This therefore directly eliminates, as a cause of what has been observed (even if it creates a significant action on the precession):

1) Any action which cannot have a significant effect other than a variation in the Coriolis force resulting from the rotation of the Earth.

This eliminates the tilt resulting from an acceleration applied to the bob which is slowly variable (period of at least several hours, to fix ideas), and which remains $\ll$ g (tilt $\ll$ 10⁻⁴ rad, to fix ideas), which is the case of all tilts which may result from the direct or indirect gravitational action of the celestial bodies, and of the resulting motion of the Earth. Indeed we have seen that, in this case, only the modification of the Foucault precession caused by the tilt was capable of having a significant action ( [14] , §5.2 b), preliminary remark; or ( [15] , §4.1.2 1), preliminary remark.

We therefore find again, but this time much more directly, and much more globally, an essential conclusion of the analysis carried out in [14] and [15] .

2) The action of the Earth magnetic field on an electrically charged bob. The action of the Lorentz force on that charge is:

$F_L=qv\wedge B$(1)

where $F_L$ is the Lorenz force, $v$ the bob speed, which is in the horizontal plane, and $B$ the Earth magnetic field. The Lorentz force is of exactly the same form as the Coriolis force $F_C=qv\wedge \Omega$ ( $\Omega$ being the rotation of the Earth), and the Coriolis force, which causes the Foucault effect, does not create ovalisation.

$\overline{{e^{\prime }}_i\left(t_j\right)}=c_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)+\cdots +c_{ni}\cos 2n\pi f_i\left(t_j-t_{0i}\right)+\cdots$(2)

By expanding $\frac{1}{N}{\displaystyle {\sum }_{j=1}^N{\overline{{e^{\prime }}_i\left(t_j\right)}}^2}$, we find a sum of terms independent of t and

the average over 1 month of a sum of sinusoids whose periods at most equal to half a day, that is at least 60 periods. As we can neglect this average, we therefore have:

$\frac{1}{N}{\displaystyle {\sum }_{j=1}^N{\overline{{e^{\prime }}_i\left(t_j\right)}}^2}\approx \frac{c_{1i}^2}{2}+\frac{c_{2i}^2}{2}+\cdots +\frac{c_{ni}^2}{2}+\cdots$(3)

${e^{\prime }}_i\left(t\right)=-\omega \eta \left(t\right)\sin \left(2\left(\theta \left(t\right)-{\theta }_A\left(t\right)\right)\right)$(4)

ω being the pulsation of the pendulum, and $\theta \left(t\right)$ the azimuth of the major axis of the ellipse described by the pendulum.

We have, for the run j:

$\overline{{e^{\prime }}_i\left(t_j\right)}=-\omega \overline{{\eta }_i\left(t\right)\sin \left(2\left(\theta \left(t\right)-{\theta }_{Ai}\left(t\right)\right)\right)}$(5)

Over the duration of a run (50 min), we can consider that ${\eta }_i\left(t\right)$ and ${\theta }_{Ai}\left(t\right)$, whose period is one day or close to one day, varies little, and confuse them with ${\eta }_i\left(t_j\right)$ and ${\theta }_{Ai}\left(t_j\right)$. Similarly, the precession of the pendulum with respect to its starting azimuth ${\theta }_L$ being small on a run, one can confuse $\theta \left(t\right)$ and ${\theta }_L$. Hence:

$\overline{{e^{\prime }}_i\left(t_j\right)}\approx -\omega {\eta }_i\left(t_j\right)\sin 2\left({\theta }_L-{\theta }_{Ai}\left(t_j\right)\right)$(6)

As we saw in §4.1 above, we can consider that, on a 360 degrees turn, ${\theta }_{Ai}\left(t_j\right)$ is a linear function of time. As ${\theta }_L$ is fixed, $\sin 2\left({\theta }_L-{\theta }_{Ai}\left(t_j\right)\right)$ is a sinusoid the period of which is $T_i/2$¹¹.

