Annus MirabilisEinstein in 1905
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The equation explained

dW=f(A1A2…α1α2…)dA1dA2…dα1dα2…d W = f(A_1 A_2 \ldots \alpha_1 \alpha_2 \ldots) d A_1 d A_2 \ldots d \alpha_1 d \alpha_2 \ldots
In wordsThe probability that a repeated expansion gives values within given small ranges equals a density of all the amplitudes and all the phases, times the product of the widths of all the ranges.OverviewRepeat the expansion from randomly chosen starting moments and the amplitudes and phases come out differently each time. The footnote gives each combination of values a statistical probability, a density times the product of small ranges.Full explanationImagine the expansion carried out at the same point again and again, each time from a starting moment chosen at random. The amplitudes AνA_\nu and phases αν\alpha_\nu come out with different values each time, and the footnote supposes that there are statistical probabilities dWd W for how often the various combinations of values occur. It writes them in the usual form for quantities that vary continuously: the probability of finding every amplitude AνA_\nu within its own small range dAνd A_\nu and every phase αν\alpha_\nu within its range dανd \alpha_\nu, all at once, is a function f of all the values times the product of all the small ranges. Nothing is yet assumed about the form of f; the next display states the condition on it.Every stepAν,  αν(ν=1,2,3…)A_\nu, \; \alpha_\nu \quad (\nu = 1, 2, 3 \ldots)Each repetition of the expansion yields one value for every amplitude and every phase. The paper lists them one by one, as A and α with the subscripts 1, 2 and so on.dAν,  dανd A_\nu, \; d \alpha_\nuAsk for the chance that every value falls within its own small range at once. For quantities that vary continuously, that chance is proportional to the product of the widths of all the ranges.dW=f∏νdAν dανd W = f \prod_\nu d A_\nu \, d \alpha_\nuThe factor of proportionality, f, depends on where all the values lie: it is the probability density.dW=f(A1A2…α1α2…)dA1dA2…dα1dα2…d W = f(A_1 A_2 \ldots \alpha_1 \alpha_2 \ldots) d A_1 d A_2 \ldots d \alpha_1 d \alpha_2 \ldotsThe same thing written out term by term, which is the printed line. The paper sets the arguments of f without commas between them.Also

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