ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 159 



Using the first approximation in the terms on the right, we deduce 



(n-lY(n+ l)-(n _)( + ) n'(n + 2)*(n-s+ !)( + 8 + 1) 

 nfr.il) 2( S - (2n-l)(2n+l) (2n + 1) (2 + 3) 



X *=*- + - -**- (n + !)( + 2 



* 



This formula is found to lead to-the roots of the period-equation with a surprising 

 degree of accuracy. 



Our analysis is only applicable when /* is small in comparison with a, but subject to 

 this limitation the approximations of the present section will improve as hg/<a z a? 

 increases, that is, as the depth of the water increases or the angular velocity of rotation 

 diminishes. They will give good results even with small values of n when w is 

 sufficiently small, and they may be used to determine the limiting values assumed 

 by the roots when the angular velocity of rotation is indefinitely reduced. 



We see then that, the roots of the period-equation are of two classes, which may 

 be distinguished by their limiting forms when the rotation is annulled. The roots of 

 the former class are such that the values of X remain finite when w = 0, their limitino- 



3 ft 



values being given by the formula 



x = 



n (n + 1) hg n 



There will be an equal number of positive and negative roots of this class, but 

 though these approach the same limiting values their numerical values will not be 

 equal as in the case where s 0, and the positive and negative roots must therefore 

 be determined independently. 



The roots of the second class are all positive and are such that the values of X/o> 

 tend to finite limits when w is reduced to zero, the limiting values being given by 



the formula 



\ _ 2s 

 to n (n + 1) 



whereas X. will tend to the limit zero. 



The analogue of the types of motion which correspond with the former roots will 

 still be oscillatory when the rotation is annulled, but the types of motion corre- 

 sponding with the roots of the second class will cease to exist as oscillations when 

 the angular velocity of rotation is reduced to zero. These types of motion will 

 have their equivalent in steady motions, but an infinitesimal amount of rotation 

 would immediately convert such steady motions into oscillatory motions of very 

 long period. 



For the particular case s = the roots of the second class are all zero even 

 when the angular velocity of rotation is finite. Hence steady motions can exist 

 on a rotating globe, but these are necessarily of zonal type. We have in fact 