${\eta }_i\left(t_j\right)=k_{0i}+k_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)+k_{2i}\cos 4\pi f_i\left(t_j-t_{0i}\right)+\cdots$(7)

Hence, from Equations (6) and (7):

$\frac{1}{N}{\displaystyle {\sum }_{j=1}^N{\overline{{e^{\prime }}_i\left(t_j\right)}}^2}=\frac{{\omega }^2}{2}{\displaystyle {\sum }_{j=1}^N\left({\eta }_i{\left(t_j\right)}^2+B_j\right)}$(8)

with

$B_j={\eta }_i{\left(t_j\right)}^2\cos 4\left({\theta }_L-{\theta }_{Ai}\left(t_j\right)\right)$(9)

1) We find in the development of ${\eta }_i{\left(t_j\right)}^2$:

$\left(a\right)=k_{0i}^2+k_{1i}^2/2+k_{2i}^2/2+\cdots$(10)

and:

$\left(b\right)=\left(k_{1i}^2/2\right)\cos 4\pi f_i\left(t_j-t_{0i}\right)+\left(k_{2i}^2/2\right)\cos 8\pi f_i\left(t_j-t_{0i}\right)+\cdots$(11)

$\left(c\right)=2k_{0i}\left(k_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)+k_{2i}\cos 4\pi f_i\left(t_j-t_{0i}\right)+\cdots \right)$(12)

and:

$\left(d\right)=2k_{1i}{k}_{2i}\cos 2\pi f_i\left(t_j-t_{0i}\right)\cos 4\pi f_i\left(t_j-t_{0i}\right)+\cdots$(13)

Let us group (b) and (c):

$\begin{array}{c}\left(b\right)+\left(c\right)=2k_{0i}{k}_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)\\ +\left(\left(k_{1i}^2/2\right)+2k_{0i}{k}_{2i}\right)\cos 4\pi f_i\left(t_j-t_{0i}\right)+\cdots \end{array}$(14)

The average of (b) + (c) is nil for $t_j=\infty$. Over 30 days the maximum of the absolute value of this average is:

$\frac{1}{30}\left(2k_{0i}{k}_{1i}\right)+\frac{1}{60}\left(\left(k_{1i}^2/2\right)+2k_{0i}{k}_{2i}\right)+\cdots$

The average of (d) is nil for $t_j=\infty$. Over 30 days the maximum of the absolute value of this average is:

$\frac{2}{60}\left(k_{1i}{k}_{2i}\right)+\frac{2}{90}\left(k_{1i}{k}_{3i}\right)+\cdots$

2) Ultimately the uncertainty on (b) + (c) + (d) remains sufficiently small in front of (a) that we can neglect it. Hence:

$\frac{1}{N}{\displaystyle {\sum }_{j=1}^N{\eta }_i{\left(t_j\right)}^2}\approx \left(a\right)=k_{0i}^2+k_{1i}^2/2+k_{2i}^2/2+\cdots$(15)

3) Furthermore, as we saw above, $\cos 4\left({\theta }_L-{\theta }_{Ai}\left(t_j\right)\right)\approx$ sinusoid whose frequency is $2f_i$.

From Equation (9), which provides the expression of $B_j$, it follows that only the product of this sinusoid by the sinusoid of the same frequency $2f_i$ in ${\eta }_i{\left(t_j\right)}^2$ may have a non-zero average.

From Equation (14), it results that the maximum value of this average, which corresponds to the case where the 2 sinusoids are in phase, is:

$\frac{1}{2}\left(\left(k_{1i}^2/2\right)+k_{0i}{k}_{2i}\right)=k_{1i}^2/4+k_{0i}{k}_{2i}/2$(16)

4) In the end, in Equation (8):

$\frac{1}{N}{\displaystyle {\sum }_{j=1}^N{\overline{{e^{\prime }}_i\left(t_j\right)}}^2}=\frac{{\omega }^2}{2}\left(k_{0i}^2+k_{1i}^2/2+k_{2i}^2/2+\cdots \right)=\frac{{\omega }^2}{2}\overline{{\eta }^2}$(17)

where $\overline{{\eta }^2}$ is the square of the root mean square of ${\eta }_i\left(t_j\right)$ over the duration of the observation, as deduced from the expression of ${\eta }_i\left(t_j\right)$, which is given by Equation (7).

