348 TIDAL WAVES. [CHAP. VIII 



The formulae for the component displacements (f , 77, say), can 

 be written down from the relations u ^ v = rj, or u = ia%> v = lo " 7 7- 

 It appears that in all cases of periodic disturbing forces the fluid 

 particles describe ellipses having their principal axes along the 

 meridians and the parallels of latitude, respectively. 



Substituting from (7) in (4) we obtain the differential equation 



1 d j h sin e (d? s 

 sin d dO I/ 2 - cos 2 6 (d0 + f^ C D 



~ f cot 9 | + s 2 f ' cosec 2 0\ + 4 waf = - 4ma f 



.................. (8). 



In the case of the/ree oscillations we have f = 0. The manner 

 in which the boundary -conditions (if any), or the conditions of 

 finiteness, then determine the admissible values of/, and thence of 

 <7, will be understood by analogy, in a general way, from Arts. 191, 

 193. For further details we must refer to the paper cited below*. 

 A practical solution of the problem, even in the case (s = 0) of 

 symmetry about the axis, with uniform depth, has not yet been 

 worked out. 



The more important problem of the forced oscillations, though 

 difficult, can be solved for certain laws of depth, and for certain 

 special values of <r which correspond more or less closely to the 

 main types of tidal disturbance. To this we now proceed. 



209. It is shewn in the Appendix to this Chapter that the 

 tide-generating potential, when expanded in simple-harmonic 

 functions of the time, consists of terms of three distinct types. 



The first type is such that the equilibrium tide-height would 

 be given by 



The corresponding forced waves are called by Laplace the f Oscilla- 

 tions of the First Species ' ; they include the lunar fortnightly 



* Sir W. Thomson, " On the General Integration of Laplace's Differential 

 Equation of the Tides," Phil. Mag., Nov. 1875. 



t In strictness, 6 here denotes the geocentric latitude, but the difference between 

 this and the geographical latitude may be neglected in virtue of the assumptions in- 

 troduced in Art. 207. 



