Annus MirabilisEinstein in 1905
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The equation explained

∂φ∂ϱ=1T.\frac{\partial \varphi}{\partial \varrho} = \frac{1}{T}.
In wordsThe rate at which the entropy density grows with the energy density, at any frequency, equals one over the absolute temperature.OverviewComparing the two expressions for the entropy's increase gives the common rate: ∂φ/∂ρ is one over the temperature, at every frequency. The paper calls this the law of black radiation; it lets φ be found from the radiation law and the other way round.Full explanationThe two displays before this one give the same increase of entropy dSd S for the same increase of energy dEd E, so their factors must agree. The rate at which the entropy per unit volume and frequency grows with the density is one over the absolute temperature, at every frequency. The paper calls this 'the law of black radiation': given φ\varphi, it fixes how ϱ\varrho depends on ν\nu and TT, and given the radiation law, φ\varphi follows by integrating over ϱ\varrho, starting from φ=0\varphi = 0 at ϱ=0\varrho = 0. Section 4 does exactly that with Wien's law. Doubling the temperature halves the rate.Every stepdS=∂φ∂ϱdEd S = \frac{\partial \varphi}{\partial \varrho} d EFrom the density: the second display of the paragraph.dS=1TdEd S = \frac{1}{T} d EFrom thermodynamics: the preceding display.∂φ∂ϱdE=1TdE\frac{\partial \varphi}{\partial \varrho} d E = \frac{1}{T} d EBoth are the same dSd S.∂φ∂ϱ=1T\frac{\partial \varphi}{\partial \varrho} = \frac{1}{T}Divide both sides by dEd E, which is not zero: the printed law.Also

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