1^90.] Harmonic Analysis of Tidal Observations. 289 



In order to minimise the disturbance due to the lunar tide M. 2 , we 

 have to make the n + 1 tides cover an exact number of semi-lunations, 

 namely, the same period as that involved in the evaluation of M.,. 

 The elimination of the M 2 tide is adequate, although rtot so com- 

 plete as the elimination of the effect of the fej, tide on M 2 , because Mj 

 is nearly three times as large as S 2 . 



A Long Series of Observations. Suppose that there is a half year of 

 observations, or two periods of six semi-lunations, each of which 

 periods contains exactly the same number of tides. 



Then each of these periods is to be reduced independently with the 

 assumption that 7 = 0'272and *:, = K" '. If this assumption is found 

 subsequently to be very incorrect, it might be necessary to amend 

 these reductions by multiplying \ n by H"-f-O272H,, and by adding 

 Kg K" to w ; but such repetition will not usually be necessary. From 

 these reductions we get independent values of H, cos * H, sin K, from 

 each quarter year, and the mean of these is to be adopted, from which 

 to compute H, and K,. It remains to evaluate H" and K" ' . 



The factor f" and the angle 2v" vary so slowly that the change may 

 be neglected from one quarter to the next, although each quarter is 

 supposed to have been reduced with its proper values. 



Let h and h' be the sun's mean longitude at the two epochs ; they 

 will clearly differ by nearly 90, and we put 2h' = 2h + 7r + 28h. 

 Hence it is clear that the value of ia in tha second quarter is 



Thus the four equations, such as (20), appertaining to the two 

 quarters, may be written 



A, = nH,cos *, + . f"H"cos (-*"), 

 7 



B t = nHj sin K t --- f"H" sin (ID K"), 



.... (21), 



B/ = n'H,8in/t,+ .f"H" sin 

 7 



where the accented symbols apply to the second quarter, and where 



A,, sini(n + l)o A ct . a 

 = -- ^ ' fa = 656, a constant. 

 7 (n + l)sm0 



From (2*1), 



, = 2 .f"H" 



- B, + B,' + (0 -n')H, sin *, = 2^- . f"H" cos oh sin ( + 2A- -c"). 



7 



