## ‘Sur les équations du mouvement relatif des systèmes de corps,’ pp. 142-154 in: Journal de l'École Royale Polytechnique, Cahier XXIV, Tome XV.

Paris: Bachelier, 1835.

First edition, complete journal issue (‘cahier’) in original printed wrappers, of the first formulation of the ‘Coriolis force’. “On a rotating earth the Coriolis force acts to change the direction of a moving body to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This deflection is not only instrumental in the large-scale atmospheric circulation, the development of storms, and the sea-breeze circulation, it can even affect the outcome of baseball tournaments: a ball thrown horizontally 100m in 4s in the United States will, due to the Coriolis force, deviate 1.5cm to the right” (Persson, p. 1373). In 1829 Coriolis published *Calcul de l’Effet des Machines*, which fir the first time gave the correct expression for kinetic energy, *mv*^{2}/2, and its relation to mechanical work. “During the following years Coriolis worked to extend the notion of kinetic energy and work to rotating systems. The first of his papers, ‘Sur le principe des forces vives dans les mouvements relatifs des machines’, was read to the Académie des Sciences in 1832. Three years later came the paper that would make his name famous, ‘Sur les équations du mouvement relatif des systèmes de corps’. Coriolis’s papers do not deal with the atmosphere or even the rotation of the earth, but with the transfer of energy in rotating systems like waterwheels. The 1832 paper established that the relation between potential and kinetic energy for a body affected by a force is the same in a rotating system as in a non-rotating one …Three years later, in 1835, Coriolis went back to analyze the relative motion associated with the system, in particular the centrifugal force. It is directed perpendicular to the moving body’s trajectory (seen from a fixed frame of reference), which for a stationary body is radially out from the center of rotation. For a moving body this is not the case; it will point off from the center of rotation. The centrifugal force can therefore be decomposed into one radial centrifugal force, and another, the ‘Coriolis force.’ It is worth noting that Coriolis called the two components ‘forces centrifuges composées’ and was interested in ‘his’ force only in combination with the radial centrifugal force to be able to compute the total centrifugal force” (*ibid*., p. 1378). No copies listed on ABPC/RBH.

“The first ideas concerning the influence of the Earth’s diurnal rotation on terrestrial objects came with the debate on the very existence of that rotation. This was inevitable, for Aristotelian physics offered a seemingly weighty argument against it: everything lifted into the sky, such as birds or clouds, would no longer share the Earth’s rotational movement and hence should drift away to the west at a gigantic speed. Since this is not what we observe, the conclusion had to be that the Earth does not rotate. To move beyond this Aristotelian argument, a different notion of inertia was needed. It was provided by Galilei who put forward the idea that objects persist in their horizontal movement, by which he meant that they would continue their circular trajectory; thus, a stone thrown up in the air does not suddenly lose its rotational motion but keeps on moving along with the Earth.

“In 1668, Giovanni Borelli, a member of the Accademia del Cimento in Florence, examined what this principle of inertia implies for objects that are dropped from a tower. On the rotating Earth, the top of a tower describes a larger zonal circle than its foot, and thus must have a higher (circular) velocity to the east. Borelli supposed that an object falling from the top must retain this excess in eastward velocity and hence land slightly to the east of the tower’s foot … he found the right direction and the right order of magnitude [but] concluded that the effect was too small to be measured, as the deflection is easily dwarfed by other perturbing effects. More than a century would pass before significant progress was made on this problem. In 1803, Laplace and Gauss independently derived an expression for the eastward deflection of freely falling objects …

“A related but more complicated problem is that of the deflection of bodies launched in an arbitrary direction. Riccioli in his *Almagestum Novum* (1651) argued that, if the Earth rotates, a projectile fired northward should deflect to the east: the projectile comes from a latitude whose zonal circle is larger than the one to which it goes; the accompanying excess in eastward velocity would be retained by the projectile, amounting to a deflection to the east. By contrast, Riccioli (incorrectly) expected no deflection for eastwardly fired objects. His reasoning implies that a systematic difference between northwardly and eastwardly fired objects should have been detected; he took the absence of any such evidence as an argument against the Earth’s rotation. Once again, it was Laplace who put the problem on a solid mathematical footing; in the fourth volume of his *Mécanique Céleste* he derived the equations governing the deflection of projectiles. He showed, in particular, that a body launched vertically upward would land slightly westward.

