NEWTON S PRINCIPIA. 389 



the motion is nearly the same at all depths ; if much greater 

 the water near the bottom is almost at rest. Each particle 

 of water describes an ellipse. Those at the bottom oscillate 

 in straight lines, and in very deep water those at the sur 

 face describe circles. 



When the channel is not uniform the question of the 

 motion is much more difficult. Professor Kelland has in 

 vestigated one case of this kind, for which we refer our 

 readers to the Edin. Trans, for 1841. 



The motion of any series of oscillatory waves will, of 

 course, tend to subside from the effects of internal friction. 

 The rate at which this takes place depends very much on 

 the length of the wave. The magnitude of the wave is 

 found to depend on the factor * 



C = C Q . X 2 



where X is the length of the wave, p/ a numerical quantity 

 depending on the amount of internal friction in the fluid, 

 c Q , c the values of the factor at the times zero and t. The 



value of V it/ for water is -0564, an inch and a second 

 being the units of space and time. &quot; Suppose, first, that A 

 is two inches and t ten seconds, then 16 7r 2 // t A~ 2 = 

 1-256 and c : c :: 1 : 0-2848, so that the height of the 

 waves which varies as c is only about a quarter of what 

 it was ; accordingly the ripples excited on a small pool by 

 a puff of wind rapidly subside when the exciting cause 

 ceases to act. Now suppose that A is 40 fathoms, or 2880 

 inches, and that t is 86,400 seconds, or a whole day. In 

 this case 16 7r 2 ^ t A~ 2 is equal to only 0-005232, so that 

 by the end of an entire day, in which time waves of this 

 length would travel 574 English miles, the height would 

 be diminished by little more than one two-hundredth 

 part in consequence of friction. Accordingly the long 

 swells of the ocean are but little allayed by friction, and 



* Prof. Stokes, Camb. Phil. Trans, vol. ix., part 2. 

 cc 3 



