# On the bearing of the Principle of Relativity on Gravitational Astronomy

On the bearing of the Principle of Relativity on Gravitational Astronomy.

By W. de Sitter, Assoc. R.A.S.

1. The principle of relativity, first developed in connection with the electromagnetic theory of light, has in recent years been more and more considered as of universal application, and the claim has been made that the whole of our physical sciences should be framed in conformity with it. In this paper are considered some of the results of introducing the principle in the laws of planetary motion. Poincaré has made the remark that some of the difficulties of the problem of three bodies are due to the fact that in the undisturbed motion the nodes and perihelia are fixed. Now, with a law of attraction which is in conformity with the principle of relativity, the perihelion has a motion even in the problem of two bodies, and it is probable that such questions as the existence of periodic solutions and the convergence of series entirely change their aspect. In the present paper, however, these interesting mathematical questions are not touched upon, and I consider the problem exclusively from the point of view of the practical astronomer, investigating only such effects as may be expected to yield the possibility of an empirical verification of the principle.

All natural phenomena are described in terms of four variable quantities; three "coordinates" ${\displaystyle x,y,z,}$ and the "time" ${\displaystyle t}$. If we introduce other variables ${\displaystyle x',y',z',t',}$ the formulæ describing any event will generally be altered. Also the physical laws, which are differential equations defining ${\displaystyle x,y,z,}$ as functions of ${\displaystyle t}$ and certain constants, are in general altered by a change of the system of reference. Physical science has, however, been so built up that there are certain transformations which leave the form of the equations unaltered. In the classical, or Newtonian, mechanics these are all orthogonal transformations of the three coordinates ${\displaystyle x,y,z,}$ combined with an arbitrary uniform velocity of these axes and an arbitrary change of the units of length and time.

The principle of relativity can be enunciated as the postulate that the transformations, with respect to which the laws of nature shall be invariant, are "Lorentz-transformations."[1] A Lorentz-transformation is defined by a modulus ${\displaystyle q}$ and an axis. Taking the latter as the axis of ${\displaystyle x}$, the transformation-formulæ are—

 ${\displaystyle x'={\frac {x-qct}{\sqrt {1-q^{2}}}},\quad y'=y,\quad z'=z,\quad ct'={\frac {ct-qx}{\sqrt {1-q^{2}}}},}$

where ${\displaystyle c}$ is a universal constant, which, according to the electromagnetic theory, is equal to the velocity of light in free space. In Newtonian mechanics the value of this constant is ${\displaystyle c=\infty }$. Putting, then, ${\displaystyle q=v/c}$, so that ${\displaystyle q=0}$, the Lorentz-transformation degenerates into a "Newton-transformation,"—

 ${\displaystyle x'=x-vt,\quad y'=y,\quad z'=z,\quad t'=t}$

To both Lorentz- and Newton-transformations may be added an arbitrary orthogonal transformation of coordinates.

The physical meaning of the principle has been very clearly explained in these pages by Messrs. Plummer and Whittaker,[2] and need not be repeated here. The mathematical formulæ are all that is required for our purpose.

The literature of the subject is very extensive, and it is hardly possible for an outsider to be even superficially acquainted with it. Also I do not claim originality for any of the formulæ or results given below. The starting-point of my investigations has been the papers by Poincaré and Minkowski.[3] The manner in which the equations of motion are derived below is entirely derived from the last section of Poincaré’s paper. I also owe much to conversations with and advice from my colleague Professor Lorentz.

2. Let there be two systems of reference:—

the "general" system ${\displaystyle (x',y',z',t')}$,

and the "special" system ${\displaystyle (x,y,z,t)}$.

The first is an absolutely arbitrary system of reference. The three variables ${\displaystyle x',y',z'}$ define what Newton would call absolute space, and ${\displaystyle t'}$ corresponds to absolute time.

The second is a system of reference of which the origin and the direction of the axes are chosen according to the needs of the problem in hand. The transformation from ${\displaystyle x,y,z,t}$ to ${\displaystyle x',y',z',t',}$ or inversely, is a Lorentz-transformation.

The chief consequence of the principle of relativity is that it is impossible to observe any but relative motions. The "general" system only differs from any other possible system in that no convention is made as to its origin or the direction of its axes. It is thus not possible, from the point of view of the principle of relativity, to speak of "absolute" velocity or position otherwise than as velocity or position relative to any, not specified, "general" system of reference. If the word "absolute" is used in this sense, it is unobjectionable, but unnecessary. The laws of nature must, of course, be primarily framed with respect to the general system of reference. This does not mean that they assert anything about "absolute" motion or "absolute" time, but only that they must be so built up as to be true in whatever system of reference we choose to use.

We have thus to consider two problems:—

(a) What is the law of force that must replace Newton’s law, and what is the motion of a planet under this law? So far as this differs from ordinary Keplerian motion, we shall have to consider the question whether the differences are large enough to be verified by observation.

(b) A motion being given in any "special" system, how is it expressed in the "general" system, or in any other special system? This only involves the working out of the formulæ of transformation for the required cases.

A third question then arises, viz. whether the "astronomical system," i.e. the coordinates and the time actually used by practical astronomers, coincides or not with any "special" system as defined above. This question is practically not affected by the introduction of the principle of relativity, and we need refer to it only very briefly.

3. I will now begin by stating the necessary formulæ for a Lorentz-transformation in the form which I find most convenient for my purpose. Very probably most of them have already been published elsewhere in the same form, but I found it easier to work them out for myself than to search for them in an unfamiliar literature.

The formulæ become more symmetrical if the unit of time is so chosen that ${\displaystyle c=1}$. From an astronomical point of view, however, it is more convenient to retain the ordinary unit of time. Both advantages are combined by using ${\displaystyle ct}$ as variable instead of ${\displaystyle t}$. Derivatives with respect to ${\displaystyle ct}$ are denoted by Greek letters, thus—

 ${\displaystyle \xi ={\frac {dx}{cdt}},\quad \eta ={\frac {dy}{cdt}},\quad \zeta ={\frac {dz}{cdt}}}$.
Further, we put
 ${\displaystyle r^{2}=x^{2}+y^{2}+z^{2},\quad \phi ^{2}=\xi ^{2}+\eta ^{2}+\zeta ^{2}}$

The modulus of the transformation is ${\displaystyle q}$. We also introduce ${\displaystyle q_{1}=1-{\sqrt {1-q^{2}}}}$. If ${\displaystyle q}$ is a small quantity of the first order, then ${\displaystyle q_{1}}$ is of the second order. The cosines of the angles which the axis of the transformation makes with the axes of ${\displaystyle x,y,z}$ are denoted by ${\displaystyle \alpha ,\beta ,\gamma ,}$ so that ${\displaystyle \alpha ^{2}+\beta ^{2}+\gamma ^{2}=1}$.

The transformation-formulæ are then—

 ${\displaystyle x'=x+{\frac {\alpha \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},}$ ${\displaystyle y'=y+{\frac {\beta \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},}$ ${\displaystyle z'=z+{\frac {\gamma \left[q_{1}r_{q}-qct\right]}{\sqrt {1-q^{2}}}},}$ ${\displaystyle ct'={\frac {ct-qr_{q}}{\sqrt {1-q^{2}}}}.}$ (1)

We find easily

 ${\displaystyle \xi '={\frac {dx'}{cdt'}}={\frac {\xi {\sqrt {1-q^{2}}}+\alpha \left[q_{1}\phi _{q}-q\right]}{1-q\phi _{q}}},}$ (2)

and similarly for ${\displaystyle \eta '}$ and ${\displaystyle \zeta '}$.

Further,

 ${\displaystyle 1-\phi '^{2}={\frac {\left(1-\phi ^{2}\right)\left(1-q^{2}\right)}{\left(1-q\phi _{q}\right)^{2}}}.}$

In these formulæ ${\displaystyle r_{q}}$ and ${\displaystyle \phi _{q}}$ are the projections of ${\displaystyle \rho }$ and ${\displaystyle \phi }$ on the axis of transformation:—

 ${\displaystyle r_{q}=\alpha x+\beta y+\gamma z,}$ ${\displaystyle \phi _{q}=\alpha \xi +\beta \eta +\gamma \zeta .}$

If we put

 ${\displaystyle (\kappa )={\frac {1}{\sqrt {1-\phi ^{2}}}},}$ ${\displaystyle (\xi )={\frac {\xi }{(\kappa )}},\ (\eta )={\frac {\eta }{(\kappa )}},\ (\zeta )={\frac {\zeta }{(\kappa )}},}$ ${\displaystyle (\phi )^{2}=(\xi )^{2}+(\eta )^{2}+(\zeta )^{2},}$

we can easily verify that the transformation-formulas for ${\displaystyle (\xi ),(\eta ),(\zeta ),(\kappa )}$ are the same as those for ${\displaystyle x,y,z,t}$, viz.—

 ${\displaystyle (\xi )'=(\xi )+{\frac {\alpha \left[q_{1}\left(\phi \right)_{q}-q(\kappa )\right]}{\sqrt {1-q^{2}}}},}$ ${\displaystyle (\kappa )'={\frac {(\kappa )-q(\phi )_{q}}{\sqrt {1-q^{2}}}}.}$ (3)
The invariants of the transformation are all of the form—,
 ${\displaystyle x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}-ct_{1}ct_{2},}$ (4)

where ${\displaystyle x_{1},\ y_{1},\ z_{1},\ ct_{1}}$, or ${\displaystyle x_{2},\ y_{2},\ z_{2},\ ct_{2},}$ may be replaced by any set of quantities, which are transformed by the same formulæ, such as ${\displaystyle (\xi ),(\eta ),(\zeta ),(\kappa ),}$ etc.

The equation

 ${\displaystyle (\kappa )^{2}-(\phi )^{2}=1}$

is thus not altered by the transformation. If now we define a new variable ${\displaystyle c\tau }$ by the equations

 ${\displaystyle {\frac {d\tau }{dt}}={\sqrt {1-\phi ^{2}}}={\frac {1}{(\kappa )}},\ {\text{or}}\ {\frac {dt}{d\tau }}=(\kappa )={\sqrt {1-(\phi )^{2}}},}$ (5)

this variable is the same function of ${\displaystyle x',y',z',t'}$ as of ${\displaystyle x,y,z,t,}$ and is consequently independent of the system of reference. We have, of course,

 ${\displaystyle (\xi )={\frac {dx}{cd\tau }},\quad (\eta )={\frac {dy}{cd\tau }},\quad (\zeta )={\frac {dz}{cd\tau }},\quad (\kappa )={\frac {dct}{cd\tau }}.}$

The variable ${\displaystyle \tau }$ is called by Minkowski the "Eigenzeit" of the point whose coordinates are ${\displaystyle x,y,z,}$ which may be translated by "proper-time".[4] In many problems it is more convenient as an independent variable than ${\displaystyle t}$.

Every point has thus its own proper-time, which is independent of the system of reference, but depends on the state of motion of the point and on its previous history. The proper-time of a point rigidly connected with the axes of the system of reference (${\displaystyle x,y,z,t}$) is ${\displaystyle t}$ itself. As a convenient abbreviation, we may speak of "heliocentric time," "geocentric time," etc., meaning the proper-time of the Sun, the Earth, etc.

4. A set of values of ${\displaystyle m,x,y,z,t,}$ defining the position of a particle of mass ${\displaystyle m}$ in the system of reference (${\displaystyle x,y,z,t}$), may be called an "event." Two events are called simultaneous if their values of ${\displaystyle t}$ are the same. Two events which are simultaneous in one system (${\displaystyle x,y,z,t}$) are in general not simultaneous in another system (${\displaystyle x',y',z',t'}$). And, within certain restrictions, which are of no importance for our purpose, a system can always be found in which two arbitrarily given events are simultaneous.

