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 assuming that f, α, F, u, etc.. are subject to the following conditions and the ones deduced by symmetry:

Regarding the integral J, it must be extended:

I° in relation to the volume element dτ = dx dy dz over the whole space;

2° in relation to time t, over the interval between the limits t = t0, t = t1.

According to the principle of least action, the integral J must be a minimum, if one sets the various quantities which appear in:

1° the conditions (2);

2° the condition that the state of the system is determined by both limiting times t = t0, t = t1.

This last condition allows us to transform our integral by partial integration with respect to time. If we have indeed an integral of the form

$$\int dt\ d\tau A\frac{dB\delta C}{dt},$$

where C is a quantity that defines the system state and its variation δC, it will be equal to (by partial integration with respect to time):

$$\int d\tau\mid AB\delta C\mid{t=t_{1}\atop t=t_{0}}-\int dt\ d\tau\frac{dA}{dt}dB\delta C.$$

Since the system state is determined by both limiting times, it is δC = 0 for t = t0, t = t1, so the first integral which is related to these two periods is zero, and the 2nd one remains.

We can also integrate by parts with respect to x, y or z, we have indeed

$$\int A\frac{dB}{dx}dx\ dy\ dz\ dt=\int AB\ dy\ dz\ dt-\int B\frac{dA}{dx}dx\ dy\ dz\ dt.$$

Our integrations are extended to infinity, it must be $$x=\pm\infty$$ in the first integral on the right-hand side; so, since we always assume that all our functions vanish at infinity, this integral will be zero and it follows

$$\int A\frac{dB}{dx}d\tau\ dt=-\int B\frac{dA}{dx}d\tau\ dt.$$

If the system is supposed to be subject to bindings, the binding conditions should be connected to the conditions imposed on the various quantities appearing in the integral J.

Let us first give to F, G, H the increasements δF, δG, δH; where:

$$\delta\alpha=\frac{d\delta H}{dy}-\frac{d\delta G}{dz}.$$

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 We should have

$$\delta J=\int dt\ d\tau\left[\sum\alpha\left(\frac{d\delta H}{dy}-\frac{d\delta G}{dz}\right)-\sum u\delta F\right]=0,$$

or, integrating by parts,

whence, by setting the arbitrary coefficient δF equal to zero,

This relationship gives us (by partial integration):

or

$$\int\sum Fud\tau=\int\sum\alpha^{2}d\tau,$$

hence finally:

Now, thanks to equation (3), δJ is independent from δF and thus δα; let us vary now the other variables

It follows, by returning to expression (1) of J,

$$\delta J=\int dt\ d\tau\left(\sum f\delta f-\sum F\delta u\right).$$

But f, g, h are first subject to conditions (2), so that

and for convenience we write:

The principles of variation calculus tells us that we must do the calculation as if ψ is an arbitrary function, as if δJ is represented by (6), and as if the changes were no longer subject to the condition (5).

We have in addition:

$$\delta u=\frac{d\delta f}{dt}+\delta\rho\xi,$$

whence, after partial integration,

If we assume at first that the electrons do not undergo a variation,

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 δρ = δρξ = 0 and the second integral is zero. Because δJ must vanish, we should have:

It remains in the general case:

It remains to determine the forces acting on the electrons. To do this we must suppose that a supplementary force -Xdτ, -Ydτ, -Zdτ applies to each element of an electron, and write that this force is in equilibrium with the forces of electromagnetic origin. Let U, V, W be components of the displacement of the element dτ of the electron, where the displacement is counted from an arbitrary initial position. Let δU, δV, δW be the variations of this displacement; the virtual work corresponding to the supplementary force is:

so that the equilibrium condition about which we have spoken can be written:

It's about the transformation of δJ. To begin the search for the continuity equation, we express how the charge of an electron is preserved by the variation.

Let x0, y0, z0 be the initial position of an electron. Its current position is:

We also introduce an auxiliary variable ε, which produces changes in our various functions, so that for any function A we have:

It is indeed convenient to switch from the notation of variation calculus to that of ordinary calculus, or vice versa.

Our functions should be regarded: 1° as dependent on five variables x, y, z, t, ε, so that we can remain at the same place when ε and t vary alone: we then indicate their derivatives by the ordinary d; 2° as dependent on five variables x0, y0, z0, t, ε so that we may always follow a single electron when t and ε vary alone, then we denote their derivatives by ∂. We will have then:

Denote now by Δ the functional determinant of x, y, z with respect to

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 x0, y0, z0:

If ε, x0, y0, z0 remain constant, we give to t an increasement ∂t; to x, y, z the increasements ∂x0, ∂y0, ∂z0 will result; and to Δ the increasement ∂Δ, and there will be:

hence

We deduce:

The mass of each electron is invariable, we have:

where:

These are the different forms of the continuity equation with respect of variable t. We find similar forms with respect to the variable ε. Either:

it follows:

Note the difference between the definition of $$\delta U=\tfrac{dU}{d\epsilon}\delta\epsilon$$ and that of $$\delta\rho=\tfrac{d\rho}{d\epsilon}\delta\epsilon$$, we note that it is this definition of δU that suits to formula (10).

This equation will allow us to transform the first term of (9); we find in fact:

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 or, by partial integration,

Let us propose now to determine

$$\delta(\rho\xi)=\frac{d(\rho\xi)}{d\epsilon}\delta\epsilon$$.

Note that ρΔ does not depend on x0, y0, z0; indeed, if we consider an electron whose initial position is a rectangular parallelepiped whose edges are dx0, dy0, dz0, the charge of this element is

$$\rho\Delta dx_{0\ }dy_{0\ }dz_{0\ }$$

and this charge should remain constant, then:

We deduce:

Now we know that for any function A, we have by the continuity equation,

$$\frac{1}{\Delta}\frac{\partial A\Delta}{\partial t}=\frac{dA}{dt}+\sum\frac{dA\xi}{dx}$$

and also

$$\frac{1}{\Delta}\frac{\partial A\Delta}{\partial\epsilon}=\frac{dA}{d\epsilon}+\sum\frac{dA\frac{\partial U}{\partial\epsilon}}{dx}$$

We thus have:

The right-hand sides of (17) and (17bis) must be equal, and if one remembers that

$$\frac{\partial U}{\partial t}=\xi,\quad\frac{\partial U}{\partial\epsilon}\delta\epsilon=\delta U,\quad\frac{d\rho\xi}{d\epsilon}\delta\epsilon=\delta\rho\xi$$

we get:

Transforming now the second term of (9); we get:

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 The right-hand side becomes by partial integration:

$$\int dt\ d\tau\left[-\sum\rho\delta U\frac{dF}{dt}-\sum\rho\eta\delta U\frac{dF}{dy}-\sum\rho\zeta\delta U\frac{dF}{dz}+\sum\rho\xi\delta V\frac{dF}{dy}+\sum\rho\xi\delta W\frac{dF}{dz}\right].$$

Now note, that:

$$\sum\rho\xi\delta V\frac{dF}{dy}=\sum\rho\zeta\delta U\frac{dH}{dx},\quad\sum\rho\xi\delta W\frac{dF}{dz}=\sum\rho\eta\delta U\frac{dG}{dx}$$

If, indeed, we develop Σ on the two sides of these relations, they become identities; and remember that

$$\frac{dH}{dx}-\frac{dF}{dz}=-\beta,\quad\frac{dG}{dx}-\frac{dF}{dy}=\gamma$$

the right-hand side in question will become:

$$\int dt\ d\tau\left[-\sum\rho\delta U\frac{dF}{dt}-\sum\rho\gamma\eta\delta U-\sum\rho\beta\zeta\delta U\right],$$

so that finally:

$$\delta J=\int dt\ d\tau\sum\rho\delta U\left(\frac{d\psi}{dx}+\frac{dF}{dt}+\beta\zeta-\gamma\eta\right)=\int dt\ d\tau\sum\rho\delta U\left(-f+\beta\zeta-\gamma\eta\right).$$

Equating the coefficient of δU on both sides of (10) we get:

$$X=f-\beta\zeta+\gamma\eta\,$$

This is equation (2) of the preceding §.

§ 3. — The transformation and the principle of least action
Let us see if the principle of least action gives us the reason for the success of the transformation. We must look at the transformation of the integral:

$$J=\int dt\ d\tau\left(\frac{\sum f^{2}}{2}-\frac{\sum\alpha^{2}}{2}\right)$$

(formula 4 of § 2).

We first find

$$dt^{\prime}\ d\tau^{\prime}=l^{4}dt\ d\tau$$

because x', y', z', t' are related to x, y, z, t by linear relations whose determinant is equal to l4; then we have:

{{MathForm1|(1)|$$\left\{ \begin{align} l^{4}\sum f^{\prime2} & =f^{2}+k^{2}(g^{2}+h^{2})+k^{2}\epsilon^{2}(\beta^{2}+\gamma^{2})+2k^{2}\epsilon(g\gamma-h\beta)\\ l^{4}\sum\alpha^{\prime2} & =\alpha^{2}+k^{2}(\beta^{2}+\gamma^{2})+k^{2}\epsilon^{2}(g^{2}+h^{2})+2k^{2}\epsilon(g\gamma-h\beta) \end{align}\right.$$}}

(formula 9 of § 1), hence:

$$l^{4}\left(\sum f^{\prime2}-\sum\alpha^{\prime2}\right)=\sum f^{2}-\sum\alpha^{2}$$

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 so that if we set

$$J^{\prime}=\int dt^{\prime}\ d\tau^{\prime}\left(\frac{\sum f^{\prime2}}{2}-\frac{\sum\alpha^{\prime2}}{2}\right),$$

we get:

J' = J.

However, to justify this equality, the integration limits have to be the same; so far we have assumed that t varies from t0 to t1, and x, y, z from ∞ to + ∞. On this account the integration limits would be affected by the transformation, but nothing prevents us from assuming t0 =- ∞, t1 = + ∞; with those conditions the limits are the same for J and J'.

We then compare the following two equations analogues to equation (10) of § 2:

{{MathForm1|(2)|$$\left\{ \begin{align} \delta J & =-\int\sum X\delta U\ d\tau\ dt\\ \delta J' & =-\int\sum X^{\prime}\delta U^{\prime}\ d\tau^{\prime}\ dt^{\prime}. \end{align}\right.$$}}

For this, we must first compare δU with δU.

