Henri Poincaré

“ Planck were true, "the physical phenomena would cease obeying laws expressed by differential equations, and it would be, undoubtedly, the greatest and most profound revolution that natural philosophy suffered since Newton". ”

“ Let us admit with Poincaré that, for values given of h and β, the function Θ has a maximum for α = α0, ω = ω0 and passes through the point α0, the place of the maximum, the line l whose distance α0 in the beginning could be selected at will. As the exponent n is very high, the maximum of Θn is extremely pronounced and the only elements of the integrals which we have to take into account, are those who are in the immediate vicinity of α0 and of ω0. ”

“ It is immediately seen that only a tension directed towards the interior is appropriate; we will determine the magnitude by the condition of an electron at rest and which has consequently the shape of a sphere, and it must be in equilibrium with the electrostatic repulsions. So when the particle is set into motion, the stress of Poincaré is united with the electrodynamic actions, and will inevitably produce the oblateness which is required by the principle of relativity. ”

“ I will reproduce the reasoning of Poincaré, but I will at first say that in the formulas that we will encounter, α indicates a complex variable of which the real part is always positive. In the representation we will limit ourselves to the half of plane α characterized by , and in integrations in respect to α we will follow a straight line l perpendicular to the axis of real α, and prolonged indefinitely on the two sides. The values of the integrals will be independent of the length of the distance of this line at the origin of α. ”

“ The first assumption corresponds to l = 1, the second with . Let us immediately add that the first value is the only one which is compatible with the postulate of relativity.
If one wants to realize the persistence and the balance of an electron while making use of the ordinary notions of mechanics, it is obviously not sufficient to consider the electrodynamic actions. The particle - that we consider here as a sphere carrying a surface charge - would immediately explode because of the mutual repulsions or, which is to same, of the stresses of Maxwell exerted on its surface. ”

“ Ultimately, we can write
where the factor C is the same in both cases. We do not have to deal with it because it is sufficient to determine the ratio of X to Y.
Now we obtain the Planck formula - which can be regarded as an expression of reality - if we make on the function ω the following hypothesis, which is consistent with quantum theory.
Let ε be the magnitude of the quantum of energy which is specific to the resonators considered, and denote by δ an infinitely small quantity [2] . ”

“ For other physical quantities such as electric and magnetic forces, it is necessary to follow a less direct method; we will seek, perhaps with a little groping, the formulas of transformation suitable to ensure the invariance of the electromagnetic equations. ”

“ By considering these questions, Poincaré starts by seeking the invariants of the group of relativity; indeed, it is clear that it must be possible to represent the phenomena by equations which contain only these invariants. However, the problem is undetermined. It is natural to admit that the propagation velocity is equal to that of light and that the variations of the law of Newton must be of second-order magnitude in respect to the velocities. ”

“ He initially supposed that there is only the connection represented by the equation
r is the semi-axis of the electron, rθ its equatorial radius, b and m variables that remain constant when r and θ (or one of these quantities) vary with the translation speed v. This granted, we know for any value of v the dimensions of the electron - because we know that - and by the ordinary formulas of electromagnetic field, the energy, momentum and the Lagrange function can be calculated. Between these values, considered as functions of v, there must be well known relations. ”

“ One can consider all the transformations which are represented by formulas (1) , with different values for speed -ε, and the corresponding values of k and l, this last coefficient has to be regarded as a function of ε; we can add to it other similar transformations which we deduce from (1) by changing the directions of the axes, and finally by arbitrary rotations. The postulate of relativity requires that all these transformations form a group and that is only possible if l has the constant value 1. ”