102 Sir William Thomson on Vortex Statics. 



Let a be the radius of the circle thus formed by the axis of the 

 closed helix ; let r denote the radius of the cross section of the 

 ideal toroid on the surface of which the helix lies, supposed 

 small in comparison with a ; and let 6 denote the inclination 

 of the helix to the normal section of the toroid. We have 



a 27ra a 



tan o = ^ — x — = s=-, 



because -^p is, as it were, the step of the screw, and 2irr is 



the circumference of the cylindrical core on which any short 

 part of it may be approximately supposed to be wound. 



Let k be the cyclic constant, I the given force resultant of 

 the impulse, and ijl the given rotational moment. We have 

 (§28) approximately 



I = K7T0 2 , \l = K^TTT^a . 



Hence 



tan#= \f . 





11. Suppose now, instead of a single thread wound spirally 

 round a toroidal core, we have two separate threads forming, 

 as it were, a "two-threaded screw," and let each thread make 

 a whole 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 applicable to each thread, if a; 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 con- 

 venient to take N for the number of turns of both threads (so 

 that the number of turns of one thread alone is \ N), and k 

 the cyclic constant for either thread alone, and thus for very 

 high steady modes of the double vortex ring, 



1 = 2 K7ra 2 } fju = /cN7rr 2 a, 

 tan0= A /-fi^l. 



V JN/X/C 2 7Ta 



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 ultra- 

 transcendent character, except for very great values of N, or 



