For long wavelengths, Rayleigh's 1900 heuristic formula approximately meant that energy was proportional to temperature, U λ = const. T. It is known that ⁠ dS / dU λ ⁠ = ⁠ 1 / T ⁠ and this leads to ⁠ dS / dU λ ⁠ = ⁠ const. / U λ ⁠ and thence to ⁠ d S / dU λ ⁠ = − ⁠ const. / U λ ⁠ for long wavelengths. But for short wavelengths, the Wien formula leads to ⁠ 1 / T ⁠ = − const. ln U λ + const. and thence to ⁠ d S / dU λ ⁠ = − ⁠ const. / U λ ⁠ for short wavelengths. Planck perhaps patched together these two heuristic formulas, for long and for short wavelengths, to produce a formula d 2 S d U λ λ 2 = α α U λ λ ( β β + U λ λ ) .