Consider an electron whose initial coordinates are x0, y0, z0; its coordinates at the instant t are

$$x=x_{0}+U,\quad y=y_{0}+V,\quad z=z_{0}+W.$$

If one considers the electron after the corresponding transformation, it will have as coordinates

$$x^{\prime}=kl\left(x+\epsilon t\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz$$

where

$$x^{\prime}=x_{0}+U^{\prime},\quad y^{\prime}=y_{0}+V^{\prime},\quad z^{\prime}=z_{0}+W^{\prime}$$

but it will only attain these coordinates at the instant

$$t^{\prime}=kl\left(t+\epsilon x\right).$$

If we subject our variables to the variations δU, δV, δW, and when we give at the same time t an increasement δt, the coordinates x, y, z will experience a total increasement

$$\delta x=\delta U+\xi\delta t,\quad\delta y=\delta V+\eta\delta t,\quad\delta z=\delta W+\zeta\delta t.$$

We will also have:

$$\delta x^{\prime}=\delta U^{\prime}+\xi^{\prime}\delta t^{\prime},\quad\delta y^{\prime}=\delta V^{\prime}+\eta^{\prime}\delta t^{\prime},\quad\delta z^{\prime}=\delta W^{\prime}+\zeta^{\prime}\delta t^{\prime},$$

and in virtue of the transformation:

$$\delta x^{\prime}=kl\left(\delta x+\epsilon\delta t\right),\quad\delta y^{\prime}=l\delta y,\quad\delta z^{\prime}=l\delta z,\quad\delta t^{\prime}=kl\left(\delta t+\epsilon\delta x\right).$$

hence, assuming δt = 0, the relations:

{{MathForm1||$$\left\{ \begin{align} \delta x^{\prime} & =\delta U^{\prime}+\xi^{\prime}\delta t^{\prime} & =kl\delta U,\\ \delta y^{\prime} & =\delta V^{\prime}+\eta^{\prime}\delta t^{\prime} & =l\delta V,\\ \delta t^{\prime} & =kl\epsilon\delta U. \end{align}\right.$$}}

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 Note that

It follows, by replacing δt' by its value

If we recall the definition of k, we draw from this:

and also

hence

Now, in virtue of equations (2) we must have:

By replacing ΣXδU by its value (3) and by identifying, it follows:

These are the equations (11) of § 1. The principle of least action leads us to the same result as the analysis of § 1.

If we turn to formulas (1), we see that Σf² - Σα² is not affected by the transformation, except one constant factor; it is not the case with expression Σf² + Σα² which represents the energy. If we confine ourselves to the case where ε is sufficiently small, so that the square can be neglected so that k = 1, and if we also assume l = 1, we find:

or by addition

§ 4. — The group
It is important to note that the transformations form a group.

Indeed, if we set:

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 and in addition

$$x^{\prime\prime}=k^{\prime}l^{\prime}(x^{\prime}+\epsilon^{\prime}t^{\prime}),\quad y^{\prime\prime}=l^{\prime}y^{\prime},\quad z^{\prime\prime}=l^{\prime}z^{\prime},\quad t^{\prime\prime}=k^{\prime}l^{\prime}(t^{\prime}+\epsilon^{\prime}x^{\prime}),$$

with

$$k^{-2}=1-\epsilon^{2},\quad k^{\prime-2}=1-\epsilon^{\prime2}$$

it follows:

$$x^{\prime\prime}=k^{\prime\prime}l^{\prime\prime}(x+\epsilon^{\prime\prime}t),\quad y^{\prime\prime}=l^{\prime\prime}y,\quad z^{\prime\prime}=l^{\prime\prime}z,\quad t^{\prime\prime}=k^{\prime\prime}l^{\prime\prime}(t+\epsilon^{\prime\prime}x),$$

with

$$\epsilon^{\prime\prime}=\frac{\epsilon+\epsilon^{\prime}}{1+\epsilon\epsilon^{\prime}},\quad l^{\prime\prime}=ll^{\prime},\quad k^{\prime\prime}=kk^{\prime}(1+\epsilon\epsilon^{\prime})=\frac{1}{\sqrt{1-\epsilon^{\prime\prime2}}}.$$

If we take for l the value 1, and we suppose ε infinitely small,

$$x^{\prime}=x+\delta x,\quad y^{\prime}=y+\delta y,\quad z^{\prime}=z+\delta z,\quad t^{\prime}=t+\delta t$$

it follows:

$$\delta x=\epsilon t,\quad\delta y=\delta z=0,\quad\delta t=\epsilon x.$$

This is the infinitesimal generator of the transformation group, which I call the transformation T1, and which can be written in 's notation:

$$t\frac{d\varphi}{dx}+x\frac{d\varphi}{dt}=T_{1}.$$

If we assume ε = 0 and l = 1 + δl, we find instead

$$\delta x=x\delta l,\quad\delta y=y\delta l,\quad\delta z=z\delta l,\quad\delta t=t\delta l$$

and we would have another infinitesimal transformation t0 of the group (assuming that l and ε are regarded as independent variables) and we would have with 's notation:

$$T_{0}=x\frac{d\varphi}{dx}+y\frac{d\varphi}{dy}+z\frac{d\varphi}{dz}+t\frac{d\varphi}{dt}.$$

But we could give the y- or z-axes the special role, which we gave the x-axis; thus we have two further infinitesimal transformations:

which also would not alter the equations of.

We can form combinations devised by, such as

$$\left[T_{1},T_{2}\right]=x\frac{d\varphi}{dy}-y\frac{d\varphi}{dx},$$

but it is easy to see that this transformation is equivalent to a coordinate change, the axes are rotating a very small angle around the z-axis. We

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 should not be surprised if such a change does not alter the form of the equations of, obviously independent of the choice of axes.

We are thus led to consider a continuous group which we call the  group and which admit as infinitesimal transformations:

1° the transformation T0 which is permutable with all others;

2° the three transformations T1, T2, T3;

3° the three rotations [T1, T2], [T2, T3], [T3, T1].

Any transformation of this group can always be decomposed into a transformation of the form:

$$x^{\prime}=lx,\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=lt$$

and a linear transformation which does not change the quadratic form

$$x^{2}+y^{2}+z^{2}-t^{2}\,$$

We can still generate our group in another way. Any transformation of the group may be regarded as a transformation of the form:

preceded and followed by a suitable rotation.

But for our purposes, we should consider only a part of the transformations of this group; we must assume that l is a function of ε, and it is a question of choosing this function in such a way that this part of the group that I call P still forms a group.

Let's rotate the system 180° around the y-axis, we should find a transformation that will still belong to P. But this amounts to a sign change of x, x', z and z'; we find:

So l does not change when we change ε into -ε.

On the other hand, if P is a group, then the inverse substitution of (1)

must also belong to P; it will therefore be identical with (2), that is to say that

$$l=\frac{1}{l}.$$

We must therefore have l = 1.

§ 5. — waves
has put the formulas that define the electromagnetic field produced by the motion of a single electron in a particularly elegant form.

Let us remember the equations

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 We know we can integrate by the retarded potentials and we have:

In these formulas we have:

$$d\tau_{1}=dx_{1}dy_{1}dz_{1},\quad r^{2}=(x-x_{1})^{2}+(y-y_{1})^{2}+(z-z_{1})^{2},$$

whereas ρ1 and ξ1 are the values of ρ and ξ at the point x1, y1, z1 and the instant

$$t_{1}=t-r\,$$

x0, y0, z0 being coordinates of a molecule of the electron at the instant t;

$$x_{1}=x_{0}+U,\ y_{1}=y_{0}+V,\ z_{1}=z_{0}+W$$

being its coordinates at the instant t1;

U, V, W are functions of x0, y0, z0, so that we can write:

$$dx_{1}=dx_{0}+\frac{dU}{dx_{0}}dx_{0}+\frac{dU}{dy_{0}}dy_{0}+\frac{dU}{dz_{0}}dz_{0}+\xi_{1}dt_{1};$$

and if we assume t to be constant, as well as x, y and z:

$$dt_{1}=+\sum\frac{x-x_{1}}{r}dx_{1}.$$

We can therefore write:

$$dx_{1}\left(1+\xi_{1}\frac{x_{1}-x}{r}\right)+dy_{1}\xi_{1}\frac{y_{1}-y}{r}+dz_{1}\xi_{1}\frac{z_{1}-z}{r}=dx_{0}\left(1+\frac{dU}{dx_{0}}\right)+dy_{0}\frac{dU}{dy_{0}}+dz_{0}\frac{dU}{dz_{0}}$$

so that the other two equations can deduced by circular permutation.

We therefore have:

we set

$$d\tau_{0}=dx_{0}dy_{0}dz_{0}\,$$

Consider the determinants that appear in both sides of (3) and at the begin of the first part; if we seek to develop, we see that the terms of the 2d and 3rd degree from ξ1, η1, ζ1 disappear and that the determinant is equal to

$$1+\xi_{1}\frac{x_{1}-x}{r}+\eta_{1}\frac{y_{1}-y}{r}+\zeta_{1}\frac{z_{1}-z}{r}=1+\omega,$$

ω designates the radial component of the velocity ξ1, η1, ζ1, that is to say, the component directed along the radius vector indicating from point x, y, t to point x1, y1, z1.

In order to obtain the second determinant, I look at the coordinates of different molecules of the electron at instant t', which is the same for all molecules, but in such a way that for the molecule considered we have $$t_{1}=t'_{1}$$. The coordinates of a molecule will then be:

$$x'_{1}=x_{0}+U',\ y'_{1}=y_{0}+V',\ z'_{1}=z_{0}+W'$$

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 U', V', W' is what become of U, V, W, when we replace t1 by t'1; since t'1 is the same for all molecules, we have:

$$dx_{1}^{\prime}=dx_{0}\left(1+\frac{dU^{\prime}}{dx_{0}}\right)+dy_{0}\frac{dU^{\prime}}{dy_{0}}+dz_{0}\frac{dU^{\prime}}{dz_{0}}$$

and therefore

$$d\tau_{1}^{\prime}=d\tau_{0}\mid1+\frac{dU^{\prime}}{dx_{0}},\quad\frac{dU^{\prime}}{dy_{0}},\quad\frac{dU^{\prime}}{dz_{0}}\mid$$

by setting

$$d\tau_{1}^{\prime}=dx_{1}^{\prime}dy_{1}^{\prime}dz_{1}^{\prime}.$$

But the element of electric charge is

$$d\mu_{1}=\rho_{1}d\tau_{1}^{\prime}$$

and moreover for the molecule considered, we have t1 = t'1, and therefore $$\tfrac{dU^{\prime}}{dx_{0}}=\tfrac{dU}{dx_{0}}$$ etc..; we can write:

$$d\mu_{1}=\rho_{1}d\tau_{0}\mid1+\frac{dU}{dx_{0}},\quad\frac{dU}{dy_{0}},\quad\frac{dU}{dz_{0}}\mid$$

so that equation (3) becomes:

$$\rho_{1}d\tau_{1}(1+\omega)=d\mu_{1}\,$$

and equations (2):

$$\psi=\frac{1}{4\pi}\int\frac{d\mu_{1}}{r(1+\omega)},\quad F=\frac{1}{4\pi}\int\frac{\xi_{1}d\mu_{1}}{r(1+\omega)}.$$

If we are dealing with a single electron, our integrals are reduced to a single element, provided we consider only the points x, y, x which are sufficiently remote so that r and ω have substantially the same value for all points of the electron. The potentials ψ, F, G, H depend on the position of the electron and also its velocity, because not only ξ1, η1, ζ1 show up in the numerator of F, G, H, but the radial component ω shows up in the denominator. It is of course its position and its velocity at the instant t1.

The partial derivatives of φ, F, G, H with respect to t, x, y, z (and therefore the electric and magnetic fields) will also depend on its acceleration. Moreover, they depend linearly, since the acceleration in these derivatives is introduced as a result of a single differentiation.

was thus led to distinguish the electric and magnetic field terms which do not depend on the acceleration (this is what he calls the velocity wave) and those that are proportional to acceleration (that is what he calls the acceleration wave).

