Atom and the Charge of Electricity carried by it. 517 



When the vorticity is uniformly distributed over the cross 

 section of the cylinder, 



pm 2 



When the vorticity is all as near to the surface of the 

 cylinder as possible, 



16tt I a 2 \ a z & b 2 — a 2 Jj 



We may expand the right-hand side of the last equation, 

 and get 



pm 2 f 1 a 2 1 a 4 1 a 6 \ 



«tt J 3 b 2 -a 2 4 (b 2 -a 2 ) 2 + 5 (b 2 -a 2 )' 6 ' " f ' 



so that when a is small compared with b, 



_ pm 2 a 2 

 UttV 2 ' 



In this case the tension in the cylinder is very small com- 

 pared with its value in the two previous cases : the value of 

 A in the first case is greater than that in the second ; the 

 more the vortex filaments are concentrated at the axis the 

 greater will be the value of A. Now let us suppose that a 

 Faraday tube contains a given amount of vorticity distributed 

 among irrotationally moving liquid ; the axes of the vortex 

 filaments being parallel to the axis of the tube. The moment 

 of momentum of the fluid in the tube about its axis will 

 depend upon the distribution of vorticity in the tube : the 

 more the vorticity is concentrated near the axis of the tube 

 the greater will be the moment of the momentum. Now 

 suppose we apply a couple to the Faraday tube, the couple 

 acting in such a direction as to increase the moment of 

 momentum ; this couple will cause the vortex filaments to 

 concentrate more at the axis of the tube, and will consequently 

 increase the tension in the tube. If, however, the couple on 

 the Faraday tube acts in the opposite direction to the moment 

 of momentum of the fluid in the tube, the action of the couple 

 will cause the vortex filaments to spread out and get nearer 

 the boundary of the tube: this will diminish the value of A, 

 and consequently diminish the tension in the tube. If we 

 suppose that the solid on which the tube abuts is an atom 

 containing gyrostats, then when the gyrostats are rotating in 



