t 192 ] 



XXVI. Hydrodynamic Problems in reference to the Theory of 

 Ocean Current*. By K. ZoPPMTZ*. 



THE aim of the following pages is to show what are the 

 motions admitted by an unlimited stratum of liquid under 

 external influences acting only upon the surface, supposing 

 that friction takes place in the liquid (as it does in water and 

 all other known liquids to a greater or a less amount). 



The differential equations for the motion of frictional liquids 

 are as follows : — 



at fju 0d' fjb 7 



*L + 1|2 _y-^a, = 0. 



at fioy H< 



dio 1 "dp rj k 

 clt (Adz J£ 



d# ~dy "dz ~ ' 



in which u, v, iv are the velocities in the directions of the rect- 

 angular coordinates x, y, z ; X, Y, Z are the components of 

 the external forces ; p is the pressure, jll the density, k the 

 coefficient of friction of the liquid, and A the symbol for the 

 sum of the three partial differential quotients of the second 

 order according to x } y, and z. 



The surface-conditions can be expressed most simply thus : — 

 (1) The particles of the surface of the liquid will always remain 

 in contact with those of the adjacent body ; that is, both must 

 have the same velocity-components perpendicular to the sur- 

 face. Naming these v and i> l5 then must at any time 



v — 1^ = 0. 



This condition includes, if the adjacent body is itself liquid or 

 gaseous, the necessity that the pressure normal to the surface 

 be equal on both sides, because otherwise the connexion ex- 

 pressed by the preceding equation would be broken. (2) The 

 difference of velocity of the particles of the liquid against those 

 of the body in contact, parallel to the plane of contact, is pro- 

 portional to the tangential component T of the internal pressure- 

 forces acting upon the surface, and has the opposite direction. 

 Accordingly, if t and T\ denote the tangential velocities of the 



* Translated from a separate impression, communicated by the Author, 

 from the Annalen der Physik und Chemie, new series, vol. iii. pp. 582-007, 



