Sir William Thomson on Vortex Statics. 101 



resultant of impulse. The modes successively described above 

 are successive solutions of the maximum-minimum problem of 

 § 4 — a determinate problem with the multiple solutions indi- 

 cated above, but no other solution, when the vorticity is given 

 in a single simple ring of the liquid. 



9. The problem of steady motion, for the case of a vortex- 

 line with infinitely thin core, bears a close analogy to the fol- 

 lowing purely geometrical problem : — 



Find the curve whose length shall be a minimum with given 

 resultant projectional area, and given resultant areal moment 

 (§27 below). This would be identical with the vortex prob- 

 lem if the energy of an infinitely thin vortex ring of given 

 volume and given cyclic constant were a function simply of 

 its apertural circumference. The geometrical problem clearly 

 has multiple solutions answering precisely to the solutions of 

 the vortex problem. 



10. The very high modes of solution are clearly very nearly 

 identical for the two problems (infinitely high modes iden- 

 tical), and are found thus : — 



Take the solution derived in the manner explained above, 

 from a regular polygon of N sides, when N is a very great 

 number. It is obvious that either problem must lead to a form 

 of curve like that of a long regular spiral spring of the ordinary 

 kind bent round till its two ends meet, and then having its 

 ends properly cut and joined so as to give a continuous end- 

 less helix with axis a circle (instead of the ordinary straight 

 line-axis), and IN" turns of the spiral round its circular axis. 

 This curve I call a toroidal helix, because it lies on a toroid*, 

 just as the common regular helix lies on a circular cylinder. 



* 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 fee 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 pass- 

 ing 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 sec- 

 tional 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. 



