352 TIDAL WAVES. [CHAP. VIII 



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

 p=l. Hence, from (2), 



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 



and by successive applications of this we find 



(2j+2)(2y + 3) (2j + 4)(2j 



, . 1 . ,_ _ +&c 



2j(2j + lV (2j+2)(2j+3) + (2/+4.)(% +B) 



.................. (21), 



on the present supposition that Nj+k tends with increasing k to the 

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

 formula determines the value of N lf Now 



1 = J\r iJ 8_ 1 = _2^ 1 # / , 

 and the equation (11) then gives 



NtH .................. (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 1} 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 1} B 3 , ... in succession in the above 



