ON HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 335 



consecutive, and one series beginning a quarter-lunar-anomalistic period after the 

 other. By subtracting the equations (33) for consecutive sets in each series, Mj is 

 eliminated from each series, and the results are then combined so as to eliminate 

 first N2 and then L.,. In these final equations it is assumed that all other 

 constituents may be neglected, and even for N, and Lj that the terms involving 

 (1 — (rrl(r„) may be neglected, while those of the type (36) may be replaced by terms 

 of the type (37). 



To find Sj, Ro cos ((r„ t — e„) is taken as S, and N is chosen so that the tides 

 considered cover an exact number of semi-lunations. All constituents except K^ 

 and T„ are then neglected in the summations. It is assumed that the amplitude 

 of K, bears to that of S^ the equilibrium ratio, and that their lags are equal, while 

 Tj is supposed simply proportional to S^. In the summations involving K^, the 

 terms containing (1 — ffr/tro) are neglected. 



To find Ko, Ro cos (a^t — e„) is again taken as S, and two sets of the same number of 

 tides are taken so that one begins 3 months after the other. Again T, is taken 

 proportional to Sj, the factor of proportionality changing a little from one set to the 

 other. By subtracting corresponding equations (33) for the two sets, the terms 

 involving Sj become small, and when the value of S2 already found is used, we have 

 two equations for the amplitude and lag of Ko. All other constituents are again 

 neglected in the summations. 



In finding K, and O, the procedure is analogous to that for finding N, and L.^, 

 M, taking the place of Mj and a semi-lunar period taking the place of a semi-lunar 

 anomalistic period. This time, however, the contributions of Mj and Sj to the 

 summations are accurately computed with the values of these constituents already 

 found, while K, and P, are supposed to have their amplitudes in the equilibrium 

 ratio and their lags equal. 



The finding of Pj is analogous to that of K„ the process for K, and 0, taking the 

 place of that for Sj. 



Only the constituents Mj, Sj, Nj, K^, L,^ and K,, 0„ P, are considered. 



Correlation with Generating Potential. 



12. The amplitudes of some of the harmonic constituents of the generating 

 potential are really variable, though the variations are very slow. Also, some of 

 their speeds have not quite the values scheduled, though again the deviations are 

 small. In both cases the effects take a long time to accumulate and the changes 

 may be neglected over a year, while a longer record is not subjected to a single 

 analysing process. 



When the amplitude and phase of a constituent observed at any particular time 

 are connected with the generating potential, the deviations mentioned have to be 

 taken into account. There is one tj'pe of case, however, in which this does not 

 appear to have been adequately done. It is the case in which shallow water 

 constituents have speeds equal to those of constituents of other origin. When the 

 ratio of the amplitudes and the difference of the phases of two constituents are 

 slowly changing, it is clear that true tidal constants cannot be obtained until the 

 constituents have been separated in some way. Such separation has never been 

 made, and this fact may possibly account for some of the irregularities in the results 

 from year to year, especially for some of the long period constituents.* 



Historical Account. 



13. In 1866 Prof. Thomson (afterwards Sir William Thomson and Lord Kelvin) 

 applied to the Admiralty for a year's record of any trustworthy tide-gauge, and 

 received that of Ramsgate for 1864. This he began to analyse harmonically with the 

 aid of Messrs. E. Maclean, J. Smith and W. Ross of the University of Glasgow, at 

 first using three-hourly heights but afterwards hourly heights. Approximate values 

 for M, and S^ were found and the existence of Ssa and Sa detected. 



* In the preliminary report it was stated that the inconsistencies in the results 

 for long period constituents are not due to defects in the method of analysis. This 

 was based on the fact that residues sre allowed for ; the subject of the present 

 section was not there considered , 



