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The equation explained

p v = R \frac{n}{N} T.

In wordsThe pressure times the volume equals the gas constant R times the number n of points over the number N, times the absolute temperature.OverviewThe gas law: pressure times volume equals R times n/N times the temperature. It holds equally for the osmotic pressure of a dilute solution, and no assumption about how the molecules move went into it.Full explanationFrom the footnote's chain, the work

p \, dv

equals

R \frac{n}{N} T \frac{dv}{v}

, with the factor

T

that the printed chain drops. Divide by

dv

and multiply by

v

. The result is the Boyle-Gay-Lussac law: at a fixed temperature

pv

stays constant, and at a fixed volume the pressure grows in proportion to the absolute temperature.

n/N

is the amount in gram-molecules. For a dilute solution, where the points are the dissolved molecules, the same equation is the law of osmotic pressure that the paper calls of the same form. The paper calls it remarkable that deriving these laws needs no assumption about the law by which the molecules move.Every step

p d v = R \frac{n}{N} T \frac{d v}{v}

The work and the last member of the preceding display, with the factor

T

restored.

p = R \frac{n}{N} \frac{T}{v}

Divide both sides by

dv

p v = R \frac{n}{N} T

Multiply both sides by

v

: the printed line.AlsoOverviewFull explanationEvery step

Read it in the paper