NOTES

The fundamental electro-magnetic equations of Maxwell for stationary media are:—

curl H = 1/c(∂D/∂t+ ρν) (1)curl E = -1/c∂B/∂t(2)div D = ρB = μHdiv B = 0D = kE

curl H = 1/c(∂D/∂t+ ρν) (1)

curl E = -1/c∂B/∂t(2)

div D = ρB = μHdiv B = 0D = kE

div D = ρ

B = μH

div B = 0

D = kE

According to Hertz and Heaviside, these require modification in the case of moving bodies.

Now it is known that due to motion alone there is a change in a vectorRgiven by

(∂R/∂t) due to motion =u. div R + curl [Ru]

whereuis the vector velocity of the moving body and [Ru] the vector product of R andu.

Hence equations (1) and (2) become

ccurl H = ∂D/∂t+udiv D + curl Vect. [Du] + ρν (1·1)

and

-ccurl E = ∂B/∂t+udiv B + curl Vect. [Bu] (2·1)

which gives finally, for ρ = 0 and div B = 0,

∂D/∂t+udiv D =ccurl (H - 1/cVect. [Du]) (1·2)∂B/∂t= -ccurl (E - 1/cVect. [uB]) (2·2)

∂D/∂t+udiv D =ccurl (H - 1/cVect. [Du]) (1·2)

∂B/∂t= -ccurl (E - 1/cVect. [uB]) (2·2)

Let us consider a beam travelling along thex-axis, with apparent velocityv(i.e., velocity with respect to the fixed ether) in medium moving with velocityux=uin the same direction.

Then if the electric and magnetic vectors are proportional toeiA(x-vt), we have

∂/∂x=iA, ∂/∂t= -iAv, ∂/∂y= ∂/∂z= 0,uy=uz= 0Then ∂D_y/∂t= -c∂Hz/∂x-u∂Dy/∂z... (1·21)and ∂Bz/∂t= -c∂Ey/∂x-u∂Bz/∂x(2·21)

∂/∂x=iA, ∂/∂t= -iAv, ∂/∂y= ∂/∂z= 0,uy=uz= 0

Then ∂D_y/∂t= -c∂Hz/∂x-u∂Dy/∂z... (1·21)

and ∂Bz/∂t= -c∂Ey/∂x-u∂Bz/∂x(2·21)

Since D = KE and B = μH, we have

iAv(κEy) = -ciA(Hz+uKEy) (1·22)iAv(μHz) = -ciA(Ey+uμHz) (2·22)orv(K -u)Ey=cHz(1·23)μ(v-u)Hz=cEy(2·23)

iAv(κEy) = -ciA(Hz+uKEy) (1·22)

iAv(μHz) = -ciA(Ey+uμHz) (2·22)

orv(K -u)Ey=cHz(1·23)

μ(v-u)Hz=cEy(2·23)

Multiplying (1·23) by (2·23)

μK(v-u)² =c²

Hence (v-u)² =c²/μk=v₀²

∴v=v₀+u,

making Fresnelian convection co-efficient simply unity.

Equations (1·21) and (2·21) may be obtained more simply from physical considerations.

According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is δD/δt.

Consider a slab of matter moving with velocityuxalong thex-axis, then even in a stationary field of electrostatic polarisation, that is, for a field in which δD/δt= 0, there will be some change in the polarisation of the body due to its motion, given byux(δD/δx). Hence we must add this term to a purely temporal rate of change δD/δt. Doing this we immediately arrive at equations (1·21) and (2·21) for the special case considered there.

Thus the Hertz-Heaviside form of field equations givesunityas the value for the Fresnelian convection co-efficient. It has been shown in the historical introduction how this is entirely at variance with the observed optical facts. As a matter of fact, Larmor has shown (Aether and Matter) that 1 - 1/μ² is not only sufficient but is also necessary, in order to explain experiments of the Arago prism type.

A short summary of the electromagnetic experiments bearing on this question, has already been given in the introduction.

According to Hertz and Heaviside the total polarisation is situated in the medium itself and is completely carried away by it. Thus the electromagnetic effect outside a moving medium should be proportional to K, the specific inductive capacity.

Rowlandshowed in 1876 that when a charged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.

Röntgen(Annalen der Physik 1888, 1890) found that if the dielectric is rotated while the condenser remains stationary, the effect is proportional to K - 1.

