Prof. Helmholtz on Integrals expressing Vortex -motion. 511 



The above version of one of the most important recent investiga- 

 tions in mathematical physics was made long ago for my own use, 

 and does not pretend to be an exact translation. Professor Helm- 

 holtz has been kind enough to revise it; and it may therefore be ac- 

 cepted as representing the spirit of the original. A portion of the 

 contents of the paper had been anticipated by Professor Stokes in 

 various excellent papers in the Cambridge Philosophical Transac- 

 tions ; but the discovery of the nature and motions of vortex-fila- 

 ments is entirely novel, and of great consequence. Sir W. Thomson 

 has recently propounded a very singular speculation as to the ulti- 

 mate nature of matter, mainly founded on the properties of the Helm- 

 holtz ring. I append an extract from a letter I have just received 

 from him, which fills an important gap towards the end of Professor 

 Helmholtz's work. — P. G. Tait. 



" Following as nearly as may be Helmholtz's notation, let g be 

 the radius of the circular axis of a uniform vortex-ring, and a the 

 radius of the section of its core (which will be approximately circular 

 when a is small in comparison with g), the vortex motion being so 

 instituted that there is no molecular rotation in any part of the fluid 

 exterior to this core, and that in the core the angular velocity of 

 the molecular rotation is approximately w, or rigorously 



9 



for any fluid particle at distance )/ from the straight axis. 



" I find that the velocity of translation is approximately equal to 



<^7i g^_r\ 



2g V & a 4/ 



(quantities of the same order as this multiplied by — being neglected.) 



" The velocity of the liquid at the surface of the core is approxi- 

 mately constant and equal to wa. At the centre of the ring it is 



7Ttott 2 



9 

 " If these be denoted by Q and W respectively, and if T be the 

 velocity of translation, we therefore have 



2g \ " a 4 J 



log -i — - 



a 4 



W. 



2tt 

 Hence the velocity of translation is very large in comparison "with 



