^{page 60 note *} One of the Helmholtz's now well-known fundamental theorems shows that, from the molecular rotation at every point of an infinite fluid the velocity at every point is determinate, being expressed synthetically by the same formula as those for finding the “magnetic resultant force” of a pure electro-magnet.

— Thomson's Reprint of Papers on Electrostatics and Magnetism.

^{page 61 note *} My first series of papers on vortex motion in the “Transactions of the Royal Society of Edinburgh,” will be thus referred to henceforth.

^{page 61 note †} First or gravest, and second, and third, and higher modes of steady motion to be regarded as analogous to the first, second, third, and higher fundamental modes of an elastic vibrator, or of a stretched cord, or of steady undulatory motion in an endless uniform canal, or in an endless chain of mutually repulsive links.

^{page 63 note *} I call a circular toroid a simple ring generated by the revolution of any singly-circumferential closed plane curve round any axis in its plane not cutting it. A “tore,” following French usage, is a ring generated by the revolution of a circle round any line in its plane not cutting it. Any simple ring, or any solid with a single hole through it, may be called a toroid; but to deserve this appellation it had better be not very unlike a tore.

The endless closed axis of a toroid is a line through its substance passing somewhat approximately through the centres of gravity of all its cross sections. An apertural circumference of a toroid is any closed line in its surface once round its aperture. An apertural section of a toroid is any section by a plane or curved surface which would cut the toroid into two separate toroids. It must cut the surface of the toroid in just two simple closed curves, one of them completely surrounding the other on the sectional surface: of course, it is the space between these curves which is the actual section of the toroidal substance, and the area of the inner one of the two is a section of the aperture.

A section by any surface cutting every apertural circumference, each once and only once, is called a cross section of the toroid. It consists essentially of a simple closed curve.

^{page 66 note *} The first of these was given in § 58 of my paper on vortex motion It has since become known far and wide by being seen on the back of the “Unseen Universe.”