Eichenwald(Annalen der Physik 1903, 1904) rotated together both condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K. This is of course quite consistent with Rowland and Röntgen.

Blondlot(Comptes Rendus, 1901) passed a current of air in a steady magnetic field Hy, (H = Hz= 0). If this current of air moves with velocityuxalong thex-axis, an electromotive force would be set up along thez-axis, due to the relative motion of matter and magnetic tubes of induction. A pair of plates atz= ±a, will be charged up with density ρ = Dz= KE = K.usHy/c. But Blondlot failed to detect any such effect.

H. A. Wilson(Phil. Trans. Royal Soc. 1904) repeated the experiment with a cylindrical condenser made of ebony, rotating in a magnetic field parallel to its own axis. He observed a change proportional to K — 1 and not to K.

Thus the above set of electro-magnetic experiments contradict the Hertz-Heaviside equations, and these must be abandoned.

[P. C. M.]

Lorentz. Versuch einer theorie der elektrischen und optischen Erscheinungen im bewegten Körpern.

(Leiden—1895).

Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also notes 73, 86, pages 318, 328.

Lorentz wanted to explain the Michelson-Morley null-effect. In order to do so, it was obviously necessary to explain the Fitzgerald contraction. Lorentz worked on the hypothesis that an electron itself undergoes contraction when moving. He introduced new variables for the moving system defined by the following set of equations.

x¹= β(x-ut),y¹=y,z¹=z,t¹= β(t- (u/c²)·x)

and for velocities, used

vx¹ = β²vx+u,vy¹ = βvy,vz¹ = βvzand ρ¹ = ρ/β.

With the help of the above set of equations, which is known as the Lorentz transformation, he succeeded in showing how the Fitzgerald contraction results as a consequence of “fortuitous compensation of opposing effects.”

It should be observed that the Lorentz transformation is not identical with the Einstein transformation. The Einsteinian addition of velocities is quite different as also the expression for the “relative” density of electricity.

It is true that the Maxwell-Lorentz field equations remainpracticallyunchanged by the Lorentz transformation, but theyarechanged to some slight extent. One marked advantage of the Einstein transformation consists in the fact that the field equations of a moving system preserveexactlythe same form as those of a stationary system.

It should also be noted that the Fresnelian convection coefficient comes out in the theory of relativity as a direct consequence of Einstein’s addition of velocities and is quite independent of any electrical theory of matter.

[P. C. M.]

See Lorentz, Theory of Electrons (English edition), § 181, page 213.

H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo matematico di Palermo 21 (1906).

[P. C. M.]

Lorentz showed that the Maxwell-Lorentz system of electromagnetic field-equations remained practically unchanged by the Lorentz transformation. Thus the electromagnetic laws of Maxwell and Lorentzcan be definitely proved“to be independent of the manner in which they are referred to two coordinate systems which have a uniform translatory motion relative to each other.” (See “Electrodynamics of Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are concerned, the principle of relativitycan be proved to be true.

But it is not known whether this principle will remain true in the case of other physical laws. We can always proceed on the assumption that it does remain true. Thus it is always possible to construct physical laws in such a way that they retain their form when referred to moving coordinates. The ultimate ground for formulating physical laws in this way is merely a subjective conviction that the principle of relativity is universally true. There is noa priorilogical necessity that it should be so. Hence the Principle of Relativity (so far as it is applied to phenomena other than electromagnetic) must be regarded as apostulate, which we have assumed to be true, but for which we cannot adduce any definite proof, until after the generalisation is made and its consequences tested in the light of actual experience.

[P. C. M.]

See “Electrodynamics of Moving Bodies,” p. 5-8.

Equations (i) and (ii) become when expanded into Cartesians:—

∂mz/∂y- ∂my/∂z- ∂ex/∂τ = ρνx}∂mx/∂z- ∂mz/∂x- ∂ey/∂τ = ρνy} ... (1·1)∂my/∂x- ∂mx/∂y- ∂ez/∂τ = ρνz}

∂mz/∂y- ∂my/∂z- ∂ex/∂τ = ρνx}

∂mx/∂z- ∂mz/∂x- ∂ey/∂τ = ρνy} ... (1·1)

∂my/∂x- ∂mx/∂y- ∂ez/∂τ = ρνz}

and ∂ex/∂x+ ∂ey/∂y+ ∂ez/∂z= ρ (2·1)

Substitutingx₁,x₂,x₃,x₄andx,y,z, andiτ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρνx, ρνy, ρνz,iρ, wherei= √(-1).

