Annus MirabilisEinstein in 1905
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The equation explained

φ(ϱ,ν)=−ϱβν{lg⁡ϱαν3−1}.\varphi(\varrho, \nu) = -\frac{\varrho}{\beta \nu} \left\{ \lg \frac{\varrho}{\alpha \nu^3} - 1 \right\}.
In wordsThe entropy per unit volume and unit frequency equals minus the energy density over the constant β times the frequency, times the natural logarithm of the density over the constant α times the frequency cubed, less one.OverviewIntegrating that rate from zero density gives the entropy density, at each frequency, of radiation that obeys Wien's law.Full explanationSection 3's law says ∂φ∂ϱ=1T\frac{\partial \varphi}{\partial \varrho} = \frac{1}{T}, and the preceding line gives 1T\frac{1}{T} as a function of ϱ\varrho at a fixed frequency. So φ\varphi is found by integrating it over ϱ\varrho, and section 3 also fixed where to start: φ\varphi vanishes for ϱ=0\varrho = 0. The integral of a logarithm is the variable times the logarithm, less the variable, which gives the bracket with its −1-1. Where the density is small the bracket is negative and φ\varphi positive.Every step∂φ∂ϱ=−1βνlg⁡ϱαν3\frac{\partial \varphi}{\partial \varrho} = -\frac{1}{\beta \nu} \lg \frac{\varrho}{\alpha \nu^3}Section 3's law, with 1T\frac{1}{T} taken from the preceding display.φ(ϱ,ν)=−1βν∫0ϱlg⁡ϱαν3dϱ\varphi(\varrho, \nu) = -\frac{1}{\beta \nu} \int_0^{\varrho} \lg \frac{\varrho}{\alpha \nu^3} d \varrhoIntegrate over the density at a fixed frequency, starting from ϱ=0\varrho = 0, where section 3 says φ\varphi vanishes.∫lg⁡ϱαν3dϱ=ϱlg⁡ϱαν3−ϱ\int \lg \frac{\varrho}{\alpha \nu^3} d \varrho = \varrho \lg \frac{\varrho}{\alpha \nu^3} - \varrhoThe rule for integrating a logarithm: differentiating the right-hand side gives back lg⁡ϱαν3\lg \frac{\varrho}{\alpha \nu^3}. At ϱ=0\varrho = 0 it is zero, since ϱ\varrho times its logarithm goes to zero there.φ(ϱ,ν)=−ϱβν{lg⁡ϱαν3−1}\varphi(\varrho, \nu) = -\frac{\varrho}{\beta \nu} \left\{ \lg \frac{\varrho}{\alpha \nu^3} - 1 \right\}Multiply by −1βν-\frac{1}{\beta \nu} and take out the factor ϱ\varrho: the printed line.Also

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