214] SEMI-DIURNAL TIDES: UNIFORM DEPTH. 359 



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

 of the coefficients of i$ and i/ 2 -*&quot; 2 in the expansion of (1 z&amp;gt; 2 )*. We 

 infer that, unless B 4 have such a value as to make N ao = 0, the 

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

 those of (1 v&quot;)%, so that we may write 



? = L + (l-v*)*M ..................... (15), 



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

 the equator (y = 1) this makes 



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 .Z? 4 so that N x = 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__ J3 _ 



_ 2j(2j + 6) (2; + 2)(2/+8) (2j+4)(2y + 10) n , 



j &quot; &quot; 



_ 



2J + 6 2J + 8 2J + 10 k 



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

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

 use this symbol for B^jH &quot;. We have then 



B t = N,H &quot;, B e =N,B t , B a = N 3 B e ,.... 



Finally, writing f = f + f, we obtain 



. l vf&amp;gt; + N 1 N,N s * + ......... (19). 



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

 tion is to assume an approximate value for JV} +1 , where j is a 

 moderately large number, and then to deduce Nj, JV}_,,... N a , A^ 

 in succession by means of the formula (17). 