We get,

∂mz/∂x₂- ∂my/∂x₃-i(∂ex/∂x₄) = ρνx{ = ρ₁ }- ∂mz/∂x₁+ ∂mx/∂x₃-i(∂ey/∂x₄) = ρνy= ρ₂ } ... (1·2)∂my/∂x₁- ∂mx/∂x₂-i(∂ez/∂x₄) = ρνz{ = ρ₃ }

∂mz/∂x₂- ∂my/∂x₃-i(∂ex/∂x₄) = ρνx{ = ρ₁ }

- ∂mz/∂x₁+ ∂mx/∂x₃-i(∂ey/∂x₄) = ρνy= ρ₂ } ... (1·2)

∂my/∂x₁- ∂mx/∂x₂-i(∂ez/∂x₄) = ρνz{ = ρ₃ }

and multiplying (2·1) by i we get

∂iex/∂x₁+ ∂iey/∂x₂+ ∂iez/∂x₃=iρ = ρ₄ ... ... (2·2)

Now substitute

mx=f₂₃= -f₃₂andiex=f₄₁ = -f₁₄my=f₃₁= -f₁₃iey=f₄₂ = -f₂₄mz=f₁₂= -f₂₁iez=f₄₃ = -f₃₄

mx=f₂₃= -f₃₂andiex=f₄₁ = -f₁₄

my=f₃₁= -f₁₃iey=f₄₂ = -f₂₄

mz=f₁₂= -f₂₁iez=f₄₃ = -f₃₄

and we get finally:—

∂f₁₂/∂x₂+ ∂f₁₃/∂x₃+ ∂f₁₄/∂x₄= ρ₁ }∂f₂₁/∂x₁+ ∂f₂₃/∂x₃+ ∂f₂₄/∂x₄= ρ₂ } ... (3)∂f₃₁/∂x₁+ ∂f₃₂/∂x₂+ ∂f₃₄/∂x₄= ρ₃ }∂f₄₁/∂x₁+ ∂f₄₂/∂x₂+ ∂f₄₃/∂x₃= ρ₄ }

∂f₁₂/∂x₂+ ∂f₁₃/∂x₃+ ∂f₁₄/∂x₄= ρ₁ }

∂f₂₁/∂x₁+ ∂f₂₃/∂x₃+ ∂f₂₄/∂x₄= ρ₂ } ... (3)

∂f₃₁/∂x₁+ ∂f₃₂/∂x₂+ ∂f₃₄/∂x₄= ρ₃ }

∂f₄₁/∂x₁+ ∂f₄₂/∂x₂+ ∂f₄₃/∂x₃= ρ₄ }

Page 12—refer also to page 6, of Einstein’s paper.

One of the two fundamental Postulates of the Principle of Relativity is that the velocity of light should remain constant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocityu, it should always remain the centre of spherical waves expanding outwards with velocityc.

At first sight, it may not appear clear why the velocity should remain constant. Indeed according to the theory of Ritz, the velocity should becomec+u, when the source of light moves towards the observer with the velocityu.

Prof. de Sitter has given an astronomical argument for deciding between these two divergent views. Let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit. Let the observer be in the plane of the orbit, at a great distance Δ.

Experiment.

The light emitted by the star when at the position A will be received by the observer after a time, Δ/(c+u) while the light emitted by the star when at the position B will be received after a time Δ/(c-u). Let T be the real half-period of the star. Then the observed half-period from B to A is approximately T - 2Δu/c²and from A to B is T + 2Δu/c². Now if 2uΔ/c²be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2uΔ/c²are not only of the same order as T, but are mostly much larger. For example, ifu= 100km/sec, T = 8 days, Δ/c= 33 years (corresponding to an annual parallax of ·1″), then T - 2uΔ/c²= 0. The existence of the Spectroscopic binaries, and the fact that they follow Kepler’s Law is therefore a proof thatcis not affected by the motion of the source.

In a later memoir, replying to the criticisms of Freundlich and Günthick that an apparent eccentricity occurs in the motion proportional tokuΔ₀,u₀being the maximum value ofu, the velocity of light emitted being

u₀=c+ku,k= 0 Lorentz-Einsteink= 1 Ritz.

u₀=c+ku,

k= 0 Lorentz-Einstein

k= 1 Ritz.

