61 
of Edinburgh, Session 1875-76. 
section of the rotational shell are properly shaped, but certainly 
unstable if the shell be too thin. In this case also the energy is 
maximum-minimum for given vorticity and given impulse. 
6. In these examples of steady motion, the “ resultant impulse” 
(V. M.* § 8) is a simple impulsive force, without couple; the cor- 
responding rigid body of example 3 is a circular toroid, and its 
motion is purely translational and parallel to the axis of the toroid. 
5. We have also exceedingly interesting cases of steady motion 
in which the impulse is such that, if applied to a rigid body, it 
would be reducible, according to Poinsofs method, to an impulsive 
force in a determinate line, and a couple with this line for axis. 
To this category belong certain distributions of vorticity giving 
longitudinal vibrations, with thickenings and thinnings of the core 
travelling as waves in one direction or the other round a vortex 
ring, which will be investigated in a future communication to the 
Koyal Society. In all such cases, the corresponding rigid body of 
§ 2 example (2) has both rotational and translational motion. 
7. To find illustrations, suppose, first, the vorticity (defined below, 
§ 24) and the force resultant of the impulse to be (according to the 
conditions explained below, § 29) such that the cross section is 
small in comparison with the aperture. Take a ring of flexible 
wire (a piece of very stout lead wire with its ends soldered together 
answers well), bend it into an oval form, 
and then give it a right-handed twist 
round the long axis of the oval, so that 
the curve comes to be not in one plane 
(fig. 1). A properly-shaped twisted 
ellipse of this kind [a shape perfectly 
determinate when the vorticity, the 
force resultant of the impulse, and the rotational moment of the 
impulse (V. M. § 6), are all given] is the figure of the core in what 
we may call the first f steady mode of single and simple toroidal 
* My first series of papers on vortex motion in the “ Transactions of the 
Royal Society of Edinburgh,” will he thus referred to henceforth. 
t First or gravest, and second, and third, and higher modes of steady mo- 
tion to be regarded as analogous to the first, second, third, and higher funda- 
mental 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. 
