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  Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set

  a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain

  It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get

  It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set v = c – κ, w = c – λ, κ and λ being positive and less than c, then

  It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain

  We might also have obtained the formula for V, for the case when v and w have the same direction, by compounding two transformations in accordance with § 3. If in addition to the systems K and k figuring in § 3 we introduce still another system of co-ordinates k′ moving parallel to k, its initial point moving on the axis of X with the velocity w, we obtain equations between the quantities x, y, z,t and the corresponding quantities of k′, which differ from the equations found in § 3 only in that the place of “v” is taken by the quantity

  from which we see that such parallel transformations—necessarily— form a group.

  We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.

  II. ELECTRODYNAMICAL PART

  § 6. TRANSFORMATION OF THE MAXWELL-HERTZ EQUATIONS FOR EMPTY SPACE. ON THE NATURE OF THE ELECTROMOTIVE FORCES OCCURRING IN A MAGNETIC FIELD DURING MOTION

  Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have

  where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force.

  If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations

  where

  Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force—(X′, Y′, Z′) and (L′, M′, N′)—of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations:—

  Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for system K. Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a factor ψ(v), which is common for all functions of the one system of equations, and is independent of ξ, η, ζ and τ but depends upon v. Thus we have the relations

  If we now form the reciprocal of this system of equations, firstly by solving the equations just obtained, and secondly by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity – v, it follows, when we consider that the two systems of equations thus obtained must be identical, that ψ(v)ψ(– v) = 1. Further, from reasons of symmetry* ψ(v) = ψ(– v). and therefore

  and our equations assume the form

  As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude “one” when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude “one” when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (X, Y, Z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (X′, Y′, Z′). Consequently the first three equations above allow themselves to be clothed in words in the two following ways:—

  1. If a unit electric point charge is in motion in an electromagnetic field, there acts upon it, in addition to the electric force, an “electromotive force” which, if we neglect the terms multiplied by the second and higher powers of v/c, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.)

  2. If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)

  The analogy holds with “magnetomotive forces.” We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of co-ordinates.

  Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the “seat” of electrodynamic electromotive forces (unipolar machines) now have no point.

  § 7. THEORY OF DOPPLER’S PRINCIPLE AND OF ABERRATION

  In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations

  where

  Here (X0, Y0, Z0) and (L0, M0, N0) are the vectors defining the amplitude of the wave-train, and l, m, n the direction-cosines of the wave-normals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system k.

  Applying the equations of transformation found in § 6 for electric and magnetic forces, and those found in § 3 for the co-ordinates and the time, we obtain directly

  where

  From the equation for ω′ it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency n, in such a way that the connecting line “source—observer” makes the angle ϕ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency v′ of the light perceived by the observer is given by the equation

  This is Doppler’s principle for any velocities whatever. When ϕ = 0 the equation assumes the perspicuous form

  We see that, in contrast with the customary view, when v = – c, v′ = ∞.

  If we call the angle between the wave-normal (direction of the ray) in the moving system and the connecting line “source—observer” ϕ′, the equation for l′ assumes the form

  This equation expresses the law of aberration in its most general form. If , the equation becomes simply

  We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A′ respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain

  which equation, if ϕ = 0, simplifies into

  It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

  § 8. TRANSFORMATION OF THE ENERGY OF LIGHT RAYS. THEORY OF THE PRESSURE OF RADIATION EXERTED ON PERFECT REFLECTORS

  Since A2/8π equals the energy of light per unit of volume, we have to regard A′2/8π
, by the principle of relativity, as the energy of light in the moving system. Thus A′2/A2 would be the ratio of the “measured in motion” to the “measured at rest” energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If l, m, n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:—

  We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k.

  The spherical surface—viewed in the moving system—is an ellipsoidal surface, the equation for which, at the time τ = 0, is

  If S is the volume of the sphere, and S′ that of this ellipsoid, then by a simple calculation

  Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and E′ when measured in the moving system, we obtain

  and this formula, when ϕ = 0, simplifies into

  It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.

  Now let the co-ordinate plane ξ = 0 be a perfectly reflecting surface, at which the plane waves considered in § 7 are reflected. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion.

