This paper contained a discussion of the consequences of the assumption of continuity of motion throughout a perfect fluid; one of the bases of von Helmholtz's grand investigation, on which W. Thomson founded his theory of vortex-atoms. It is entirely on the assumed absence of finite slip that von Helmlioltz deduces the action of a rotating element on any other element of the fluid, and that Thomson calculates the action of one vortex-atom or part of such an atom on another atom, or on the remainder of itself. The creation of a single vortex-atom, in the sense in which it is defined by Thomson, involves action applied simultaneously to all parts of the fluid mass, not to the rotating portion alone.