  An ordinary differential field $(F,d)$ of characteristic zero, a subgroup $H$
of affine group $ GL(n,C)\propto C^n$ with respect to its identical
representation in $F^n$ and the following two fields of differential rational
functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, d>^H=\{f^d< x> \in C<
x, d> : f^d< hx+ h_0> = f^d< x> {for any} (h,h_0)\in H \},$$ $$C< x,
d>^{(F^*,H)}=\{f^d< x> \in C< x, d> : f^{g^{-1}d}< hx+ h_0> = f^d< x> {for any}
  g\in F^* {and} (h,h_0)\in H \}$$ are considered, where $C$ is the constant
field of $(F,d)$ and $C< x, d>$ is the field of differential rational functions
in $x_1,x_2,...,x_n$ over $C$. The field $C< x, d>^H$ ($C< x, d>^{(F^*,H)}$) is
an important tool in the equivalence problem of paths(respect. curves) in
Differential Geometry with respect to the motion group $H$. In this paper an
pure algebraic approach is offered to describe these fields. The field $C< x,
d>^{(F^*,H)}$ and its relation with $C< x, d>^H$ are investigated. It is shown
also that $C< x, d>^H$ can be derived from some algebraic (without derivatives)
invariants of $H$.
  Key words: Differential field, differential rational function, invariant,
differential transcendent degree.
  2000 Mathematics Subject Classification: 12H05, 53A04, 53A55
