This paper is concerned with the constitutive behavior of a particular orthogonal fiber network under biaxial dead loading. We also describe a new kind of biaxial, dead-loading machine which is applicable to anisotropic materials. The machine does not require that the loads are exerted along symmetry axes of the material. A specimen of the cloth was loaded by different loading paths to the same equibiaxial dead-load, and two different final deformations were observed. A related observation was reported by Treloar for rubber in 1948. In order to understand this instability, we experimentally determined the energy function for the cloth. The energy function is then used in a variational calculation to explain this instability.