The calculation of these two waves is facilitated by the transformation. Indeed, we can apply this transformation to the system, so that the velocity of the single electron under consideration becomes zero. We will use for the x-axis the direction of the velocity before the transformation, so that, at the instant t1,

$$\eta_{1}=\zeta_{1}=0\,$$

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 and we will take ε = -ξ, so that

$\xi'_{1}=\eta'_{1}=\zeta'_{1}=0\,$

We can therefore reduce the computation of the two waves to the case where the electron velocity is zero. Let's start with the velocity wave, we first note that this wave is the same as if the electron motion was uniform.

If the electron velocity is zero, then:

$$\omega=0,\quad F=G=H=0,\quad\psi=\frac{\mu_{1}}{4\pi r},$$

μ1 is the electrical charge of the electron. The speed was reduced to zero by the transformation, we have now:

$$F^{\prime}=G^{\prime}=H^{\prime}=0,\quad\psi^{\prime}=\frac{\mu_{1}}{4\pi r^{\prime}},$$

r' is the distance from point x', y', z' at point x'1, y'1, z'1, and therefore:

Now let us carry out the reverse transformation to find the true field corresponding to the velocity ε, 0, 0. We find, with reference to equations (9) and (3) of § 1:

We see that the magnetic field is perpendicular to the x-axis (direction of velocity) and the electric field, and the electric field is directed to the point:

If the electron continues to move in a rectilinear and uniform way with the velocity it had at the instant t1, that is to say, with the velocity -ε, 0, 0, the point (5) would be the one occupied at the instant t.

Taking the acceleration wave, we can, through the transformation, reduce its determination to the case of zero velocity. This is the case if we imagine an electron whose oscillation amplitude is very small, but very fast, so that the displacements and velocities are much smaller, but the accelerations are finished. We thus come back to the field that has been studied in the famous work by entitled Die Kräfte elektrischer Schwingungen nach der Maxwell'schen Theorie, and that for a point at great distance. In these conditions:

I° Both electric and magnetic fields are equal.

2° They are perpendicular to each other.

3° They are perpendicular to the normal of the spherical wave, that is to say to the sphere whose center is the point x1, y1, z1.

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 I say that these three properties will remain, even when the velocity is not zero, and for this it is enough to show that they are not altered by the transformation.

Indeed, let A be the intensity common to both fields, let

These properties expressed through the equalities

which means again that

are the direction cosines of three rectangular directions, and we deduce the relations:

or

with the equations that we can deduce by symmetry.

If we take the equations (3) of § 1, we find:

We found above in § 3:

So

On the other hand, from equations (9) of § 1, we get:

This shows that

I say now that

Indeed, by virtue of equations (7) (and equations 9, § 1) the first parts of equations (8) are written respectively:

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 They then vanish in virtue of equations $$\sum f\left(x-x_{1}\right)=\sum\alpha\left(x-x_{1}\right)=0$$ and in virtue of equations (6). Yet this is precisely what was demonstrated.

We can also achieve the same result by considerations of homogeneity.

Indeed, ψ, F, G, H are functions of $$x-x_{1},\ y-y_{1},\ z-z_{1},\ \xi_{1}=\frac{dx_{1}}{dt_{1}},\ \eta_{1}=\frac{dy_{1}}{dt_{1}},\ \zeta_{1}=\frac{dz_{1}}{dt_{1}}$$ being homogeneous of degree -1 with respect to x, y, z, t, x1, y1, z1, t1 and their differentials.

So the derivatives of ψ, F, G, H with respect to x, y, z, t (and hence also the two fields f, g, h; α, β, γ) will be homogeneous of degree -2 with respect to the same quantities, if we remember also that the relation

$$t-t_{1}=r=\sqrt{\sum\left(x-x_{1}\right)^{2}}$$

is homogeneous with respect to these quantities.

But these derivatives depend on these fields of x - x1, the velocities $$\frac{dx_{1}}{dt_{1}}$$, and the accelerations $$\frac{d^{2}x_{1}}{dt_{1}^{2}}$$; they consist of a term independent of accelerations (velocity wave) and a term linear in respect to accelerations (acceleration waves). But $$\frac{dx_{1}}{dt_{1}}$$ is homogeneous of degree 0 and $$\frac{d^{2}x_{1}}{dt_{1}^{2}}$$ is homogeneous of degree -1; hence it follows that the velocity wave is homogeneous of degree -2 with respect to x - x1, y - y1, z - z1, and the acceleration wave is homogeneous of degree -1. So in a very distant point an acceleration wave is predominant and can therefore be regarded as being assimilated to the total wave. In addition, the law of homogeneity shows that the acceleration wave is similar to itself at a distance and at any point. It is therefore, at any point, similar to the total wave at a remote point. But in a distant point the disturbance can propagate as plane waves, so that the two fields should be equal, mutually perpendicular and perpendicular to the direction of propagation.

I shall refer for more details to a work by in the Journal de Physique (Year 1905).

§ 6. — Contraction of electrons
Suppose a single electron in rectilinear and uniform motion. From what we have seen, we can, through the transformation, reduce the study of the field determined by the electron to the case where the electron is motionless; the  transformation replaces the real electron in motion by an ideal electron without motion.

Let α, β, γ, f, g, h be the real field; let α', β', γ', f', g', h'  be the

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 field after the transformation, so the ideal field α', f' corresponds to the case where the electron is motionless; we have:

and the actual field (in virtue of the formulas 9 of § 1):

We now determine the total energy due to the motion of the electron, the corresponding action, and the electromagnetic momentum, in order to calculate the electromagnetic mass of the electron. For a distant point, it suffices to consider the electron as reduced to a single point; we are thus brought back to the formulas (4) of the preceding § which generally can be appropriate. But here they do not suffice, because the energy is mainly located in the ether parts nearest to the electron.

On this subject we can make several hypotheses.

According to that of, the electrons are spherical and not deformable.

So when we apply the transformation when the real electron is spherical, the electron becomes a perfect ellipsoid. The equation of this ellipsoid is based on § 1:

But here we have:

so that the equation of the ellipsoid becomes:

If the radius of the real electron is r, the axes of the ideal electron would therefore be:

In 's hypothesis, however, the moving electrons are deformed, so that the real electron would become a ellipsoid, while the ideal electron is still always a perfect sphere of radius r; the axes of the real electron will then be:

We denote by

the longitudinal electric energy; by

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 the transverse electric energy; by

$$C=\frac{1}{2}\int\left(\beta^{2}+\gamma^{2}\right)d\tau$$

the transverse magnetic energy. There is no longitudinal magnetic energy, since α = α' = 0. We denote by A', B', C' the corresponding quantities in the ideal system. We first find:

$$C'=0,\ C=\epsilon^{2}B$$

In addition, we can observe that the actual field depends only on x = εt, y, and x, and write:

$$d\tau=d(x+\epsilon t)dy\ dz$$

$$d\tau'=dy'dy'dz'=kl^{3}d\tau\,$$

hence

$$A^{\prime}=kl^{-1}A,\quad B^{\prime}=k^{-1}l^{-1}B,\quad A=\frac{lA^{\prime}}{k},\quad B=klB^{\prime}.$$

In 's hypothesis we have B' = 2A', and A ' (being inversely proportional to the radius of the electron) is a constant independent of the velocity of the real electron; we get for the total energy:

$$A+B+C=A'lk\left(3+\epsilon^{2}\right)$$

and for the action (per unit time):

$$A+B-C=\frac{3A^{\prime}l}{k}.$$

Now calculate the electromagnetic momentum; we find:

$$D=\int\left(g\gamma-h\beta\right)d\tau=-\epsilon\int\left(g^{2}+h^{2}\right)d\tau=-2\epsilon B=-4\epsilon klA^{\prime}.$$

But there must be some relation between the energy E = A + B + C, the action per unit time H = A + B - C, and the momentum D. The first of these relations is:

$$E=H-\epsilon\frac{dH}{d\epsilon},$$

the second is

$$\frac{dD}{d\epsilon}=-\frac{1}{\epsilon}\frac{dE}{d\epsilon};$$

hence

The second of equations (2) is always satisfied; but the first is so only if

$$l=\left(1-\epsilon^{2}\right)^{\frac{1}{6}}=k^{-\frac{1}{3}},$$

that is to say if the volume of the ideal electron is equal to that of the real electron; or if the volume of the electron is constant; that's the hypothesis of.

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 This is in contradiction with the results of § 4 and with the result obtained by by another way. That is the contradiction which is to be explained.

Before addressing this explanation, I note that whatever is the hypotheses we have adopted

$$H=A+B-C=\frac{l}{k}(A^{\prime}+B^{\prime}),$$

or, because of C' = 0,

We can compare the result of the equation J = J' obtained in § 3.

We have indeed:

$$J=\int H\ dt,\quad J^{\prime}=\int H^{\prime}\ dt^{\prime}.$$

We observe that the state of the system depends only on x + εt, y and z, that is to say on x', y', z', and we have:

$$t^{\prime}=\frac{l}{k}t+\epsilon x^{\prime}$$

By comparing equations (3) and (4) we find J = J'.

Let us consider any hypothesis, which may be either that of, or that of , or that of , or an intermediate hypothesis.

Let

$$r,\ \theta r,\ \theta r$$

the three axes of the real electron; that of the ideal electron will be:

$$klr,\ \theta lr,\ \theta lr$$

Then A' + B' is the electrostatic energy of an ellipsoid with axes klr, θlr, θlr.

Let us suppose that the electricity is spread on the surface of the electron as it is known from an inductor, or uniformly distributed within the electron; than this energy will be of the form:

$$A^{\prime}+B^{\prime}=\frac{\varphi\left(\frac{\theta}{k}\right)}{klr},$$

where φ is a known function.

The hypothesis of is to assume:

$$r = const.,\ \theta = 1.$$

That of :

$$l = 1,\ kr = const.,\ \theta = k.$$

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 That of :

$$l=k^{-\frac{1}{3}},\quad k=\theta,\quad klr=const..$$

We then find:

$$H=\frac{\varphi\left(\frac{\theta}{k}\right)}{k^{2}r}.$$

found, in different notation (Göttinger Nachrichten, 1902, p. 37)

$$H=\frac{a}{r}\frac{1-\epsilon^{2}}{\epsilon}\log\frac{1+\epsilon}{1-\epsilon},$$

a is a constant. However, in the hypothesis of, we have θ = 1; then:

which defines the function φ.

This granted, imagine that the electron is subject to a binding, so there is a relation between r and φ; in the hypothesis of this relation would be φr = const., in that of  φ²r² = const. We assume in a more general way

$$r=b\theta^{m}\,$$

b is a constant; hence:

$$H=\frac{1}{bk^{2}}\theta^{-m}\varphi\left(\frac{\theta}{k}\right).$$

What is the shape of the electron when the velocity become -εt, if we do not suppose the involvement of forces other than the binding forces? Its form will be defined by the equality:

or

$$-m\theta^{-m-1}\varphi+\theta^{-m}k^{-1}\varphi^{\prime}=0$$

or

$$\frac{\varphi^{\prime}}{\varphi}=\frac{mk}{\theta}.$$

If we want equilibrium to occur so that θ = k, it is necessary that $$\tfrac{\theta}{k}=1$$, the logarithmic derivative of φ is equal to m.

