ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 41 



II. On the Treatment of a Short Seriks of Tidal Observations and on 



Tidal Prediction. 



§ 1. Harmonic Analysis. 



Having been asked to write an article on the tides in a new edition of 

 the ' Admiralty Scientific Manual,' now in the press, I thought it would 

 be useful to show how harmonic analysis might be applied to the reduc- 

 tion of a short series of tidal observations, such as might be made when a 

 ship lies for a fortnight or a month in a port. 



The process of harmonic analysis, as applicable to a year of continuous 

 observation, needs some modification for a short series, and as it was not 

 possible to explain the reasons for the rules laid down within the limits of 

 the article, it seems desirable to place on record an explanation of the 

 instructions given. 



The observations to be treated are supposed to consist of hourly ob- 

 servations extending over a fortnight or a month. In the reduction of a 

 long series of observations the various tides are disentangled from one 

 another by means of an appropriate grouping of the hourly observations. 

 When, however, the series is short, the method of grouping is not suflB- 

 cient in all cases. 



With the amount of observation supposed to be available, a determina- 

 tion of the elliptic tides was not possible, and it was therefore proposed to 

 consider only the tides Mj, Sg, K,, Kj, 0, P — that is to say, the principal 

 lunar, solar, and luni-solar semidiurnal tides, and the luni-solar, lunar, 

 and solar diurnal tides. The luni-solar and solar semidiurnal tides have, 

 however, so nearly the same speed that we cannot hope for a direct 

 separation of them by the grouping of the hourly values, and we must 

 have recourse to theory for completing the process ; and the like is true 

 of the luni-solar and solar diurnal tides. 



Also, the tides K, and P have very nearly half the speed of S2 ; hence 

 the diurnal tides K; and P will appear together as the diurnal constituent, 

 whilst S2 and Kg will appear as the semi-diurnal constituent, from the 

 harmonic analysis of the same table of entries. 



It thus appears that three difierent harmonic analyses will suffice to 

 determine the six tides, viz. : — 



First, an analysis for M2 ; second, an analysis for ; third, an analysis 



for S2, K2, K„ P. 



The rules therefore begin with instructions for drawing up three 

 schedules, to be called M, O, S, for the entry of hourly tide-heights. 

 Each schedule consists of twenty-four hour columns, and a number of 

 rows for the successive days. In M and O certain squares are marked, 

 in which two successive hourly entries are to be put. The instructions 

 for drawing up the schedules are simply rules for preparing part of the 

 first page of the series M, O, S of the computation forms for a year of 

 observation. 



In order to minimise the vitiation of the results derived from the 

 M sheet by the S2 tide, and vice versa, and similarly to minimise the 

 vitiation of the results from the O sheet by the Kj tide, it is important to 

 choose the proper number of entries in each of the three sheets. 



It was shown in Section III. of the Tidal Report to the British Asso- 

 ciation for 1885 how these periods were to be determined. The equation 



