In the literature two definitions of incompressible solid can be found. The first one is: Poisson's ratio =0.5, and the second is: the third invariant of the Cauchy-Green deformation tensor J=1. I wish to understand the connections between these two definitions. To me, here is something strange. Poisson's ratio relates strains to stresses,and J as the combination of strain tensor components, doesn't depend on Poisson's ratio. I'll be grateful to you, if you explain me how these two definitions are connected.