Prof. de Sitter admits the validity of the criticisms. But he remarks that an upper value ofkmay be calculated from the observations of the double star β-Aurigae. For this star, the parallax π = ·014″,e= ·005,u₀= 110km/sec, T = 3·96,

Δ > 65 light-years,kis < ·002.

Δ > 65 light-years,

kis < ·002.

For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol. 35, p. 163.

[M. N. S.]

If ρ is the volume density in a moving system then ρ√(1 -u²) is the corresponding quantity in the corresponding volume in the fixed system, that is, in the system at rest, and hence it is termed the rest-density of electricity.

[P. C. M.]

As we had already occasion to mention, Sommerfeld has, in two papers on four dimensional geometry (vide, Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of Minkowski’s space-time vector of the first kind, he uses the more expressive term ‘four-vector,’ thereby making it quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis.

The representation of the plane (defined by two straight lines) is much more difficult. In three dimensions, the plane can be represented by the vector perpendicular to itself. But that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a single line, but an infinite number of lines constituting a plane. This difficulty has been overcome by Minkowski in a very elegant manner which will become clear later on. Meanwhile we offer the following extract from the above mentioned work of Sommerfeld.

(Pp. 755, Bd. 32, Ann. d. Physik.)

“In order to have a better knowledge about the nature of the six-vector (which is the same thing as Minkowski’s space-time vector of the2ndkind) let us take the special case of a piece of plane, having unit area (contents), and the form of a parallelogram, bounded by the four-vectorsu,v, passing through the origin. Then the projection of this piece of plane on thexyplane is given by the projectionsux,uy,vx,vyof the four vectors in the combination

φxy=uxvy-uyv{x}.

Let us form in a similar manner all the six components of this plane φ. Then six components are not all independent but are connected by the following relation

φyzφxl+ φzxφyl+ φxyφzl= 0

Further the contents | φ | of the piece of a plane is to be defined as the square root of the sum of the squares of these six quantities. In fact,

| φ |² = φyz² + φzx² + φxy² + φxl² + φyl² + φzl².

Let us now on the other hand take the case of the unit plane φ*normal to φ; we can call this plane the Complement of φ. Then we have the following relations between the components of the two plane:—

φyz*= φxl, φzx*= φyl, φxy*= φzlφzl*= φyx...

The proof of these assertions is as follows. Letu*,v*be the four vectors defining φ*. Then we have the following relations:—

ux*ux+uy*uy+uz*uz+ul*ul= 0ux*vx+uy*vy+uz*vz+ul*vl= 0vx*ux+vy*uy+vz*uz+vl*ul= 0vx*vx+vy*vy+vz*vz+vl*vl= 0

ux*ux+uy*uy+uz*uz+ul*ul= 0

ux*vx+uy*vy+uz*vz+ul*vl= 0

vx*ux+vy*uy+vz*uz+vl*ul= 0

vx*vx+vy*vy+vz*vz+vl*vl= 0

If we multiply these equations byvl,ul,vs, and subtract the second from the first, the fourth from the third we obtain

ux*φxl+uy*φyl+uz*φzl= 0vx*φzl+vy*φyl+vz*φzl= 0

ux*φxl+uy*φyl+uz*φzl= 0

vx*φzl+vy*φyl+vz*φzl= 0

multiplying these equations byvx*.ux*, or byvy*.uy*, we obtain

φxz*φxl+ φyz*φyl= 0 and φxy*φxl+ φzx*φzl= 0

from which we have

φyz*: φxy*: φzx*= φxl: φzl: φyl

In a corresponding way we have

φyz: φxy: φzx= φxl*: φzl*: φyl*.

i.e.φik*= λφ(ik)

when the subscript (ik) denotes the component of φ in the plane contained by the lines other than (ik). Therefore the theorem is proved.

We have (φ φ*) = φyzφyz*+ ...= 2 (φyzφzl+ ...)= 0

We have (φ φ*) = φyzφyz*+ ...

= 2 (φyzφzl+ ...)

= 0

The general six-vectorfis composed from the vectors φ, φ*in the following way:—

f= ρφ + ρ*φ*,

ρ and ρ*denoting the contents of the pieces of mutually perpendicular planes composingf. The “conjugate Vector”f*(or it may be called the complement off) is obtained by interchanging ρ and ρ*.

