in reference to the Theory of Ocean Currents. 207 



aperiodic variable changes in the forces acting upon the 

 surface are propagated into the depths with extreme slow- 

 ness, the periodic with very rapidly diminishing amplitude. 

 From the combination of these two propositions it follows that 

 the motion of the main body of a sheet of water subject to 

 periodically variable surface-forces is determined by the mean 

 velocity of the surface, and that the periodic changes are per- 

 ceptibly only in a proportionally very thin superficial stratum. 



It is hence obvious that hitherto the influence of the fric- 

 tion has in one direction been underrated, and overrated in 

 another : — underrated, inasmuch as it has been believed that 

 that influence could not be regarded as penetrating to such 

 depths ; overrated, inasmuch as it has been customary to attri- 

 bute to friction much too considerable an influence in regard 

 to the propagation of the motions of variable currents. Its 

 action has been still more overrated in another direction, of 

 which the following investigation will give some explanation. 



Up to the present time the liquid layer was presupposed un- 

 limited in two dimensions ; but the case of laterally bounded 

 currents is also capable of treatment*, and therewith the influ- 

 ence of the sides (the shores) upon the flow can be recognized. 



Let the liquid have again the depth A, but be bounded by 

 two parallel perpendicular sides with the distance 2b between 

 them, which are wetted by the liquid, and at which, therefore, 

 the velocity is = 0. If the X-axis be situated as before, but 

 the axis of Y placed perpendicular to the two sides, and the 

 point of origin in the centre of the upper-surface line of the 

 cross section, then the velocity ic parallel to the sides is given 

 by the differential equation 



dw _ k fb 2 w c) 2 iv\ 

 ~di~1x Xd^ + V ' 

 and the conditions : — 



For # = 0, — ■-= — \-p^= z P ( f>{t J y)\ for # = /i, io = 0; 



fory = 6, w = ; fory= — b, io = 0. 



For the case of stationary motion (which shall exclusively 

 be further pursued here), we have 



"d 2 iv "d 2 w 



d* 2 + §^ 



moreover the velocity of the contiguous medium must be in- 



* The corresponding thermal problem Lame* has taught us to handle in 

 the briefest and most elegant form in his Lemons sur la TMorie Analytioue 

 de la Chaleur, p. 327. 



^ + ^=0; 



