in reference to the Theory of Ocean Currents. 199 



sents the influence of the change of velocity 7 propagated in 

 the interior. If t and 6 are enormously great, but their dif- 

 ference little, the second member approximates to the value 



y(t-e). 



The influence of that alteration is then at every depth propor- 

 tional to its duration and amount. 



If the velocity of the contiguous medium is, at the surface, 

 a periodic function of the time — for example, 



cf)(t)= cos (at—b), — 



then, if t— \=/> be introduced as a new variable, becomes 



1 *-■*«**-*) cos (a\—b)d\ = cos (at — b) 1 e~ m2 ^ cos ap dp 



+ sin (at—b) j e' m2a § sin ap dp. 



After an indefinitely long time these two integrals become 

 constant with respect to the time, and consequently the velo- 

 city at every depth becomes a periodic function of the time, 



of the same period ( — j as that of the contiguous medium, 



but of changed amplitude, dependent on „r, and with shifted 

 period of occurrence of maxima and minima. If <f>(t) is a pe- 

 riodic function of general character, it can be represented under 

 the form 



(f>(t) = w 1 + w > \ cos (at— bi) + iv' 2 cos (2at — b 2 ) + . . . 



Putting this value in the integral, we obtain for w an ex- 

 pression, the first member of which changes, for £ = 00 , into 



PQ l ~ x ) w 



upon which follows a series of members with cos (nat—b) and 

 sin (nat—b), of the same sort as in the previous more simple 

 case for n=l. 



If we wish to calculate the mean velocity during a period 



T = — .all terms affected with sine and cosine vanish from the 

 a 



time-integral, and there remains 



27r ] u * ph + i 



— I wdt — 



as the mean velocity, consequently the same as with the sta- 

 tionarv motion, 



