Annus MirabilisEinstein in 1905
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The equation explained

δ∫0∞ϱdν=0.\delta \int_0^\infty \varrho d \nu = 0.
In wordsThe change in the sum, over every frequency, of the energy density is zero: the total energy per unit volume is held fixed.OverviewThe side condition: the maximum is sought only among spectra with the same total energy, so the small changes of the density must leave the energy per unit volume unchanged.Full explanationThe entropy is to be a maximum at a given energy, so only those alterations of the spectrum count that leave the total energy unchanged. The energy per unit volume is ∫0∞ϱdν\int_0^\infty \varrho d \nu, so its variation must vanish: whatever energy is taken from some frequencies must be put back at others.Every step∫0∞ϱdν\int_0^\infty \varrho d \nuThe radiation's energy per unit volume: the density added up over every frequency, as in section 1.∫0∞δϱdν\int_0^\infty \delta \varrho d \nuIts change when the density at each frequency changes by δϱ\delta \varrho.δ∫0∞ϱdν=0\delta \int_0^\infty \varrho d \nu = 0The energy is given, so the change must be zero: the printed side condition.Also

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