318 TIDAL WAVES. [CHAP. VIII 



the coefficients being the normal coordinates of the system. Again, since the 

 products of these coefficients must disappear from the expressions for the 

 kinetic and potential energies, we are led to the 'conjugate' properties of 

 spherical harmonics quoted in Art. 88. The actual calculation of the potential 

 and kinetic energies will be given in the next Chapter, in connection with an 

 independent treatment of the same problem. 



The effect of a simple-harmonic disturbing force can be 

 written down at once from the formula (12) of Art. 165. If 

 the surface-value of O be expanded in the form 



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



where fl n is a surface-harmonic of integral order n, then the 

 various terms are normal components of force, in the generalized 

 sense of Art. 133; and the equilibrium value of f corresponding 

 to any one term H n is 



? 0. ............................ (9). 



c/ 



Hence, for the forced oscillation due to this term, we have 



where <r measures the ' speed ' of the disturbing force, and <r n that 

 of the corresponding free oscillation, as given by (5). There is no 

 difficulty, of course, in deducing (10) directly from the equations 

 of the preceding Art. 



192. We have up to this point neglected the mutual attrac- 

 tion of the parts of the liquid. In the case of an ocean covering 

 the globe, and with such relations of density as we meet with in the 

 actual earth and ocean, this is not insensible. To investigate its 

 effect in the case of the free oscillations, we have only to sub- 

 stitute for fl n , in the last formula, the gravitation-potential of the 

 displaced water. If the density of this be denoted by p, whilst p 

 represents the mean density of the globe and liquid combined, we 

 have* 



and g = ^7rap, (12), 



* See, for example, Kouth, Analytical Statics, Cambridge 1892, t. ii. pp. 91-92. 



