Although the parametric equations for the equilibrium curve involve the linear mass density, the fits for A and B don't appear to be affected by changes in the value for the linear mass density, λ. All the values for the minimum agree to 15 decimal places.
At the minimum gλs+B=0 so B=-g λs
min making the argument of the sinh
-1u equal to u=gλ(s-s
min)/A so there are two parameters, A/gλ and s
min appear to be independent.
What we've been calling the equilibrium equations may in fact be the parametric equations for a catenary.
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