ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 141 



method of determining the higher roots is given. The approximations will not 

 however be sufficiently close for the earlier roots, and consequently it is necessary 

 to evaluate these earlier roots by trial and error. The method of procedure is 

 indicated by numerical examples, and several of the more important roots are- 

 tabulated for four different depths of the ocean. The most interesting result is the 

 existence of a second class of free oscillations besides those whose existence may be 

 at once inferred by analogy from the simpler problem of the oscillations of an ocean 

 covering a non-rotating, globe. The characteristics of the oscillations of this class 

 are discussed in 11. 



In 12 a general analytical solution of the problem of the forced vibrations due to 

 any disturbing force is given, but as the analytical expressions obtained are too 

 intricate to afford much indication of the nature of the forced tides, the various 

 types of oscillation which occur on the earth are afterwards treated numerically. 



In certain cases, intimately associated with those actually occurring, the analytical 

 expressions however admit of considerable reductions. These cases are discussed 

 in 14, where theorems due to LAPLACE and Professor DARWIN are obtained and 

 generalized. 



15-18 contain numerical examples of the evaluation of the semi-diurnal and 

 diurnal tidal constituents. The arithmetic is considerably simplified when the 

 period of the disturbing force is rigorously equal to half a sidereal day or a sidereal 

 day, and consequently these cases are first dealt with and the results compared 

 with those of LAPLACE. Additional examples are however also given to illustrate 

 the effects of the departure of the periods from exact coincidence with half a sidereal 

 day and a sidereal day respectively, the cases selected for investigation corresponding 

 with the leading lunar constituents. 



1. Properties of Tesseral Harmonics. 



Let P (/A) denote the zonal harmonic of order n. Then P,, is the solution which 

 remains finite when p, = 1 of the differential equation 



Let 



P; (p.) = (1 /A 3 )* 8 ^7" (2). 



Then, on differentiating the equation (I) s times, we obtain 



or, 



(1 _ ^v *. {( t _ ^-i. P.} - 2 (, + 1) ,* (1 - ^ ^ {(1 



4. ( n - s) (n + s + 1) P;, = 0, 



