The average orthodox mathematician who writes or reads books about hydrodynamics is so keen about getting a peg on which he can hang what he calls examples that he quite forgets to apply his formulae to compressible fluids. It appeared to me on reading Colonel De Villamil's letter that a useful purpose would be served by writing the present note; it contains nothing but what anyone could write out for himself, but it was only recently that I made the attempt myself and I should be surprised if many readers have seen the same thing before. We know that in the irrotational motion of an incompressible fluid the velocity potential satisfies Laplace's equation, the present object is to find the corresponding equation for a compressible fluid, to grind it out at full length, and (what is more important than grinding it out) to interpret the result.