thus the experiment of Fizeau, in which he compared the velocities of light when going with, and against, a stream of water, was interpreted by Fresnel as indicating a certain entrainment of the ether. This interpretation was based on Fresnel's theory of refraction, which assumed that the etherial density was increased in material media; and it is only the excess ether which must be carried by the matter. On the electron theory, however (and indeed on any resonance theory of dispersion and refraction) there is no excess density of the ether in ponderable bodies; and it is not difficult to see that Fizeau's experiment requires a stationary ether. A result which leads to the same view has been obtained in electro-dynamics by H. A. Wilson in measuring the electric force produced by moving an insulator in a magnetic field. All such experiments upon the effects of relative motion, so far as I know, give positive results which may be predicted from the hypothesis of a fixed ether, and the magnitude of the effects observed is in general of the same order as the fraction $$\scriptstyle{\frac{v}{\text{V}}}$$, where $$\scriptstyle{v}$$ is the relative velocity involved, and $$\scriptstyle{\text{V}}$$ the velocity of light. The theory of a stagnant ether leads us, however, in a no less direct manner to expect certain modifications in the phenomena of light and electricity when there in no relative motion of

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494 material objects, but when all the apparatus concerned as well as the observer are carried through the ether with the velocity $$\scriptstyle{v}$$. The effects to be expected are of the order of $$\scriptstyle{\left(\frac{v}{\text{V}}\right)^{2}}$$; this is a very small fraction even when $$\scriptstyle{v}$$ is the velocity of the earth in its orbit, but the possible accuracy of certain optical experiments is so great that these effects could certainly be found if they existed without some compensating effect to mask them. As is well known, these effects have never been found; the first conclusively negative results were obtained in the celebrated experiments of Michelson and Morley, and several other optical investigations have also failed to show the expected results. On the electrical side the problem has been attacked by Trouton and Noble, who hung up an electrical condenser by a torsion wire and looked for a torque which, on the theory of a stagnant ether, ought to exist when the condenser is carried along by the earth. Although the sensitiveness of their experimental arrangement was ample for the observation of the expected second order effect, their result was also negative. The most obvious interpretation of these results is that the ether near the earth has the same velocity as the earth; but, as has been stated, it appears to be impossible to reconcile this view with the great mass of optical and electro-dynamic evidence. The only satisfactory way out of this difficulty which has hitherto been suggested is a hypothesis put forward in 1892 by Lorentz, and which had been independently suggested but not published by FitzGerald. According to this hypothesis, when any material body moves relatively to the ether its linear dimensions parallel to the direction of motion are contracted in the ratio of $$\scriptstyle{\sqrt{1-\frac{v^{2}}{\text{V}^{2}}}}$$ to 1, while the dimensions perpendicular to the direction of motion remain unchanged. If this contraction takes place in the interferometer of Michelson and Morley and in the condenser of Trouton and Noble, their negative results are entirely explained on the theory of a stationary ether. As Lorentz points out, this contraction will be very small in any motions of material bodies which we can observe; for example the diameter of the earth in the direction of its orbital path will be diminished by only 6·5$cm$ by its motion. It would moreover be impossible to detect the shrinkage, however great it might be, by ordinary

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Rh measurements, since the standards of length must shrink in the same ratio as the bodies to be measured. It would be quite misleading, however, to leave the impression that this hypothesis depends for its credibility altogether upon the fact that it enables us to evade a serious difficulty and that it cannot be disproved by ordinary means. The electrical forces between charged bodies (electrons) are modified by motion through the ether; and they are modified in precisely such a way that if a given system of charges were in equilibrium under these forces in a certain configuration when at rest, it would when in motion be in equilibrium in a configuration obtained from the first by the application of the Lorentz-FitzGerald shrinkage. Now it is a fundamental theorem in electrostatics, that a charged system cannot be in equilibrium under the electrical forces alone; in the case of a collocation of electrons or atoms in equilibrium, the electrical forces must be balanced by other forces. If these inter-electronic forces are ethereal in origin and subject to the same laws as electro-magnetic forces, then the Lorentz-FitzGerald contraction would be expected à priori; and from this point of view the absence of the second order effects is evidence for the ethereal nature of inter-atomic and inter-molecular forces. Forces of this character would suffice to account for the changed dimensions of moving bodies even if the electrons themselves were left unaltered by the motion. But, as Lorentz has pointed out, we must also bring in dynamical considerations which show that for complete absence of second-order effects the electrons themselves must suffer the same contraction. The experiments of Lord Rayleigh and of Brace have shown that there is no double refraction due to the convection of transparent bodies by the earth. This implies that the periods of vibration of the electrons in the line of motion and perpendicular to it must be equal; and in order that this may be so, the longitudinal and the transverse masses of the electron must be altered by the motion in the same manner as the forces in these directions. An electron which does not change its shape (such as the rigid spherical electron of Abraham) will not have this property; nor will an electron which alters its form in any other manner than that described above for material bodies (such as the constant-volume electron of Bucherer). The electron proposed by Lorentz obviates these difficulties. If we assume that it is, when at rest, a sphere of radius, $$\scriptstyle{a}$$, it must when in motion with velocity $$\scriptstyle{v}$$, become an ellipsoid of revolution with its shorter axis in the direction of the motion and equal to $$\scriptstyle{a\sqrt{1-\frac{v^{2}}{\text{V}^{2}}}}$$, the

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496 perpendicular to the motion remaining the same. If $$\scriptstyle{m_1}$$ and $$\scriptstyle{m_2}$$ are its longitudinal and transverse masses, and $$\scriptstyle{m_0}$$ the mass for infinitesimal velocities, we shall have where, for brevity, $$\scriptstyle{\beta}$$ has been put for $$\scriptstyle{\frac{v}{\text{V}}}$$. With this electron Lorentz has shown that no optical or electrical effects of motion through the ether can be detected. The subject has been approached from a different standpoint, and treated in a very interesting and instructive manner by Einstein. His fundamental postulate amounts to a denial that it is possible to observe any effects of uniform convection through the ether in which all the bodies concerned (including the observer) take part. This he calls the Principle of Relativity; the significance of the name is that only relative motion of one portion of matter with respect to another, or of one electrical charge with respect to another, can produce any observable effect; uniform motion, relative to the ether alone, becomes as impotent, if not as meaningless, as absolute motion. Einstein considers two sets of coördinate axes, one at rest in the ether ($$\scriptstyle{x}$$, $$\scriptstyle{y}$$, $$\scriptstyle{z}$$), while the other moves with the constant velocity $$\scriptstyle{v}$$ in the x direction ($$\scriptstyle{\xi}$$, $$\scriptstyle{\eta}$$, $$\scriptstyle{\zeta}$$). He defines carefully the meaning of "time" ($$\scriptstyle{t}$$ in the fixed system, $$\scriptstyle{\tau}$$ in the moving system) by means of clocks distributed at various points, some at rest with the fixed axes, and some moving with the moving axes. The clocks are supposed to be synchronized by light signals. By kinematic considerations he shows that, in order for the principle of relativity to hold, we must have, $\begin{align}&\scriptstyle{\xi=\frac{1}{\sqrt{1-\beta^{2}}}(x-vt)}\\&\scriptstyle{\eta=y}\\&\scriptstyle{\zeta=z}\\&\scriptstyle{\tau=\frac{1}{\sqrt{1-\beta^{2}}}\left(t-\frac{v}{\text{V}^{2}}x\right)}\end{align}$|undefined where, as before, $$\scriptstyle{\beta=\frac{v}{\text{V}}}$$.

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