124 THE MECHANICS OP THE EARTH'S ATMOSPHERE. 



portional to the sectional area. Therefore r t corresponds to that con- 

 stant diminution of the sectional area which causes the current to 

 diminish just as the irregular obstruction by the waves does. 



For a constant value of a x and a 2j respectively, since A, H u and H, 

 remain unchanged, the condition that a minimum of (& + L) should 

 exist gives 



d{$+L)=d&- S ^a\6r l —~a 2 2 dr 2 =0 (46) 



The other minimum condition in which the a are to be replaced by 



P 



a= u 



H—r 



is 



> *. *i , # '*i *2 ., $n 



which agrees perfectly with that first found. 



The quantities r x and r 2 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 i/- x and i/- 2 . 



Horizontal transportation of the superficial layer. — The quantity of 

 flow \\ and p 2 of the two fluids is no longer the same as it would be 

 over plane surfaces of water for equal values of the velocities a x and 

 a 2 , but it is smaller than before in the upper medium by the quantity 

 *"!«! and in the lower medium by the quantity r% a®. 



Imagine now the velocity (— a 2 ) added to both sides so that the lower 

 medium comes to rest, but the waves progress with the velocity {—(h). 

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

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

 —a 2 r 2 , and thus the wind in the upper region travels not with a uni- 

 form velocity (a,i+a 2 ), but just above the billowy surface there occurs 

 a diminution of the flow of air to the amount of a x r x . 



These two currents cause the 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 velocities a x and a 2 over plane boundary surfaces, 

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

 direction of the wind, is 



M = s 2 a 2 r 2 — s 1 air ] (5). 



This can only be equal to zero when 



Si a 2 r 2 =si oxti (5a), 



or, if we introduce w, the velocity of the wind, 



w = a t + a 2 . , . „ . . . . (56), 



