209-210] TIDES OF LONG PERIOD. 351 



It is to be noticed that (12) may be included under the typical 

 form (13), provided we write B^ 2H'. 



These equations determine B l} B s , ... B 2j+1) ... in succession, in 

 terms of A, and the solution thus obtained would be appropriate, 

 as already explained, to the case of a zonal sea bounded by two 

 parallels in equal N. and S. latitudes. In the case of an ocean 

 covering the globe, it would, as we shall prove, give infinite 

 velocities at the poles, except for one definite value of A, to be 

 determined. 



Let us write 



B 2j+l /B 2j ^ = N j+l ..................... (14); 



we shall shew, in the first place, that as j increases Nj must tend 

 either to the limit or to the limit 1. The equation (13) may be 

 written 



* 



N- -i- 



Hence, when j is large, either 



approximately, or N j+l is not small, in which case N j+2 will be 

 nearly equal to 1, and the values of Nj +s , Nj +4) ... will tend more 

 and more nearly to 1, the approximate formula being 



_ 



Hence, with increasing j, Nj tends to one or other of the forms 

 (16) and (17). 



In the former case (16), the series (8) will be convergent for 

 /*= + !, and the solution is valid over the whole globe. 



In the other event (17), the product N! . N a . ... N j+1) and 

 therefore the coefficient B Z j +1 , tends with increasing j to a finite 

 limit other than zero. The series (8) will then, after some finite 

 number of terms, become comparable with 1 -f yu, 2 -f yu, 4 + ..., or 

 (1 /A 2 )" 1 , so that we may write 



1 dty T M /IQ\ 



* s-*^r-= L + * ~i ( L ^)> 



