ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 165 



By a comparison of the approximate values given in these tables with the true 

 values we see that for extreme cases here tabulated, the error involved in the 

 approximation does not amount to more than about 3 per cent., even with a depth as 

 small as 7260 feet. We have here a justification of the statements made in the 

 last section as to the approximation to the higher roots. 



10. Oscillations of the First Class. Determination of the Type. 



We shall describe as oscillations of the first class those whose periods remain finite 

 when the rotation-period is indefinitely prolonged, that is, those for which the roots 

 of the period-equation are of the first class. The types of motion whose periods 

 become infinitely long with the rotation-period will be called oscillations of the 

 second class. 



The determination of the type involves the determination of the constants Cj, 

 Dj+i, Q +2 , &c. (supposing for convenience that we are dealing with symmetrical 

 types). For this purpose we may either make use of the formula} of 6, or we may 

 make use of the formulae of 5 for the determination of the C's, after which the D's 

 must be computed from equation (49). The latter method will be closely analogous 

 to that used in 9 of Part I., but the former is the more convenient when the 

 numerical determination of the constants is required. 



If we make use of the notation (64), the following relations may be deduced from 

 the equations (52) : 



C^_ In + 1 _Ci_ 2+_l_ ~] 



D " + *) ( - D s D',+1 " " (n - s + 1) (n + 2)' L I 



But we have seen in the last section how the quantities e,f, E, F may be deter- 

 mined numerically. Hence the above formulas allow us to compute the ratios of the 

 constants C, D. One of these constants must be regarded as arbitrary, and the 

 ratios of the others to it can then be computed. When the type under examination 

 is that which corresponds to a root of the period-equation approximating to a root of 

 M^ = 0, we select as the arbitrary constant of integration the quantity C*, as the 

 continued fractions c, f, E, F required to determine the ratios of the remaining 

 constants to this one will then be free from singularities. 



When these ratios have been determined we may substitute their values in the 

 formulas 



