'Perpetual motion' paper by George Biddell Airy Airy wrote a paper (maybe deceptively) titled On certain Conditions under which a Perpetual Motion is possible (Cambridge Philosophical Transactions, December 14, 1829). Alternative site (in more than one sense) with JPEG images: https://www.besslerwheel.com/airy/. I've OCRed the (very legible) images and I can post the text here if required.
As a jack of all trades and master of none (I have read and understood The Theoretical Minimum by Susskind and Hrabovsky), I found the paper rather dense. Can somebody summarize it in modern language?
 A: He considers the differential equation
$$\phi''(t)=-e \phi(t)-g\phi(t-c),$$
which models a body acted upon by a force whose magnitude depends on the body's "position at some time preceding that action" (quoted from the second paragraph).  He is inspired by an analysis of vocal chords by someone named Mr. Willis.
If $g=0$, this is a run-of-the-mill second order linear differential equation, and it has the solution $$\phi(t)=a\sin(t\sqrt{e}+b)$$ for $a$ and $b$ arbitrary constants.
But with $g>0$ it is difficult to solve, and I'm unaware of a theorem that proves a solution even exists.  He refers to another paper that (presumably; I didn't check it myself) gives a formula for a small-$g$ approximation, and then he describes, using their notation, how to calculate the increase in amplitude from cycle to cycle.  This corresponds to an increase in energy in the system.
Commentary. A way to think about this differential equation is that it is modeling a driven simple harmonic oscillator without damping and with driving force $-g\phi(t-c)$.  If $g$ is small, then in the short term the system has the characteristics of an undriven simple harmonic oscillator, thus operates at its natural frequency.  The driving force is a phase-shifted version of its position, which in particular is at the natural frequency, and driving a simple harmonic oscillator at its natural frequency causes resonance.
The ability of the system to drive itself is probably where the energy is coming from, since the system is having to do work to apply this force through time.  What I mean is, it takes external energy to apply this force.
In short: it is a paper about an approximate solution to a second-order differential equation that was inspired by some contemporaneous research.
