214] SEMI-DIURNAL TIDES: UNIFORM DEPTH. 359 



approximately. This is the same as the limiting form of the ratio 

 of the coefficients of T$ and i$~* in the expansion of (1 z/ 2 )*. We 

 infer that, unless B have such a value as to make N m = 0, the 

 terms of the series (10) will become ultimately comparable with 

 those of (1 i^)*, so that we may write 



? = L + (l-vtyM ..................... (15), 



where L, M are functions of v which do not vanish for v 1. Near 

 the equator (y = 1) this makes 



(16). 



Hence, by Art. 208 (7), u would change from a certain finite 

 value to an equal but opposite value as we cross the equator. 



It is therefore essential, for our present purpose, to choose the 

 value of B 4 so that N^ = 0. This is effected by the same method 

 as in Art. 210. Writing (13) in the form 



we see that Nj must be given by the converging continued fraction 



3 



2;(2y+6) (2j+2)(2;+8) (2j+4)(27 + 10) . 



l\j = '. ^ rt . * -g.. K ... (iO> 



2; + 3 2; + 5_ 2^ + 7 ~ 



2jT6~ 2J + 8" 2JTTO 



This holds from j = 2 upwards, but it appears from (12) that it 

 will give also the value of N^ (not hitherto defined), provided we 

 use this symbol for B^fH'". We have then 



Finally, writing f= f + f 7 , we obtain 



...(19). 



As in Art. 210, the practical method of conducting the calcula- 

 tion is to assume an approximate value for Nj +l , where j is a 

 moderately large number, and then to deduce Nj t Nj- lt ... N^, N t 

 in succession by means of the formula 