We have

 ${\displaystyle c\left(t'_{1}-t'_{2}\right)={\frac {c\left(t_{1}-t_{2}\right)-q\Delta _{q}}{\sqrt {1-q^{2}}}},}$

where

 ${\displaystyle \Delta _{q}=\alpha \left(x_{1}-x_{2}\right)+\beta \left(y_{1}-y_{2}\right)+\gamma \left(z_{1}-z_{2}\right).}$
Thus, if we wish the two events to be simultaneous with respect to the time ${\displaystyle t',}$ or ${\displaystyle t'_{1}=t'_{2}}$, we have
 ${\displaystyle c\left(t_{1}-t_{2}\right)=q\Delta _{q}.}$

Denoting the coordinates of simultaneous events by non-italic letters, we find thus—

 ${\displaystyle \mathrm {x} '_{1}-\mathrm {x} '_{2}=x_{1}-x_{2}+{\frac {\alpha \left[q_{1}\Delta _{q}-qc\left(t_{1}-t_{2}\right)\right]}{\sqrt {1-q^{2}}}},}$

or, remarking that ${\displaystyle q_{1}-q^{2}=-q_{1}{\sqrt {1-q^{2}}},}$

 ${\displaystyle \mathrm {x} '_{1}-\mathrm {x} '_{2}=x_{1}-x_{2}-\alpha q_{1}\Delta _{q},}$ (6)

and similarly for the other coordinates.

We find easily, denoting the distance between simultaneous positions by ${\displaystyle {\boldsymbol {r}}}$:—

 ${\displaystyle {\boldsymbol {r}}'_{q}=\Delta '_{q}=\Delta _{q}-q_{1}\Delta _{q}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)=\left(1-q_{1}\right)\Delta _{q}=\Delta _{q}{\sqrt {1-q^{2}}}.}$

Therefore (6) can be written—

 ${\displaystyle x_{1}-x_{2}=\mathrm {x} '_{1}-\mathrm {x} '_{2}+\alpha {\frac {q_{1}{\boldsymbol {r}}'_{q}}{\sqrt {1-q^{2}}}}.}$ (7)

5. If we consider the action on ${\displaystyle m_{1}}$ at time ${\displaystyle t_{1}}$ of a force emanating from ${\displaystyle m_{2}}$ at time ${\displaystyle t_{2}}$, we will suppose—

 ${\displaystyle c\left(t_{1}-t_{2}\right)=\Delta ={\sqrt {\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}}}.}$ (8)

The expression

 ${\displaystyle c^{2}\left(t_{1}-t_{2}\right)^{2}-\Delta ^{2}}$

is of the general form (4), and is thus an invariant of the transformation. We have thus also ${\displaystyle c\left(t'_{1}-t'_{2}\right)=\Delta '}$. The equation (8) states that the force is propagated through space with the velocity of light. This is, of course, an arbitrary assumption, which is not a necessary consequence of the principle of relativity. The velocity of propagation might be defined by any invariant of the transformation containing ${\displaystyle c\left(t_{1}-t_{2}\right)}$, put equal to zero. But it is a natural assumption, and the most simple which can be made.[5]

Denoting now the simultaneous relative coordinates by letters of another type, ${\displaystyle {\boldsymbol {x,y,z}}}$, we have, for time ${\displaystyle t_{1}}$,

 ${\displaystyle {\boldsymbol {x}}=\mathrm {x} _{1}-\mathrm {x} _{2}=x_{1}-x_{2}+\Delta \xi _{2}-\delta x_{2},}$ (9)

where

 ${\displaystyle \delta x_{2}={\tfrac {1}{2}}\Delta ^{2}{\frac {d^{2}x_{2}}{c^{2}dt^{2}}}+{\tfrac {1}{6}}\Delta ^{3}{\frac {d^{3}x_{2}}{c^{3}dt^{3}}}+\dots }$
The values of ${\displaystyle \xi _{2}}$ and the higher differential coefficients must be taken for the time ${\displaystyle t_{2}}$, defined by (8). We consider ${\displaystyle 1/c}$ as a small quantity of the first order, and we wish our formulæ to be exact to the second order inclusive.

We find easily

 ${\displaystyle \Delta ^{2}\left(1-\phi _{2}^{2}\right)={\boldsymbol {r}}^{2}+2{\boldsymbol {r}}\epsilon _{2}\Delta +2{\boldsymbol {r}}\chi _{2},}$

where

 {\displaystyle {\begin{aligned}{\boldsymbol {r}}\epsilon _{1}&={\boldsymbol {x}}\xi _{1}+{\boldsymbol {y}}\eta _{1}+{\boldsymbol {z}}\zeta _{1},\\{\boldsymbol {r}}\epsilon _{2}&={\boldsymbol {x}}\xi _{2}+{\boldsymbol {y}}\eta _{2}+{\boldsymbol {z}}\zeta _{2},\\{\boldsymbol {r}}\chi _{2}&={\boldsymbol {x}}\delta x_{2}+{\boldsymbol {y}}\delta y_{2}+{\boldsymbol {z}}\delta z_{2}.\end{aligned}}}

${\displaystyle \epsilon _{1}}$ and ${\displaystyle \epsilon _{2}}$ are of the first order, ${\displaystyle \chi _{2}}$ is of the second order.

In the equations of motion there appear the invariants—

 {\displaystyle {\begin{aligned}\mathrm {A} &=c\left(t_{2}-t_{1}\right)(\kappa )_{1}-\left(x_{2}-x_{1}\right)(\xi )_{1}-\left(y_{2}-y_{1}\right)(\eta )_{1}-\left(z_{2}-z_{1}\right)(\zeta )_{1},\\\mathrm {B} &=c\left(t_{1}-t_{2}\right)(\kappa )_{2}-\left(x_{1}-x_{2}\right)(\xi )_{2}-\left(y_{1}-y_{2}\right)(\eta )_{2}-\left(z_{1}-z_{2}\right)(\zeta )_{2},\\\mathrm {C} &=(\kappa )_{1}(\kappa )_{2}-(\xi )_{1}(\xi )_{2}-(\eta )_{1}(\eta )_{2}-(\zeta )_{1}(\zeta )_{2}.\end{aligned}}}

To express these in simultaneous relative coordinates we have—

 {\displaystyle {\begin{aligned}\mathrm {A} {\sqrt {1-\phi _{1}^{2}}}&=-\Delta \left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)+{\boldsymbol {r}}\epsilon _{1},\\\mathrm {B} ^{2}\left(1-\phi _{2}^{2}\right)&={\boldsymbol {r}}^{2}\left(1-\phi _{2}^{2}+\epsilon _{2}^{2}\right)+2{\boldsymbol {r}}\chi _{2},\end{aligned}}} (10)
 ${\displaystyle \mathrm {C} ={\frac {1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}}{\sqrt {\left(1-\phi _{1}^{2}\right)\left(1-\phi _{2}^{2}\right)}}}.}$

6. The equations of motion can be given in three forms:—

 ${\displaystyle mc{\frac {d(\xi )}{d\tau }}=m{\frac {d^{2}x}{d\tau ^{2}}}=(\mathrm {X} )={\frac {\mathrm {X} }{\sqrt {1-\phi ^{2}}}},}$ (11)
 ${\displaystyle mc{\frac {d(\xi )}{d\tau }}=m{\frac {d}{dt}}\left({\frac {dx}{d\tau }}\right)=\mathrm {X} =(\mathrm {X} ){\sqrt {1-\phi ^{2}}},}$ (12)
 ${\displaystyle m{\frac {d^{2}x}{dt^{2}}}=\left\{\mathrm {X} -\xi (\mathrm {X} \xi +\mathrm {Y} \eta +\mathrm {Z} \zeta )\right\}{\sqrt {1-\phi ^{2}}},}$ (13)

and similarly for the other coordinates.

${\displaystyle \mathrm {X} }$ is the force according to the ordinary definition, or "Newtonian" force; ${\displaystyle (\mathrm {X} )}$ is called the "Minkowskian" force.[6] The mass ${\displaystyle m}$ is a constant.

Differentiating the formula ${\displaystyle (\kappa )^{2}=1+(\phi )^{2}}$, we derive a fourth equation analogous to (11) or (12), viz.:

 ${\displaystyle mc{\frac {d(\kappa )}{d\tau }}=m{\frac {d^{2}(ct)}{d\tau ^{2}}}=(\mathrm {T} )={\frac {\mathrm {T} }{\sqrt {1-\phi ^{2}}}},}$ (11′)
or
 ${\displaystyle mc{\frac {d(\kappa )}{dt}}=m{\frac {d}{dt}}\left({\frac {dct}{d\tau }}\right)=\mathrm {T} =(\mathrm {T} ){\sqrt {1-\phi ^{2}}},}$ (12′)

where

 ${\displaystyle \mathrm {T} =\mathrm {X} \xi +\mathrm {Y} \eta +\mathrm {Z} \zeta .}$

Minkowski gives the name "kinetic energy" to the quantity

 {\displaystyle {\begin{aligned}\mathrm {E} &=mc^{2}\left[(\kappa )-1\right]={\tfrac {1}{2}}mc^{2}\phi ^{2}+{\tfrac {3}{8}}mc^{2}\phi ^{4}+\dots ,\\{\text{or}}\qquad \mathrm {E} &={\tfrac {1}{2}}m{\mathsf {\mathrm {v} }}^{2}\left(1+{\tfrac {3}{4}}\phi ^{2}+\dots \right).\end{aligned}}}

The equation (12′) thus turns out to be the equation of energy,

 ${\displaystyle {\frac {d\mathrm {E} }{dt}}=\mathrm {X} {\frac {dx}{dt}}+\mathrm {Y} {\frac {dy}{dt}}+\mathrm {Z} {\frac {dz}{dt}}.}$

If we use Minkowskian velocities and forces, and put

 ${\displaystyle (\mathrm {E} )={\tfrac {1}{2}}mc^{2}(\phi )^{2},}$

we find similarly from (11′)

 ${\displaystyle {\frac {d(\mathrm {E} )}{d\tau }}=(\mathrm {X} ){\frac {dx}{d\tau }}+(\mathrm {Y} ){\frac {dy}{d\tau }}+(\mathrm {Z} ){\frac {dz}{d\tau }}.}$

The law of the force must be such that the form of the equations of motion is not changed by a Lorentz-transformation. Therefore

 ${\displaystyle (\mathrm {X} ),\ (\mathrm {Y} ),\ (\mathrm {Z} ),\ (\mathrm {T} )}$

must be transformed by the same formulæ as

 ${\displaystyle {\begin{array}{cccc}x,&y,&z,&t,\\(\xi ),&(\eta ),&(\zeta ),&(\kappa ),\end{array}}}$

or, in other words, ${\displaystyle \mathrm {(X),(Y),(Z)} }$ must be linear functions of

 ${\displaystyle {\begin{array}{ccccccc}x_{1},&y_{1},&z_{1},&&x_{2},&y_{2},&z_{2},\\(\xi )_{1},&(\eta )_{1},&(\zeta )_{1},&&(\xi )_{2},&(\eta )_{2},&(\zeta )_{2},\end{array}}}$

the coefficients being invariants of the transformation.

For zero velocities the equations of Newtonian mechanics must be reproduced, therefore the coordinates can only enter by their differences ${\displaystyle x_{1}-x_{2}}$, etc.

