Sir William Thomson on Vortex Statics. 103 



for values of 6 infinitely nearly equal to a right angle (this 

 latter limitation leading to the case of infinitely small trans- 

 verse vibrations). 



12. The gravest steady mode of the double vortex-ring cor- 

 responds to N=2. This with the single vortex-core gives the 

 case of the twisted ellipse (§7). With the double core it 

 gives a system which is most easily un- Fig- 5. 

 derstood by taking two plane circular 

 rings of stiff metal linked together. 

 First, place them as nearly coincident 

 as their being linked together permits 

 (fig. 5). Then separate them a little, 

 and incline their planes a little, as shown 

 in the diagram. Then bend each into 

 an unknown shape determined by the strict solution of the 

 transcendental problem of analysis to which the hydro-kinetic 

 investigation leads for this case. 



13. Go back now to the supposition of § 11, and alter it to 

 this : — 



Let each thread make one turn and a half, or any odd num- 

 ber of half turns, round the toroidal core : thus each thread 

 will have an end coincident with an end of the other. Let 

 these coincident ends be united. Thus there will be but one 

 endless thread making an odd number N of turns round the 

 toroidal core. The cases of N = 3 and N=9 are represented 

 in the annexed diagrams (figs. 8 and 9)*. 



Imagine now a three-threaded toroidal helix, and let N 

 denote the whole number of turns round the toroidal core; we 

 have 



I = 3/c7r<2 2 , fi = K^irr 2 a } 





Nju, /e 2 7j-2 



Suppose now N to be divisible by 3 ; then the three threads 

 form three separate endless rings linked together. The case 

 of N = 3 is illustrated by the annexed diagram (fig. 6), which 

 is repeated from the diagram of V. M. § 58. If N be not di- 

 visible by 3, the three heads run together into one, as illus- 

 trated for the case of N = 14 in the annexed diagram 

 (fig. 7). 



'or 



* 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. 