“Laplace’s work on the deflection of projectiles was a spin-off of his earlier treatise on tides, which was published in its definitive form in the first volume of his *Mécanique Céleste*. For a hypothetical ocean covering the entire planet, he examined their modes of propagation. To do this, he first had to derive the equations of motion including the effects of the Earth’s rotation. Adopting the usual geographical coordinates, he demonstrated that these equations contain four terms that represent a deflecting force due to the Earth’s rotation; moving bodies are subject to a deflecting force in a direction perpendicular to – and in magnitude proportional to – their velocity …

“In 1835, Coriolis derived ‘his’ force in a theoretical treatise on the forces acting in rotating devices [in the offered paper]. He called it the ‘force centrifuge composée’; the association with the name of Coriolis became common only by the late 19th century. In fact, his paper passed unnoticed in the first few decades after its publication. Notably, Poisson in his papers on the deflection of projectiles in 1837/1838 drew on Laplace’s *Mécanique Céleste* and shows no awareness of Coriolis’ paper … It is only natural that Laplace was much more influential than Coriolis. Firstly, because Laplace’s work concerned the *Earth’s* rotation, which, ever since Borelli, was the primary object of study in effects of deflection; by contrast, Coriolis’ work was more abstract and the applications he envisaged were rooted in the industrial revolution, with rotating devices such as waterwheels – a wholly different context. Secondly, Laplace’s derivation of the deflecting force preceded that by Coriolis by four decades. Historically, it would certainly be more accurate to speak of the ‘Laplace force’. But, then, Laplace has already got so many things named after him, that it should be considered a matter of fairness to speak of the ‘Coriolis force’! …

“In 1851 Foucault carried out his famous experiment in which he observed a slow rotation of the vertical plane of the pendulum’s oscillation. This effect is due to the Coriolis force, and Foucault’s experiment was the first direct mechanical proof of the Earth’s rotation. “There is an ironical twist here, for in fact the experiment had already been inadvertently done at a time when the Earth’s rotation itself was still under debate. In notes from the 1660s, members of the Accademia del Cimento mention their experiments on pendulums and report a clockwise rotation of the vertical plane. They did not realize that the Earth’s rotation may have something to do with it. For them, it was a nuisance; and afterwards they fixed the pendulum at two ropes to preclude this annoying perturbation!” (Gerkema & Gostiaux).

The importance of the Coriolis force in meteorology was first pointed out by Ferrel in 1856. He clarified the way in which a particle of air which moves to the east is subject to an extra centrifugal force (since its own eastward velocity adds up to the already existing eastward velocity associated with the Earth’s rotation). Ferrel constructed a schematic view of the general circulation of the atmosphere. In each hemisphere, he distinguished three zones (the trade winds near the equator, the westerlies at mid-latitudes, and the polar easterlies), which are separated by belts of low or high pressure (from equator to poles: the doldrums, subtropical high, subpolar low). Then, Ferrel introduced what we now know as the ‘geostrophic balance’, according to which the winds follow the isobars rather than being perpendicular to them.

Coriolis was born in Paris on 21 May 1792 to a small aristocratic family that had been impoverished by the French Revolution. The young Coriolis showed remarkable mathematical talents at an early age. At 16 he was admitted to the Polytechnic School where he later became a teacher. During the 1830 Revolution Augustin-Louis Cauchy (1789-1857) left France for a period and Coriolis was invited to take his place. In 1832 he became associate professor with Navier at École des Ponts et Chaussées. In 1838 he became director of the Polytechnic School, succeeding Pierre Louis Dulong (1785-1838) and accumulated teaching and research activities. Coriolis died in 1843 at the age of 51 in Paris.

Gerkema & Gostiaux, ‘A brief history of the Coriolis force,’ *Europhysics News* 43 (2012), pp. 14-17. Persson, ‘How do we understand the Coriolis force?’ *Bulletin of the American Meteorological Society *79 (1998), pp. 1373-1385.

4to (280 x 222 mm), pp. [iv], [1-2], 3-278, [1] (upper margin of a few leaves soiled). Original blue printed wrappers (wrappers creased and a bit soiled, closed tear to front wrapper not affecting printing, bottom of spine worn with slight loss), uncut and mostly unopened.

Item #5153

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Price:
$2,500.00
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