Example: Non relativistic limit

In this case, $T \rightarrow \frac{1}{2} m \beta^2$, $\vec{p} \rightarrow m \vec{\beta}$, $\vec{p}/T \rightarrow 2\vec{\beta} /\beta^2$, and $F / \left( \sum _{i=1} ^n m_i \right)^2 \rightarrow 0$, then

$$\begin{eqnarray*}E_{\rm d} &=& \frac{ \sum _{i<j} \left( m_i + m_j \right) E_{ij... ...{m_i + m_j} \left( \vec{\beta} _i - \vec{\beta} _j \right) ^2 . \end{eqnarray*}$$