$P _{12} V _2^\gamma= P _{34} V_3^\gamma \quad \dots(i)$

$P _{34} V_4^\gamma= P _{12} V_1^\gamma \quad \dots(ii)$

Multiply $(i)$ and $(ii)$

$V _2^\gamma \cdot V _4^\gamma= V _1^\gamma \cdot V _3^\gamma$

$V _2 V _4= V _1 V _3 \Rightarrow\left(\frac{ V _2}{ V _3}=\frac{ V _1}{ V _4}\right)= K _4$

$\eta=\frac{\text { Work done }}{ Q _{\text {suppl. }}}=1-\frac{ Q _{\text {rej }}}{ Q _{\text {supp }}}=1-\frac{ n _{ C / P } \Delta T _{34}}{ n _{ C / P } \Delta T _{12}}$

$=1-\frac{\Delta( PV )_{34}}{\Delta( PV )_{12}}=1-\frac{ P _{34}\left( V _3- V _4\right)}{ P _{12}\left( V _2- V _1\right)}$