ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 149 



the determination of the free and forced vibrations has been fully discussed in that 

 paper. A strictly analogous procedure might be adopted in the present case, but it is 

 found convenient to modify the previous treatment in some respects. We shall 

 therefore only indicate briefly the course of procedure when our analysis corresponds 

 with that given in Part I., giving greater detail as regards those points where a 

 different method has been found desirable. 



4. Extension to the Case of Variable, Depth. 



The formulse developed in the preceding sections apply only to the case where the 

 depth is uniform. We may however obtain a relation of the same nature as the 

 equation (31) when the depth is a function of the latitude given by the formula 



h = k + l(l-p*) ......... (33), 



that is, when both the internal and external surfaces of the ocean are spheroids of 

 revolution. 



The above expression for h may be written in the form 



X- - 



(34)> 



where 



Substituting the expression (34) for h in the equation (14) we obtain 

 (L - ^)= (D + 



+ [(D + <rp) (D - 07.) - ( 2 - cr 

 which, with the aid of (11), becomes 



= < D + ^ (D ~ 



The right-hand member is of the same form as that of the equation (14), except 

 that h is replaced by K. We may therefore introduce two auxiliary functions j, 

 * 2 , defined by (16), (18), and proceed as in 2, and we shall obtain in place of (20) 

 the equation 



