On the assumption of a conservative system of external forces, prove Hehmholtz's formulae
$$
\begin{gathered}
\frac{d}{d l}\binom{\mathbf{w}}{\rho} \cdots \frac{\mathbf{w}}{\rho} \cdot \nabla \mathbf{v} \\
\frac{d^2}{d t^2}\left(\frac{\mathbf{w}}{\rho}\right) \cdots\left(\frac{d}{d l} \frac{\mathbf{w}}{\rho}\right) \cdot \nabla \mathbf{v}+\frac{\mathbf{w}}{\rho} \cdot \frac{d}{d l} \nabla \mathbf{v} .
\end{gathered}
$$
i
ii
Taking the curl of both members of (7) Art. 37 we have
$$
\dot{d w}: c u r l(w \cdot v) \quad 0 .
$$
Expanding the second term by (18) of Art. 8 , and using the formula
(1) of this chapter, we may write the result
$$
\begin{aligned}
0 & =\frac{d \mathbf{w}}{d t}+\mathbf{w} \operatorname{div} \mathbf{v}-\mathbf{w} \cdot \nabla \mathbf{v} \\
& =\frac{d \mathbf{w}}{d t}-\frac{\mathbf{w}}{\rho} \frac{d \rho}{d t}-\mathbf{w} \cdot \nabla \mathbf{v}
\end{aligned}
$$
by the equation of continuity. On division by $\rho$ this becomes
$$
\frac{d}{d t}\left(\frac{\mathbf{w}}{\rho}\right)=\frac{\mathbf{w}}{\rho} \cdot \nabla \mathbf{v}
$$
By differentiating this we arrive at the second formula.