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WALTER RITZ

#### Annales 267 (Oeuvres 419)

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

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, asapplied 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.

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

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 arethe 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

(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.

* Translation Editor�s note (Bakman).Ritz used k=6.4 to reconcile his formula with the observed anomaly for Mercury (41'') however recent data give 43.1'', which leads to k=7. Substituting this result into Ritz�s formula yields exactly the general relativity formula .