State and prove Bernoulli's theorem for a time-dependent irrotational flow of an inviscid fluid.

A large vessel is part-filled with inviscid liquid of density $\rho$. The pressure in the air above the liquid is maintained at the constant value $P+p_{a}$, where $p_{a}$ is atmospheric pressure and $P>0$. Liquid can flow out of the vessel along a cylindrical tube of length $L$. The radius $a$ of the tube is much smaller than both $L$ and the linear dimensions of the vessel. Initially the tube is sealed and is full of liquid. At time $t=0$ the tube is opened and the liquid starts to flow. Assuming that the tube remains full of liquid, that the pressure at the open end of the tube is atmospheric and that $P$ is so large that gravity is negligible, determine the flux of liquid along the tube at time $t$.