$(1)$
\[\frac{1}{\rho}\frac{\partial p}{\partial s}+\frac{1}{2}\frac{\partial (V^2)}{\partial s}+\frac{\partial V}{\partial t}+g\frac{\partial z}{\partial s}=0\]
$(2)$
\[pv^\kappa=Cより\]
\[p(\frac{1}{\rho})^\kappa=C\]
\[p\rho^{-\kappa}=C\]
$(3)$
定常だから
\[\frac{1}{\rho}\frac{\partial p}{\partial s}+\frac{1}{2}\frac{\partial (V^2)}{as}+g\frac{\partial z}{\partial s}=0\\
p\rho^{-\kappa}=Cより\\
\frac{1}{\rho}=C'p^{\frac{1}{\kappa}}\\
C'=\frac{p^{\frac{1}{\kappa}}}{\rho}
\]
\[\therefore C'\int{p^{-\frac{1}{\kappa}}}dp+\frac{1}{2}\int dV^2+g\int dz=C\]
\[
C'\frac{1}{1+\frac{1}{\kappa}}p^{1+\frac{1}{\kappa}}+\frac{V^2}{2}+gz=C\\
C'\frac{\kappa}{\kappa+1}p^{\frac{\kappa+1}{\kappa}}+\frac{V^2}{2}+gz=C\\
\frac{p^{-\frac{1}{\kappa}}}{\rho}\frac{\kappa}{\kappa+1}p^{\frac{\kappa+1}{\kappa}}+\frac{V^2}{2}+gz=C\\
\frac{\kappa}{\kappa+1}\frac{p}{\rho}+\frac{V^2}{2}+gz=C\\
\]