$\sqrt{\overline{{\eta }_i^2}}=\frac{1}{\omega }\sqrt{c_{1i}^2+c_{2i}^2+\cdots +c_{ni}^2+\cdots }$(18)

During a run of duration Δt, the average precession speed is given ( [1] or [2] , p. 212, Table XIII) by Formula (3), where the 2nd member must in fact be divided by 2¹³.

Hence, on a given run:

$\overline{{{\Phi }^{\prime }}_l}=\frac{3}{64}{p}^2\overline{{\alpha }^2}\Delta t{\epsilon }_l\overline{\sin 2\left(X_l-\Phi \right)}$(19)

where:

$X_l$ = azimuth of the Moon.

$\Phi$ = azimuth of the major axis of the ellipse described by the pendulum.

$\overline{{{\Phi }^{\prime }}_l}$ = average velocity of the precession attributable to the Moon.

$p=\sqrt{g/l}$.

$\alpha$ = angular major axis.

${\epsilon }_l$ = average value over the run of the coefficient of anisotropy (with Allais’s definition) of the anisotropy resulting from the Moon. It takes into account the average influence, over the run, of the elevation of the Moon.

$\epsilon =4\eta$.

$X_l$ = azimuth of the direction of anisotropy ${\theta }_A$ associated with the Moon.

With notations of [14] or [15] , where $\theta =\Phi$, $\omega =p$, $\alpha =\alpha$, $\beta =\beta$, $X_l={\theta }_{Ai}$, Equation (19) becomes:

$\overline{{{\theta }^{\prime }}_i}=\frac{3}{16}{\omega }^2\overline{{\alpha }^2}\Delta t{\eta }_i\overline{\sin 2\left({\theta }_{Ai}-\theta \right)}$(20)

Which can also be written:

$\overline{{{\theta }^{\prime }}_i}=K{\eta }_i\overline{\sin 2\left({\theta }_{Ai}-\theta \right)}$(21)

with:

$K=\frac{3}{16}{\omega }^2\overline{{\alpha }^2}\text{Δ}t$(22)

Over a run, $\left({\theta }_L-\theta \right)$ remains small. Hence $\overline{\sin 2\left({\theta }_{Ai}-\theta \right)}\approx \sin 2\left({\theta }_{Ai}-{\theta }_L\right)$, where ${\theta }_L$ is the starting azimuth.

Hence, considering the run j:

$\overline{{{\theta }^{\prime }}_i\left(t_j\right)}\approx K{\eta }_i\left(t_j\right)\sin 2\left({\theta }_{Ai}-{\theta }_L\right)$(23)

The influence of the astral source $S_i$ on the precession of the pendulum is:

$\overline{{\theta }_i\left(t_j\right)}=a_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)+\cdots +a_{ni}\cos 2n\pi f_i\left(t_j-t_{0i}\right)+\cdots$(24)

Hence:

$\overline{{{\theta }^{\prime }}_i\left(t_j\right)}=b_{1i}\cos 2\pi f_i\left(t_j-t_{0i}\right)+\cdots +b_{ni}\cos 2n\pi f_i\left(t_j-t_{0i}\right)+\cdots$(25)

With $b_{ni}=2n\pi \frac{a_{ni}}{T_i}$

To calculate the root mean square of ${\eta }_i\left(t\right)$, we therefore use the approach¹⁴ which led to Equation (8), then ultimately to Equation (17).

In Equation (18), we replace $c_{ni}$ with $b_{ni}$, and ω with K. Hence:

$\sqrt{\overline{{\eta }_i^2}}=\frac{2\pi }{K{T}_i}\sqrt{a_{1i}^2+4a_{2i}^2+\cdots +n^2{a}_{ni}^2+\cdots }$(26)

The calculation was carried out by considering, in Equations (18) and (26), only the amplitudes of harmonics 1 and 2 (the amplitudes of harmonics 3 are significantly smaller).

See Tables 1-3 .

The Horodnic average concerns the 2 pendulums of the same observation, while the Allais’ average concerns different observations.

We limited ourselves to cases where, on the one hand, an eclipse effect had been observed and where, on the other hand, the ovalization had been measured. There are in fact very few eclipses for which these conditions have been met¹⁷.

