842 



condition, that the work of these forces during a small angular dis- 

 placement equals tiie increase of the kinetic energy ^7ni;/ of the 

 ring, we find, when the density of the liquid is n, 

 d«, / dF 



d di 



= — (i mvr') = 2.Tr COS 6 . rde . dr . (i 



Irtr cos s . rde . r cos g 



Jr ] 2.T (r + rfr)' cos'e . ds 



bur 



èt 



bt 



dF QF dv, d-ar 



According to the elementary laws of internal friction the force F 

 is proportional to the velocity-gradient in the direction of the radius ; 

 in determining this slope we must only take into account the gradient 

 which is due to the change of the angular velocity with r'). The 



velocity-gradient thus becomes equal to r ccw f v^l —- 1, and therefore 



1) The gradient of velocity which is the consequence of a uniform rotation of 

 the liquid does not produce any friction. In tlie classical hydrodynamical theory 

 this results from the circumstance that in a uniform rotation there is no deformation 

 and consequently no stress. (Note added in the translation). 