Introducing further the condition that the resulting equations must not contain the velocities in the first degree, and certain other simplifications, Poincaré is led to take (l.c., page 174)—

 ${\displaystyle (\mathrm {X} )_{1}={\frac {k^{2}m_{1}m_{2}}{B_{1}^{3}}}\left\{x_{2}-x_{1}-(\xi )_{2}{\frac {\mathrm {A} _{1}}{\mathrm {C} }}\right\}.}$ (14)
Similarly we have for the force acting on ${\displaystyle m_{2}}$ from ${\displaystyle m_{1}}$,
 ${\displaystyle (\mathrm {X} )_{2}={\frac {k^{2}m_{1}m_{2}}{A_{2}^{3}}}\left\{x_{1}-\mathrm {x} _{2}-(\xi )_{1}{\frac {\mathrm {B} _{2}}{\mathrm {C} }}\right\}.}$ (15)

We will now introduce simultaneous coordinates. Let these be for time ${\displaystyle t}$

 ${\displaystyle \mathrm {x_{1},\ y_{1},\ z_{1}} }$ and ${\displaystyle \mathrm {x_{2},\ y_{2},\ z_{2}} .}$

In the equations of motion of ${\displaystyle m_{1}}$, i.e. in the expression (14), we must use the coordinates and velocities of ${\displaystyle m_{2}}$ for the time ${\displaystyle t_{2}}$ defined by

 ${\displaystyle c\left(t-t_{2}\right)=\Delta _{2}={\sqrt {\left(\mathrm {x} _{1}-x_{2}\right)^{2}+\left(\mathrm {y} _{1}-y_{2}\right)^{2}+\left(\mathrm {z} _{1}-z_{2}\right)^{2}}},}$

and we have

 ${\displaystyle {\boldsymbol {x}}=\mathrm {x} _{1}-\mathrm {x} _{2}=\mathrm {x} _{1}-x_{2}-\Delta _{2}\xi _{2}-\delta x_{2}.}$

In (15) we must use the coordinates and velocities of ${\displaystyle m_{1}}$ for the time ${\displaystyle t_{1}}$ defined by

 ${\displaystyle c\left(t-t_{1}\right)=\Delta _{1}={\sqrt {\left(\mathrm {x} _{2}-x_{1}\right)^{2}+\left(\mathrm {y} _{2}-y_{1}\right)^{2}+\left(\mathrm {z} _{2}-z_{1}\right)^{2}}},}$

and we have

 ${\displaystyle {\boldsymbol {x}}=x_{1}-\mathrm {x} _{2}+\Delta _{1}\xi _{1}-\delta x_{1}.}$

Further, we have for use in (14)—

 {\displaystyle {\begin{aligned}&\mathrm {B} _{1}^{2}\left(1-\phi _{2}^{2}\right)={\boldsymbol {r}}^{2}\left(1-\phi _{2}^{2}+\epsilon _{2}^{2}\right)+2{\boldsymbol {r}}\chi _{2},\\&\mathrm {A} _{1}{\sqrt {1-\phi _{1}^{2}}}=-\Delta _{2}\left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)+{\boldsymbol {r}}\epsilon _{1},\end{aligned}}}

and in (15)—

 {\displaystyle {\begin{aligned}&\mathrm {A} _{2}^{2}\left(1-\phi _{1}^{2}\right)={\boldsymbol {r}}^{2}\left(1-\phi _{1}^{2}+\epsilon _{1}^{2}\right)-2{\boldsymbol {r}}\chi _{1},\\&\mathrm {B} _{1}{\sqrt {1-\phi _{2}^{2}}}=-\Delta _{1}\left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)-{\boldsymbol {r}}\epsilon _{2}.\end{aligned}}}

The expression for ${\displaystyle \mathrm {C} }$ is the same in both cases.

We find then,

 ${\displaystyle {\begin{array}{ccl}&&(\mathrm {X} )_{1}={\dfrac {k^{2}m_{1}m_{2}}{B_{1}^{3}}}\left\{-{\boldsymbol {x}}-\delta x_{2}-{\dfrac {{\boldsymbol {r}}\epsilon _{1}\xi _{2}}{1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}}}\right\},\\(I.)\\&&(\mathrm {X} )_{2}={\dfrac {k^{2}m_{1}m_{2}}{A_{2}^{3}}}\left\{{\boldsymbol {x}}-\delta x_{1}+{\dfrac {{\boldsymbol {r}}\epsilon _{1}\xi _{2}}{1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}}}\right\}.\end{array}}}$ (16)

This form of the equations is not unique. We can multiply by any power of ${\displaystyle \mathrm {C} }$, or make more complicated alterations, for which the reader is referred to Poincaré.

Multiplying by ${\displaystyle \mathrm {C} }$ we get—

 ${\displaystyle (\mathrm {X} )_{1}={\frac {k^{2}m_{1}m_{2}}{\mathrm {B} _{1}^{3}{\sqrt {\left(1-\phi _{1}^{2}\right)\left(1-\phi _{2}^{2}\right)}}}}\left\{-\left({\boldsymbol {x}}+\delta x_{2}\right)\left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)-{\boldsymbol {r}}\epsilon _{1}\xi _{2}\right\}.}$ (17)
 (II.)${\displaystyle \quad (\mathrm {X} )_{2}={\frac {k^{2}m_{1}m_{2}}{A_{2}^{3}{\sqrt {\left(1-\phi _{1}^{2}\right)\left(1-\phi _{2}^{2}\right)}}}}\left\{\left({\boldsymbol {x}}-\delta x_{1}\right)\left(1-\xi _{1}\xi _{2}-\eta _{1}\eta _{2}-\zeta _{1}\zeta _{2}\right)+{\boldsymbol {r}}\epsilon _{2}\xi _{1}\right\}.}$ (17)

This last form (II.) is preferred by Lorentz (l.c., p. 1239), because the corresponding Newtonian force, ${\displaystyle \mathrm {X} _{i}=(\mathrm {X} )_{i}{\sqrt {1-\phi _{i}^{2}}}}$, does not contain the velocity of ${\displaystyle m_{i}}$. (I.) is the law adopted by Minkowski, presumably because it gives the simplest result for a planet of infinitesimal mass.

As has already been remarked, the values of ${\displaystyle \xi _{2},\ \eta _{2},\ \zeta _{2},}$ in ${\displaystyle (\mathrm {X} )_{1}}$, and those of ${\displaystyle \xi _{1},\ \eta _{1},\ \zeta _{1},}$ in ${\displaystyle (\mathrm {X} )_{2}}$ differ from the values at the time ${\displaystyle t}$ for which the simultaneous coordinates are taken. If we are content to neglect third orders, however, we can assume all velocities to correspond to the time ${\displaystyle t}$. If the motion were quasi-stationary, i.e. if the accelerations could be neglected, the velocities would be constant, and also ${\displaystyle \delta x_{2}}$ and ${\displaystyle \delta x_{1}}$ would disappear. The equations (16) and (17) would in that case be rigorous.

7. We will first consider the law (I.). Introducing the developments of ${\displaystyle \mathrm {B} _{1}}$ and ${\displaystyle \mathrm {A} _{2}}$ we find—

 {\displaystyle {\begin{aligned}(\mathrm {X} )_{1}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}-\delta x_{2}-{\boldsymbol {r}}\epsilon _{1}\xi _{2}+{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{2}^{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{2}\right\},\\(\mathrm {X} )_{2}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{{\boldsymbol {x}}-\delta x_{1}+{\boldsymbol {r}}\epsilon _{2}\xi _{1}-{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{1}^{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{1}\right\}\end{aligned}}} (18)

We thus have by (13)—

 {\displaystyle {\begin{aligned}m_{1}{\frac {d^{2}\mathrm {x} _{1}}{dt^{2}}}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}-\delta x_{2}+3{\boldsymbol {\frac {x}{r}}}\chi _{2}+{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{2}^{2}-{\boldsymbol {r}}\epsilon _{1}\xi _{2}+{\boldsymbol {r}}\epsilon _{1}\xi _{1}+{\boldsymbol {x}}\phi _{1}^{2}\right\}\\m_{2}{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}&={\frac {k^{2}m_{1}m_{2}}{{\boldsymbol {r}}^{3}}}\left\{{\boldsymbol {x}}-\delta x_{1}+3{\boldsymbol {\frac {x}{r}}}\chi _{1}-{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{1}^{2}+{\boldsymbol {r}}\epsilon _{2}\xi _{1}-{\boldsymbol {r}}\epsilon _{2}\xi _{2}-{\boldsymbol {x}}\phi _{2}^{2}\right\}\end{aligned}}} (19)

Now we have, to second orders,—

 {\displaystyle {\begin{aligned}\delta x_{2}&={\tfrac {1}{2}}\Delta _{2}^{2}{\frac {d^{2}x_{2}}{c^{2}dt^{2}}}={\tfrac {1}{2}}{\frac {k^{2}m_{1}}{c^{2}}}\cdot {\boldsymbol {\frac {x}{r}}}\\\delta x_{1}&={\tfrac {1}{2}}\Delta _{1}^{2}{\frac {d^{2}x_{1}}{c^{2}dt^{2}}}=-{\tfrac {1}{2}}{\frac {k^{2}m_{2}}{c^{2}}}\cdot {\boldsymbol {\frac {x}{r}}}\\\chi _{2}={\boldsymbol {\frac {x}{r}}}\delta x_{2}&+{\boldsymbol {\frac {y}{r}}}\delta y_{2}+{\boldsymbol {\frac {z}{r}}}\delta z_{2}={\tfrac {1}{2}}{\frac {k^{2}m_{1}}{c^{2}}}\end{aligned}}} ${\displaystyle \chi _{1}=-{\tfrac {1}{2}}{\frac {k^{2}m_{2}}{c^{2}}}.}$

Further, if we put

 ${\displaystyle \mathrm {M} =m_{1}+m_{2},\quad m_{1}=\mu \mathrm {M} ,\quad m_{2}=(1-\mu )\mathrm {M} ,}$ ${\displaystyle \lambda ^{2}={\frac {k^{2}\mathrm {M} }{c^{2}}},}$
we find
 {\displaystyle {\begin{aligned}{\frac {d^{2}\mathrm {x} _{1}}{dt^{2}}}&={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}(1-\mu )\left\{-{\boldsymbol {x}}+\mu \lambda ^{2}{\boldsymbol {\frac {x}{r}}}+\left(\xi _{1}-\xi _{2}\right){\boldsymbol {r}}\epsilon _{1}+{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{2}^{2}+{\boldsymbol {x}}\phi _{1}^{2}\right\},\\{\frac {d^{2}\mathrm {x} _{2}}{dt^{2}}}&={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\mu \left\{{\boldsymbol {x}}+(1-\mu )\lambda ^{2}{\boldsymbol {\frac {x}{r}}}+\left(\xi _{1}-\xi _{2}\right){\boldsymbol {r}}\epsilon _{2}-{\tfrac {3}{2}}{\boldsymbol {x}}\epsilon _{1}^{2}-{\boldsymbol {x}}\phi _{2}^{2}\right\}.\end{aligned}}} (20)

In these equations the system of reference is arbitrary. We can always, by a Lorentz-transformation, make the velocity of any point equal to zero. If there were a point in the system having no acceleration, like the centre of gravity in Newtonian mechanics, the equations could be simplified by taking that point as origin of the system of reference. Now the integrals of the motion of the centre of gravity do not exist in this simple form under the new law, but it is still possible to find a point without acceleration. The centre of gravity has the coordinates ${\displaystyle x_{0},\ y_{0}\ z_{0}}$, defined by—

 ${\displaystyle x_{0}=\mu x_{1}+(1+\mu )x_{2},\dots }$

We have thus

 ${\displaystyle \xi _{0}=\mu \xi _{1}+(1-\mu )\xi _{2}.}$

If we denote relative velocities by letters without index, thus—

 ${\displaystyle \xi =\xi _{1}-\xi _{2},}$ ${\displaystyle \phi ^{2}=\xi ^{2}+\eta ^{2}+\zeta ^{2}}$,${\displaystyle {{\boldsymbol {r}}\epsilon }={{\boldsymbol {x}}\xi }+{{\boldsymbol {y}}\eta }+{{\boldsymbol {z}}\zeta },}$ etc.

