Annus MirabilisEinstein in 1905
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The equation explained

Z=∑ν=1ν=∞Aνsin⁡(2πνtT+αν),Z = \sum_{\nu = 1}^{\nu = \infty} A_\nu \sin\left(2 \pi \nu \frac{t}{T} + \alpha_\nu\right) ,
In wordsThe Z-component of the electric force at one point is a sum, over the frequency numbers ν = 1, 2, 3 and so on, of an amplitude times the sine of two π ν times the time divided by the long interval of the expansion, plus a phase.OverviewThe footnote's first step in defining radiation as disordered as possible: over a long time, one component of the electric force at a point is written as a Fourier series, a sum of sine waves, each with its own amplitude and phase.Full explanationPlanck's relation assumes radiation 'as disordered as possible', and this footnote says what that means. Take the ZZ-component of the electric force at an arbitrary point of the space and expand it, between the times t=0t = 0 and t=Tt = T, in a Fourier series. TT is a time very long compared with every period of oscillation that matters. Each term is a sine wave that runs through ν\nu whole periods in the time TT, so its frequency is ν\nu divided by TT. Its amplitude AνA_\nu is never negative, and its phase αν\alpha_\nu lies between 00 and 2π2 \pi. The series is only the description; the footnote's condition comes in the next two displays, about how the amplitudes and phases are distributed when the expansion is repeated.Every stepAνsin⁡(2πνtT+αν)A_\nu \sin\left(2 \pi \nu \frac{t}{T} + \alpha_\nu\right)One term of the series: a sine wave of amplitude AνA_\nu and phase αν\alpha_\nu. As tt runs from 00 to TT, the angle 2πνtT2 \pi \nu \frac{t}{T} runs from 00 to 2πν2 \pi \nu, so the wave completes ν\nu whole periods in the interval.Aν≧0,0≦αν≦2πA_\nu \geqq 0, \quad 0 \leqq \alpha_\nu \leqq 2 \piThe ranges the paper sets. A negative amplitude would be the same wave as a positive one with its phase turned by π\pi, so the amplitudes can be taken non-negative and the phases over one full turn.Z=∑ν=1ν=∞Aνsin⁡(2πνtT+αν)Z = \sum_{\nu = 1}^{\nu = \infty} A_\nu \sin\left(2 \pi \nu \frac{t}{T} + \alpha_\nu\right)Adding the terms for ν=1\nu = 1, 22, 33 and so on without end gives the field at that point over the whole interval, as the paper writes it.Also

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