Maximum and Minimum Energy in Vortex Motion. 529 



much larger body of experimental evidence. Further inves- 

 tigations in the required direction are now in progress in the 

 laboratory of the Finsbury Technical College. 



Postscript. — Since writing the foregoing paper, the detailed 

 evidence which has led to the conclusion that the ethyl- 

 derivative ofm.p. 187° is a diazo-compound has appeared in 

 a communication to the Chemical Society ( Journ. Chem. Soc, 

 Trans. 1887, p. 434). Additional evidence of the production of 

 mixed compounds on the decomposition of mixed diazoamido- 

 derivatives is given in a recent paper by Heumann and Oeco- 

 nomides (Ber. 1887, p. 904). These authors find that the 

 mixed compound C 6 H 5 . N 3 H . C 7 H 7 , on being heated with 

 phenol, gives a mixture of aniline and p-toluidine together 

 with oxyazobenzene and p-tolueneazophenol. 



LXI. On the Stability of Steady and of Periodic Fluid Motion 

 (continued from May number). — Maximum and Minimum 

 Energy in Vortex Motion*. By Sir William Thomson, 



f.r's. 



10. ri^HE condition for steady motion of an incompressible 

 JL inviscid fluid filling a finite fixed portion of space 

 (that is to say, motion in which the velocity and direction of 

 motion continue unchanged at every point of the space within 

 which the fluid is placed) is that, with given vorticity, the 

 energy is a thorough maximum, or a thorough minimum, or 

 a minimax. The further condition of stability is secured, by 

 the consideration of energy alone, for any case of steady 

 motion for which the energy is a thorough maximum or a 

 thorough minimum ; because when the boundary is held fixed 

 the energy is of necessity constant. But the mere consi- 

 deration of energy does not decide the question of stability 

 for any case of steady motion in which the energy is a 

 minimax. 



11. It is clearf that, commencing with any given motion, 

 the energy maybe increased indefinitely by properly-designed 

 operation on the boundary (understood that the primitive 

 boundary is returned to). Hence, with given vorticity, but 

 with no other condition, there is no thorough maximum 

 of energy in any case. There may also, except in the case 

 of irrotat ; onal circulation in a multiplexly continuous vessel 



* Being a communication read before the British Association, Section A, 

 at the Swansea Meeting, Saturday, August 28, 1880, and published in the 

 Report for that year, p. 473 ; and in ' Nature,' Oct. 28, 1880. Reprinted 

 now wdth corrections, amendments, and additions. 



t See also §§ 3 to 9 above. 



