65 
of Edinburgh, Session 1875 - 76 . 
number of turns round the toroidal core. The two threads, each 
endless, will be two helically tortuous rings linked together, and 
will constitute the core of what will now be a double vortex ring. 
The formulae just now obtained for a single thread would be appli- 
cable to each thread, if k denoted the cyclic constant for the circuit 
round the two threads, or twice the cyclic constant for either, and 
N the number of turns of either alone round the toroidal core. 
But it is more convenient to take N for the number of turns of 
both threads (so that the number of turns of one thread alone is 
JN), and k the cyclic constant for either thread alone, and thus for 
very high steady modes of the double vortex ring 
I = 2/c7ra 2 , fx = KN7rr 2 a, 
tan 0 
= 11 
sj N 
m 
N/XK^7r5 * 
Lower and lower steady modes will correspond to smaller and 
smaller values of N, but in this case, as in the case of the single 
vortex core, the form will be a curve of some ultratranscendent 
character, except for very great values of N, or for values of 6 in- 
finitely nearly equal to a right angle (this latter limitation leading 
to the case of infinitely small transverse vibrations). 
12. The gravest steady mode of the double vortex ring corre- 
sponds to N = 2. This with the single vortex core gives the case 
of the twisted ellipse (§ 7). With the double core it gives a sys- 
tem which is most easily understood 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 deter- 
mined 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 : — 
