Analysis to the Dynamical Theory of the Tides. 237 



the C's and obtain an equation for the determination of the periods 

 of free oscillation of zonal type. This equation will in general be of 

 a transcendental character, and will be expressible by means of non- 

 terminating continued fractions. The convergence of the continued 

 fractions which occur is, however, very rapid so long as the depth of 

 the water is not materially less than that which occurs on the earth, 

 and this fact enables us to approximate to the roots numerically by a 

 process of trial and error with great facility. The type of oscillation 

 as indicated by the height of the surface-waves is then determined 

 by calculating the ratios of successive C's by means of the formula 

 (a). Numerical results are given in the paper for four different 

 values of li, corresponding to depths of about 7260, 14.520, 29,040, 

 and 58,080 feet. For these depths, the longest periods of free oscil- 

 lation of symmetrical type (corresponding to even values of the 

 suffixes n) are found to be — 



18 hrs. 3 mins., 15 hrs. 11 mius., 12 hrs. 29 mins., 9 hrs. 52 mins., 



while for the unsymmetrical types the longest periods are — 



30 hrs. 29 mins., 25 hrs. 28 mins., 21 hrs. mins., 16 hrs. 52 mins. 



3. If we retain the 7's in the equations (a), these equations will 

 serve to determine the C's in terms of the 7's, and thus to evaluate 

 the height of the forced tides resulting from a given disturbing force. 

 The most important application is that in which all the 7's are zero 

 except 70, and where X is small in comparison with w. The method 

 of procedure is then similar to that adopted by Professor Darwin* for 

 the discussion of the long-period tides, and the numerical results are 

 found on comparison to agree with his, but the analytical form in 

 which they appear is different. The series of zonal harmonics by 

 which the tide-heights are expressed always converge with greater 

 rapidity than the power-series of Professor Darwin, while we have 

 the additional advantage of being able to include in the analysis 

 the effect of the gravitational attraction of the water. Some further 

 numerical results relatively to the forced tides in an ocean of variable 

 depth are also given in the paper. 



4. The series obtained for the forced tides of long period indicate 

 that the tides do not tend towards their " equilibrium " values 

 when the period of the disturbing force is prolonged. This cir- 

 cumstance, which at first sight appears to be at variance with the 

 general laws of oscillating systems, has been explained by Professor 

 Lambf as a consequence of the fact that the system is capable of 

 free oscillations of infinitely long period, or free steady motions. 

 We probably have examples of these free steady motions in the large 



* 'Encyc. Brit.,' art. Tides, § 18. 

 f ' Hydrodynamics,' § 198. 



