354 TIDAL WAVES. [CHAP. VIII 



and, at the equator (/* = 0), 



Again, for = 10, or a depth of 29040 feet, we get 

 S/H = 2359 - l-OOOO/* 2 + -589V - 1623/t 6 



+ -0258/t 8 - -0026/* 10 + -0002/Lt 12 ....... (26). 



This makes, at the poles, 



?=-Jtf x 470, 

 and, at the equator, 



For /3 = 5, or a depth of 58080 feet, we find 



?/# - 2723 - rOOOO/i 2 + -340V- 



- -0509/A 6 + 0043/A 8 - -000 V ......... (27). 



This gives, at the poles, 



f=-f# x -651, 

 and, at the equator, 



?= JJI x-817. 



Since the polar and equatorial values of the equilibrium tide are 

 \H and \H , respectively, these results shew that for the depths 

 in question the long-period tides are, on the whole, direct, though 

 the nodal circles will, of course, be shifted more or less from the 

 positions assigned by the equilibrium theory. It appears, more 

 over, that, for depths comparable with the actual depth of the sea, 

 the tide has less than half the equilibrium value. It is easily 

 seen from the form of equation (5), that with increasing depth, 

 and consequent diminution of /3, the tide height will approximate 

 more and more closely to the equilibrium value. This tendency is 

 illustrated by the above numerical results. 



It is to be remarked that the kinetic theory of the long- 

 period tides was passed over by Laplace, under the impression 

 that practically, owing to the operation of dissipative forces, 

 they would have the values given by the equilibrium theory. 

 He proved, indeed, that the tendency of frictional forces must 

 be in this direction, but it has been pointed out by Darwin* 

 that in the case of the fortnightly tide, at all events, it is 

 doubtful whether the effect would be nearly so great as Laplace 

 supposed. We shall return to this point later. 



* I.e. ante p. 350. 



