For a system whose lagrangian function does not contain time explicitly, let n+1 variables $q_1,...,q_n,t$ be given as functions of some parameter $\tau$. The action integral becomes $$A=\int_{\tau_1}^{\tau_2} L\left(q_1,...,q_n;\frac{q'_1}{t'},...,\frac{q'_n}{t'}\right)~t'~d\tau$$
In configuration space the kinetic energy of the system can be written as the kinetic energy of a single particle of mass 1: $$T=\frac{1}{2}\left(\frac{ds}{dt}\right)^2$$ but since our independent variable is no longer t but $\tau$ we get $$T=\frac{1}{2}\left(\frac{ds}{d\tau}\right)^2~/~t'^2$$
This seems like a simple change of variables but I'm getting caught up on it. I was thinking that I need to do $$\frac{ds}{dt}=\frac{ds}{d\tau}\frac{d\tau}{dt'}\frac{dt'}{dt}$$
but I'm not seeing how to do $\frac{d\tau}{dt'}$ or $\frac{dt'}{dt}$. Any help would be greatly appreciated!