in reference to the Theory of Ocean Currents. 201 



then is 

 %P P ^^ «— ! °<An= v/W {e^ Ve~*dz + r+* V^dz}. 



The second integral is obtained from this by integrating ac- 

 cording to x from to as. The resulting double integrals are 

 reduced by partial integration to single ones ; and we get 



a „ r°° sin mx 



= s/ve* {«■■* Ce~*dz - e~'^ flr*dz + 2 f d&-*-?\. 



I Jt+I Jt-« Jo J 



With this becomes 



ar 



w^^^L^+P^l-^dz + e-P 2 *^ e-*de\. (12) 



To obtain from this the more general expression for w 1 ^= $(\) ? 

 we have only to take the second part with the plus sign, insert 

 (t— A.) in the place of t, multiply by (p(\)d\, and integrate 

 from to t. 



The velocity in the surface-stratum is obtained from equa- 

 tion (12) by putting therein # = : — 



L V7T .)„*/„* J 



The second part of this expression approximates, for in- 

 creasing time, to the reciprocal ofps/irat, and therewith to 

 Putting this in words, it means : — The velocity of the surface- 

 stratum continually approaches nearer to that of the conti- 

 guous medium. 



This necessary inference of the theory, however, draws a 

 limit to its applicability to the theory of ocean-currents. Ex- 

 perience teaches that nowhere do the surface-waters of the sea 

 take the mean velocity of the masses of air blowing over them, 

 because, when the wind is rather high, periodic motions of 

 those strata (waves) arise, on the sides of which the wind acts 

 in quite another manner, namely by pressure on the side- 

 surfaces, so that, with still more augmented difference of ve- 

 locity between air and water, breaches of connexion take place, 

 discontinuous, turbulent movements. Therefore the surface- 

 condition above laid down can only for very inconsiderable 

 velocities correspond to the reality ; for greater velocities the 

 water-surface as a whole cannot fulfil it. In this case it is 



