Annus MirabilisEinstein in 1905
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The equation explained

−d(E−TS)=pdv=TdS=RnNdvv;- d(E - T S) = p d v = T d S = R \frac{n}{N} \frac{d v}{v} ;
In wordsWhen the volume grows by dv at a fixed temperature, the decrease of the system's energy less the temperature times the entropy equals the pressure times dv, which equals the temperature times the growth of the entropy. The footnote sets this equal to the gas constant R times the number n over the number N times dv over the volume, where the next line needs a further factor of the temperature.OverviewThe footnote derives the gas law from the entropy difference. At a fixed temperature the work p dv equals T dS, and the entropy difference gives dS. As printed, the last member lacks a factor T that the next line needs.Full explanationThe footnote, with EE the energy of the system, reads the entropy difference thermodynamically. At a fixed temperature the work p dvp \, dv the system does as its volume grows equals the decrease of its free energy E−TSE - TS. That decrease is T dST \, dS if the energy does not change when only the volume does. For points that do not act on one another, as section 5 assumes, the energy does not change, and the footnote uses this without stating it. Differentiating the entropy difference with respect to vv gives dSdS. As printed, the last member reads RnNdvvR \frac{n}{N} \frac{dv}{v}, which is dSdS and not T dST \, dS. The factor TT is needed for the next line, pv=RnNTpv = R \frac{n}{N} T, to follow.Every step−d(E−TS)=pdv- d(E - T S) = p d vAt a fixed temperature, the work the system does when its volume grows by dvdv equals the decrease of E−TSE - TS.−d(E−TS)=−dE+TdS- d(E - T S) = - d E + T d SAt a fixed temperature the change of TSTS is T dST \, dS.−dE+TdS=TdS- d E + T d S = T d SThe footnote's next equality holds when the energy does not change as only the volume changes, which is true for points that do not act on one another. The footnote uses this without saying so.dS=RnNdvvd S = R \frac{n}{N} \frac{d v}{v}Differentiate the entropy difference of the preceding display with respect to vv, with S0S_0 and v0v_0 fixed. The rate of change of lg⁡v\lg v is 1v\frac{1}{v}.TdS=RnNTdvvT d S = R \frac{n}{N} T \frac{d v}{v}Multiply by TT. The printed line drops this TT from its last member. The receipt records the dropped letter as typo-dropped-T-footnote-p142, and the page image shows clean space where it would stand. With TT restored the next line follows.Also

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