A certain solid material has an isothermal compressibility ${}_T$ given by

${}_T=-\frac{1}{V}(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$

Prove that adiabatic volume and pressure changes are related by

$\frac{dV}{V}=-{}_T\frac{C_V}{C_P}dP$

where $C_V$ and $C_P$ are the specific heats of the material at constant volume and pressure respectively.