Page:Deux Mémoires de Henri Poincaré.djvu/15

 Finally, the formula (12) will provide us the function of probability ω for an unspecified positive value of η. It is true that the unspecified factor of function Φ(α) will be found in ω, but such a factor does not have any importance.

We can thus say that the probability ω is entirely given as soon as we know the distribution of energy for all temperatures. There is only one function ω for a distributions which is given as a function of the temperature. Consequently, the assumptions that we made on ω and which lead to the law of Planck are the only ones that we can admit.

That is the reasoning by which Poincaré established the necessity of the quantum hypothesis.

We see that the conclusion depends on the assumption that Planck's formula is an accurate image of reality. This could be drawn into question, and the formula could only be approximate. It is for this reason that Poincaré takes up the problem by abandoning Planck's law and using only the relationship that this physicist has found between the energy of a resonator and that of black body radiation. The reassessment led to the conclusion that the total energy of the radiation will be infinite unless the integral $$\int_{0}^{\eta_{0}}\omega\ d\eta $$ does not tend to zero with η0. The function ω must have at least one discontinuity (for η = 0), similar to those given by quantum theory.