1.10.2 Example how to find Lie group $(x,y)$ given canonical coordinates $R,S$

Given$R=x,S=\frac{y}{x}$ find Lie group $\overline{x},\overline{y}$.  Solving for $x,y$ from $R,S$ gives

$$\begin{array}{rl}x& =R\\ y& =SR\end{array}$$

Hence

$$\begin{array}{rl}\overline{x}& =\overline{R}\\ \overline{y}& =\overline{S}\overline{R}\end{array}$$

But $\overline{S}=S+ϵ$ by definition of canonical coordinates and $\overline{R}=R$ by definition of canonical coordinates. Hence the above becomes

$$\begin{array}{rl}\overline{x}& =R\\ \overline{y}& =(S+ϵ)R\end{array}$$

Using the values given for $R,S$ in terms of $x,y$ the above becomes

$$\begin{array}{rl}\overline{x}& =x\\ \overline{y}& =(\frac{y}{x}+ϵ)x\\ & =y+ϵx\end{array}$$