24 



Prof. J. J. Thomson. 



molecules in B will, on the whole, he moving faster than those in A, 

 and so may he supposed to be at a higher temperature, since the 

 smaller the radius of a vortex ring the greater its velocity. Thus B 

 and A might he the hot and cold chambers respectively of a heat engine, 

 and in this way work might be derived from the gas which was origi- 

 nally at a uniform temperature, so that this arrangement would not 

 obey the second law of thermodynamics. 



If the molecule on the vortex atom theory of matter consisted of a 

 single ring its velocity of translation would be a function only of its 

 radius. It is, however, for several reasons advisable to take a more 

 general case, and to suppose that the molecule consists of several 

 rings linked through each other, the rings being nearly equal in 

 radius, and also nearly coincident in position ; or what is perhaps 

 better, we may suppose that the vortex core forms an endless chain, 

 but that instead of being a single loop like the simple ring, it is 

 looped into a great many coils nearly equal in radius and nearly co- 

 incident in position. We may realise this way of arranging the 

 vortex core if we take a cylindrical rod whose length is great com- 

 pared with its radius, and describe on its surface a screw with n 

 threads so that the threads make min turns in the length of the rod, 

 where m is an integer not divisible by n. Then bend the rod into a 

 circle and join the ends, the threads of the screw will form an endless 

 chain with n loops, and we may suppose that this represents the way 

 in which the vortex rings are arranged ; it is shewn, however, in my 

 " Treatise on the Motion of Yortex Rings " that this way of arranging 

 the vortex core is unstable if n be greater than six. When the vortex 

 core is arranged in the way just described, the velocity of translation is 

 no longer a function of the size of the ring alone ; at the same time when 

 a vortex ring of this kind moves about in a fluid where the velocity is 

 not uniform, the change in the velocity of the ring will be due chiefly 

 to the change in its radius. For the velocity at a small distance d 

 from the circular axis of a vortex ring whose radius is a and strength 

 m is — 



m -t 8a 

 log- — , 



2™ ° d' 



so that as 8a/d is very large, a change £a in the radius of the ring 

 produces a change in the velocity approximately equal to — 



da m t 8a 



- • Q ^g _, 



a 27ra d 



while a change &d in the distance of the point from the circular axis 

 of the ring produces a change in the velocity equal to — 



8d m 

 d 2ira 



