210] NUMERICAL SOLUTION. 353 



manner ; for this would require us to start with exactly the right 

 value of A, and to observe absolute accuracy in the subsequent 

 stages of the work. The only practical method is to use the 

 formulas 



BJH = - 2^, B 3 = N 2 B 1} B 5 = N 3 B 3 , 



or BJH = -2N lf BJH = - 2^ 2 , 



where the values of N lt N 2y N 3 , ... are to be computed from tlfjtf m 

 continued fraction (21). It is evident a posteriori that the solutio^f X % 

 thus obtained will satisfy all the conditions of the problem, anji r &quot; 

 that the series (9) will converge with great rapidity. The mo^i U. * 

 convenient plan of conducting the calculation is to assumes +J 

 roughly approximate value, suggested by (16), for one of tkfe j 

 ratios JV} of sufficiently high order, and thence to compute 



N } ^,N^,... N. z , N t 



in succession by means of the formula (20). The values of the 

 constants A, B l} B 3 , ..., in (9), are then given by (22) and (23). 

 For the tidal elevation we find 



- N,N, . . . Ni_ t (1 -f*N,) yli- ...... (24). 



In the case of the lunar fortnightly tide, / is the ratio of a 

 sidereal day to a lunar month, and is therefore equal to about ^, 

 or more precisely 0365. This makes f 2 = 00133. It is evident 

 that a fairly accurate representation of this tide, and a fortiori of 

 the solar semi-annual tide, and of the remaining tides of long 

 period, will be obtained by putting /= 0; this materially shortens 

 the calculations. 



The results will involve the value of /3, = 4n*a*/gr. For 

 /3 = 40, which corresponds to a depth of 7260 feet, we find in 

 this way 



f/JT^-1515- T0000yu, 2 + l-5153yu, 4 - l 2120/* 8 + 6063/A 8 - -2076/I- 10 

 -f -0516/* 12 - -0097/4 14 + -001 V 5 - -0002y* 18 ...... (25) *, 



whence, at the poles (/JL = 1), 



* The coefficients in (25) and (26) differ only slightly from the numerical 

 values obtained by Prof. Darwin for the case /= -0365. 



L. 23 