We have

 ${\displaystyle \xi _{1}=\xi _{0}+(1-\mu )\xi ,}$${\displaystyle \epsilon _{1}=\epsilon _{0}+(1-\mu )\epsilon ,}$ ${\displaystyle \xi _{2}=\xi _{0}-\mu \xi ,}$etc.

We find at once from (20)—

 ${\displaystyle {\frac {d^{2}x_{0}}{dt^{2}}}={\frac {k^{2}\mathrm {M} \mu (1-\mu )}{{\boldsymbol {r}}^{3}}}\left\{-(1-2\mu )\lambda ^{2}{\boldsymbol {\frac {x}{r}}}+{\boldsymbol {r}}\xi \left(\epsilon _{1}+\epsilon _{2}\right)-{\tfrac {3}{2}}{\boldsymbol {x}}\left(\epsilon _{1}^{2}-\epsilon _{2}^{2}\right)+{\boldsymbol {x}}\left(\phi _{1}^{2}-\phi _{2}^{2}\right)\right\}.}$ (21)

The right-hand member is of the second order, and is thus the same for the laws I. and II. In it we can, for the relative coordinates and velocities, use their values in ordinary Keplerian motion. If then we take the orbital plane for the plane of (${\displaystyle x,y}$) and the axis of ${\displaystyle x}$ through the perihelion, the equations (21) become—

 {\displaystyle {\begin{aligned}{\frac {dc\xi _{0}}{dt}}&=\mathrm {P} +\mathrm {Q} .c\xi _{0}+\mathrm {R} .c\eta _{0}\\{\frac {dc\eta _{0}}{dt}}&=\mathrm {P} '+\mathrm {Q} .c\xi _{0}+\mathrm {R} '.c\eta _{0}\\{\frac {dc\zeta _{0}}{dt}}&=0.\end{aligned}}}
The coefficients ${\displaystyle \mathrm {P,Q,R,P',Q',R'} }$ are periodic functions of ${\displaystyle t}$ devoid of constant terms.[7] Consequently ${\displaystyle c\xi _{0},\ c\eta _{0}}$ are also periodic, and ${\displaystyle c\zeta _{0}}$ is constant. The point the components of whose velocity id are the non-periodic parts of ${\displaystyle c\xi _{0},\ c\eta _{0},\ c\zeta _{0}}$ therefore has no acceleration. If we perform a Lorentz-transformation to a new system having this point as its origin, then the mean values of ${\displaystyle \xi _{0},\ \eta _{0}}$ and ${\displaystyle \zeta _{0}}$ are zero. Thus, if we neglect the periodic terms (which are, moreover, of the second order, and would introduce into (20) only terms of the fourth order, and those multiplied by ${\displaystyle \mu '}$), we have—
 ${\displaystyle \xi _{1}=(1-\mu )\xi ,\quad \xi _{2}=-\mu \xi ,\quad {\text{etc.}}}$

By subtracting the second equation (20) from the first we find then—

 ${\displaystyle {\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}}={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\left\{-{\boldsymbol {x}}+{{\boldsymbol {r}}\epsilon \xi }+{{\boldsymbol {x}}\phi ^{2}}\right\}+{\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\mu '\left\{2\lambda ^{2}{\boldsymbol {\frac {x}{r}}}-2{{\boldsymbol {r}}\xi }\epsilon -3{\boldsymbol {x}}\phi ^{2}+{\tfrac {3}{2}}\mu ''{\boldsymbol {x}}\epsilon ^{2}\right\},}$ (22)

where

 ${\displaystyle \mu '=\mu (1-\mu )={\frac {m_{1}m_{2}}{\mathrm {M} ^{2}}},\ \mu ''=1-2\mu ={\frac {m_{2}-m_{1}}{\mathrm {M} }}.}$

If the mass of the planet is neglected we have ${\displaystyle \mu '=0}$, and we find—

 ${\displaystyle {\frac {d^{2}{\boldsymbol {x}}}{dt^{2}}}+k^{2}\mathrm {M} {\frac {\boldsymbol {x}}{{\boldsymbol {r}}^{3}}}={\frac {k^{2}\mathrm {M} }{{\boldsymbol {r}}^{3}}}\left\{{{\boldsymbol {r}}\epsilon \xi }+{\boldsymbol {x}}\phi ^{2}\right\}.}$ (23)
Comparing (13) and (11) we find that (23) is equivalent to
 ${\displaystyle {\frac {d^{2}{\boldsymbol {x}}}{d\tau ^{2}}}+k^{2}\mathrm {M} {\frac {\boldsymbol {x}}{{\boldsymbol {r}}^{3}}}=0.}$ (24)

This is the equation given by Minkowski (l.c., p. 110). It could, of course, have been derived by taking ${\displaystyle \mu =0}$ from the beginning. In that case the acceleration of ${\displaystyle m_{2}}$ is zero, and we can take ${\displaystyle m_{2}}$ as the origin of our system of reference. Then ${\displaystyle \xi _{2}=\eta _{2}=\zeta _{2}=0}$, ${\displaystyle \mathrm {B} _{1}={\boldsymbol {r}},\ \mathrm {x} _{1}={\boldsymbol {x}}}$, and (24) is derived at once from (14).

In (24) ${\displaystyle \tau }$ is the proper-time of the planet, or planeto-centric time; in (23) ${\displaystyle t}$ is the time of the system of reference, which has been chosen with its origin in the mean position of the centre of gravity. Practically ${\displaystyle t}$ will coincide with the proper-time of the Sun, or heliocentric time.

The equations for the other two coordinates are, of course, quite similar.

The equations for the law II. are derived from those for the law I. by multiplying the right-hand members by ${\displaystyle \mathrm {C} }$. Introducing the relative velocities we find, neglecting fourth orders,

 ${\displaystyle \mathrm {C} ={\frac {1}{\sqrt {1-{\boldsymbol {\phi }}^{2}}}}.}$

The only difference between the laws II. and I. is therefore that in (22) and (23,) the term ${\displaystyle {{\boldsymbol {x}}\phi }^{2}}$ is changed to ${\displaystyle {\tfrac {1}{2}}{{\boldsymbol {x}}\phi }^{2}}$. The simple form (24) is no longer possible with the law II. We have, however, in accordance with (12), for this law—

 ${\displaystyle {\frac {d}{dt}}{\frac {d{\boldsymbol {x}}}{d\tau }}+k^{2}{\frac {\mathrm {M} }{{\boldsymbol {r}}^{3}}}=0.}$ (25)

The only previous investigations on the same subject which have come to my knowledge, beyond the quoted papers of Poincaré and Minkowski, are those by Mr. Wilkens ("Zur Elektronentheorie," V.J.S. 1904, p. 209) and by Mr. Wacker ("Ueber Gravitation und Elektromagnetismus," Inaugural Dissertation. (Tübingen), 1909). Mr. Wilkens finds secular terms in all elements. Some of these have amounts which could not long remain undetected. Thus, e.g., his secular perturbation of the mean distance corresponds, for the Earth, to a shortening of the year by 19 seconds per century. His formulæ, however, are not in conformity with the principle of relativity, and his results are not confirmed here.

Mr. Wacker gives the equations (25). His equations are thus in conformity with the principle of relativity, and he adopts the law II., and neglects the mass of the planet.

8. The equation (24) is the ordinary equation of Keplerian motion. Consequently, if we neglect the mass of the planet and use planeto-centric time, the motion is Keplerian. Thus, e.g., the integrals of areas are[8]

 ${\displaystyle x{\frac {dy}{d\tau }}-y{\frac {dx}{d\tau }}=\mathrm {G} ,\ {\text{etc}}.}$

Or, introducing heliocentric time, to second orders—

 ${\displaystyle x{\frac {dy}{dt}}-y{\frac {dx}{dt}}=\mathrm {G} \left\{1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{p}}\left(1+e^{2}+2e\ \cos \ v\right)\right\}}$[9]. (26)

In (22), on the other hand, it is not advantageous to introduce the proper-time of one of the bodies, since we should thereby lose the symmetry gained by the introduction of relative velocities and coordinates. In the second-order terms We can introduce the ordinary Keplerian motion. We find then, taking the orbital plane for plane of (${\displaystyle x,y}$), for the two laws equally—

 ${\displaystyle x{\frac {dy}{dt}}-y{\frac {dx}{dt}}={\text{const. }}-(1-2\mu ')\mathrm {G} _{0}{\frac {\lambda ^{2}}{p}}e\ \cos \ v,}$ (27)

where

 ${\displaystyle \mathrm {G} _{0}=a^{2}n{\sqrt {1-e^{2}}}=k{\sqrt {\mathrm {M} }}{\sqrt {p}},}$

and v is the true anomaly.

Similarly we find for the vis-viva integral from (24)—

 ${\displaystyle \left({\frac {dx}{d\tau }}\right)^{2}+\left({\frac {dy}{d\tau }}\right)^{2}=k^{2}\mathrm {M} \left({\frac {2}{r}}-{\frac {1}{a}}\right),}$

or in heliocentric time—

 ${\displaystyle \left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}={\frac {k^{2}\mathrm {M} \left({\frac {2}{r}}-{\frac {1}{a}}\right)}{1+\lambda ^{2}\left({\frac {2}{r}}-{\frac {1}{a}}\right)}}.}$ (28)

From (22), on the other hand, we find—

 {\displaystyle {\begin{aligned}\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}&=2{\frac {k^{2}\mathrm {M} }{r}}+{\text{const.}}-4k^{2}\mathrm {M} {\frac {\lambda ^{2}}{a}}\left({\frac {a}{r^{2}}}-{\frac {1}{r}}\right)+\\&+2k^{2}\mathrm {M} \mu '{\frac {\lambda ^{2}}{a}}\left[{\frac {4a}{r^{2}}}-{\frac {5}{r}}+{\frac {1}{2}}\mu ''\left\{{\frac {ap}{r^{3}}}-3{\frac {a}{r^{2}}}+{\frac {3}{r}}\right\}\right].\end{aligned}}} (29)

For the law II. we must add to (28) and (29) the term—

 ${\displaystyle +k^{2}\mathrm {M} {\frac {\lambda ^{2}}{a}}\left({\frac {a}{r^{2}}}-{\frac {1}{r}}\right)+{\text{const.}}}$
The equation (28) can then also be written in the form given by Wacker (l.c., page 55),
${\displaystyle \mathrm {E} ={\frac {k^{2}\mathrm {M} }{r}}+{\text{const.}},}$

where

 ${\displaystyle \mathrm {E} ={\frac {1}{\sqrt {1-\phi ^{2}}}}-1={\tfrac {1}{2}}{\bigg [}\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}{\bigg ]}\left(1+{\tfrac {3}{4}}\phi ^{2}+\dots \right).}$

9. The equations (24) of course give, taking the orbital plane as plane of (${\displaystyle x,y}$),—

 ${\displaystyle x=a(\cos \ u-e),}$${\displaystyle y=a{\sqrt {1-e^{2}}}\sin \ u,}$ ${\displaystyle r=a(1-e\ \cos \ u),}$ (30)

where

 ${\displaystyle u-e\ \sin \ u=n\tau ,\quad a^{3}n^{2}=k^{2}\mathrm {M} .}$ (31)

To introduce heliocentric time we have

 ${\displaystyle {\frac {dt}{d\tau }}={\sqrt {1+(\phi )^{2}}}=1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}{\frac {1+e\ \cos \ u}{1-e\ \cos \ u}},}$
 ${\displaystyle d\tau ={\frac {1-e\ \cos \ u}{n}}du,}$

therefore

 ${\displaystyle ndt=\left(1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right){\bigg (}1-{\frac {1-{\frac {1}{2}}{\frac {\lambda ^{2}}{a}}}{1+{\frac {1}{2}}{\frac {\lambda ^{2}}{a}}}}e\ \cos \ u{\bigg )}du}$

or if we put

 {\displaystyle {\begin{aligned}n'&=n\left(1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right)={\frac {k^{2}\mathrm {M} }{a^{3/2}}}\left(1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right),\\e'&=e\left(1-{\frac {\lambda ^{2}}{a}}\right),\end{aligned}}} (32)

we have

 ${\displaystyle u-e'\ \sin \ u=n't.}$ (33)

Consequently for the law I. the coordinates of a planet of infinitesimal mass are expressed by the ordinary formulæ (30) of elliptical motion, but to express the excentric anomaly in heliocentric time we must in the equation of Kepler (33) use a slightly different excentricity, and also Kepler’s third law is no longer quite exact. The deviations from pure Keplerian motion are periodic and have ${\displaystyle \lambda ^{2}}$ as a factor. Taking the semi-major axis of the Earth’s orbit as unit of length, the value of ${\displaystyle \lambda ^{2}}$ is

 ${\displaystyle \lambda ^{2}=9.85\times 10^{-9}.}$

All these periodic terms are thus entirely insensible. The length of the year appears to be different when measured in geocentric or in heliocentric time. This, of course, is a purely formal change of unit, and has nothing to do with the number of days in the year.

