205-206] BASIN OF VARIABLE DEPTH. 343 



1, 1 and 0, and -f 1 and -f GO, respectively. As a numerical example, in 

 the case of s=l, =5, corresponding to the values 



2, 6, 40 



of 4n 2 a 2 /<7/i , we find 



|+ 2-889 +1-874 +1-180, 



o-/2n = <- 0-125 -0-100 -0'037, 



(-2-764 -1-774 -1-143. 



The first and last root of each triad give positive and negative waves of a 

 somewhat similar character to those already obtained in the case of uniform 

 depth. The smaller negative root gives a comparatively slow oscillation which, 

 when the angular velocity n is infinitely small, becomes a steady rotational 

 motion, without elevation or depression of the surface. 



The most important type of forced oscillations is such that 



(x). 



We readily verify, on substitution in (ii), that 



-J.. ...(xi). 



We notice that when o- = 2n the tide-height has exactly the equilibrium- 

 value, in agreement with Art. 204. 



If 01, o- 2 denote the two roots of (vii), the last formula may be written 



Tides on a Rotating Globe. 



206. We proceed to give some account of Laplace's problem 

 of the tidal oscillations of an ocean of (comparatively) small depth 

 covering a rotating globe*. In order to bring out more clearly the 

 nature of the approximations which are made on various grounds, 

 we shall adopt a method of establishing the fundamental equations 

 somewhat different from that usually followed. 



When in relative equilibrium, the free surface is of course a 

 level-surface with respect to gravity and centrifugal force; we 

 shall assume it to be a surface of revolution about the polar axis, 

 but the ellipticity will not in the first instance be taken to be 

 small. 



* " Eecherches sur quelques points du systeme du monde," Mem. de VAcad. roy. 

 des Sciences, 1775 [1778] and 1776 [1779] ; Oeuvres Completes, t. ix. pp. 88, 187. 

 The investigation is reproduced, with various modifications, in the Mecanique 

 Celeste, Livre 4 me , c. i, (1799). 



