in reference to the Theory of Ocean Currents. 205 



cated problem, v must then be representable by a series of 

 cosines and sines of the same period. By making use of for- 

 mula (14) we obtain the following series * : — 



from which we see immediately that the velocity at every 

 depth has the same period 2g, but that the amplitudes, 



diminish as the depth increases, In sea-water 



v 



^ = 14-77. 



k 



If x be expressed in centimetres and 2g in seconds, for 

 #=100 metres and the period of one year (consequently for 

 ra=l, 2^ = 1 year) the exponent of e becomes about —26; 

 therefore the amplitude of this oscillation already diminishes 



to a vanishing fraction, with #=10 metres to e~ 2 ' 6 = t-qT?* 



If the depths diminish in an arithmetical series, the ampli- 

 tudes of oscillation diminish in a geometrical series, so that in 

 four depths # 1? x 2 > x zi #4 so situated that # 4 — # 3 = # 2 — x\ the 

 amplitudes ly 2 , 3 , 4 are in the ratio 



04 : #3 — 02 : #1- 

 A maximum and its succeeding minimum of an oscillation 

 of the duration 2y are ever simultaneous in depths separated 

 by a distance 



v ^ 



which for 2^=1 year gives 



#2—^=11-9 metres. 



In order to get a numerical representation of the time re- 

 quired by a surface-velocity w commencing at time £=0, and 

 remaining constant, to introduce at the bottom of a previously 

 still ocean of the finite depth h the state of velocity opposite 



to the stationary (when w = w , ), the above formula (13) 



must be employed, in which/(^) must be put =0 and <j>(X) =iv . 

 We then get 



w = 2w — n> ^ l ( — l) n - 1 nsm— r -\ e va// J d\, 

 M 2 » h Jo 



* See the expansion in Riemann's Vorlesungen, p. 137. 