If we develop $$\frac{1}{k}$$ and the right-hand side of (5) in powers of ε, equation (5) becomes:

$$\varphi\left(1-\frac{\epsilon^{2}}{2}\right)=a\left(1-\frac{\epsilon^{2}}{3}\right)$$

neglecting higher powers of ε.

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 By differentiating, we get:

$$-\epsilon\varphi^{\prime}\left(1-\frac{\epsilon^{2}}{2}\right)=\frac{2}{3}\epsilon a.$$

For ε = 0, that is to say when the argument of φ is equal to 1, these equations become:

We must therefore have $$m=-\tfrac{2}{3}$$ in conformity with the hypothesis of.

This result should come nearer to that which is connected to the first equation (a), and from which actually it does not differ. Indeed, suppose that every element dτ of the electron is subjected to a force Xdτ parallel to the x-axis, X is the same for all elements; we will then have, in conformity with the definition of momentum:

$$\frac{dD}{dt}=\int Xd\tau.$$

In addition, the principle of least action gives us:

$$\delta J=\int X\delta U\ d\tau\ dt,\quad J=\int H\ dt,\quad\delta J=\int D\delta U\ dt,$$

δU is the displacement of the center of gravity of the electron; H depends on θ and on ε if we assume that r is related to θ by the equation of binding; we have thus:

$$\delta J=\int\left(\frac{\partial H}{\partial\epsilon}\delta\epsilon+\frac{\partial H}{\partial\theta}\right)dt.$$

In addition $$\delta\epsilon=-\tfrac{d\delta U}{dt}$$; where, by integrating by parts:

$$\int D\delta\epsilon\ dt=\int D\delta U\ dt$$

or

$$\int\left(\frac{\partial H}{\partial\epsilon}\delta\epsilon+\frac{\partial H}{\partial\theta}\delta\theta\right)dt=\int D\delta\epsilon\ dt,$$

hence

$$D=\frac{\partial H}{\partial\epsilon},\quad\frac{\partial H}{\partial\theta}=0.$$

But the derivative $$\tfrac{dH}{d\epsilon}$$, contained in the right-hand side of equation (2), is the derivative taken by supposing θ as a function of ε, so that

$$\frac{dH}{d\epsilon}=\frac{\partial H}{\partial\epsilon}+\frac{\partial H}{\partial\theta}\frac{d\theta}{d\epsilon}.$$

Equation (2) is therefore equivalent to equation (6).

The conclusion is that if the electron is subject to a binding between its three axes,

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 and if no other force intervenes except the binding forces, the shape of that electron, when it is given a uniform velocity, may be such that the ideal electron corresponds to a sphere, except the case where the binding is such that the volume is constant, in conformity with the hypothesis of.

We are led in this way to pose the following problem: what additional forces, other than the binding forces, are necessary to intervene to account for the law of or, more generally, any law other than that of ?

The simplest hypothesis, and the first that we should consider, is that these additional forces are derived from a special potential depending on the three axes of the ellipsoid, and therefore on θ and on r; let F(θ, r) be the potential; in which case the action will be expressed:

$$J=\int\left[H+F(\theta,r)\right]dt$$

and the equilibrium conditions are written:

If we assume r and θ are connected by r = bθm, we can look at r as a function of θ, consider F as depending only on θ, and retain only the first equation (8) with:

$$H=\frac{\varphi}{bk^{2}\theta^{m}},\quad\frac{dH}{d\theta}=\frac{-m\theta}{bk^{2}\theta^{m+1}}+\frac{\varphi^{\prime}}{bk^{3}\theta^{m}}$$

For k = θ we need equation (8) to be satisfied; which gives, taking into account equations (7):

$$\frac{dF}{d\theta}=\frac{ma}{b\theta^{m+3}}+\frac{2}{3}\frac{a}{b\theta^{m+3}}$$

where:

$$F=\frac{-a}{b\theta^{m+2}}\frac{m+\frac{2}{3}}{m+2}$$

and in the hypothesis of, where m = -1:

$$F=\frac{a}{3b\theta}.$$

Now suppose that there is no connection and, considering r and θ as independent variables, retain the two equations (H); it follows:

$$H=\frac{\varphi}{k^{2}r},\quad\frac{dH}{d\theta}=\frac{\varphi^{\prime}}{k^{3}r},\quad\frac{dH}{dr}=\frac{-\varphi}{k^{2}r^{2}},$$

Equations (8) must be satisfied for k = θ, r = bθm; which gives:

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 One way to satisfy these requirements is to pose:

A, α, β are constants, the equations (9) must be satisfied for k = θ, r = bθm, which gives:

$$A\alpha b^{\alpha-1}\theta^{m\alpha-m+\beta}=\frac{a}{b^{2}\theta^{2m+2}},\quad A\beta b^{\alpha}\theta^{m\alpha+\beta-1}=\frac{2}{3}\frac{a}{b\theta^{m+3}}.$$

By identifying we find

But the volume of the ellipsoid is proportional to r³θ², so that the additional potential is proportional to the power γ of the volume of the electron.

In the hypothesis of, we have m = 1, γ = 1.

We thus come back to the hypothesis of, under the condition of adding an additional potential proportional to the volume of the electron.

The hypothesis of corresponds to γ = ∞.

§ 7. — Quasi-stationary motion
It remains to see if this hypothesis of the contraction of electrons reflects the inability to demonstrate absolute motion, and I will begin by studying the quasi-stationary motion of an isolated electron, or which is subject only to the action of other distant electrons.

It is known that what is called quasi-stationary motion is the motion where the velocity changes are slow enough so that the electric and magnetic energy due to motion of the electron differ little from what they would be in uniform motion; we know also that derived the transverse and longitudinal electromagnetic masses from the notion of quasi-stationary motion.

I think I should clarify. Let H be our action per unit time:

$$H=\frac{1}{2}\int\left(\sum f^{2}-\sum\alpha^{2}\right)d\tau,$$

where we consider for the moment only the electric and magnetic fields due to the motion of an electron. In the preceding §, by considering the motion as uniform, we regarded H as dependent from the velocity ξ, η, ζ of the electrons' center of gravity (the three components in the preceding §, had as values -ε, 0, 0) and the parameters r and θ that define the shape of the electron.

But if the motion is more uniform, H depend not only on the values of ξ, η, ζ, r, θ at the instant in question, but on values of these quantities at other instants which may differ in quantities of the same order as the time

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 by light to travel from one point to another of the electron; in other words, H depend not only on ξ, η, ζ, r, θ, but on their derivatives of all orders with respect to time.

Well, the motion is said to be quasi-stationary when the partial derivatives of H with respect to the successive derivatives of ξ, η, ζ, r, θ are negligible compared to the partial derivatives of H with respect to the quantities ξ, η, ζ, r, θ themselves.

The equations of such a motion can be written:

In these equations, F has the same meaning as in the preceding §, X, Y, Z are the components of the force acting on the electron: this force is solely due to electric and magnetic fields produced by other electrons.

Note that H is independent of ξ η ζ through the combination

that is to say, the magnitude of the velocity; therefore we still call D the momentum:

where:

with

If we take the current direction of the velocity as the x-axis, we get:

equations (2) and (2bis) become:

and the last three equations (1):

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 This is why gave $$\tfrac{dD}{dV}$$ the name longitudinal mass and $$\tfrac{D}{V}$$ the name transverse mass; recall that $$D=\tfrac{dH}{dV}.$$

In the hypothesis of, we have:

$$\tfrac{\partial H}{\partial V}$$ represent the derivative with respect to V, after r and θ were replaced by their values as functions of V from the first two equations (1); we will also have, after the substitution,

We choose units so that the constant factor A is equal to 1, and I pose $$\sqrt{1-V^{2}}=h$$, hence:

We will pose again:

and we find the equation for quasi-stationary motion:

Let's see what happens to these equations by the transformation. We will pose: $$1+\xi\epsilon=\mu$$, and we have first:

from which we derive easily

We also have

where:

where again:

and

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Let us return now to equations (11bis) of § 1; we can regard X1, Y1, Z1 as having the same meaning as in equations (5). On the other hand, we have l = 1 and $$\frac{\rho^{\prime}}{\rho}=k\mu$$; these equations then become:

We calculate ΣX1ξ using equation (5), we find:

$$\Sigma X_{1}\xi=h^{-3}M,$$

where:

Comparing equations (5) (6), (7) and (9), we finally find:

This shows that the equations of quasi-stationary motion are not altered by the transformation, but it still does not prove that the hypothesis of  is the only one that leads to this result.

To establish this point, we will restrict ourselves, as did, to certain particular cases; it will be obviously sufficient for us to show a negative proposal.

How do we first extend the hypotheses underlying the above calculation?

1° Instead of assuming l = 1 in the transformation, we assume any l.

2° Instead of assuming that F is proportional to the volume, and hence that H is proportional to h, we assume that F is any function of θ and r, so that [after replacing θ and r with their values as functions of V, from the first two equations (1)] H is any function of V.

I note first that, assuming H = h, we must have l = 1; and in fact the equations (6) and (7) remain, except that the right-hand sides will be multiplied by $$\tfrac{1}{l}$$; so do equations (9), except that the right-hand sides will be multiplied by $$\tfrac{1}{l^{2}}$$; and finally the equations (10), except that the right-hand sides will be multiplied by $$\tfrac{1}{l}$$. If we want that the equations of motion are not altered by the  transformation

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 that is to say that the equations (10) only differ from equations (5) by the accentuation of the letters, it must be assumed:

l = 1.

Suppose now that we have η = ζ = 0, where ξ = V, $$\frac{d\xi}{dt}=\frac{dV}{dt}$$; the equations (5) take the form:

We can also pose:

$$\frac{dD}{dV}=f(V)=f(\xi),\quad\frac{D}{V}=\varphi(V)=\varphi(\xi).$$

If the equations of motion are not altered by the transformation, we must have:

and therefore:

But we have:

$$\frac{d\xi^{\prime}}{dt^{\prime}}=\frac{d\xi}{dt}\frac{1}{k^{3}\mu^{3}},\quad\frac{d\eta^{\prime}}{dt^{\prime}}=\frac{d\eta}{dt}\frac{1}{k^{2}\mu^{2}},$$

where:

whence, by eliminating l², we find the functional equation:

$$k^{2}\mu^{2}\frac{\varphi\left(\frac{\xi+\epsilon}{1+\xi\epsilon}\right)}{\varphi(\xi)}=\frac{f\left(\frac{\xi+\epsilon}{1+\xi\epsilon}\right)}{f(\xi)},$$

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 or by posing

that is:

an equation that must be satisfied for all values of ξ and ε. For ζ = 0 we find:

where:

A is a constant, and I set $$\Omega(0)=\frac{1}{m}$$.