We have

f*= ρ*φ + ρφ*

We can verify that

fy z*=fx letc.

andf²= ρ² + ρ*², (ff*) = 2ρρ*.

|f|² and (ff*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates.

[M. N. S.]

Page 23, and Electro-dynamics of Moving Bodies, p. 17.

Puttingv=c-x, andw=c- λ, we get

V = (2c- (x+ λ))/(1 + (c-x)(c- λ)/c²) = (2c- (x+ λ))/(c²+c²- (x+ λ)c+xλ/c²)=c(2c- (x+ λ))/(2c- (x+ λ) +xλ/c)

V = (2c- (x+ λ))/(1 + (c-x)(c- λ)/c²) = (2c- (x+ λ))/(c²+c²- (x+ λ)c+xλ/c²)

=c(2c- (x+ λ))/(2c- (x+ λ) +xλ/c)

Thusvlt;c, so long as |xλ | > 0.

Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W >c.

Let A and B be separated by distancel, and let velocity of a “signal” in the system S be W >c. Let the (observing) system S′ have velocity +vwith respect to the system S.

Then velocity of signal with respect to system S′ is given by W′ = (W -v)/(1 - Wv/c²)

Thus “time” from A to B as measured in S′, is given byl/W′ =l(1 - Wv/c²)/(W -v) =t′(1)

Now ifvis less thanc, then W being greater thanc(by hypothesis) W is greater thanv,i.e., W >v.

Let W =c+ μ andv=c- λ.

Then Wv= (c+ μ)(c- λ) =c²+ (μ + λ)c- μλ.

Now we can always choosevin such a way that Wvis greater thanc², since Wvis >c²if (μ + λ)c- μλ is > 0, that is, if μ + λ > μλ/c; which can always be satisfied by a suitable choice of λ.

Thus for W >cwe can always choose λ in such a way as to make Wv>c²,i.e., λ - Wv/c²negative. But W -vis always positive. Hence with W >c, we can always maket′, the time from A to B in equation (1) “negative.” That is, the signal starting from A will reach B (as observed in system S′) in less than no time. Thus the effect will be perceived before the cause commences to act,i.e., the future will precede the past. Which is absurd. Hence we conclude that W >cis an impossibility, there can be no velocity greater than that of light.

It isconceptuallypossible to imagine velocities greater than that of light, but such velocities cannot occur in reality. Velocities greater thanc, will not produce any effect. Causal effect of any physical type can never travel with a velocity greater than that of light.

[P. C. M.]

We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes the reciprocal matrix

| ω₁ || ω₂ || ω₃ || ω₄ |

| ω₁ |

| ω₂ |

| ω₃ |

| ω₄ |

It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]

[ω,s] The vector-product of the four-vector ω andsmay be represented by the combination

[ωs] = [=ω]s-ṡω

It is now easy to verify the formulaf¹ = A⁻¹fA. Supposing for the sake of simplicity thatfrepresents the vector-product of two four-vectors ω,s, we have

f¹= [ω¹s¹] = [[=ω]¹s¹- [=s]1ω1]= [A⁻¹ [=ω]sA - A⁻¹s[=ω]A]= A⁻¹[[=ω]s-s[=ω]]A = A⁻¹fA.

f¹= [ω¹s¹] = [[=ω]¹s¹- [=s]1ω1]

= [A⁻¹ [=ω]sA - A⁻¹s[=ω]A]

= A⁻¹[[=ω]s-s[=ω]]A = A⁻¹fA.

Now remembering that generally

f= ρφ + ρ*φ*.

Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that

f1= A⁻¹fA.

This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.

“We can also form a vectorial combination of a four-vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type,i.e., a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3-space volumes can always be added together by the vectorial addition of their components. So by the addition of two 3-space volumes, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc. cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector-multiplication of a vector of the first, and a vector of the second type (i.e., a polar vector), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.

Let A = (Ax, Ay, Az) denote a vector of the first type, B = (By z, Bz x, Bx y) denote a vector of the second type. From this last, let us form three special vectors of the first kind, namely—

Bx= (Bx x, Bx y, Bx z) }By= (By x, By y, By z) } (Bi k= - Bk i, Bi i= 0).Bz= (Bz x, Bz y, Bz z) }

Bx= (Bx x, Bx y, Bx z) }

By= (By x, By y, By z) } (Bi k= - Bk i, Bi i= 0).

