1. In the teaching of hydrodynamics many points of difficulty arise, both for teacher and for student. The subject abounds in subtleties even in its very elements, and the advanced student frequently finds himself in a state of doubt as to fundamental questions which crop up unexpectedly in connection with various problems. The following notes contain a discussion of a few of these fundamental matters: for example, they deal with some theorems of energy which have been found difficult by students, perhaps mainly through want of perfectly explicit statement of their scope and purpose. Finally will be demonstrated a theorem of vortex-motion, particular cases of which have been given by various writers, but which I have not seen stated elsewhere in the same generality. This will be found to lead to Lord Kelvin's well-known and far-reaching theorem of circulation, and to other theorems of the vortex-motion of a perfect fluid, some of which are already known.