The difference between heliocentric and geocentric time, apart from this change of unit, is purely periodic. We find

 ${\displaystyle t=\tau \left(1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right)+{\frac {\lambda ^{2}}{an}}e\ \sin \ u,}$ (34)

which is the formula given by Minkowski, l.c., page 111. The value of the coefficient of ${\displaystyle \sin u}$ is, for the Earth, ${\displaystyle 0^{\mathrm {s} }.00083}$.

10. For the integration of the equations (22), or the corresponding equations for the law II., the easiest way seems to be to use the Keplerian motion as a first approximation, and to compute "perturbations" by the method of variation of elements, introducing the radial, transversal, and orthogonal perturbing forces. Taking the orbital plane for plane of (${\displaystyle x,y}$), these forces are found to be

 for law I.: ${\displaystyle \mathrm {S} =\mathrm {S} _{1}+\mathrm {S} _{2}+\mu '\{-3\mathrm {S} _{1}-(2-{\tfrac {3}{2}}\mu '')\mathrm {S} _{2}+\mathrm {S} _{3}\},}$ ${\displaystyle \mathrm {T} =(1-2\mu ')\mathrm {T} _{1},}$ ${\displaystyle \mathrm {W} =0,}$ for law II.: ${\displaystyle \mathrm {S} ={\tfrac {1}{2}}\mathrm {S} _{1}+\mathrm {S} _{2}+\mu '\{-3\mathrm {S} _{1}-(2-{\tfrac {3}{2}}\mu '')\mathrm {S} _{2}+\mathrm {S} _{3}\},}$ ${\displaystyle \mathrm {T} =(1-2\mu ')\mathrm {T} _{1},}$ ${\displaystyle \mathrm {W} =0,}$

where

 ${\displaystyle \mathrm {S} _{1}=k^{2}\mathrm {M} {\frac {\phi ^{2}}{r^{2}}},\quad \mathrm {S} _{2}=k^{2}\mathrm {M} {\frac {\epsilon ^{2}}{r^{2}}},\quad \mathrm {S} _{3}=2{\frac {\lambda ^{2}}{r^{3}}},}$ ${\displaystyle \mathrm {T} _{1}=k^{2}\mathrm {M} {\frac {x\eta -y\xi }{r^{3}}}\epsilon .}$

In the perturbing forces we can use the undisturbed or Keplerian motion. We find, then, taking the axis of ${\displaystyle x}$ towards the (unperturbed) perihelion,—

 {\displaystyle {\begin{aligned}\mathrm {S} _{1}&={\frac {\lambda ^{2}}{r^{3}}}(1+e\ \cos \ u)={\frac {\lambda ^{2}}{r^{2}}}\cdot {\frac {1+e^{2}+2e\ \cos \ v}{p}},\\\mathrm {S} _{2}&={\frac {\lambda ^{2}}{r^{4}}}ae^{2}\sin ^{2}u={\frac {\lambda ^{2}}{r^{2}}}\cdot {\frac {e^{2}\sin ^{2}v}{p}},\\\mathrm {S} _{3}&={\frac {\lambda ^{2}}{r^{4}}}ae{\sqrt {1-e^{2}}}\sin \ u={\frac {\lambda ^{2}}{r^{3}}}e\ \sin v.\end{aligned}}}

In the well-known equations for the variation of elements, we introduce as independent variable, instead of the time, either the excentric or the true anomaly, or the radius-vector, by—

 ${\displaystyle dt={\frac {r}{an}}du={\frac {r^{2}}{a^{2}n{\sqrt {1-e^{2}}}}}dv={\frac {\operatorname {cosec} v}{ane{\sqrt {1-e^{2}}}}}dr.}$
With the true anomaly as independent variable, the equations are then—
 ${\displaystyle da={\frac {2a^{2}r^{2}}{p}}\left[\mathrm {S} e\ \sin \ v+\mathrm {T} {\frac {p}{r}}\right]dv,}$ {\displaystyle {\begin{aligned}de&=r^{2}{\Big [}\mathrm {S} \ \sin \ v+\mathrm {T} \ (\cos \ u+\cos \ v){\Big ]}dv,\\ed\varpi &=r^{2}{\Big [}-\mathrm {S} \ \cos \ v+\mathrm {T} (1+{\frac {r}{p}}\sin \ v){\Big ]}dv,\end{aligned}}} ${\displaystyle d\epsilon _{1}=-{\frac {2r^{3}}{a{\sqrt {1-e^{2}}}}}\mathrm {S} dv,}$ ${\displaystyle \delta \epsilon =\delta \epsilon _{1}+{\frac {e^{2}}{1+{\sqrt {1-e^{2}}}}}\delta \varpi .}$

We find the following "perturbations":—

in ${\displaystyle \delta a}$: from ${\displaystyle \mathrm {S} _{1}:-\ 2\lambda ^{2}{\frac {a^{2}}{r^{2}}}e\ \cos \ u}$
 (35)
" ${\displaystyle \mathrm {S} _{2}:{\tfrac {2}{3}}\lambda ^{2}\left(1-e^{2}\right){\frac {a^{3}}{r^{3}}}-2\lambda ^{2}{\frac {a^{2}}{r^{2}}}e\ \cos \ u,}$
" ${\displaystyle \mathrm {S} _{3}:{\frac {4\lambda ^{2}}{\left(1-e^{2}\right)^{2}}}\left(e\ \cos \ v+{\tfrac {1}{4}}e^{2}\ \cos \ 2v\right),}$
" ${\displaystyle \mathrm {T} _{1}:-\ {\tfrac {2}{3}}\lambda ^{2}\left(1-e^{2}\right){\frac {a^{3}}{r^{3}}}}$;
in ${\displaystyle \delta e}$: from ${\displaystyle \mathrm {S} _{1},\ \mathrm {S} _{2},\ \mathrm {S} _{3}:{\tfrac {1}{2}}{\frac {1-e^{2}}{ae}}\times }$ the corresponding terms in ${\displaystyle \delta a}$
" ${\displaystyle \mathrm {T} _{1}:{\tfrac {1}{2}}{\frac {1-e^{2}}{ae}}.\delta a+{\frac {\lambda ^{2}}{a}}\cos \ v;}$
in ${\displaystyle e\delta \varpi }$: from ${\displaystyle \mathrm {S} _{1}:-\ {\frac {\lambda ^{2}}{p}}\left[\left(1+e^{2}\right)\sin \ v+{\tfrac {1}{2}}e\ \sin \ sv+ev\right],}$
" ${\displaystyle \mathrm {S} _{2}:-\ {\frac {\lambda ^{2}}{p}}.{\tfrac {1}{3}}e^{2}\sin ^{3}v,}$
" ${\displaystyle \mathrm {S} _{3}:-\ {\frac {\lambda ^{2}}{p}}\left[2\ \sin \ v-{\tfrac {1}{2}}e\ \sin \ 2v-ev\right],}$
" ${\displaystyle \mathrm {T} _{1}:\quad {\frac {\lambda ^{2}}{p}}\left[-{\tfrac {1}{2}}e\ \sin \ 2v+{\tfrac {1}{3}}e^{2}\sin ^{3}v+ev\right];}$
in ${\displaystyle \delta \epsilon _{1}}$: from ${\displaystyle \mathrm {S} _{1}:\quad 2{\frac {\lambda ^{2}}{a}}u-4{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}v,}$
" ${\displaystyle \mathrm {S} _{2}:\quad 2{\frac {\lambda ^{2}}{a}}u-2{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}(v-e\ \sin \ v),}$
" ${\displaystyle \mathrm {S} _{3}:-\ 4{\frac {\lambda ^{2}}{a{\sqrt {1-e^{2}}}}}v.}$
The periodic terms, all of which have very short periods, can be entirely neglected. For the law I. we find that there are no secular terms not multiplied by ${\displaystyle \mu '}$, which is in accordance with what was found above. The terms which have ${\displaystyle \mu '}$ as a factor also give nothing secular in ${\displaystyle \delta a}$ or ${\displaystyle \delta e}$. In ${\displaystyle \delta \varpi }$ we have—
 ${\displaystyle \delta \varpi =2\mu '{\frac {\lambda ^{2}}{p}}v.}$

For the law II. we find this same term, but also a secular term not multiplied by ${\displaystyle \mu }$, viz.—

 ${\displaystyle \delta \varpi ={\frac {1}{2}}{\frac {\lambda ^{2}}{p}}v.}$ (36)

With the law II. we have thus a secular motion of the perihelia, even for a planet of infinitesimal mass.[10] The motion in one century is—

 For Mercury[11] : ${\displaystyle \delta \varpi }$ = + 7″.15 ${\displaystyle e\delta \varpi }$ = + 1″.470 (37) " Venus : 1.43 0.009 " the Earth : 0.639 0.011 " Mars : 0.225 0.021 " Jupiter : 0.0104 " Saturn : 0.0023 " Uranus : 0.00040 " Neptune : 0.00013
 For ☄ Encke : ${\displaystyle \delta \varpi }$ = + 0″.308 " ☄ Halley : 0.0071

The effect of the secular terms in ${\displaystyle \delta \epsilon _{1}}$ is, of course, only to make the observed mean motion disagree with the value derived from Kepler's third law. As we have no sufficiently accurate means of independently determining the mean distances, this can never be detected.

The formulæ (35) are valid for all excentricities, and can thus also be used for comets. The perturbations in ${\displaystyle \delta \varpi }$ remain of the same order as for planets. Those in ${\displaystyle \delta e}$ and ${\displaystyle \delta \epsilon _{1}}$ become zero for a parabola. Only the perturbation in ${\displaystyle \delta a}$ would appear to become very large at perihelion. We have, however,

 ${\displaystyle \delta p=\left(1-e^{2}\right)\delta a-2ae\delta e,}$

from which we find, for both laws,

 ${\displaystyle \delta p=-2\lambda ^{2}\left(1-2\mu '\right)e\ \cos \ v,}$

which does remain small for ${\displaystyle e=1}$.

The two laws I. and II. are the only ones that have been actually proposed, but we can, without violating the principle of relativity, multiply the forces by any power of ${\displaystyle \mathrm {C} }$, and consequently any (positive or negative or even fractional) multiple of the quantities (37) will be in agreement with that principle.

