II. 



ON THE INTEGRALS OF THE HYDRO-DYNAMIC EQUATIONS THAT 

 REPRESENT VORTEX-MOTIONS,* 



By Prof. Hermann von Hki,mholtz. 



Hitherto the integrals of the hydro-dynamic equations have been 

 sought almost exclusively under the assumption that the rectangular 

 components of the velocity of every particle of liquid can be put equal 

 to the differential quotients in the correspoudiug directions of a certain 

 definite function that we will call the velocity potential. 



On the one hand Lagranget had proven that this assumption is al- 

 lowable whenever the movement of the mass of water has arisen and 

 is maintained under the influence of forces that can be expressed as 

 the differential quotients of a force potential, and even that the influ- 

 ence of moving solid bodies that come in contact with the liquid do not 

 affect the applicability of the assumption. Since now most of the forces 

 of nature that are easily expressed mathematically can be presented as 

 the differential quotients of a force potential, therefore also by far the 

 majority of the cases of fluid motion that are treated mathematically 

 fall into the category of those for which a velocity potential exists. 



On the other hand, even Euler J had called attention to the fact that 

 there are cases of fluid motion where no velocity potential exists; e. g., 

 the rotation of a fluid with equal angular velocities in all its parts about 

 an axis. The magnetic forces that act upon a fluid permeated by electric 

 currents, and especially the friction of fluid particles on each other and 

 on solid bodies, belong to the forces that can give rise to such forms of 

 motion. The influence of friction on fluids could not hitherto be mathe- 

 matically defined, and yet it is very large in all cases where we are not 

 treating of infinitely small vibrations, and causes the most important 

 deviations between theory and nature. The difficulty of defining this 

 influence and of finding methods for its measurement certainly lay 



* Crelle's Journal fur die reive und angewandte Mathematik, 1858, vol. lv, p. 25- 

 85. Helmholtz, Wissenschaftliche Abhandlungen, 1882, vol. I, pp. 101-134. London, 

 Edinburgh, and Duhlin Philosophical Magazine, June, 1867 (4), xxin, pp. 485-510 



t Me'canique Analytique, Paris, 1815, vol. II, p. 304. 



X Histoire de VAcade'mie des Sciences de Berlin, auuo 1755, p. 292. 



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