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#### §16. –GRAVITATION

WALTER RITZ

##### Translated from Recherches
critiques sur l'Ėlectrodynamique Générale,

*Annales de Chimie et de Physique,* Vol. 13, p. 145, 1908.

Are the preceding theories applicable to gravitation, and can one admit
that the latter propagates with the speed of light and follows the laws that we
have admitted? The answer is affirmative: the perturbations** **are, as in
the theory of Lorentz, of the second order.

*
But it seems that, in addition***,-** it is possible with these new
formulas to make the most notable difference that exists in astronomy between
calculation and observation to disappear, namely the slow rotation of the
ellipse described by Mercury, a
rotation that exceeds by 41*''** of
arc per century what the planetary perturbations would anticipate.*

Let us take as the *x-y* plane the plane of the orbit, the immobile
Sun being at the origin of the coordinates. We

Annales 268

draw from (13)
the equations of motion
[Translation editor’s note. Though some quantities other than charges
must be introduced in (13), Ritz’s idea of retardation, as applied to gravitational interactions, is
rather clear.]

(52) _{}

where μ is a
constant independent of the planet under consideration, and _{} is the distance to
the Sun. Multiplying by *y* and –*x*, and adding, we obtain the
equation of the areas

_{}

or

_{}

(Oeuvres
420) In polar coordinates, and in
neglecting powers of _{} greater than the
second, this may be written

(53) _{}

Let us next remove the quantities of the second order from the first and
second of the equations (52)

_{}

and

_{}

If we add to the equations these same quantities where _{}is replaced by its value of first approximation _{}, and _{} by _{}, we will in the final analysis have introduced only terms in
_{}, which are absolutely negligible.

Annales 269

Multiplying the
new equations obtained by _{} adding and
integrating, we obtain the equation of energy

_{}

Introducing polar coordinates, and** **eliminating *dt* by the
relation (53), then resolving in relation to _{}, we always obtain in terms of about _{},

_{}

(Oeuvres
421) The maximum and minimum of *r*,
or the axes of the ellipse, are the roots of the second factor of the member on
the right; but the ellipse itself turns slowly in its plane. Indeed, if _{} **=** *p*, we
could write

_{}

Annales 270

If we start from
one of the two maximum or minimum values of *p*, corresponding to a root
of the radical, we see that this same value will be recovered, not after a half
revolution, φ being augmented by π, but when φ is augmented by _{}; the corrective term being very small, we will thus
obviously have an ellipse turning in the plane. Let N be the number of
revolutions per century, the angle that the ellipse will have turned in this
interval of time will be

_{}.

Let

_{} be the mean distance
from the Earth to the Sun;

_{} be its mean speed
measurably equal to 30 km per second;

*a* and *e* are
the mean distance and eccentricity of the planet under consideration.

The eccentricity of the Earth being negligible here, we have

besides, we know
from the elementary theory of elliptical motion that _{} the angle sought will
thus be

_{} *

(Oeuvres 422)** **which gives:
for Mercury (k+5) 3.6*''**; for Venus
(k+5) 0.7''**; and*** **for the
Earth (k+5) 0.3''* per century.*

We could choose the arbitrary constant *k* to be equal to 6.4, *which gives for Mercury the observed anomaly*
of 41*''**, for Venus 8''**, for the Earth 3.4''**. Despite the weak eccentricity of these orbits,
these last anomalies do not seem to be admissible; to decide on the value to
give to k**, it is therefore
necessary to resume, taking into consideration the new perturbation, the
calculation of the constants of the interior planets *

Annales 271

*(masses and elements for **t**=0) and to determine them again in a
manner obtaining the most satisfactory accord possible between calculation and
observation. An influence on the motion of the Moon seems equally possible.
These perturbations only become noticeable when their effects add up over a
long time.*