159, 1GO.] WAVES IN 7 DEEP WATER. 193 



and 2=gb + $c*e-**. 



The pressure about any particle is therefore constant during the 

 motion. The form of any surface of equal pressure (5 = const.) 



is obtained by rolling a circle of radius y on the under side of 



K 



the straight line y = b j . A point fixed relatively to this circle, 



K 



at a distance r e ~ kb from its centre, will trace out the profile of 

 the surface in question. The wave-length is 



and the velocity of propagation is 



. c _ fl-&L 



v* Va 





If we differentiate (34) with respect to , we get for the 

 component velocities of a particle 



u ce~ kb cosk(a + ct), 

 v = ce~ kb sin k (a + ct), 

 and thence 



(c &quot;) 



udx + vdy = d T e kb sin k (a + d)&amp;gt; + ce 2 * 6 da, 



which is not an exact differential, so that the motion represented 

 by (34) is rotational, and cannot therefore have been generated 

 from rest by the action of ordinary natural forces. The circu 

 lation in the boundary of the parallelogram whose angular points 

 coincide with the particles 



(a, I), (a + da, fy, (a, b + db), (a + da, b + db) 

 is - ~ (ce-* kb ) da db, 



and the area of this parallelogram is 



13 



