Historic reference for Eytelwein or Capstan Equation and assumptions

Question

I'm looking for the actual first reference for what is commonly called the Eytelwein or Capstan equation: $$\frac{T_1}{T_2} = e^{ \, \mu\theta}$$

Is there a book/page where Eytelwein published this?

I'm also looking for what are the assumptions behind his original derivation? I am assuming that they are:

1. the rope is flexible.
2. the rope is inelastic.

Are there other assumptions he made?

Answer 1

The full name of that formula is the Euler-Eytelwein Formula, which means it was initially developed by Euler (what wasn't, right?) (reference). I can't point you to the exact book where this was done, however.

The Wikipedia page on the formula has some information including the assumptions:

1. The rope is on the verge of full sliding, i.e. $T_\text{load}$ is the maximum load that one can hold. Smaller loads can be held as well, resulting in a smaller effective contact angle $\phi$.
2. It is important that the line is not rigid, in which case significant force would be lost in the bending of the line tightly around the cylinder. (The equation must be modified for this case.) For instance a Bowden cable is to some extent rigid and doesn't obey the principles of the Capstan equation.
3. The line is non-elastic.

Answer 2

There are some of Eytelwein's original books on Internet Archive - see https://archive.org/search.php?query=Eytelwein&and[]=creator%3A%22eytelwein%2C%20johann%20albert%2C%201764-1848%22

My German isn't up to reading them, but this includes a likely looking diagram, and the word "friction" alongside it: https://archive.org/stream/bub_gb_M2MSAAAAIAAJ#page/n76/mode/1up

If the first reference was by Euler, it could be anywhere in the very large collection of his writings - including letters as well as published material.