On The intrinsic equation behind the Delaunay surfaces

By balancing the internal and external forces acting on axially symmetric membranes one arrives at a system of two equations describing the equilibrium states. This system allows at least two sets of analytical solutions. One of them presents the Euler's elasticas and the other one can be recognized as the class of Delaunay surfaces. The intrinsic equation describing the profile curves of the later is found and solved and this leads to new analytical formulas for these surfaces.