232] ROTATIONAL CHARACTER. 415 



The circulation in the boundary of the parallelogram whose vertices 

 coincide with the particles 



(a, 6), (a + 8a, 6), (a, b + 8b), (a+da, b+8b) 

 is,by(i), - 



and the area of the circuit is 



d (x ti) 



-^. i-T 8ctoo = (l 



<*(,&) 



Hence the angular velocity (o>) of the element (a, 6) is 



.(ii). 



This is greatest at the surface, and diminishes rapidly with increasing depth. 

 Its sense is opposite to that of the revolution of the particles in their circular 

 orbits. 



A system of waves of the present type cannot therefore be originated from 

 rest, or destroyed, by the action of forces of the kind contemplated in the 

 general theorem of Arts. 18, 34. We may however suppose that by properly 

 adjusted pressures applied to the surface of the waves the liquid is gradually 

 reduced to a state of flow in horizontal lines, in which the velocity (u f ) is 

 a function of the ordinate (/) only*. In this state we shall have #'=a, 

 while y' is a function of b determined by the condition 



d(x'^}_d(x,y] 

 d(a,b) ~ d(a, b}'" 





mi i du' du' dy' dii' 



This makes -^ = -=-, --= - 2o> -. = 



db dy' db db 



and therefore u' 



Hence, for the genesis of the waves by ordinary forces, we require as a 

 foundation an initial horizontal motion, in the direction opposite to that of 

 propagation of the waves ultimately set up, which diminishes rapidly from the 

 surface downwards, according to the law (vi), where 6 is a function of y deter- 

 mined by 



It is to be noted that these rotational waves, when established, have zero 

 momentum. 



* For a fuller statement of the argument see Stokes, Math, and Phys. Papers, 

 t. i., p. 222. 



