Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures,

(2) Let $Q_{H}=$ Heat taken by first engine

$Q_{L}=$ Heat rejected by first engine

$Q_{2}=$ Heat rejected by second engine

Work done by $1^{\text {st }}$ engine $=$ work done by $2^{\text {nd }}$ engine

$W=Q_{H}-Q_{L}=Q_{L}-Q_{2} \Rightarrow 2 Q_{L}=Q_{H}+Q_{2}$

$2=\frac{\theta_{H}}{\theta_{L}}+\frac{\theta_{2}}{\theta_{L}}$

Let $\mathrm{T}$ be the temperature of cold reservoir of first engine. Then in carnot engine.

$\frac{Q_{H}}{Q_{L}}=\frac{T_{1}}{T}$ and $\frac{Q_{L}}{Q_{2}}=\frac{T}{T_{2}}$

$\Rightarrow 2=\frac{T_{1}}{T}+\frac{T_{2}}{T}$                       using (i)

$\Rightarrow 2 T=T_{1}+T_{2} \Rightarrow T=\frac{T_{1}+T_{2}}{2}$