We then find:

Now $$\varphi(\xi^{\prime})=\varphi(\xi)\frac{k\mu}{l^{2}}$$; so we have:

As l should depend only on ε (since, if there are more electrons, l must be the same for all electrons whose velocities ξ may be different), this identity can take place only if we have:

Thus 's hypothesis is the only one consistent with the inability to demonstrate absolute motion; if we accept this impossibility, we must admit that the moving electrons contract and become ellipsoids of revolution where two axes remain constant; it must be admitted, as we have shown in the previous §, the existence of an additional potential which is proportional to the volume of the electron.

The analysis of is therefore fully confirmed, but we can better give us an account of the true reason of the fact which occupies us; and this reason must be sought in the considerations of § 4. The transformations that do not alter the equations of motion must form a group, and this can take place only if l = 1. As we do not recognize if an electron is at rest or in absolute motion, it is necessary that, when in motion, it undergoes a distortion that must be precisely that which imposes the corresponding transformation of the group.

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§ 8. — Arbitrary motion
The above results apply only to quasi-stationary motion, but it is easy to extend them to the general case; it suffices to apply the principles of § 3, that is to say, the principle of least action.

For the expression of the action

$$J=\int dt\ d\tau\left(\frac{\sum f^{2}}{2}-\frac{\sum\alpha^{2}}{2}\right),$$

it is convenient to add a term representing the additional potential F of § 6; this term will obviously have the form:

$$J_{1}=\int\sum(F)dt$$

where Σ(F) represents the sum of the additional potential due to the different electrons, each of which is proportional to the volume of the corresponding electron.

I write (F) in brackets to avoid confusion with the vector F, G, H.

The total action is then J + J1. We saw in § 3 that J is not altered by the transformation, we must show now that it is the same for J1.

We have for one electron,

$$(F)=\omega_{0}\tau\,$$

ω0 being a special coefficient of the electron and τ its volume; so I can write:

$$\sum(F)=\int\omega_{0}d\tau,$$

the integral has to be extended to the entire space, but so that the coefficient ω0 is zero outside the electrons, and that within each electron it is equal to the special coefficient of that electron. Then we have:

$$J_{1}=\int\omega_{0}d\tau\ dt$$

and after the transformation:

$$J_{1}^{\prime}=\int\omega_{0}^{\prime}d\tau^{\prime}\ dt^{\prime}.$$

Now we have ω0 = ω'0; for if a point belong to an electron, the corresponding point after the transformation still belongs to the same electron. On the other hand, we found in § 3;

$$d\tau'dt'=l^{4}d\tau\ dt$$

and since we now assume l = 1

$$d\tau'dt'=d\tau\ dt$$

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 We have therefore

$$J_{1}=J'_{1}\,$$. C.Q.F.D.

The theorem is thus general, it gives us at the same time a solution of the question we posed at the end of § 1: finding the complementary forces which are unaltered by the transformation. The additional potential (F) satisfies this condition.

So we can generalize the result announced at the end of § 1 and write:

If the inertia of electrons is exclusively of electromagnetic origin, if they are only subject to forces of electromagnetic origin, or to forces generated by the additional potential (F), no experiment can demonstrate absolute motion.

So what are these forces that create the potential (F)? They can obviously be compared to a pressure which would reign inside the electron; all occurs as if each electron were a hollow capacity subjected to a constant internal pressure (volume independent); the work of this pressure would be obviously proportional to the volume changes.

In any case, I must observe that this pressure is negative. Remember the equation (10) of § 6, according to 's hypothesis we write:

$$F=Ar^{3}\theta^{2}$$;

equations (11) of § 6 give us:

$$A=\frac{a}{3b^{4}}.$$

Our pressure is equal to A, with a constant coefficient, which is indeed negative.

Now assessing the mass of the electron – I mean the "experimental mass", that is to say the mass for low velocities – we have (cf. § 6):

$$H=\frac{\varphi\left(\frac{\theta}{k}\right)}{k^{2}r},\quad\theta=k,\quad\varphi=a,\quad\theta r=b;$$

hence

$$H=\frac{a}{bk}=\frac{a}{b}\sqrt{1-V^{2}},$$

I can write for very small V

$$H=\frac{a}{b}\left(1-\frac{V^{2}}{2}\right),$$

so that the mass, both longitudinal and transverse, will be $$\tfrac{a}{b}$$.

Now a is a numerical constant which shows that: the pressure that creates our additional potential is proportional to the 4th power of the experimental mass of the electron.

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 As 's law is proportional to the experimental mass, we are tempted to conclude that there is some relation between the cause that generates gravitation and the one that generates the additional potential.

§ 9. — Hypotheses on gravitation
Thus 's theory would completely explain the impossibility to demonstrate absolute motion, if all forces are of electromagnetic origin.

But there are forces which we can not assign an electromagnetic origin, as for example gravitation. It could happen, indeed, that two systems of bodies produce equivalent electromagnetic fields, that is to say, exerting the same action on the electrified bodies and on the currents, and yet these two systems do not exercise the same gravitational action on the ian mass. The gravitational field is thus distinct from the electromagnetic field. was thus forced to complete his hypothesis by assuming that forces of any origin, and in particular gravitation, are affected by translation (or, if preferred, by the transformation) the same way as electromagnetic forces.

It is now convenient to enter into details and look more closely at this hypothesis. If we want that the ian force is affected in this way by the transformation, we can not accept that the force depends only on the relative position of the attracting body and of the body attracted at the instant considered. It will also depend on the velocities of the two bodies. And that's not all: it is natural to assume that the force acting at time t on the attracted body, depends on the position and velocity of this body at the same time t; but it will depend, in addition, on the position and velocity of the attracting body, not at time t, but a moment earlier, as if gravitation needs a certain time to propagate.

Consider therefore the position of the attracted body at the instant t0 and, at this point, x0, y0, z0 are the coordinates, ξ, η, ζ the components of its velocity; consider the other attracting body at the corresponding time t0 + t and, at this point, x0 + x, y0 + y, z0 + z are the coordinates, ξ1, η1, ζ1 the components of its velocity.

We must first have a relationship

to define the time t. This relation will define the law of propagation of the gravitational action (I do not impose on me the condition that the propagation takes place with the same speed in all directions).

Now let X1, Y1, Z1 the 3 components of the action exerted at time t0 on the body; we have to express X1, Y1, Z1 as functions of

What are the conditions to fulfill?

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 1° The condition (1) shall not be altered by transformations of the group.

2° The components X1, Y1, Z1 will be affected by the transformations the same way as electromagnetic forces designated by the same letters, that is to say, according to equations (11bis) of § 1.

3° When two bodies are at rest, we will fall back to the ordinary law of attraction.

It is important to note that in the latter case, the relation (1) disappears, because time does not play any role if the two bodies are at rest.

The problem thus posed is obviously undetermined. We will thus seek to satisfy as many as possible other additional conditions:

4° Astronomical observations do not appear to show significant derogation to 's law, we will choose the solution that deviates the least of this law, for low velocities of two bodies.

5° We will endeavor to arrange that T is always negative; if indeed it is conceived that the effect of gravitation takes a certain time to be propagated, it would be more difficult to understand how this effect could depend on the position not yet attained by the attracting body.

There is one case where the indeterminacy of the problem disappears; it is where the two bodies are at rest relative to each other, that is to say that:

$$\xi=\xi_{1},\ \eta=\eta_{1},\ \zeta=\zeta_{1}\,$$;

this is the case we will consider first, assuming that these velocities are constant, so that the two bodies are drawn into a common translational motion, rectilinear and uniform.

We can assume that the axis of x has been taken parallel to the translation, so that η = ζ = 0, and we take ε = -ξ.

If in these circumstances we apply the transformation, after the transformation the two bodies are at rest and we have:

$$\xi'=\eta'=\zeta'=0\,$$

Then the components x'0, Y'0, Z'0 must conform to 's law and we will have a constant factor:

But we have, according to § 1:

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 We have also:

$$x+\epsilon t=x-\xi t,\quad r^{\prime2}=k^{2}(x-\xi t)^{2}+y^{2}+z^{2}$$

and

which can be written:

It seems at first sight that the indetermination remains, since we have made no hypothesis about the value of t, that is to say about the speed of transmission; and that also x is a function of t, but it is easy to see that x - ξt, y, z (which appear in our formulas) do not depend on t.

We see that if two bodies are simply in motion by a common translation, the force acting on the body is drawn normal to an ellipsoid with its center at the attracting body.

To go further we must look for the invariants of the group.

We know that the substitutions of this group (assuming l = 1) are linear substitutions which do not affect the quadratic form

$$x^2 + y^2 + z^2 - t^2.\,$$

Let on the other hand:

we see that the transformation will cause to make δx, δy, δz and δ1x, δ1y, δ1z, δ1t undergo the same linear substitutions as with x, y, z, t.

We regard

$$\begin{array}{ccccccc} x, & & y, & & z, & & t\sqrt{-1},\\ \\\delta x, & & \delta y, & & \delta z, & & \delta t\sqrt{-1},\\ \\\delta_{1}x, & & \delta_{1}y, & & \delta_{1}z, & & \delta_{1}t\sqrt{-1},\end{array}$$

as the coordinates of three points P, P', P" in a 4-dimensional space. We see that the transformation is a rotation of that space around the origin, regarded as fixed. We shall therefore have no other distinct invariants than 6 distances of the 3 points P, P', P" between them and the origin, or, if you like

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 it better, than the 2 expressions:

$$x^{2}+y^{2}+z^{2}-t^{2},\ x\delta x+y\delta y+z\delta z-t\delta t$$

or the 4 expressions of the same form, deduced from permuting (in an arbitrary way) the three points P, P', P".

But what we look for are the functions of 10 variables (2) that are invariants; so we must, among the combinations of our 6 invariants, seek those which depend only on these 10 variables, that is to say those that are homogeneous of degree 0 as compared to δx, δy, δz, δt, as compared to δ1x, δ1y, δ1z, δ1t. We will thus have 4 distinct invariants, which are:

Let us now consider the transformations undergone by the components of the force; resume the equations (11) of § 1, which relate not to the force X1, Y1, Z1, which we consider here, but to the force X, Y, Z referred to unit volume. We pose also:

$$T=\sum X\xi;$$

we see that these equations (11) can be written as (l = 1):

so that X, Y, Z, T undergo the same transformation as x, y, z, t. The invariants of the group are therefore

$$\sum X^{2}-T^{2},\quad\sum Xx-Tt,\quad\sum X\delta x-T\delta t,\quad\sum X\delta_{1}x-T\delta_{1}t.$$

But this is not X, Y, Z which we need, it is X1, Y1, Z1 with

$$T_{1}=\sum X_{1}\xi.$$

We see that

$$\frac{X_{1}}{X}=\frac{Y_{1}}{Y}=\frac{Z_{1}}{Z}=\frac{T_{1}}{T}=\frac{1}{\rho}.$$

So the transformations act on X1, Y1, Z1, T1 in the same manner as X, Y, Z, T, with the difference that these expressions are also multiplied by

$$\frac{\rho}{\rho^{\prime}}=\frac{1}{k(1+\xi\epsilon)}=\frac{\delta t}{\delta t^{\prime}}.$$

Similarly it would act on ξ, η, ζ, 1, in the same manner as δx, δy, δz, δt, with the difference that these expressions are also multiplied by the same factor:

$$\frac{\delta t}{\delta t^{\prime}}=\frac{1}{k(1+\xi\epsilon)}.$$

Consider then $$X,\ Y,\ Z,\ T\sqrt{-1}$$ as the coordinates of a fourth point

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 Q, then the invariants are functions of mutual distances of five points

and among these functions we must retain only those that are homogeneous of degree 0, on the one hand in relation to

(variables that can then be replaced by X1, Y1, Z1, T1, ξ, η, ζ, 1), on the other hand in relation to

(variables that can be replaced later by ξ1, η1, ζ1, 1).

