in reference to the Theory of Ocean Currents. 209 



tain, at least approximately, the value of p with low velocities. 

 For if we imagine a very long canal of given rectangular cross 

 section (for the sake of simplicity let it be taken as square, 

 consequently 2b = h) set in stationary motion by a gentle wind 

 with the velocity w 1 blowing parallel to its axis, the velocity 

 in the middle line of its surface (y — 0) will be 



77 • (3»+l){l+ (2w + 1)7r 6ot(2»+i>} 



Remarking that, for ?i = 0, @ot7r is already =1*0037, and for 

 higher values of n the value of @ot(2n + l)7r continually ap- 

 proaches nearer to unity, we find as a very close approxima- 

 tion : — 



_ 4w 1 j>h f 1 _1 1_ 1 1__ \ 



W °°~ IT Xph + TT 3j# + 3ff Oph + bTT '.")' 



If next we consider only two terms of this series, we get for 

 ph the quadratic equation 



p 2 Ji 2 (8iVi — 377-1%,) +ph . 4&r(8w 1 — 37nr 00 ) — $7r d ic 00 = 0. 

 according to which 



Were, for example, Wi = 4m, iv 00 — 2m } consequently w ± : w 00 = 2, 

 we should obtain ph = 0*8 8 w : or 



2-77 



and the first neglected term in the above series would be 

 = cuTa ? while the first term is = r^ — . The error would 



Zy*4: . 7T I'OO . 7T 



therefore amount to about ^ of the value, and would be re- 

 duced to about g T Q by taking a further term into account. 



It appears, however, that p between water and air is very 

 great, and on that account the ratio w 1 = 2w 00 possible only 

 when the canal is very narrow, while with a greater width iv 00 

 approaches much nearer in value to w x . Many more terms of 

 the above series would then have to be taken into account. 

 The greater the value of p the nearer does the value of tv come 

 to that of u\ ; consequently the curve which represents iv as 

 a function of the distance y from the middle approximates to 

 a straight line parallel to the axis of Y. The same can be said 

 of the velocity in any horizontal line at the depth x. It is 

 therefore evident that, in case p is very great, the banks exert 

 a very inconsiderable influence on the distribution of the ve- 

 locity. 



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



