66 
Proceedings of the Royal Society 
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 he united. Thus there will he 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 (fig. 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/<7ra 2 , fji = kN irr 2 a , 
tan 6 = 
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 Y. M. § 58. If N be not divisible 
by 3, the three threads run together into one, as illustrated for the 
case of N = 11 in the annexed diagram (fig. 7). 
14. The irrotational motion of the liquid round the rotational 
cores in all these cases is such that the fluid velocity at any point is 
equal to, and in the same direction as, the resultant magnetic force 
at the corresponding point in the neighbourhood of a closed gal- 
* The first of these was given in § 68 of my paper on vortex motion. It has 
since become known far and wide by being seen on the back of the “ Unseen 
Universe.” 