Thus we find in addition to the four invariants (5), four new distinct invariants, which are:

The last invariant is always zero, according to the definition of T1.

This granted, what are the requirements?

1° The left-hand side of relation (1), which defines the velocity of propagation must be a function of the four invariants (5)

One can obviously make a lot of hypotheses, we only look at two:

A) It may be

where t = ±r, and since t must be negative, t = -r. This means that the propagation velocity is equal to that of light. At first it seems that this hypothesis should be rejected without consideration. has indeed shown that this propagation is either instantaneous, or much faster than light. But had considered the hypothesis of finite speed of propagation, ceteris non mutatis; here, however, this hypothesis is complicated by many others, and it may happen that there is a more or less perfect compensation, as the applications of the  transformation gave us already so many examples.

B) It may be

The propagation velocity is much faster than that of light, but in some cases t may be negative, which, as we have said, seems hardly acceptable. We will add this to hypothesis (A).

2° The four invariants (7) must be functions of the invariants (5).

3° When the two bodies are in absolute rest, X1, Y1, Z1 must have the value

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 deduced from the law of, and when they are in relative rest, the value deduced from the equations (4).

Under the hypothesis of absolute rest, the first two invariants (7) must be reduced to

$$\sum X_{1}^{2},\quad\sum X_{1}x,$$

or by 's law at

$$\frac{1}{r^{4}},\quad-\frac{1}{r}$$

secondly, in hypothesis (A), the 2nd and 3rd of the invariants (5) become:

$$\frac{-r-\sum x\xi}{\sqrt{1-\sum\xi^{2}}},\quad\frac{-r-\sum x\xi_{1}}{\sqrt{1-\sum\xi_{1}^{2}}}$$

that is to say, for absolute rest, to

$$-r,\ -r$$.

We may therefore assume for example that the first two invariants (4) are reduced to

$$\frac{\left(1-\sum x\xi_{1}^{2}\right)^{2}}{\left(r+\sum x\xi_{1}\right)^{4}},\quad-\frac{\sqrt{1-\sum\xi_{1}^{2}}}{r+\sum x\xi_{1}},$$

but other combinations are possible.

We must choose between these combinations, and secondly, in order to define X1, Y1, Z1 we need a third equation. For such a choice, we must endeavor to bring us closer as much as possible to the law to. Let's see what happens when (always making t = -r ) we neglect the squares of the velocities ξ η etc.. The 4 invariants (5) then become:

$$0,\quad-r-\sum x\xi,\quad-r-\sum x\xi_{1},\quad1$$

and the 4 invariant (7):

$$\sum X_{1}^{2},\quad\sum X_{1}(x+\xi r),\quad\sum X_{1}\left(\xi_{1}-\xi\right),\quad0.$$

But to be able to compare it with the law of, another transformation is needed; here x0 + x, y0 + y, z0 + z are the coordinates of the attracting body at the instant t0 + x, and $$r=\sqrt{\sum x^{2}}$$; in the law of it is necessary to consider the coordinates x0 + x1, y0 + y1, z0 + z1 of the attracting body at the instant t0, and the distance $$r_{1}=\sqrt{\sum x_{1}^{2}}$$.

We can neglect the square of time t required for the propagation and therefore proceed as if the movement was uniform, then we have:

$$x=x_{1}+\xi_{1}t,\quad y=y_{1}+\eta_{1}t,\quad z=z_{1}+\zeta_{1}t,\quad r\left(r-r_{1}\right)=\sum x\xi_{1}t$$

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 or, since t = -r,

so that our 4 invariants (5) become:

and our 4 invariants (7):

In the second of these expressions I wrote r1 instead of r, because r is multiplied by ξ - ξ1 and I neglect the square of ξ.

On the other hand, 's law would us give for these 4 invariants (7)

So if we denote the 2nd and 3rd invariants (7) by A and B, and the 3 first invariants (7) by M, N, P, we will satisfy 's law up to terms of order of the square velocities, by:

This solution is not unique. Indeed, let C be the fourth invariant (5), C - 1 is of the order of the square of ξ, and it is equal to (A - B)².

So we could add to the 2ds members of each of equations (8) a term consisting of C - 1 multiplied by an arbitrary function of A, B, C, and a term of the form of (A - B)² also multiplied by a function of A, B, C.

At first sight, the solution (8) seems the most straightforward, it may nevertheless be adopted and in effect – since M, N, P are functions of X1, Y1, Z1 and $$T_{1}=\sum X_{1}\xi$$ – we can draw from these three equations (8) the values of X1, Y1, Z1, but in some cases these values become imaginary.

To avoid this, we will operate in another way. Let:

This is justified by the analogy with the notation

which appears in the substitution of.

In this case, and because of the condition, -r = t, the invariants (5) become:

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text:
Translation:Mikraot Gedolot/Psalms/1:1

Text
The praises of the man who did not go in the wicked’s council and in the path of sinners did not stand, and in the seat of insulters did not sit.

Rashi
The praises of the man

With ten languages of song was this book said: with nitzuakh, with niggun, with mizmor, with shir, with hallel, with t'fillah, with b'rakhah, with hoda'ah, with ashre, with hallelujah, corresponding to the ten persons who said it: Adam, Melchizedek, Abraham, Moses, David, Solomon, Asaph and the three sons of Korah. And they differ on Jeduthun: there are those who say he was a man as it is written in Chronicles (1 16:38), and there are those who say, Jeduthun that is in this book only refers to the laws and the judgments of the decrees which have gone over him and over Israel.

The praises of the man

Les felicements in the foreign language; the contentment of man and his praises are these, that he did not go; for since he did not go, he did not stand; and since he did stand, he did not sit.

Ibn Ezra
The praises of the man

Our predecessors, their memory to blessing, said that stood is harder than went and sat is harder than all of them. If so, the insulters will be worse than sinners. And Rabbi Moshe the Kohen explained to the contrary, and said that the intention of did not stand is "didn't even stand". And according to my understanding, that the word wicked are those who do not stand on one standard(?)...

מ"ג תהלים א א

title: Translation:Mikraot Gedolot/Psalms/1:1


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 On the other hand, we see that the following systems of quantities:

$$\begin{array}{ccccccc} x, & & y, &  & z, &  & -r=t\\ \\ k_{0}X_{1}, & & k_{0}Y_{1}, &  & k_{0}Z_{1}, &  & k_{0}T_{1}\\ \\ k_{0}\xi, & & k_{0}\eta, &  & k_{0}\zeta, &  & k_{0}\\ \\ k_{1}\xi_{1}, & & k_{1}\eta_{1}, &  & k_{1}\zeta_{1}, &  & k_{1} \end{array}$$

undergo the same linear substitutions when we apply the transformations of the group. We are thus led to pose:

It is clear that if α, β, γ are invariants, X1, Y1, Z1, T1 satisfy the basic condition, that is to say, it will undergo, by the effect of the transformations, a suitable linear substitution.

But for the equations (9) to be consistent, we must have:

$$\sum X_{1}\xi-T_{1}=0,$$

which, by replacing X1, Y1, Z1, T1 by their values (9) and multiplying by k0², becomes:

What we want is, if we neglect the square of speed of light, the squares of the velocities ξ, etc., as well as the product of accelerations by the distances as we did above, so that the values of X1, Y1, Z1 remain in conformity with the law of.

We can take:

$$\beta=0,\quad\gamma=-\frac{A\alpha}{C}.$$

With the order of approximation adopted, we have:

$$k_{0}=k_{1}=1,\quad C=1,\quad A=-r_{1}+\sum x\left(\xi_{1}-\xi\right),\quad B=-r_{1},$$

$$x=x_{1}+\xi_{1}t=x_{1}-\xi_{1}r\,$$

The first equation (9) becomes:

$$X_{1}=\alpha\left(x-A\xi_{1}\right)$$

But if we neglect the square of ξ, we can replace Aξ1 by -r1ξ1, or

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 by -rξ1, which gives:

$$X_{1}=\alpha\left(x+\xi_{1}r\right)=\alpha x_{1}$$

's law would give:

$$X_{1}=-\frac{x_{1}}{r_{1}^{3}}.$$

We must therefore choose, for the invariant α, one that reduces to $$-\tfrac{1}{r_{1}^{3}}$$ to the order of approximation adopted, that is to say $$\tfrac{1}{B^{3}}$$. The equations (9) become:

We first see that the corrected attraction is composed of two components, one parallel to the vector joining the positions of the two bodies, the other parallel to the velocity of the attracting body.

Recall that when we talk about the position or velocity of the attracting body, it is its position or its velocity when the gravitational wave leaves; for the body attracted, on the contrary, it is its position or its velocity when the gravitational wave reaches it, the wave is assumed to propagate with the speed of light.

I think it would be premature to push further discussion of these formulas, I will confine myself to a few remarks.

1° The solutions (11) are not unique; we can indeed replace $$\frac{1}{B^{3}}$$ which enters in the factor everywhere, by

$$\frac{1}{B^{3}}+(C-1)f_{1}(A,B,C)+(A-B)^{2}f_{2}(A,B,C)$$,

f1 and f2 are arbitrary functions of A, B, C; or we are taking β no longer as zero but adding arbitrary complementary terms to α β γ, provided they satisfy the condition (10) and are of the 2nd order with regard to ξ as far as α is concerned, and of the 1st order as far as β and γ are concerned.

2° The first equation (11) can be written:

and the quantity in brackets can, itself, written as:

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Rh CHAPTER IV.

only ceremonies they use preparatory to marriage are, in the first place, to get the parents' consent, the suitor's best claim is being a good fighter, and an expert hunter—so as to be able to protect and provide for a family. They are not at all particular as to the number of wives such men have; consequently some have five or six wives, and others none at all. If a man wishes to have a man's grown up sister for a wife, he must give his own—if he has one—in exchange; but they are very averse to marrying one of their own relations—even of a distant degree. If a woman is supposed to have a child who is not her husband's, they consider it a great disgrace; and to the infant, death is


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 so that the total force can be divided into three components corresponding to the three brackets of expression (12); the first component has a vague analogy with the mechanical force due to the electric field, the other two with mechanical forces due to a magnetic field; to complete the analogy I can, under the first point, replace $$\tfrac{1}{B^{3}}$$ by $$\tfrac{C}{B^{3}}$$ in equations (11), so that X1, Y1, Z1 only depend linearly on the velocity ξ, η, ζ of the attracted body, since C has disappeared from the denominator of (11bis).