3 observations were published: those of the solar eclipses of 11 August 1999 [18] , 31 May 2003 [19] , and 3 October 2005 [20] .

As shown in Figure 1 , ΔA was almost identical for the two pendulums, which suggested that the disruptive action was circularly symmetrical.

In [15] , the speed of the direct precession resulting from a “linear anisotropy” is given by Formula (6) of the appendix A:

${{\theta }^{\prime }}_d=2\eta \omega e\cos 2\left(\theta -{\theta }_A\right),$(32)

where ${\theta }_A$ and η are the direction and the coefficient of the total anisotropy (intrinsic anisotropy + external anisotropy) of the pendulum. The derivative $e^{\prime }$ of the ellipticity resulting from this anisotropy is given by Formula (7) of this Appendix A:

$e^{\prime }=-\eta \omega \sin 2\left(\theta -{\theta }_A\right).$(33)

If we replace $\theta$ by $\theta +\pi /2$, $\cos 2\left(\theta -{\theta }_A\right)$ and $\sin 2\left(\theta -{\theta }_A\right)$ both change sign. In the end, in Equation (32), the value of ${{\theta }^{\prime }}_d$ does not change.

Analysis of the third observation [20] , which used exactly the same pendulum, and for which the measurement of the minor axis was published, confirms that it is almost certainly what happens:

Indeed Figure 2 shows:

${\theta }^{\prime }=\frac{3}{8}\frac{ab}{l^2}\sqrt{\frac{g}{l}}$(34)

where θ is the azimuth (defined modulo 180˚) of the major axis of the ellipse described by the pendulum, g the acceleration of gravity, l the length of the equivalent simple pendulum, and a and b the half major axis and half minor axis of the ellipse. Here l = 14.21 m, and, in average over a run of 1 hour, a = 28 cm.

In formulas above, azimuths are counted positively counter-clockwise from the North. In Pr Mihaïlia’s observations, they are counted positively clockwise from the South. We therefore have ${\theta }^{\prime }=-\text{Δ}{A}^{\prime }$.

The pendulum was restarted every hour, which reset the ellipticity e to zero, and therefore, according to Equation (32), ${{\theta }^{\prime }}_d$. We see that this is indeed what happens for $\text{Δ}{A}^{\prime }$ at 12 h, 13 h and 14 h. We also note that, in general, ΔA varies in the opposite direction to e.

From the analysis of the curves in Figure 2 , between 11 h 20 min and 12 h (in local time: UTC-3), we can deduce an estimate of average value of η. Over this interval:

average value of $e=4.9\times {10}^{-3}$

average value of $e^{\prime }=4.2\times {10}^{-6}$

average value of $\text{Δ}{A}^{\prime }=-4.4\times {10}^{-6}$ rad/s

Hence, as ${{\theta }^{\prime }}_d\approx -\text{Δ}{A}^{\prime }$, and from Equations (32) and (33):

$\eta \cos 2\left(\theta -{\theta }_A\right)=5.33\times {10}^{-4}$

$\eta \sin 2\left(\theta -{\theta }_A\right)=5.01\times {10}^{-6}$

Hence estimations of the average values, between 11 h 20 min and 12 h in local time, i.e. at the average local time 11 h 40 min, of the coefficient and the direction of anisotropy :

$\eta =5.33\times {10}^{-4}$

${\theta }_A=\theta -0.5\mathrm{deg}=13.6-0.5=13.1\mathrm{deg}$, as the average value of θ is 13.6 deg¹⁸ (counted positively from the North).

1) The value of the coefficient of anisotropy is really very important (at least one hundred times greater than the total external anisotropy which can be usually observed on Horodnic pendulums).

2) The value found for the direction of anisotropy is entirely compatible with the hypothesis of an action in the direction of the eclipse.

3) It is also a similar phenomenon which very probably occurred during the eclipse of 31 May 2003 [19] .

In this case the period (which is the period in the direction of the major axis) had been measured, the report having concluded ( [19] , p. 6), to a relative variation of the period during the eclipse of 2.6 × 10⁻⁵. According to [15] (Formula 4), this can be explained by a linear anisotropy whose coefficient $\eta \ge 2.6\times {10}^{-5}$.

