# Rankine-Hugoniot equation derivation intermediate step

I'm trying to go from, this equation:

$$\frac{\gamma}{\gamma -1} \left( \frac{P}{ \rho} - \frac{P_0}{ \rho_0} \right) - \frac{1}{2} (P-P_0) \left( \frac{1}{\rho} + \frac{1}{\rho_0} \right)=0$$

to this one:

$$\frac{2 \gamma}{\gamma -1} \left( \frac{P}{P_0} - \frac{\rho}{\rho_0} \right) = \left( \frac{P}{P_0} - 1\right) \left( \frac{\rho}{\rho_0} + 1 \right)$$

from which one can then obtain the Rankine-Hugoniot equations (for physics), however, the intermediate steps are proving to be quite vexing. Any help?

To simplify, let $$N=\frac{2\gamma}{\gamma-1}$$. $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=(P-P_{0})(\frac{1}{p}+\frac{1}{p_{0}})$$ $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=(P-P_{0})(\frac{\rho+\rho_{0}}{\rho\rho_{0}})$$ $$N(\frac{P}{\rho}+\frac{P_{0}}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}}{\rho\rho_{0}}$$ $$N(\frac{P\rho\rho_{0}}{\rho}+\frac{P_{0}\rho\rho_{0}}{\rho_{0}})=P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}$$ $$N(\frac{PP_{0}\rho\rho_{0}}{P_{0}\rho}+\frac{PP_{0}\rho\rho_{0}}{P\rho_{0}})=P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho-P_{0}\rho_{0}}{P_{0}\rho_{0}}$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=\frac{P\rho+P\rho_{0}-P_{0}\rho}{P_{0}\rho_{0}}-1$$ $$N(\frac{P}{P_{0}}+\frac{\rho}{\rho_{0}})=(\frac{P}{P_{0}}-1)(\frac{p}{p_{0}}+1)$$ The equations don't look identical. Did you mix up the positive and negatives? (Or point out any errors I made)