We pose then:

it follows that C had disappeared from the denominator of (11a):

and there will also:

Then λ, μ, ν or $$\tfrac{\lambda}{B^{3}},\ \tfrac{\mu}{B^{3}},\ \tfrac{\nu}{B^{3}}$$ is a kind of electric field, while λ', μ', ν' or rather $$\tfrac{\lambda^{\prime}}{B^{3}},\ \tfrac{\mu^{\prime}}{B^{3}},\ \tfrac{\nu^{\prime}}{B^{3}}$$ is a kind of magnetic field.

3° The postulate of relativity would require us to adopt solution (11) or solution (14) or any solution that would inferred by using the first remark; but the first question that arises is whether they are compatible with astronomical observations; the discrepancy with 's law is of the order ξ², that is to say, 10000 times smaller when it were of order ξ, that is to say, if the propagation happens with the speed of light, ceteris non mutatis; it is permissible to hope that it will not be too great. But only a thorough discussion will be able to teach it to us.

 * Paris, July 1905.

H.

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52 almost certain. If again, a family increases too rapidly, for instance, if a woman has a child within twelve months of a previous one, they hold a consultation amongst the tribe she belongs to, as to whether it shall live or not; but if the father insists upon the life of the child being spared, they do not persist in its destruction, and especially if it is a female. At their confinements they receive no assistance whatever, but so soon as the child comes into the world, they wrap it up in a piece of skin rug; and, if on a journey, move on; it has no nourishment but the breast. They name them according to any circumstance that may happen; perhaps after the lake, or river they are near; or any accident or event which may have occurred—the whole family changing their names also, until another child is born—when they change again.

It will be seen by the foregoing, that jealousy is the prevalent cause of all their quarrels, for the women and the men are equally under the influences of the Green-eyed Monster. In the fights, however, which ensue, the poor women get much the worst of it, for after having had their furious combats amongst themselves the husbands think it necessary to turn too also, and thrash them into quietude.

The meetings of different tribes, I found were not solely for the purpose of exchanging food, but for the very landable purpose of bringing out their very elegant, amiable, marriageable daughters, to be seen and known, and of course, courted. By this very natural process, much ill-will and wild desperate passions, are


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Rh excited—so that wounds, not only of the heart, but of the head, and frightful murders ensue; some of them never to be forgiven, until a fearful revenge has followed.

Previous to breaking up from our present ground there was another battle, so that when the other tribe left, one of ours stole after them in the night and speared a man dead, who was sleeping in his hut beside his wife; he sent his spear right through him into the ground, for no other cause than that the murdered man had promised him his daughter years before, and had then given her to another. Having had his revenge he returned, and boasted of what he had done, upon which his relations and particular friends left the place, apparently apprehending an attack. The next morning, those who remained went to the tribe to which the murdered man belonged, and found him rolled up in his rug, ready to be tied up in a tree—a mode of disposing of the dead, who were not enemies, unknown to me before. They selected a strong, if not a lofty tree, and in the branches, about twelve feet up, they placed some logs and branches across, and sheets of bark; on these they laid the body with the face upwards, inclining toward the setting sun, and over it was placed some more bark and boughs, and then logs as heavy as the branches would bear; all this being done to protect the body from the birds of prey. Whilst this was going on, the women sat round the tree joining with the widow in the most bitter lamentations, pitiable to hear. A fire was, as usual, made all round this extraordinary


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1504 Rh Rh

It seems at first sight that the aberration of light and the associated optical phenomena will provide a means of determining the absolute motion of the Earth, or rather its motion, not in relation to other stars, but in relation to the ether. This is not the case: the experiments in which we take into account only the first order of aberration were initially unsuccessful and an explanation was easily found; but, who imagined an experiment by which terms depending on the square of the aberration could be measured, had no luck either. It seems that this inability to demonstrate absolute motion is a general law of nature.

An explanation was proposed by, who introduced the hypothesis of a contraction of all bodies in the direction of motion of the earth; this contraction would account for the Michelson-Morley experiment and all those that have been conducted to date, but leaves room for other experiments even more delicate and more easily conceived than executed,

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 which might demonstrate the absolute motion of the Earth. But if the impossibility of such a finding is considered highly probable, one can predict that these experiments, if they can ever be conducted, will give a negative result. has sought to supplement and amend his hypothesis so as to bring it into accord with the postulate of the complete impossibility of determining absolute motion. This he managed to do in his article entitled Electromagnetic phenomena in a system moving with any velocity smaller than that of Light (Proceedings de l’Académie d’Amsterdam, May 27, 1904).

The importance of this question made me determined to return to it; and the results I obtained are in agreement on all important points with those of ; I was only led to modify and complete them in a few points of detail.

The essential point, established by, is that the electromagnetic field equations are not altered by a certain transformation (which I shall call after the name of Lorentz), which has the following form:

$$x, y, z$$ are the coordinates and $$t$$ the time before the transformation, $$x', y', z'$$ and $$t'$$ after the transformation. Moreover, $$\epsilon$$ is a constant which defines the transformation

and $$l$$ is an arbitrary function of $$\epsilon$$. One can see that in this transformation the $$x$$-axis plays a particular role, but one can obviously construct a transformation in which this role would be played by any straight line through the origin. The sum of all these transformations, together with the set of all rotations of space, must form a group; but for this to occur, we need $$l = 1$$; so one is forced to suppose $$l = 1$$ and this is a consequence that has obtained by another way.

Let $$\rho$$ the electric density of the electron, $$\xi,\eta,\zeta$$ the velocity before the transformation; we obtain for the same quantities $$\rho',\xi',\eta',\zeta'$$ after processing

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text: Dictionary of National Biography, 1885-1900/Burton, Cassibelan


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 These formulas differ somewhat from those which had been found by.

Let now $$X, Y, Z$$ and $$X', Y', Z'$$ the three components of force before and after transformation, and the force is expressed in unit volume; I found

These formulas are slightly different from those of ; the additional term in $$\Sigma X\xi$$ reminds us on a result previously obtained by.

If we now denote by $$X_{1},\ Y_{1},\ Z_{1}$$ and $$X'_{1},\ Y'_{1},\ Z'_{1}$$ the components of a force, not referred to unit volume, but to unit mass of the electron, we obtain

was also led to assume that the electron in motion takes the form of an oblate spheroid; this is also the hypothesis made by, however, while assumed that two axes of the ellipsoid remain constant, which is consistent with the hypothesis $$l = 1$$,  assumed that the volume remains constant. Both authors have shown that these two hypotheses are consistent with the experiments of, as well as the original hypothesis of (spherical electron). The hypothesis of would have the advantage that it is self-sufficient, because it suffices to regard the electron as deformable and incompressible, and to explain that it takes an ellipsoidal shape when it moves. But I show, in agreement with, that it is incapable to accord with the impossibility of an experiment showing the absolute motion. As I have said, this is because $$l = 1$$ is the only case for which all the Lorentz transformations form a group.

But with the hypothesis of, the agreement between the formulas does not occur all alone; we obtain at the same time a possible explanation for the contraction of the electron, assuming that the electron, deformable and compressible, is subjected to a constant external pressure whose work is proportional to volume changes.

I show, by applying the principle of least action,

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 that under those conditions the compensation is complete, assuming that inertia is an electromagnetic phenomenon exclusively, as generally admitted since 's experiments, and apart from the constant pressure that I just mentioned and which acts on the electron, all the forces are of electromagnetic origin. We have thus the explanation of the impossibility of demonstrating absolute motion and of the contraction of all the bodies in the direction of the terrestrial motion.

But that's not all:, in the work quoted, found it necessary to complete his hypothesis by assuming that all forces, whatever their origin, are affected by translation in the same way as electromagnetic forces and, consequently, the effect produced on their components by the Lorentz transformation is still defined by equations (4).

It was important to examine this hypothesis more closely and in particular to examine what changes it would require us to make on the law of gravitation. That is what I sought to determine; I was first led to suppose that the propagation of gravitation is not instantaneous, but happens with the speed of light. This seems at odds with results obtained by, who announced that this propagation is, if not instantaneous, at least much faster than that of light. But in reality, the question posed by differs considerably from that which occupies us here. The introduction of a finite velocity of propagation was the only change Laplace introduced to 's law. Here, on the contrary, this change is accompanied by several others; it is possible, and that is indeed what happens, that a partial compensation occurs between them.

When we therefore speak of the position or velocity of the attracting body, it will be the position or the velocity at the time when the gravitational wave leaves the body; when we talk about the position or velocity of the attracted body, it will be the position or the velocity at the moment when this body was reached and attracted by the gravitational wave emanating from the other body; it is clear that the first instant precedes the second.

So if $$x, y, z$$ are the projections on the three axes of the vector joining the two positions, if the velocity of the attracted body is $$\xi,\eta,\zeta$$, and that of the attracting body $$\xi_{1},\eta_{1},\zeta_{1}$$, the three components of the attraction (which I can still call $$X'_{1},\ Y'_{1},\ Z'_{1}$$) are functions of $$x,y,z,\xi,\eta,\zeta,\xi_{1},\eta_{1},\zeta_{1}$$. I asked myself whether it was possible to determine these functions

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 in a way that they are affected by the Lorentz transformation according to equations (4) and found the ordinary law of gravitation, whenever the velocities $$\xi,\eta,\zeta,\xi_{1},\eta_{1},\zeta_{1}$$ are small enough that one can neglect the squares in respect to the square of the speed of light.

The answer must be affirmative. It is found that the corrected attraction consists of two forces, one parallel to the vector $$x, y, z$$, the other to the velocity $$\xi,\eta,\zeta$$.

The difference to the ordinary law of gravitation, as I have said, is of order $$\xi^{2}$$; if we only assume, as did, that the speed of propagation is that of light, this discrepancy is of order $$\xi$$, that is to say 10.000 times larger. It is therefore, at first sight, not absurd to assume that astronomical observations are not precise enough to detect a difference as small as the one which we imagine. But this is what only a thorough discussion will make possible to decide.

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text: Dictionary of National Biography, 1885-1900/Burton, Catharine


text:
User:Sije/Mishneh Torah/The Book of Holiness/Laws of Forbidden Intercourse

Laws of Forbidden Intercourse
They comprise thirty-seven commandments: one positive commandment and thirty-six negative commandments.

This is their enumeration: (1) Not to sexually approach one's mother. (2) Not to sexually approach a wife of one's father. (3) Not to have sexual intercourse with a sister. (4) Not to have sexual intercourse with a daughter of a wife of one's father. (5) Not to have sexual intercourse a daughter of one's son. (6) Not to have sexual intercourse with a daughter. (7) Not to have sexual intercourse with a daughter of one's daughter. (8) Not to take a woman and her daughter in marriage. (9) Not to take a woman and her son's daughter in marriage. (10) Not to take a woman and her daughter's daughter in marriage. (11) Not to have sexual intercourse with a sister of one's father. (12) Not to have sexual intercourse with a sister of one's mother. (13) Not to have sexual intercourse with a wife of one's father's brother. (14) Not to have sexual intercourse with a wife of one's son. (15) Not to have sexual intercourse with a wife of one's brother. (16) Not to have sexual intercourse with a sister of one's wife. (17) Not to lie with a beast. (18) That a woman not bring a beast upon herself. (19) Not to lie with a male. (20) Not to reveal the nakedness of one's father. (21) Not to reveal the nakedness of a brother of one's father. (22) Not to have sexual intercourse with a married woman. (23) Not to have sexual intercourse with a menstruous woman. (24) Not to marry amongst the nations. (25) That an Ammonite or Moabite may not enter the congregation of the Lord. (26) Not to keep a third generation Egyptian from entering the congregation. (27) Not to keep a third generation Edomite from entering the congregation. (28) That a bastard may not enter the congregation. (29) That a eunuch may not enter the congregation. (30) Not to castrate a male, not even a domestic animal, a beast or a fowl. (31) That the High Priest may not take a widow in marriage. (32) That a High Priest may not have sexual intercourse with a widow even without marriage. (33) That the High Priest take an adolescent virgin in marriage. (34) That a priest not take a divorced woman in marriage. (35) That he not take a prostitute in marriage. (36) That he not take a profaned woman in marriage. (37) That a person not draw near to any of the forbidden relations, and even without sexual intercourse.

