352 TIDAL WAVES. [CHAP. VIII 



where L and M are functions of & which remain finite when 

 H = 1. Hence, from (2), 



iff (1 - 



- 



which makes u infinite at the poles. 



It follows that the conditions of our problem can only be 

 satisfied if Nj tends to the limit zero ; and this consideration, as 

 we shall see, restricts us to a determinate value of the hitherto 

 arbitrary constant A. 



The relation (15) may be put in the form 



1 PJ __ AT. 



2j(2j + l) 

 and by successive applications of this we find 



ft ft 



2j(2j+l) (2j+2)(2j + 3) (2j + 4) (2j + 5) 



' &* 11 _ SfL _ +&c 



5r 

 (21), 



on the present supposition that NJ+JS tends with increasing k to the 

 limit 0, in the manner indicated by (16). In particular, this 

 formula determines the value of N lt Now 



and the equation (11) then gives 



N 1 H' .................. (22); 



in other words, this is the only value of A which is consistent with 

 a zero limit of Nj, and therefore with a finite motion at the poles. 

 Any other value of A, differing by however little, if adopted as a 

 starting-point for the successive calculation of B lt B 3 , ... will 

 inevitably lead at length to values of Nj which approximate to the 

 limit 1. 



For this reason it is not possible, as a matter of practical 

 Arithmetic, to calculate B lt B 3 , ... in succession in the above 