4) Whether the precession created by linear anisotropy is mainly direct precession, as in the case of Professor Mihaïlia’s observations, or mainly indirect precession, in all cases there must be ovalization. As we saw in §2, this directly eliminates a certain number of explanations.

Two eclipses observed with a pendulum equipped with an automatic alidade are concerned: the total lunar eclipse of 26/07/2018, where an eclipse effect seems very probable, and the solar eclipse of 1/09/2016. These observations had never been published.

The pendulum was one of the two pendulums used during the 2019 observations (see [14] , in this case the pendulum B). This one was already equipped with the ball suspension used in 2019, which is weakly anisotropic. The pendulum (equivalent length 6.38 m) was started from azimuth 65˚, with an amplitude of 450 mm, and stopped after 50 min. The observations, which were not always regularly spaced, were spread from 27/07/2018 3 a.m. to 29/07/2018 2:02 p.m. As in 2019, the quantity studied was the average value over the run of the derivative of the ellipticity $\text{d}e/\text{d}t=e^{\prime }$ (cf [14] §3).

Figure 3 shows a very clear concomittance between the eclipse and the peak of $e^{\prime }$.

Temperature and humidity had been measured: Figure 4 shows that nothing particular happened at that time.

Note further that, the anomaly highlighted in Figure 3 relating to the derivative of ellipticity, the disturbing action can only be a variation of the ovalization, which directly eliminates a certain number of explanations (§2).

At each run, an estimate of the coefficient of anisotropy η was made, using the method b described in [14] , appendix B4, which gives a very noisy result when η is low (several 10⁻⁶, to fix ideas). Figure 5 shows that there were 2 very significant peaks, one of 1.2 × 10⁻⁵ just before the eclipse, and the other (0.95 × 10⁻⁵) inside the eclipse.

This time, the pendulum was equipped with a chuck suspension (to prevent the 1 mm wire from breaking, the latter was connected to the chuck by an 8 mm diameter intermediate rod). This suspension being strongly anisotropic, the intrinsic anisotropy of the pendulum was this time the main component of its total anisotropy. Figure 6 presents both $e^{\prime }$ and η.

On each of the 3 days of the observation, a diurnal action clearly appears, a little after 12 p.m. This is most likely the lunisolar action highlighted by Allais, and found again in Horodnic in 2019. The anomaly concerned, which coincides quite well with the eclipse itself, is clearly before 12 p.m.

There was no temperature recording at that time (the device was not yet installed).

It should also be noted that, for the two eclipses concerned, the observed coefficients of anisotropy (about 10⁻⁵) were important, but not exceptional.

The observations concerned 50 days, scattered from April, 2000 till June, 2000: cf [21] , §7, p. 670, and Fig XVa, p. 686 ( Figure 7 is a copy of Fig XVa).

The pendulum (about 14.75 m long) was installed in Stefan cel Mare High School in Suceava (Romania). It was always started from the same azimuth.

In Figure 7 , the tip of the arrow pointed upwards shows the value of the higher angular deviation of the plane of oscillation (usually, this was reached after about 20 minutes). The arrow pointing downwards shows the position of the oscillation plane when the pendulum is stopped, one hour after it was set in motion. May 8th was the day of a conjunction Sun-Jupiter. Obviously something very unusual occurred this day. There was only 1 chance among 50 that it occurs so.

It appears from Formula (11) of [15] that the precession resulting from the total linear anisotopy of the pendulum (intrinsic anisotropy + anisotropy of external origin) increases quadratically from the beginning of the run. When exercised in the opposite direction of the Foucault effect, after a certain time it reverses the direction of precession. This is exactly what is observed with each run. Very probably, on May 8, 2000, external anisotropy was exceptionally important.

Indeed, the maximum of this alignment took place approximately 6 hours after that of the eclipse. Let us recall that it was on the occasion of this eclipse, during which particularly marked deviations of his pendulum were observed (see Figure 8 ) that Allais discovered the “eclipse effect”, with which his name was subsequently associated.

It cannot be said that there was then actually an influence of Jupiter, but the coincidence is so noticeable that it deserves to be reported.