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We can define the gradients $\nabla \equiv \frac{\D \ln \T}{\D \ln \P}$ and $\nabla' \equiv \frac{\D \ln \T'}{\D \ln \P}$. Here $\nabla$ represents the average temperature gradient with respect to pressure of all matter at a given level, while $\nabla'$ is the temperature gradient with respect to pressure of a rising/falling fluid element.   At first order the temperature excess of such fluid elements can therefore be written as  \begin{equation}  \Delta \T = \T'(r + \Delta r) - \T(r + \delta r) \simeq \Delta r \Bigg[ \bigg[  \frac{\D \T'}{\D r}-\frac{\D \T}{\D r}\Bigg] r}\bigg]  \end{equation}