  It is well known that $G=\langle x,y:x^2=y^3=1\rangle$ represents the modular
group $PSL(2,Z)$, where $x:z\rightarrow\frac{-1}{z},
y:z\rightarrow\frac{z-1}{z}$ are linear fractional transformations. Let
$n=k^2m$, where $k$ is any non zero integer and $m$ is square free positive
integer. Then the set
$$Q^*(\sqrt{n}):=\{\frac{a+\sqrt{n}}{c}:a,c,b=\frac{a^2-n}{c}\in
Z~\textmd{and}~(a,b,c)=1\}$$ is a $G$-subset of the real quadratic field
$Q(\sqrt{m})$ \cite{R9}. We denote $\alpha=\frac{a+\sqrt{n}}{c}$ in $
Q^*(\sqrt{n})$ by $\alpha(a,b,c)$. For a fixed integer $s>1$, we say that two
elements $\alpha(a,b,c)$, $\alpha'(a',b',c')$ of $Q^*(\sqrt{n})$ are
$s$-equivalent if and only if $a\equiv a'(mod~s)$, $b\equiv b'(mod~s)$ and
$c\equiv c'(mod~s)$. The class $[a,b,c](mod~s)$ contains all $s$-equivalent
elements of $Q^*(\sqrt{n})$ and $E^n_s$ denotes the set consisting of all such
classes of the form $[a,b,c](mod~s)$.
  In this paper we investigate proper $G$-subsets and $G$-orbits of the set
$Q^*(\sqrt{n})$ under the action of Modular Group $G$
