A Carnot heat engine operating between a high $T_{H}$ and low $T_{L}$ energy reservoirs has an efficiency given by the temperatures. Compare this to two combined heat engines, one operating between $T_{H}$ and an intermediate temperature $T_{M}$ giving out work $W_{A}$ and the other operating between $T_{M}$ and $T_{L}$ giving out work $W_{B}$. The combination must have the same efficiency as the single heat engine, so the heat transfer ratio $Q_{H} / Q_{L}=\psi\left(T_{H}, T_{L}\right)=\left[Q_{H} / Q_{M}\right]\left[Q_{M} / Q_{L}\right]$ The last two heat transfer ratios can be expressed by the same function $\psi()$ also involving the temperature $T_{M}$. Use this to show a condition that the function $\psi$ () must satisfy.