The elucidation of these commandments is in these chapters:
 * Chapter One
 * Chapter Two
 * Chapter Three
 * Chapter Four
 * Chapter Five
 * Chapter Six
 * Chapter Seven
 * Chapter Eight
 * Chapter Nine
 * Chapter Ten
 * Chapter Eleven
 * Chapter Twelve
 * Chapter Thirteen
 * Chapter Fourteen
 * Chapter Fifteen
 * Chapter Sixteen
 * Chapter Seventeen
 * Chapter Eighteen
 * Chapter Nineteen
 * Chapter Twenty
 * Chapter Twenty One
 * Chapter Twenty Two


text:
User talk:71.106.91.177

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title: User talk:71.106.91.177


text: Category:Clay animation

title: Category:Clay animation


text:
Author:Thomas Nixon Carver

Works

 * "Political Science: General Introduction" in The Harvard Classics Vol. 51 (1914)

title: Author:Thomas Nixon Carver


text: Author talk:Thomas Nixon Carver

title: Author talk:Thomas Nixon Carver


text:
Page:The Federalist (Ford).djvu/798

720 the States, by appropriate legislation, and through the action of their judicial and ministerial officers, from enforcing the delivery of fugitives from labor to the person to whom such service or labor is due.

. 5. The foreign slave-trade is hereby forever prohibited; and it shall be the duty of Congress to pass laws to prevent the importation of slaves, coolies, or persons held to service or labor, into the United States and the Territories from places beyond the limits thereof.

. 6. The first, third and fifth sections, together with this section of these amendments, and the third paragraph of the second section of the first article of the Constitution, and the third paragraph of the second section of the fourth article thereof, shall not be amended or abolished without the consent of all the States.

. 7. Congress shall provide by law that the United States shall pay to the owner the full value of his fugitive from labor, in all cases where the marshal or other officer, whose duty it was to arrest such fugitive, was prevented from so doing by violence and intimidation from mobs or riotous assemblages, or when after arrest such fugitive was rescued by like violence or intimidation, and the owner thereby deprived of the same; and the acceptance of such payment shall preclude the owner from further claim to such fugitive. Congress shall provide by law for securing to the citizens of each State the privileges and immunities of citizens in the several States.

We, the people of the Confederate States, each State acting in its sovereign and independent character, in order to form a permanent federal government, establish justice, insure domestic tranquillity, and secure the blessings of liberty to ourselves and our posterity—invoking the favor and guidance of Almighty God—do ordain and establish this Constitution of the Confederate States of America.

ARTICLE 1.

1 All legislative powers herein delegated shall be vested in a Congress of the Confederate States, which shall consist of a Senate and House of Representatives.

1 The House of Representatives shall be composed of members chosen every second year by the people of the several States; and the electors in each State shall be citizens of the Confederate States, and have the qualifications requisite for electors of the most numerous branch of the State Legislature: but no person of foreign birth, not a citizen of the Confederate States, shall be allowed to vote for any officer, civil or political, State or federal.

2 No person shall be a Representative who shall not have attained the age of twenty-five years, and be a citizen of the Confederate

title: Page:The Federalist (Ford).djvu/798


text:
Author:Charles Jesse Bullock

Works

 * "Adam Smith and the "Wealth of Nations"" in The Harvard Classics Vol. 51 (1914)

title: Author:Charles Jesse Bullock


text:
Author talk:Charles Jesse Bullock

Image
There is an image of author, though of poor reproduction in — billinghurst  sDrewth  13:08, 14 January 2011 (UTC)
 * Passport Applications, January 2, 1906 - March 31, 1925 (M1490) > 1921 - 1922 > Roll 1807 - Certificates: 108976-109349, 05 Jan 1922-07 Jan 1922 > 543

title: Author talk:Charles Jesse Bullock


text:
User talk:72.152.243.185

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text:
User talk:92.23.92.233

Welcome

Hello, and welcome to Wikisource! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

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Brithwold
Bertwald of Ramsbury this link looks like a better match for this biography…for both the article and the redirect article…JamAKiska (talk) 03:09, 15 January 2011 (UTC)

title: User talk:92.23.92.233


text:
Author:William Bennett Munro

Works

 * "The Growth of the American Constitution" in The Harvard Classics, Vol. 51 (1914)
 * "The Era of Discovery" in The Harvard Classics, Vol. 51 (1914)

title: Author:William Bennett Munro


text: Author talk:William Bennett Munro

title: Author talk:William Bennett Munro


text: On the Natural Faculties/Book Three

title: On the Natural Faculties/Book Three


text:
Author:George Pierce Baker (1866-1935)

Works

 * "Drama: General Introduction" in The Harvard Classics, Vol. 51 (1914)

title: Author:George Pierce Baker (1866-1935)


text: Category:Federal Rules

title: Category:Federal Rules


text:
Page:The Genealogy of Morals.djvu/122

98 once used against me) the product of an insane hate of knowledge, mind, and flesh? A curse on flesh and spirit in one breath of hate? An apostasy and reversion to the morbid Christian and obscurantist ideals? And finally a self-negation and self-elimination on the part of an artist, who till then had devoted all the strength of his will to the contrary, namely, the highest artistic expression of soul and body. And not only his art; of his life as well. Just remember with what enthusiasm Wagner followed in the footsteps of Feuerbach. Feuerbach's motto of "healthy sensuality" rang in the ears of Wagner during the thirties and forties of the century, as it did in the ears of many Germans (they dubbed themselves "Young Germans"), like the word of redemption. Did he eventually change his mind on the subject? For it seems at any rate that he eventually wished to change his teaching on that subject. . . and not only is that the case with the Parsifal trumpets on the stage: in the melancholy, cramped, and embarrassed lucubrations of his later years, there are a hundred places in which there are manifestations of a secret wish and will, a despondent, uncertain, unavowed will to preach actual retrogression, conversion, Christianity, mediaevalism, and to say to his disciples, "All is vanity! Seek salvation elsewhere!" Even the "blood of the Redeemer" is once invoked.

4.

Let me speak out my mind in a case like this, which has many painful elements—and it is a typical case: it is certainly best to separate an artist from his work so

title: Page:The Genealogy of Morals.djvu/122


text:
Page:The Genealogy of Morals.djvu/123

Rh that he cannot be taken as seriously as his work. He is after all merely the presupposition of his work, the womb, the soil, in certain cases the dung and manure, on which and out of which it grows—and consequently, in most cases, something that must be forgotten if the work itself is to be enjoyed. The insight into the origin of a work is a matter for psychologists and vivisectors, but never either in the present or the future for the aesthetes, the artists. The author and creator of Parsifal was as little spared the necessity of sinking and living himself into the terrible depths and foundations of mediaeval soul-contrasts, the necessity of a malignant abstraction from all intellectual elevation, severity, and discipline, the necessity of a kind of mental perversity (if the reader will pardon me such a word), as little as a pregnant woman is spared the horrors and marvels of pregnancy, which, as I have said, must be forgotten if the child is to be enjoyed. We must guard ourselves against the confusion, into which an artist himself would fall only too easily (to employ the English terminology) out of psychological "contiguity"; as though the artist' himself actually were the object which he is able to represent, imagine, and express. In point of fact, the position is that even if he conceived he were such an object, he would certainly not represent, conceive, express it. Homer would not have created an Achilles, nor Goethe a Faust, if Homer had been an Achilles or if Goethe had been a Faust. A complete and perfect artist is to all eternity separated from the "real," from the actual; on the other hand, it will be appreciated that he can at times get tired to the point of despair of this eternal "unreality" and falseness of his innermost being—

title: Page:The Genealogy of Morals.djvu/123


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Page:The Genealogy of Morals.djvu/124

100 and that he then sometimes attempts to trespass on to the most forbidden ground, on reality, and attempts to have real existence. With what success? The success will be guessed—it is the typical velleity of the artist; the same velleity to which Wagner fell a victim in his old age, and for which he had to pay so dearly and so fatally (he lost thereby his most valuable friends). But after all, quite apart from this velleity, who would not wish emphatically for Wagner's own sake that he had taken farewell of us and of his art in a different manner, not with a Parsifal, but in more victorious, more self-confident, more Wagnerian style—a style less misleading, a style less ambiguous with regard to his whole meaning, less Schopenhauerian, less Nihilistic? ..

5.

What, then, is the meaning of ascetic ideals? In the case of an artist we are getting to understand their meaning: Nothing at all. . . or so much that it is as good as nothing at all. Indeed, what is the use of them? Our artists have for a long time past not taken up a sufficiently independent attitude, either in the world or against it, to warrant their valuations and the changes in these valuations exciting interest. At all times they have played the valet of some morality, philosophy, or religion, quite apart from the fact that unfortunately they have often enough been the inordinately supple courtiers of their clients and patrons, and the inquisitive toadies of the powers that are existing, or even of the new powers to come. To put it at the lowest, they always need a rampart, a support, an already constituted authority: artists never stand by

title: Page:The Genealogy of Morals.djvu/124


text:
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Rh, standing alone is opposed to their deepest instincts. So, for example, did Richard Wagner take, "when the time had come," the philosopher Schopenhauer for his covering man in front, for his rampart. Who would consider it even thinkable, that he would have had the courage for an ascetic ideal, without the support afforded him by the philosophy of Schopenhauer, without the authority of Schopenhauer, which dominated Europe in the seventies? (This is without consideration of the question whether an artist without the milk of an orthodoxy would have been possible at all.) This brings us to the more serious question: What is the meaning of a real philosopher paying homage to the ascetic ideal, a really self-dependent intellect like Schopenhauer, a man and knight with a glance of bronze, who has the courage to be himself, who knows how to stand alone without first waiting for men who cover him in front, and the nods of his superiors? Let us now consider at once the remarkable attitude of Schopenhauer towards art, an attitude which has even a fascination for certain types. For that is obviously the reason why Richard Wagner all at once went over to Schopenhauer (persuaded thereto, as one knows, by a poet, Herwegh), went over so completely that there ensued the cleavage of a complete theoretic contradiction between his earlier and his later aesthetic faiths—the earlier, for example, being expressed in Opera and Drama, the later in the writings which he published from 1870 onwards. In particular, Wagner from that time onwards (and this is the volte-face which alienates us the most) had no scruples

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text: Index:Constitution of the Confederate States of America.djvu


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