124 THE MECHANICS OF THE EARTh's ATMOSPHERE. 



portional to the sectional area. Therefore n corresponds to that con- 

 stant diminution of tlie sectional area which causes the current to 

 diminish just as the irreguhir obstruction by the waves does. 



For a constant value of a^ and «,, respectively, since A, JTi, and H2 

 remain unchanged, the condition that a minimum of [0 -\- L) should 

 exist gives 



S[<P+L) = S0—.,aldri—:,a:6r,=O (4^) 



The other niiuinium condition in which ^/,, a-, are to be replaced by 



Pi 1 P' 



a,= ^^^ and a.=^r-. 



is 



which agrees perfectly with that tirst found. 



The quantities ri and r-i depend only on the form of the wave, and are 

 generally found by simple computations as soon as we have found the 

 form of the functions (j-i and if-o. 



Horizontal trnnsportation of the superficial layer. — The quantity of 

 flow pi and p2 of the two fluids is no longer the same as it would be 

 over plane surfaces of water for equal values of the velocities Oi and 

 «25 but it is smaller than before in the upper medium by the quantity 

 Vi «! and in the lower medium by the quantity Vo a-z. 



Imagine now the velocity' {—(h) added to both sides so that the lower 

 medium comes to rest, but the waves progress with the velocity { — a-z). 

 Then beneath plane boundary surfaces all motion disappears, but be- 

 neath billowy surfaces a general current is set up of the magnitude 

 —a-z Vz, and thus the wind in the upper region travels not with a uni- 

 form velocity (aj + rtj), but just above the billowy surface there occurs 

 a diminution of the How of air to the amount of rti Vi. 



These two currents cause tlie mass of air and water taken together to 

 have a different moment of motion in a horizontal direction than if they 

 flowed with the same velocitic. «i and az over plane boundary surfaces, 

 and this difference of moment of motion, reckoned as positive in the 

 direction of the wind, is 



jl = s-z «•> Vz — Si tti ri (o). 



This can only be equal to zero when 



s-, az rz=Si GiVi (5rt), 



or, if we introduce u\ the velocitj- of the wind, 



iv = Cti -\- Uz (5&)> 



