Is the Equilibrium Curve a Catenary?

  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.