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When and by whom was the word 'polytropic' first used to describe an equation of state or process of the form $P \propto V^n$? Why was this particular word chosen?

The earliest use of the word that I could find by searching on Google Books was in https://doi.org/10.1080/14786445408647467, which seems to be unrelated to the use I am looking for. The other 'contemporaneous' results I could find seemed to be mostly newer books mislabelled as older ones, or works in which the word was used without any explanation, suggesting it was already well-established.

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The best source of word origins and their earliest usages is the Unabridged Oxford English Dictionary. You need a library subscription.

The first meaning could be of mathematical original since n can assume several values Mathematics. Of a function: taking several different values for each value of the variable; many-valued. Cf. monotropic adj. 1. Obsolete. rare—0.

1889 Cent. Dict. at Function Polydromic or polytropic function, one which is not monotropic.

Then it defines the usage in thermodynamics and shows a 1907 paper where the authors use this term specifically in thermodynamic sense

1907 J. F. Klein tr. G. Zeuner Techn. Thermodynamics I. xxix. 152 If the initial condition is given by p1 and v1, we accordingly have pvn = p1v1n..as the equation of the sought pressure curve, which we will hereafter call the polytropic curve.

Now you should search the 1907 paper by J. F. Klein.

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