Intermediate phases in lyotropic Liquid crystalline systems provide fascinating topological links between the hexagonal and lamellar phases. The formation of intermediate phases has been shown to result from a delicate balance of interaction in amphiphilic systems. Their structures fall into one of three broad classes: ribbon, mesh, or bicontinuous. In concentrated binary mixtures of nonionic surfactants and water, the intermediate phase is observed to replace the bicontinuous cubic phase under certain circumstances. In this paper we present more detailed scattering results from the mesh intermediate phases in the C30EO9/(H2O)-H-2 and C16EO6/(H2O)-H-2 systems. We show that the space group is indeed R (3) over barm and consists of a three-connected mesh structure with no interlayer connections. Three different mathematical models for the structure, a minimal curvature surface, a rod-box model, and a simple rod model, are fitted to the revised lattice parameters. All give good values for the surface area per molecule and the aggregate thickness parameter. They show that the progressive decrease in the ethylene oxide hydration with increasing temperature causes the progressive decrease in surface curvature. Also in this paper we address the structural relationship between the R (3) over barm mesh phase and adjacent phases in both surfactant systems.