This work is concerned with explicit determination of spin tensor at any point of a moving continuous medium which measures the material time rate of change of orthonormal eigenvectors of symmetric Cauchy deformation tensor at the same point. Using Sylvester formula, the spin tensor is expressed as a sum of some antisymmetric tensors involving vorticity, and Cauchy and strain rate tensors. The scalar coefficients are obtained in terms of invariants of Cauchy tensor for three distinct cases corresponding to the type of eigenvalues of Cauchy tensor: (i) all eigenvalues are equal; (ii) two eigenvalues are distinct; (iii) all three eigenvalues are distinct. Similar expressions are also obtained for logarithmic spin tensor. It has also been shown that the objective time rate of Cauchy stress tensor based on the logarithmic spin is the only objective time rate which allows the derivation of a special hyperelastic constitutive equations depending on Hencky's logarithmic strain measure from the direct integration of hypoelastic constitutive relations of grade zero in the absence of initial stresses.