Sir William Thomson on Vortex Statics. 99 



culation outside all, is a case of motion which is steady, if the 

 outer and inner contours of the 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 

 circular given vorticity and given impulse. 



6. In these examples of steady motion, the " resultant im- 

 pulse " (V. M.* § 8) is a simple impulsive force, without 

 couple : the corresponding rigid body of example (3) is a 

 toroid; and its motion is purely translational and parallel to 

 the axis of the toroid. 



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 Poinsot's method, to an 

 impulsive force in a determinate line, and a couple ivith 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 Royal Society. In all such cases 

 the corresponding rigid body of § 2 example (2) has both ro- 

 tational and translational motion. 



7. To find illustrations, suppose, first, the vorticity (defined 

 below, § 24) and the force resultant of the impulse to be (ac- 

 cording 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 Fig. 1. 



ellipse of this kind [a shape perfectly ... N 



determinate when the vorticity, the Zlg =— — ^ ^^ 



force resultant of the impulse, and the dj ^v\\ 



rotational moment of the impulse f(\ j j) 



(V. M. § 6), are all given] is the >NL jj/ 



figure of the core in what we may call ^^ ^ ^ ^T 



the firstf steady mode of single and ***** -*''* 



simple toroidal vortex motion with 



rotational moment. To illustrate the second steady mode, 



commence with a circular ring of flexible wire, and pull it out, 



* My first series of papers on Vortex Motion in the Transactions of the 

 Royal Society of Edinburgh will be thus referred to henceforth. 



t First or greatest, and second, and third, and higher modes of steady 

 motion to be regarded as analogous to the first, second, third, and higher 

 fundamental mudes 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. 



