^{note * page 562} The sectional and equational numbers are reckoned consecutively from two previous papers “On deep-water two-dimensional waves produced by any given initiating disturbance,” §§ 1–10, Proc. Boy. Soc. Edin., February 1st, 1904, and Phil. Mag., June 1904Google Scholar; and “On front and rear of a free procession of waves in deep water,”1 §§ 11–31, Proc Hoy. Soc. Edin., June 20th, 1904Google Scholar, and Phil. Mag., October 1904.Google Scholar

^{note † page 562} This designation does not include an interesting class of canal waves of which the dynamical theory was first given by Kelland, in the Trans. Boy. Soc. Edin. for 1839Google Scholar; the case in which the wave length is very long in comparison with the depth and breadth of the canal, and the transverse section is of any shape other than rectangular with horizontal bottom and vertical sides.

^{note * page 563} “Forcive” is a very useful word introduced, after careful consultation with literary authorities, by my brother the late Prof. James Thomson, to denote any system of force.

^{note * page 565} It now seems to me certain that if any motion be given within a finite portion of an infinite incompressible liquid originally at rest, its fate is necessarily dissipation to infinite distances with infinitely small velocities everywhere; while the total kinetic energy remains constant. After many years of failure to prove that the motion in the ordinary Helmholtz circular ring is stable, I came to the conclusion that it is essentially unstable, and that its fate must be to become dissipated as now described. I came to this conclusion by extensions not hitherto published of the considerations described in a short paper entitled: “On the stability of steady arid periodic fluid motion,” in the Phil. Mag. for May 1887.