  Let the incidental light be defined by the quantities A, cosϕ, v (referred to system K). Viewed from k the corresponding quantities are

  For the reflected light, referring the process to system k, we obtain

  Finally, by transforming back to the stationary system K, we obtain for the reflected light

  The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently A2(ccos ϕ – v)/8π. The energy leaving the unit of surface of the mirror in the unit of time is A″′2(– ccos ϕ″′ + v)/8π. The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product Pv, where P is the pressure of light, we obtain

  In agreement with experiment and with other theories, we obtain to a first approximation

  All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.

  § 9. TRANSFORMATION OF THE MAXWELL-HERTZ EQUATIONS WHEN CONVECTION-CURRENTS ARE TAKEN INTO ACCOUNT

  We start from the equations

  where

  denotes 4π times the density of electricity, and (ux, uy, uz) the velocity-vector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.

  Let these equations be valid in the system K, and transform them, with the assistance of the equations of transformation given in §§ 3 and 6, to the system k. We then obtain the equations

  where

  and

  Since—as follows from the theorem of addition of velocities (§ 5)—the vector [uξ, uη, uζ) is nothing else than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of our kinematical principles, the electrodynamic foundation of Lorentz’s theory of the electrodynamics of moving bodies is in agreement with the principle of relativity.

  In addition I may briefly remark that the following important law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of co-ordinates moving with the body, its charge also remains—when regarded from the “stationary” system K—constant.

  § 10. DYNAMICS OF THE SLOWLY ACCELERATED ELECTRON

  Let there be in motion in an electromagnetic field an electrically charged particle (in the sequel called an “electron”), for the law of motion of which we assume as follows:—

  If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations

  where x, y, z denote the co-ordinates of the electron, and m the mass of the electron, as long as its motion is slow.

  Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time.

  Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the co-ordinates, and moves with the velocity v along the axis of X of the system K. It is then clear that at the given moment (t = 0) the electron is at rest relatively to a system of coordinates which is in parallel motion with velocity v along the axis of X.

  From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time (for small values of t) the electron, viewed from the system k, moves in accordance with the equations

  in which the symbols ξ, η, ζ, τ, X′, Y′, Z′ refer to the system k. If, further, we decide that when t = x = y = z = 0 then τ = ξ = η = ζ = 0, the transformation equations of §§ 3 and 6 hold good, so that we have

  With the help of these equations we transform the above equations of motion from system k to system K, and obtain

  Taking the ordinary point of view we now inquire as to the “longitudinal” and the “transverse” mass of the moving electron. We write the equations (A) in the form

  and remark firstly that εX′, εY′, εZ′ are the components of the ponderomotive force acting upon the electron, and are so indeed as viewed in a system moving at the moment with the electron, with the same velocity as the electron. (This force might be measured, for example, by a spring balance at rest in the last-mentioned system.) Now if we call this force simply “the force acting upon the electron,”* and maintain the equation—mass × acceleration = force—and if we also decide that the accelerations are to be measured in the stationary System K, we derive from the above equations

  With a different definition of force and acceleration we should naturally obtain other values for the masses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously.

  We remark that these results as to the mass are also valid for ponderable material points, because a ponderable material point can be made into an electron (in our sense of the word) by the addition of an electric charge, no matter how small.

  We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of co-ordinates of the system K along the axis of X under the action of an electrostatic force X, it is clear that the energy withdrawn from the electrostatic field has the value ∫εXdx. As the electron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron. Bearing in mind that during the whole process of motion which we are considering, the first of the equations (A) applies, we therefore obtain

  Thus, when v = c, W becomes infinite. Velocities greater than that of light have—as in our previous results—no possibility of existence.

  This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well.

  We will now enumerate the properties of the motion of the electron which result from the system of equations (A), and are accessible to exp
eriment.

  1. From the second equation of the system (A) it follows that an electric force Y and a magnetic force N have an equally strong deflective action on an electron moving with the velocity v, when Y = Nv/c. Thus we see that it is possible by our theory to determine the velocity of the electron from the ratio of the magnetic power of deflexion Am to the electric power of deflexion Ae, for any velocity, by applying the law

  This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e.g. by means of rapidly oscillating electric and magnetic fields.

  2. From the deduction for the kinetic energy of the electron it follows that between the potential difference, P, traversed and the acquired velocity v of the electron there must be the relationship

  3. We calculate the radius of curvature of the path of the electron when a magnetic force N is present (as the only deflective force), acting perpendicularly to the velocity of the electron. From the second of the equations (A) we obtain

  or

  These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move.