107 Letters from Paul Lévy to Maurice Fréchet

Henri Poincaré, 1912–2012

“…6 In the same year 1839, Cauchy sketched his so-called "third theory" of the ether, revived in 1888 in a slightly different form and renamed "labile ether" by William Thomson (Lord Kelvin). In this theory the elastic constants of the generic Cauchy-Green theory are adjusted so that the velocity of longitudinal waves (nearly) vanishes.…”

“…Maxwell had arrived at this theory in 1865 on the basis of his field-theoretical interpretation of the received laws of electricity and magnetism. In 1855, guided 6 O. Darrigol by hydrodynamic analogies and by Stokes's circulation theorem, he had obtained Cartesian-coordinate variants of the equations ∇ × H = j, ∇ × E = −∂μH/∂t, ∇ · μH = 0, ∇ · E = 0 (6) for the fields E and H, the magnetic permeability μ, the dielectric permittivity , the electric density ρ, and the (quasi-stationary) current density j. In 1861, on the basis of a mechanical model in which the magnetic field corresponded to the rotation of cells and the electric current to the flow of idle wheels between these cells, Maxwell replaced the equation ∇ × H = j with the more general equation…”

“…It was therefore necessary at least to add that the postulate was true except for certain initial conditions. 6 As von Plato comments [90] (p. 84), we have here the formulation usually given today for the ergodic principle, in order to take into account the possibility of exceptional initial conditions. Once again, although this idea was also present in Boltzmann's works, the Austrian scientist was nowhere mentioned.…”

“…Let us observe in passing that in these blessed years when it was permitted to publish mistakes, Borel erred in writing his inequality, considering ε instead of the number we called β, a fact which naturally did not change anything in the final result. Nobody seemed to have paid attention to Borel's note: when these results were rediscovered by Lévy and then Hadamard in the 1920s, neither of them had the slightest idea of its existence (on this subject see the letters from Lévy to Fréchet [6], pp. 137 to 141).…”

“…There exist a unique isometry φ 2 which maps the geodesic ]s 2 , s 3 [ into the geodesic ]s 2 , s 1 [, a unique isometry φ 4 which maps the geodesic ]s 4 , s 5 [ into the geodesic ]s 4 , s 3 [, and a unique isometry φ 6 which maps the geodesic ]s 6 , s 1 [ into the geodesic ]s 6 , s 5 [ (Figure 1). Let Γ be the group of isometries generated by φ 2 , φ 4 , φ 6 . It is not hard to see that Γ is a Fuchsian group and that the quotient space Γ \ D is nothing but 8 Poincaré seems to feel a bit uncomfortable with the precise meaning of this identification, and remains quite vague about it.…”