ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 



-145 



(16). We may therefore impose on them any other arbitrary condition not 

 inconsistent with the former. Suppose we choose them so as to satisfy the relation 



(A 



(18). 



The two equations (16), (18) serve for the complete definition of the two functions 

 <!>!, ., ; making use of the latter, (17) reduces to 



(D - 



, + (D - 07.) *,}.... (19). 



Thus on replacing \fi by its value in terms of V it V. 2 in the right-hand member 

 of (14), we deduce 



If we suppose that h is constant, the terms involving 

 virtue of (11) we shall obtain 



will disappear, while in 





or 



(20). 



We have now for the determination of the functions t/, , ^^ 

 simultaneous differential equations (15), (16), (18), (20). 



the four 



3. Integration in Series of Tesseral Harmonics. 



Let us suppose that \jj, , v, ^" ]; % are each expressible as series of tessera! 

 harmonics of the same rank s. Omitting the exponential factor e'^ ! + 5lf " , we assume 

 that 



< ] 



Then, if p denote the density of the water, and o- the mean density of the whole 

 VOL. cxci. A. u 