Bz= (Bz x, Bz y, Bz z) }

Since Bj jis zero, Bjis perpendicular to thej-axis. Thej-component of the vector-product of A and B is equivalent to the scalar product of A and Bj,i.e.,

(A Bj,) = AxBj x+ AyBj y+ AzBj z.

We see easily that this coincides with the usual rule for the vector-product;e. g., forj=x.

(ABx) = AyBxy- AzBzx.

Correspondingly let us define in the four-dimensional case the product (Pf) of any four-vector P and the six-vectorf. Thej-component (j=x,y,z, orl) is given by

(Pfj) = Pxfjx+ Pyfjy+ Pwfjz+ Pzfjl

Each one of these components is obtained as the scalar product of P, and the vectorfjwhich is perpendicular to j-axis, and is obtained fromfby the rulefj= [(fjx,fjy,fjz,fjl)fjj= 0.]

We can also find out here the geometrical significance of vectors of the third type, whenf= φ,i.e.,frepresents only one plane.

We replace φ by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane φ*, which is formed by the perpendicular four-vectors U*, V*. The components of (Pφ) are then equal to the 4 three-rowed under-determinants DxDyDzDlof the matrix

| PxPyPzPl|| || Ux*Uy*Uz*Ul*|| || Vx*Vy*Vz*Vl*|

| PxPyPzPl|

| |

| Ux*Uy*Uz*Ul*|

| |

| Vx*Vy*Vz*Vl*|

Leaving aside the first column we obtain

Dx= Py(Uz*Vl*- Ul*Vz*) + Pz(Ul*Vy*- Uy*Vl*)+ Pl(Uy*Vz*- Uz*Vy*)= Pyφzy*+ Pz*φly+ Plφ*yz.= Pyφxy+ Pzφxz+Plφxl,

Dx= Py(Uz*Vl*- Ul*Vz*) + Pz(Ul*Vy*- Uy*Vl*)

+ Pl(Uy*Vz*- Uz*Vy*)

= Pyφzy*+ Pz*φly+ Plφ*yz.

= Pyφxy+ Pzφxz+Plφxl,

which coincides with (Pφx) according to our definition.

Examples of this type of vectors will be found on page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf*, the magnetic-rest-force. The rest-ray Ω = iw[Φψ]*also belong to the same type (page 39). It is easy to show that

Ω = -i| w₁ w₂ w₃ w₄ || Φ₁ Φ₂ Φ₃ Φ₄ || ψ₁ ψ₂ ψ₃ ψ₄ |

Ω = -i| w₁ w₂ w₃ w₄ |

| Φ₁ Φ₂ Φ₃ Φ₄ |

| ψ₁ ψ₂ ψ₃ ψ₄ |

When (Ω₁, Ω₂, Ω₃) = 0, w₄ =i, Ω reduces to the three-dimensional vector

| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ || || ψ₁ ψ₂ ψ₃ |

| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |

| |

| ψ₁ ψ₂ ψ₃ |

Since in this case, Φ₁ = w₄ F₁₄ =en(the electric force)ψ₁ = -iw₄ f₂₃ =mx(the magnetic force)we have (Ω) = |exeyez||mxmymz|

Since in this case, Φ₁ = w₄ F₁₄ =en(the electric force)

ψ₁ = -iw₄ f₂₃ =mx(the magnetic force)

we have (Ω) = |exeyez|

|mxmymz|

[M. N. S.]

The four-vector φ = wF which is called by Minkowski the electric-rest-force (elektrische Ruh-Kraft) is very closely connected to Lorentz’s Ponderomotive force, or the force acting on a moving charge. If ρ is the density of charge, we have, when ε = 1, μ = 1,i.e., for free space

ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]= ρ₀/(√(1 - V²/c²)) [dx+ 1/c(v₂h₃-v₃h₂)]

ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]

= ρ₀/(√(1 - V²/c²)) [dx+ 1/c(v₂h₃-v₃h₂)]

Now since ρ₀ = ρ√(1 - V²/c²)

We have ρ₀φ₁ = ρ[dx+ 1/c(v₂h₃-v₃h₂)]

N. B.—We have put the components ofeequivalent to (dx,dy,dz), and the components ofmequivalent tohxhyhz), in accordance with the notation used in Lorentz’s Theory of Electrons.

We have therefore

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