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If the net radiative energy of frequency ν is not to continually increase or decrease, the following equality must hold

${\frac {R}{N}}T={\bar {E}}={\bar {E}}_{\nu }={\frac {L^{3}}{8\pi \nu ^{2}}}\rho _{\nu },$

or, equivalently,

$\rho _{\nu }={\frac {R}{N}}{\frac {8\pi \nu ^{2}}{L^{3}}}T.$

This condition for dynamic equilibrium not only lacks agreement with experiment, it also eliminates any possibility for equilibrium between matter and aether. The wider the range of frequencies of the resonators is chosen the bigger the radiation energy in the space becomes, and in the limit we obtain:

$\int _{0}^{\infty }\rho _{\nu }\,d\nu ={\frac {R}{N}}{\frac {8\pi }{L^{3}}}T\int _{0}^{\infty }\nu ^{2}\,d\nu =\infty \ .$

Planck's Derivation of the Fundamental Quantum

In the next section we want to show that the determination that Mr. Planck gave of the elementary quanta is to some extent independent of the "black body radiation" theory that he created.

Planck's formula ^[1] for ρ_ν that suffices for all experiments so far goes

$\rho _{\nu }={\cfrac {\alpha \nu ^{3}}{e^{\cfrac {\beta \nu }{T}}-1}},$

where

$\alpha =6.1\cdot 10^{-56},$

$\beta =4.866\cdot 10^{-11}.$

In the limit of large values of T/ν, that is for large wavelengths and radiation densities, this formula approaches the form:

$\rho _{\nu }={\frac {\alpha }{\beta }}\nu ^{2}T.$

1. M. Planck, Ann. d. Phys. 4. p.561. 1901.