112 



U. S. COAST AND GEODETIC SURVEY. 



be computed once for all. Separate sets of such coefficients must, 

 however, be computed for the effect of each short-period component 

 upon each long-period component. In the usual reductions in which 

 the effects of 3 short-period components upon 5 long-period com- 

 ponents are considered, 15 sets of 4 coefficients each, or 60 coefficients 

 in all, are required. 



The coefficients are given in the following table :^ 



MiiC). 

 iC") 

 {S'). 

 {S") 



N2(C'). 

 (C") 

 (S'). 

 (S") 



Oi(C'). 

 iC") 

 (S'). 

 (S"). 



Ssa. 



-0. 1046 

 -0. 0755 

 -0. 0035 

 +0. 0096 



-0. 0176 

 +0. 0025 

 +0. 0004 

 +0. 0002 



-0. 1934 

 -0. 1^31 

 -0. 0093 

 +0. 0180 



In the above table the sign is so taken that the values are to be 

 applied to the sums directly as indicated. 



After the clearances have been applied and the normal equations 

 (436) solved and the resulting amplitude and epoch obtained for each 

 of the long-period components, the reductions mil be completed in 

 accordance with the processes already outlined, but it must be kept 

 in mind that in this reduction the initial value of t is taken to corre- 

 spond to 11.30 a. m. on the first day of series. 



In obtaining the numerical values of such quantities as 

 S^/n cos 24(n— l)a and S?/n sin 24(7i— l)a, in order to avoid the labor 

 of separate multiplication for each day, the following abbreviations 

 have been proposed by the British authorities. 



The values of cos 24(?i— l)a and of sin 24(71 — 1)0- are divided into 

 11 groups according as they fall nearest 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 

 '0.7, 0.8, 0.9, or 1.0. The daily values are then distributed into 11 

 'Corresponding groups, so that all values in one group will be multi- 

 plied by 0, another grouj) by 0.1, etc. The cos 24(n— 1)0- and 

 sin 24 {n~l)a include negative as well as positive values. The former 

 are taken into account by changing the sign of the daily mean to 

 which the negative cosines apply. 



As a part of the routine reductions of the tidal records from the 

 principal tide stations it is the practice of the office to obtain the 

 mean sea level for each calendar month, using for the purpose the 

 hourly heights for 29 consecutive days, beginning with the first day 

 day of each month. It is therefore desirable to have a method of 

 using these means directly in the analysis for the annual and semi- 

 annual components, thus avoiding any special summation for the 

 purpose. 



The period of the annual component is approximately the length 

 of the Julian year; that is, 365.25 days. If this period were divided 



•Trom Scientific Papers by Sir George H. Darwin, Vol. I, p. 64. 