11. The final result by the method of the preceding article must, for the law I. and for ${\displaystyle \mu =0}$, of course be the same as that derived in article 9, and contained in the formulæ (30), (32), (33). We have, since ${\displaystyle \varpi _{0}=0}$,—

 ${\displaystyle x=a_{0}\left(\cos \ u_{0}-e_{0}\right)+\delta x,}$ ${\displaystyle y=a_{0}{\sqrt {1-e_{0}^{2}}}\sin \ u_{0}+\delta y,b}$ ${\displaystyle u_{0}-e_{0}\sin \ u_{0}=n_{0}t,\quad a_{0}^{3}n_{0}^{2}=k^{2}\mathrm {M} ,}$ {\displaystyle {\begin{aligned}\delta x&={\frac {x}{a}}\delta a+{\tfrac {3}{2}}{\frac {an\ \sin \ u}{r}}\int \delta adt-{\frac {a^{2}}{p}}\sin \ u{\Big [}\delta \epsilon _{1}+e\ \cos \ v\left(\delta \epsilon -\delta \varpi \right){\Big ]}\\&\qquad \qquad \qquad \qquad \qquad \qquad +{\frac {ae^{2}}{\sqrt {1-e^{2}}}}\sin \ u\delta \varpi -\left[{\frac {a^{2}\sin ^{2}u}{r}}+a\right]\delta e,\\\delta y&={\frac {y}{a}}\delta a-{\tfrac {3}{2}}{\frac {an{\sqrt {1-e^{2}}}\cos \ u}{r}}\int \delta adt\\&\qquad \qquad \qquad \qquad \qquad +{\frac {a^{2}}{p}}{\sqrt {1-e^{2}}}\cos \ u{\Big [}\delta \epsilon _{1}+e\ \cos \ v\left(\delta \epsilon -\delta \varpi \right){\Big ]}\\&\qquad \qquad \qquad \qquad -ae\delta \varpi +\left[{\frac {a^{2}{\sqrt {1-e^{2}}}}{r}}\cos \ u-{\frac {ae}{\sqrt {1-e^{2}}}}\right]\sin \ u\delta e,\end{aligned}}} (38)

where ${\displaystyle \delta a,\delta e,e\delta \varpi ,\delta \epsilon _{1}}$ must be collected from (35), taking the values for ${\displaystyle S_{1}+S_{2}+T_{1}}$, and

 ${\displaystyle e(\delta \epsilon -\delta \varpi )=e\delta \epsilon _{1}-{\sqrt {1-e^{2}}}.e\delta \varpi .}$

On the other hand, from article 9, we have—

 ${\displaystyle \delta x=-a\ \sin \ u\Delta u+(\cos \ u-e)\Delta a-a\Delta e,}$
 ${\displaystyle \delta y=a{\sqrt {1-e^{2}}}\cos \ u\Delta u+a{\sqrt {1-e^{2}}}\sin \ u\Delta a-{\frac {ae}{\sqrt {1-e^{2}}}}\sin \ u\Delta e,}$ (39)

where

 ${\displaystyle r\Delta u=-\left({\tfrac {3}{2}}\Delta a+{\tfrac {1}{2}}\lambda ^{2}\right)(u-e\ \sin \ u)+\left(a\Delta e-\lambda ^{2}e\right)\sin \ u,}$

and we have put

 ${\displaystyle a=a_{0}+\Delta a,\qquad e=e_{0}+\Delta e.}$

By equating the values of ${\displaystyle \delta x}$ and ${\displaystyle \delta y}$ given by (38) and (39) we must then find for ${\displaystyle \Delta a}$ and ${\displaystyle \Delta e}$ constant values. To the first power of ${\displaystyle e}$ I find—

 ${\displaystyle \Delta a=\lambda ^{2},\qquad \Delta e={\tfrac {3}{2}}{\frac {\lambda ^{2}}{a}}.e.}$
12. In the motion of the Moon we have first the same terms as in the planetary motion, but ${\displaystyle \mathrm {M} }$ in the expression for ${\displaystyle \lambda ^{2}}$ is now the sum of the masses of the Earth and Moon. The motion of the perigee in one century for the law II. is found from (36)—
 ${\displaystyle \delta \varpi =+0''.0101.}$

But, in addition to this, the disturbing force of the Sun is changed. To find the effect of this, we must use the first equation of (20), first taking for ${\displaystyle m_{1}}$ the Moon and for ${\displaystyle m_{2}}$ the Sun, and then for ${\displaystyle m_{1}}$ the Earth and for ${\displaystyle m_{2}}$ the Sun. The difference of the two equations thus derived—without the first term, which gives the ordinary Newtonian perturbation—will furnish the new terms in the acceleration of the Moon with respect to the Earth.

We can neglect the masses of the Moon and the Earth, and the acceleration of the Sun. Then, if we take the origin of the system of reference in the Sun, we find for the additional terms—

 ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=k^{2}m\left\{{\frac {\Delta \epsilon _{1}\xi _{1}}{\Delta ^{3}}}+{\frac {x_{1}\phi _{1}^{2}}{\Delta ^{3}}}-{\frac {\rho \epsilon _{2}\xi _{2}}{\rho ^{3}}}-{\frac {x_{2}\phi _{2}^{2}}{\rho ^{3}}}\right\},}$

where the suffix 1 refers to the Moon and 2 to the Earth, and where

 ${\displaystyle \Delta }$ = distance Sun — Moon, ${\displaystyle \rho }$ = ⁠" Sun — Earth,

and ${\displaystyle m}$ is the Sun's mass. For the law II. the terms involving ${\displaystyle \phi _{1}^{2}}$ and ${\displaystyle \phi _{2}^{2}}$ are halved.

A development in series cannot be avoided. We will retain only the terms of the first order in the parallax, and in the inclination and excentricities, and of these only those which can give secular perturbations, i.e. which do not contain the Sun's longitude. I find—

 {\displaystyle {\begin{aligned}{\frac {d^{2}x}{dt^{2}}}&=-{\frac {k^{2}m\nu }{\rho }}\left[{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\cos \ l+{\tfrac {3}{4}}a\nu e+{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)e\ \cos \ 2l\right]\\&+{\frac {k^{2}m\nu }{\rho }}\left[\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)\cos \ l+{\tfrac {3}{4}}a\nu e+\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{4}}a\nu \right)e\ \cos \ 2l\right],\\{\frac {d^{2}y}{dt^{2}}}&=-{\frac {k^{2}m\nu }{\rho }}\left[{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\sin \ l+{\tfrac {1}{2}}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)e\ \sin \ 2l\right]\\&+{\frac {k^{2}m\nu }{\rho }}\left[\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{2}}a\nu \right)\sin \ l+\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {1}{4}}a\nu \right)e\ \sin \ 2l\right],\\{\frac {d^{2}z}{dt^{2}}}&={\tfrac {1}{2}}{\frac {k^{2}m\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-a\nu \right)\sin \ i\ \sin \left(l-\Omega \right)\\&-{\frac {k^{2}m\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-{\tfrac {3}{2}}a\nu \right)\sin \ i\ \sin \left(l-\Omega \right).\end{aligned}}}
The second lines must be halved for law II. We have put—

{\displaystyle {\begin{aligned}&\lambda '={\frac {k{\sqrt {\mathrm {M} }}}{c}},\quad \mathrm {M} ={\text{mass of Earth + Moon,}}\\&\nu =n'/c,\quad n'={\text{mean motion of Sun.}}\\&l={\text{the Moon's mean anomaly.}}\end{aligned}}}

I find then for the two laws—

 I. II. ${\displaystyle S={\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\lambda '\nu }{\rho {\sqrt {a}}}}(1+e\ \cos \ l),}$ ${\displaystyle {\frac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a\nu ^{2}}{\rho }}\left(1-e\ \cos \ l\right),}$ ${\displaystyle T=-{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\lambda '\nu }{\rho {\sqrt {a}}}}e\ \sin \ l,}$ ${\displaystyle 0,}$ ${\displaystyle W=-{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-2a\nu \right)i\ \sin \ (l-\Omega )}$ ${\displaystyle {\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a\nu ^{2}}{\rho }}i\ \sin \ (l-\Omega ).}$

There are no secular perturbations in ${\displaystyle a}$, ${\displaystyle e}$, and ${\displaystyle i}$. In the other elements we find—

 I. II. ${\displaystyle \delta \varpi =-{\frac {m}{\mathrm {M} }}{\frac {a^{3/2}\lambda '\nu }{\rho }}.nt,}$ ${\displaystyle -{\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,}$ ${\displaystyle \delta \epsilon _{1}=-{\frac {m}{\mathrm {M} }}{\frac {a^{3/2}\lambda '\nu }{\rho }}.nt,}$ ${\displaystyle -{\tfrac {1}{2}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,}$ ${\displaystyle \delta \Omega =-{\tfrac {1}{4}}{\frac {m}{\mathrm {M} }}{\frac {a^{2}\nu }{\rho }}\left({\frac {\lambda '}{\sqrt {a}}}-2a\nu \right).nt,}$ ${\displaystyle +{\tfrac {1}{8}}{\frac {m}{\mathrm {M} }}{\frac {a^{3}\nu ^{2}}{\rho }}.nt,}$

The numerical values for one century are—

 I. II. ${\displaystyle \delta \varpi =-1''.272}$ ${\displaystyle -0''.024}$ ${\displaystyle \delta \Omega =-0.270}$ ${\displaystyle +0.011}$

These quantities are too small to be detected from observation. Even the value of ${\displaystyle \delta \varpi }$ for law I. is well within the limits of uncertainty of the observed value.

13. We are thus left with the motion of the perihelion of Mercury as the only effect which reaches an appreciable amount. Unfortunately this same motion presents the well-known excess of observation on theory, which has been explained by Seeliger by the attraction of the masses forming the zodiacal light.[12] Until we have some independent means of accurately determining this mass — which seems a very remote possibility indeed — any motion of the perihelion of Mercury within reasonable limits can be so explained. It will be remarked that the value (37) of ${\displaystyle e\delta \varpi }$ for Mercury is only little larger than the part of the motion (1″.203), which Seeliger explains by a rotation of the astronomical system of coordinates with respect to the "inertial system," i.e. the system to which the equations of motion are referred. If this rotation were rejected, and the term (37) were added, most of Seeliger's residuals would be very little changed. There would then be appreciable residuals only for the nodes of Mercury and Venus, and the perihelion of Mars. The node of Venus could be put right by increasing the density of Seeliger's second ellipsoid to about three times the value adopted by Seeliger (which would still leave it less than 1/2000 of that of the inner ellipsoid). Seeliger's rotation is practically determined from the motions of the node of Mercury and the perihelion of Mars, the excesses of which over their theoretical values, as determined by Newcomb, are 1.2 and 2.1 times their respective probable errors. If the density of the second ellipsoid were so determined as to represent the node of Venus exactly, the residual in the perihelion of Mars would be reduced to 1.6 times its p.e. The other residuals would not be much affected,[13] and the representation would thus be on the whole very satisfactory.

