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