  The purpose of the paper is to develop further a projection variational
approach in relativistic hydrodynamics. The approach, previously proposed in
[gr-qc/9908032], is based on the variation of the vector field and the
projection tensor (instead of the given metric tensor) and their first partial
derivatives. The previously proved property of non-commutativity of the
variation and the partial derivative in respect to the projection tensor has
been used to find all the variations. Subsequently, motivated by some analogy
with the well-known (3+1) ADM projection formalism, an assumption has been made
about a zero-covariant derivative of the projection tensor in respect to the
projection connection. The combination of the equations for the variations of
the projective tensor with covariant and contravariant indices has lead to the
derivation of an important and concisely written relation: the derivative of
the vector field length is equal to the ''twice'' projected along the vector
field initial Christoffell connection. The result is of interest due to the
following reasons: 1. It is a more general one and contains in itself a
well-known formulae in affine differential geometry for the so called
equiaffine connections (admitting covariantly conserved tensor fields), for
which the trace of the connection is equal to the gradient of the logarithm of
the vector field length. 2. The additional term is the projected (with the
projection tensor) initially given connection and accounts for the influence of
the reference system on the change of the vector field's length, measured in
this system. 3. The formulae has been obtained within the proposed formalism of
non-commuting variation and partial derivative.