14. We now come to the effect of a Lorentz-transformation on the elements of the orbit. First consider the orbital plane. In the system (${\displaystyle x,y,z,t}$), of which the origin is in the Sun, or in the mean centre of gravity if the mass of the planet is not negligible, this plane is—

 ${\displaystyle ax+by+cz=0}$

We must transform to simultaneous coordinates by (6) or (7). Since ${\displaystyle x_{2}=y_{2}=z_{2}=0}$ and ${\displaystyle \xi _{2}=\eta _{2}=\zeta _{2}=0}$ we have ${\displaystyle \Delta _{q}=r_{q},\ x_{1}-x_{2}=x,}$ etc. The transformed plane is thus—

 ${\displaystyle a\left(x'_{1}-x'_{2}\right)+b\left(y'_{1}-y'_{2}\right)+c\left(z'_{1}-z'_{2}\right)+{\frac {q_{1}}{1-q_{1}}}r'_{q}(a\alpha +b\beta +c\gamma )=0,}$

or

 ${\displaystyle a\left(x'_{1}-x'_{2}\right)+b\left(y'_{1}-y'_{2}\right)+c\left(z'_{1}-z'_{2}\right)=0,}$

where

 ${\displaystyle a'=a\left(1-q_{1}\right)+q_{1}\mathrm {C} \alpha ,}$${\displaystyle b'=b\left(1-q_{1}\right)+q_{1}\mathrm {C} \beta ,}$${\displaystyle c'=c\left(1-q_{1}\right)+q_{1}\mathrm {C} \gamma ,}$ ⁠${\displaystyle \mathrm {C} =a\alpha +b\beta +c\gamma .}$
The orbit thus remains fixed and plane after the transformation. Since
 {\displaystyle {\begin{aligned}&a=\sin \ i\ \sin \ \Omega ,\\&b=-\sin \ i\ \cos \ \Omega ,\\&c=\cos \ i,\end{aligned}}}

we find easily

 {\displaystyle {\begin{aligned}\sin \ &id\ \Omega =q_{1}\mathrm {C} \left(\alpha \ \cos \ \Omega +\beta \ \sin \ \Omega \right)\\&di=q_{1}\mathrm {C} \left[\left(\alpha \ \sin \ \Omega -\beta \ \cos \ \Omega \right)\cos \ i-\gamma \ \sin \ i\right]\end{aligned}}} (40)

If ${\displaystyle \mathrm {L} }$ and ${\displaystyle \mathrm {B} }$ are the longitude and latitude of the positive half of the axis of the transformation we have—

 ${\displaystyle \alpha =\cos \ \mathrm {L} \ \cos \ \mathrm {B} ,\qquad \beta =\sin \ \mathrm {L} \ \cos \ \mathrm {B} ,\qquad \gamma =\sin \ \mathrm {B} ,}$ ${\displaystyle \mathrm {C} =\sin \ i\ \cos \ \mathrm {B} \ \sin(\Omega -\mathrm {L} )+\cos \ i\ \sin \ \mathrm {B} ,}$ ${\displaystyle \alpha \ \cos \ \Omega +\beta \ \sin \Omega =\cos \ \mathrm {B} \ \cos(\Omega -\mathrm {L} ),}$ ${\displaystyle \alpha \ \sin \ \Omega -\beta \ \cos \Omega =\cos \ \mathrm {B} \ \sin(\Omega -\mathrm {L} ).}$

If the plane to be transformed is the plane of (${\displaystyle x,y}$) itself, we have in the system (${\displaystyle x,y,z,t}$) ${\displaystyle i_{0}=0}$. The transformed position of the plane is then defined by

 ${\displaystyle \tan \ i'_{0}={\frac {q_{1}\gamma {\sqrt {\alpha ^{2}+\beta ^{2}}}}{1-q_{1}\left(1-\gamma ^{2}\right)}},\qquad \tan \ \Omega '_{0}={\frac {\alpha }{-\beta }},}$

or

 ${\displaystyle \tan \ i'_{0}={\tfrac {1}{2}}q_{1}{\frac {\sin \ 2\mathrm {B} }{1-q_{1}\cos ^{2}\mathrm {B} }},\qquad \Omega '_{0}=\mathrm {L} +90^{\circ }.}$ (41)

Let the plane of (${\displaystyle x,y}$) be the ecliptic, and consider another plane of which the inclination and node in the system (${\displaystyle x,y,z,t}$) are ${\displaystyle i}$ and ${\displaystyle \Omega }$. In the system (${\displaystyle x',y',z',t'}$) its inclination and node on the plane of (${\displaystyle x',y'}$) are ${\displaystyle i+di}$ and ${\displaystyle \Omega +d\Omega }$, as given by the formulæ (40). Let its inclination on the transformed ecliptic be ${\displaystyle i+\Delta i,\Omega +\Delta \Omega }$. Taking unity for the denominator of ${\displaystyle \tan i'_{0}}$ in (41) we find easily—

 {\displaystyle {\begin{alignedat}{2}\sin \ i\Delta \Omega &=\sin \ id\ \Omega &&-q_{1}\gamma \cos \ i(\alpha \ \cos \ \Omega +\beta \ \sin \Omega ),\\\Delta i&=di&&+q_{1}\gamma \cos \ i(\beta \ \cos \ \Omega -\alpha \ \sin \Omega ),\end{alignedat}}}

or

 ${\displaystyle \Delta \Omega =q_{1}\cos ^{2}\mathrm {B} \ \sin(\Omega -\mathrm {L} )\cos(\Omega -\mathrm {L} )}$ {\displaystyle {\begin{aligned}\Delta i=q_{1}\sin \ i[\cos ^{2}\mathrm {B} \ \sin ^{2}(\Omega -\mathrm {L} )-\ &\sin ^{2}B\ \cos \ i],\\&-q_{1}\ \sin ^{2}i\ \sin \ \mathrm {B} \ \cos \ \mathrm {B} \ \sin \ (\Omega -\mathrm {L} ).\end{aligned}}}

If the transformed plane be the equator, we have—

 ${\displaystyle \Omega =\Upsilon =pt,\qquad i=\epsilon ,}$
where ${\displaystyle p}$ is the constant of general precession in longitude. We have thus—
 ${\displaystyle \Delta \Upsilon =-{\tfrac {1}{2}}q_{1}\cos ^{2}\mathrm {B} \ \sin \ 2\mathrm {L} +q_{1}\ \cos ^{2}\mathrm {B} \ \cos \ 2\mathrm {L} .pt.}$

If we take for the axis of the transformation the direction of the Sun’s motion relatively to the fixed stars, and for ${\displaystyle q}$ the velocity of this motion, divided by the velocity of light, so that in the system (${\displaystyle x',y',z',t')}$ the mean velocity of the stars is zero, we have—

 ${\displaystyle \mathrm {L} =270^{\circ },\quad \mathrm {B} =+55^{\circ },\quad q=0.000070.}$

We find—

 ${\displaystyle \Delta \Upsilon =-0''.0000044{\text{ per century.}}}$ (42)

This is, on the basis of the principle of relativity, the theoretical difference between the constant of procession as determined from the fixed stars (system ${\displaystyle x',y',z',t'}$) and from the motions in the solar system (system ${\displaystyle x,y,z,t}$).

15. If we take the orbital plane in the system (${\displaystyle x,y,z,t}$) as the plane of (${\displaystyle x,y}$), and choose the axis of ${\displaystyle x}$ perpendicular to the axis of the transformation, we have—

 ${\displaystyle \alpha =0,\qquad r_{q}=\beta y,}$

and the formulæ for transformation to simultaneous relative coordinates in the system (${\displaystyle x',y',z',t'}$) become—

 ${\displaystyle x'=x,\quad y'=y\left(1-\beta ^{2}q_{1}\right),\quad z'=-\beta \gamma q_{1}y,\quad ct'={\frac {ct-\beta qy}{\sqrt {1-q^{2}}}}.}$

The problem is now, if ${\displaystyle x,y}$ are given as functions of ${\displaystyle t}$, to find the expression of ${\displaystyle x',y',z'}$ as functions of ${\displaystyle t'}$. Knowing that the orbit remains plane, we can eliminate ${\displaystyle z'}$ by combining with our Lorentz-transformation a rotation round the axis of ${\displaystyle x'}$ by an angle given by—

 {\displaystyle {\begin{aligned}\mathrm {K} \ \sin \ \mathrm {I} &=\ -\ \gamma \beta q_{1},\\\mathrm {K} \ \cos \ \mathrm {I} &=1-\beta ^{2}q_{1},\end{aligned}}}

from which

 ${\displaystyle \mathrm {K} ^{2}=1-2q_{1}+2q_{1}^{2}=1+2q_{1}{\sqrt {1-q^{2}.}}}$

The new plane of ${\displaystyle x'_{1},\ y'_{1}}$ is thus the transformed orbital plane, and we have—

 ${\displaystyle x'_{1}=x'=x,\quad y'_{1}=y'\ \sec \ \mathrm {I} =\mathrm {K} y.}$

If, to take a simple example, the orbit in (${\displaystyle x,y,z,t}$) were a circle—

 ${\displaystyle x=a\ \cos \ nt,\qquad y=a\ \sin \ nt,}$
we find
 ${\displaystyle n't'=nt-{\frac {an\beta q}{c}}\sin \ nt,\quad n'=n{\sqrt {1-q^{2}}},}$

or putting

 ${\displaystyle \epsilon ={\frac {an\beta q}{c}},}$

we have

 ${\displaystyle \phi -\epsilon \ \sin \phi =n't',}$ ${\displaystyle x'_{1}=a\ \cos \ \phi ,\qquad y'_{1}=a\mathrm {K} \ \sin \phi .}$ (43)

If the orbit in (${\displaystyle x,y,z,t}$) is a "perturbed" ellipse, the formulæ of transformation are more complicated, but the differences between the real motion and pure elliptic motion remain of the second order and periodic in terms of ${\displaystyle t',}$ if they were so in terms of ${\displaystyle t}$.

16. One of the consequences of the principle of relativity is that it must be impossible by observations on bodies belonging to one and the same system to detect a motion of the whole system. Suppose in the system of reference (${\displaystyle x,y,z,t}$) the Sun to be at rest, and a planet to describe a circle with uniform velocity. This, of course, is a dynamically possible state of motion under the new law as well as under the old. In the system (${\displaystyle x',y',z',t'}$) derived from (${\displaystyle x,y,z,t}$) by a Lorentz-transformation with the axis (${\displaystyle \alpha ,\beta ,\gamma }$) and the modulus ${\displaystyle q}$, the Sun and the planet have a common velocity ${\displaystyle -cq}$ in the direction (${\displaystyle \alpha ,\beta ,\gamma }$), and the relative orbit is no longer circular, but is defined by (43). Let, again, the orbital plane be chosen as plane of (${\displaystyle x,y}$) in the first system of reference, and let the axis of ${\displaystyle x}$ be perpendicular to the axis of the transformation. The observer belonging to the system has no means of ascertaining the position of the plane of (${\displaystyle x',y'}$), he can only observe the plane of the orbit, i.e. the plane of ${\displaystyle \left(x'_{1},\ y'_{1}\right)}$. To him the velocity of the Sun is thus in the direction (${\displaystyle \alpha ',\beta ',\gamma '}$), where—

 ${\displaystyle \alpha '=0,\qquad \gamma '={\sqrt {1-\beta '^{2}}}}$ ${\displaystyle \beta '=\cos(\mathrm {A} +\mathrm {I} ),{\text{ if }}\beta =\cos \ \mathrm {A} ,}$

or

 ${\displaystyle \beta '\mathrm {K} =\beta {\sqrt {1-q^{2}}}}$

Let the observer be on the Sun, and let a signal be sent him, from the planet every time when this latter crosses the axis of ${\displaystyle y}$.

In the system (${\displaystyle x,y,z,t}$) the intervals between these crossings are equal, and also the times required by the signal to reach the observer are equal: he will observe signals at equal intervals.

In the system (${\displaystyle x',y',z',t'}$) the intervals between the crossings are unequal, and the aberration-times are unequal, and these two effects must cancel each other.

