ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 167 



as before, retaining D^ as the arbitrary constant of integration when the type 

 under consideration is that whose period is approximately given bv the formula 



n(n + 1) 2o)S/\ = 0. 



For we may anticipate that this quantity will predominate over the others, at least 

 when the depth is sufficiently large or the angular velocity of rotation sufficiently 

 small. The ratios of the remaining constants to D' may then be computed from 

 the formulae (67), and on substituting these ratios in the equations 



,,., ,-, -- , ~ ,^ . . . 



' ' ' "x ' ' j 



iv i n 



JJ +2 p> | 



~Dr 



[. . . + ^.P;., + ^^ +I P. + . .] . 



we obtain expressions for the height of the surface-waves and the velocity-com- 

 ponents. The values of \ for the oscillations of this class being always positive, the 

 direction of the wave-propagation will always be westwards. 



By way of numerical illustration 1 have computed the series for corresponding to 

 the case n = 3, s = 1, and to the values ^g-, -$, -f$, for hg/4aPa*. In these four 

 cases the series within the square brackets are found to be 



- 0-1880P] + 0'3753P^ - OD916PJ + 0'01153P! 



0-00093P! 2 + 0-000052PJ 4 0'000002P1 + . . . 



- 1-4735PJ 0-3260P1 + 0'11690PJ 0'01309PJ + O'OOOSOP},, 



0-000032P1, + 0-000001 P{ 4 ... 



- 07296PJ - 0-2248P1 + 0-03159PJ - 0-00168PJ + 0'00005lP! 



-0'000001P! 2 + . .. 



- 0-3617P.1 - 0-1277P1 + 0-00814PJ - 0'0002lPJ + -000003P! - ... 



It will be seen that as the depth increases or 01 diminishes the coefficients all 

 become smaller, while the convergence of the series improves. It may be inferred as 

 in the last section that Cj/Dj + 1 will tend towards the limit zero when r is less than 



