  Different phenomenological RG transformations based on scaling relations for
the derivatives of the inverse correlation length and singular part of the
free-energy density are considered. These transformations are tested on the 2D
square Ising and Potts models as well as on the 3D simple-cubic Ising model.
Variants of RG equations yielding more accurate results than Nightingale's RG
scheme are obtained. In the 2D case the finite-size equations which give the
{\it exact} values of the critical point or the critical exponent are found.