The times of crossing are—

 ${\displaystyle \phi ={\frac {\pi }{2}},\qquad {\frac {3\pi }{2}},\qquad {\frac {5\pi }{2}},\qquad {\text{ etc.}}}$
or
 ${\displaystyle n't'={\frac {\pi }{2}}-\epsilon ,\qquad {\frac {3\pi }{2}}+\epsilon ,\qquad {\frac {5\pi }{2}}-\epsilon ,\qquad {\text{ etc.}}}$

The intervals are therefore—

 ${\displaystyle {\frac {\pi +2\epsilon }{n'}},\qquad {\frac {\pi -2\epsilon }{n'}},\qquad {\text{ etc.}}}$

or

 ${\displaystyle {\frac {\pi }{n'}}+{\frac {2a\beta q}{c{\sqrt {1-q^{2}}}}},\qquad {\frac {\pi }{n'}}-{\frac {2a\beta q}{c{\sqrt {1-q^{2}}}}},\qquad {\text{ etc.}}}$

Let the aberration-time be called ${\displaystyle b/c}$. We find easily—

 ${\displaystyle b^{2}=\left(a\mathrm {K} \pm \beta 'qb\right)^{2}+q^{2}b^{2}\left(1-\beta '^{2}\right),}$

the upper sign being taken for the crossings through the positive axis of ${\displaystyle y}$ (first and third epoch), and the lower sign for those in which ${\displaystyle y}$ is negative (middle epoch). We find then—

 ${\displaystyle b_{2}-b_{1}=b_{2}-b_{3}=-{\frac {2\beta 'a\mathrm {K} q}{1-q^{2}}}=-{\frac {2a\beta q}{\sqrt {1-q^{2}}}}}$

The corrections to be applied to the true intervals to derive the observed intervals are—

 ${\displaystyle {\frac {b_{2}-b_{1}}{c}},\qquad {\text{ and}}\qquad {\frac {b_{3}-b_{2}}{c}}.}$

The observed intervals are thus seen to be equal, as in the system (${\displaystyle x,y,z,t}$).

Dr. C. V. Burton,[14] following an idea of Maxwell, has proposed to determine the velocity of the solar system "with respect to the aether" from observations of eclipses of Jupiter’s satellites. Apart from the complications introduced by the excentricities and inclinations, and by the observer being on the Earth instead of on the Sun, his method is to determine ${\displaystyle \beta q}$ from the observed inequality in the intervals. If the principle of relativity be true, the result must necessarily be nil. The method could thus be used to verify the principle of relativity.

Mathematically speaking, the dynamical aspect of the principle is not directly concerned in this verification. The observable effect would be nil whatever the motions of the observer and the signaller, which are subjected to the same Lorentz-transformation, may be, even if they were dynamically impossible, so long as the principle is true for the geometrical relations of time, space, and rays of light. But in practice the coordinates are always derived from the integration of equations of motion, and they could not be transformed by a Lorentz-transformation unless the right-hand members of these equations of motion satisfy the conditions imposed by the principle of relativity.

If we take for ${\displaystyle cq}$ the Sun’s velocity relative to the fixed stars, the effect to be expected, in case the principle of relativity were not true, would be equivalent to an inequality in the time of eclipse of ${\displaystyle 0^{\mathrm {s} }.10}$ with a period of 12 years, corresponding to

 ${\displaystyle 0^{\circ }.00023,\quad 0^{\circ }.00012,\quad 0^{\circ }.00006,\quad 0^{\circ }.00002,}$

in the longitudes of the four satellites. Our observations, as well as our knowledge of the principal constants of the theory, must become many times more accurate than they are now, before it will be possible to detect the existence of an inequality of this amount.

17. The velocity of the solar system relatively to the fixed stars, on the other hand, is the same in any system of reference. If in the system (${\displaystyle x',y',z',t'}$) in which the Sun has the velocity ${\displaystyle -cq}$, the star has no velocity (${\displaystyle \xi '_{\star }=\eta '_{\star }=\xi '_{\star }=0}$), then, transforming back to the system (${\displaystyle x,y,z,t}$), in which the Sun has no velocity, we find for the star the velocity ${\displaystyle \xi _{\star }=\alpha cq,\ \eta _{\star }=\beta cq,\ \zeta _{\star }=\gamma cq,}$ (neglecting terms of the second order).

If in the system (${\displaystyle z',y',z',t'}$) two groups of stars have any systematic motion relatively to each other, or all stars have a systematic motion relatively to the axes of reference, the same will be the case in the system (${\displaystyle x,y,z,t}$), apart from small changes (of the second order) in the constants defining the direction of motion relatively to the ecliptic, which are, however, of no importance, as they are constant corrections to quantities whose values must be derived from observation. If no such systematic motion exists in (${\displaystyle x',y',z',t'}$), neither does it exist in (${\displaystyle x,y,z,t}$).

This brings us to the question of the astronomical system of coordinates. Let this be (${\displaystyle \mathrm {x,y,z,t} }$). The observations are referred to a system of axes ${\displaystyle \mathrm {x',y',z'} }$, which is the system of the fundamental catalogue. This system is reduced to (${\displaystyle \mathrm {x,y,z} }$) by a rotation—

 {\displaystyle {\begin{aligned}&\mathrm {x} =\mathrm {x} '\ \cos \ pt+\mathrm {y} '\ \sin \ pt,\\&\mathrm {y} =\mathrm {y} '\ \cos \ pt+\mathrm {x} '\ \sin \ pt,\\&\mathrm {z} =\mathrm {z} ',\end{aligned}}}

where ${\displaystyle p}$ is the constant of precession. This constant is so determined that the fixed stars shall have no systematic rotation with respect to the system (${\displaystyle \mathrm {x,y,z} }$), and the assumption is then made that (${\displaystyle \mathrm {x,y,z} }$) coincides with (${\displaystyle x,y,z}$), the system to which the equations of motion are referred. Whether this is so or not can only be decided by comparison with the motions in the solar system. The problem has been exhaustively treated by Anding (Encyclopädie der Math. Wissensch., vi., 2, page 3) and Seeliger ("Ueber die sogenannte absolute Bewegung," Münchener Sitzungsber., 1906, page 85) and others. Since the nodes are fixed under the new law, and also the perihelia are practically fixed, with the exception of Mercury (for which see art. 13), and the correction (42) to the constant of precession is insensible, the problem is not changed in aspect by the introduction of the principle of relativity, and we need not further refer to it here.

18. Astronomical time is defined by the rotation of the Earth, and is measured by the hour-angle of the axis of ${\displaystyle x}$. Such deviations from uniformity as are derived from the action of known causes—which are, however, negligible, and remain negligible under the new law—are allowed for, and the thus corrected rotation is supposed to be uniform. Now a motion which is in a system (${\displaystyle x,y,z,t}$) described by the formulæ of uniform rotation, is no longer so described in another system (${\displaystyle x',y',z',t'}$). The definition of astronomical time would thus be dependent on the system of reference.

But the axis of ${\displaystyle x}$ cannot be observed. Time is determined by the observation of the hour-angles of stars. In the new system (${\displaystyle x',y',z',t'}$) the direction in which we see the stars is also changed, and a compensation is effected.

Determinations of time are made at intervals, and are used to regulate the clock. Equal intervals on the clock are then supposed to be equal intervals of "time." The construction of the clock depends on the laws of the simple pendulum, which swings under the action of gravitation. We would thus be led to identify astronomical time with the proper-time of the pendulum-bob. Instead of this we can, of course, take the proper-time of the transit-room, or of the centre of the Earth. In this time as variable the motion of the Earth, adopting the law I. and neglecting its mass, is strictly Keplerian. But if we wish to consider the whole solar system, it is better to introduce heliocentric (or "barocentric") time. The difference has been found to be insensible [equation (34)]. The whole question of the definition of astronomical time is thus only of academic interest. So far as the interpretation of observations is concerned, we can identify astronomical time with the variable ${\displaystyle t}$ of any system of reference we wish to use.

1. This name has been first introduced by Poincaré in the paper quoted below.
2. H. C. Plummer. "On the Theory of Aberration and the Principle of Relativity" (M.N., Jan, 1910). E.T.W.: "Recent Researches on Space, Time, and Force" (Report of the Council, M.N., Feb. 1910). Both authors make free use of the word "aether." As there are many physicists nowadays who are inclined to abandon the aether altogether, it may be well to point out that the principle of relativity is essentially independent of the concept of an aether, and, indeed, is considered by some to lead to a negation of its existence. Astronomers have nothing to do with the aether, and it need not concern them whether it exists or not. All Mr. Plummer’s results remain true, and retain their full value, if the "aether" is eliminated from his terminology. And also in Mr. Whittaker’s note the word "aether" is not essential, except, of course, from an historical point of view.
3. Poincaré: "Sur la dynamique de l'électron," Rendiconti del circolo matematico di Palermo, vol. xxi. p. 129 (Dec. 1905). Minkowski: "Die Grundgleichungen für die electromagnetische Vorgänge in bewegten Körpern," Göttinger Nachrichten, Math. physik. Klasse, 1908, page 53.
4. It should be remarked that this is not the same as "local" time, as originally defined by Lorentz.
5. It is, of course, not allowed to put ${\displaystyle c\left(t_{1}-t_{2}\right)=0}$, which would mean instantaneous action at a distance, since the expression ${\displaystyle c\left(t_{1}-t_{2}\right)}$ by itself is not invariant.
6. See Lorentz, Göttingen Lectures, October 1910 (Physikalische Zeitschrift, vol. xi. p. 1234). The names quoted above occur on page 1237.
7. If the true anomaly ${\displaystyle v}$ is introduced as independent variable instead of ${\displaystyle t}$, I find—
 {\displaystyle {\begin{aligned}&\quad {\frac {dc\xi _{0}}{dv}}=\mathrm {P} _{1}+\mathrm {Q} _{1}.c\xi _{0}+\mathrm {R} _{1}.c\eta _{0},{\text{ etc.}}\\\mathrm {P} _{1}&={\frac {k{\sqrt {\mathrm {M} }}}{p^{3/2}}}\lambda ^{2}\mu '\mu ''\{e\ \cos \ 2v+e^{2}\ \cos \ v-{\tfrac {3}{2}}e^{2}\ \sin ^{2}\ v\ \cos \ v\},\\\mathrm {P} '_{1}&={\frac {k{\sqrt {\mathrm {M} }}}{p^{3/2}}}\lambda ^{2}\mu '\mu ''\{e\ \cos \ 2v+2e^{2}\ \cos \ v-{\tfrac {3}{2}}e^{2}\ \sin ^{3}\ v\},\\\mathrm {Q} _{1}&=-{\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \sin \ 2v-3e\ \cos ^{2}\ v\ \sin \ v\right\},\\\mathrm {R} _{1}&=\mathrm {Q} '_{1}={\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \cos \ 2v+2e\ \cos \ v-3e\ \sin ^{2}\ v\ \cos \ v\right\},\\\mathrm {R} '_{1}&={\frac {\lambda ^{2}\mu '}{p}}\left\{2\ \sin \ 2v+4e\ \sin \ v-3e\ \sin ^{3}\ v\right\}.\end{aligned}}}

These are not developments in powers of ${\displaystyle e}$, but are rigorous for all values of ${\displaystyle e}$. It may be mentioned that, if the first term within the brackets in (21), which arises from the non-stationary character of the motion, were neglected, ${\displaystyle \mathrm {P} _{1}}$ would have a constant term, and we should find an acceleration of the centre of gravity.

8. As we shall now use only relative coordinates, the distinction between the different types of letters has become unnecessary, and is dropped.
9. Mr. Wacker has a similar formula, l. c., page 34.
10. This was also found by Mr. Wacker, who gives for Mercury 7".2 per century (l.c., page 58).
11. The value 6″.69 quoted by Professor Lorentz in his Göttingen lecture (l.c., page 1239) was a first approximation.
12. "Das Zodiakallicht und die empirischen Glieder in der Bewegung der innern Planeten," Münchener Sitzungsberichte, 1906, p. 595.
13. The residual for the node of Mars, which is found by Seeliger about equal to the probable error, would almost disappear.
14. Phil. Mag., 1910 March, page 417.

This work is in the public domain in the United States because it was published before January 1, 1923.

The author died in 1934, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 80 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.