350 TIDAL WAVES. [CHAP. VIII 



latitude. The constants A, B are then determined by the 

 conditions that u = at each of these parallels. If the boundaries 

 in question are symmetrically situated on opposite sides of the 

 equator, the constant B will be zero, and the odd function f(fi) 

 may be disregarded ab initio. By supposing the boundaries to 

 contract to points at the poles we pass to the case of an unlimited 

 ocean. If we address ourselves in the first instance to this latter 

 form of the problem, the one arbitrary constant (A) which it is 

 necessary to introduce is determined by the condition that the 

 motion must be finite at the poles. 



210. The integration of the equation (5) has been treated by 

 Lord Kelvin* and Prof. G. H. Darwin -f. 



We assume 



-^*^=B&+B t p*+...+Bj +l p,*+ ......... (8). 



/* / a/t 



This leads to 



......... (9), 



where A is arbitrary ; and makes 



(10). 



Substituting in (5), and equating coefficients of the several powers 

 of fju, we find 



(11), 



= (12), 



and thenceforward 



* Sir W. Thomson, " On the ' Oscillations of the First Species ' in Laplace's 

 Theory of the Tides," Phil. Mag., Oct. 1875. 



t " On the Dynamical Theory of the Tides of Long Period," Proc. Roy. Soc., 

 Nov. 5, 1886 ; Encyc. Britann., Art, " Tides," 



