M. K. Zoppritz on the Theory of Ocean Currents. 193 



liquid and the contiguous body, the equation 

 T = X(r 1 ~r) 



will subsist, in which \ is the constant of the external friction, 

 depending only on the nature of the two bodies in contact. If 

 wetting of a part of the surface takes place, we must there put 

 t = ti, consequently \= go . 



Let the liquid be spread over a solid plane. Of external 

 forces gravitation only acts perpendicular to this plane. Let 

 its direction be the positive X-direction ; then is 



X = <7, Y=Z = 0. 



Let the solid plane be wetted, so that the liquid particles 

 adjacent to it always remain at rest. This requires that for 

 them u = v = w = always. The body in contact with the other 

 surface will at every point have equal velocity and in the same 

 direction, but in general dependent on the time. If the initial 

 motion of the liquid was parallel to that of the contiguous 

 medium, or was also = 0, then, in accordance with the second 

 surface-condition, only motions in the same direction can at 

 any time take place. Consequently, if we place the plane of 

 XZ parallel to this direction, v is always = 0. The differen- 

 tial equations then become 



du 1 "dp k ,, 



dt fM^X ^ fJU 



dw 1 "ftp __ k 

 dt fju'dz fjb 



Aw = 



'dx ~dz 



The conditions for the particles in contact with the solid 

 surface are 



v-v! = 0, T=X(t!-t). 



If the angles made by the normal to the surface with the 

 X- and Z-axes be denoted by (n, x) and (n, z) respectively, 

 the 1st equation gives 



(u — Ui) cos (ft, x) + (iv — u? x ) cos (n, z) = ; 



while the 2nd splits up into the two following : — 



T sin (ft, as) — \{u x ~ u ) \ T sin (ft , z) = \(iv 1 —io). 



These equations are satisfied if we assume the surface to be 

 horizontal — that is, put 



cos (ft, ob) = I, cos (ft, y) = cos {ii, z) = 0, 

 Phil Mag, S. 5. Vol. 6. No. 36. Sept. 1878. 



