OS U. S. COAST AND GEODETIC SURVEY 
277. Formulas (424) and (425) represent the clearances for any 
long-period constituent A due to any short-period constituent Ag. 
The first must be subtracted from terms corresponding to 
Yy,cos 24(n—1)a and the latter from terms corresponding to 
Ly, sin 24(n—1)a of formula (417) before solving the latter. 
278. In (424) and (425) the coefficients of A’, and A’’,, which 
for brevity we may designate as C’, C’’, S’, and S”’, respectively, 
contain only values that are constant for all series and may therefore 
be computed once for all. Separate sets of such coefficients must, 
however, be computed for the effect of each short-period constituent 
upon each long-period constituent. In the usual reductions in which 
the effects of 3 short-period constituents upon 5 long-period con- 
stituents are considered, 15 sets of 4 coefficients each, or 60 coefficients 
in all, are required. 
The coefficients are given in the following table: * 
Long-period constituents 
| Mm Mf MSf Sa Ssa 
—_—_—— —_— — |! = = cca ahd Was ———$ — | —_ ——- —. 
W716} (CO) aes Se See See ete ook oe —0. 0556 +0. 0030 +5. 739 —0. 1041 —0. 1046 
(CRE ARE Re. SOE eee eee —0. 1704 —0. 0377 —2. 923 —0. 0762 —0. 0755 
OSD ilee eee ee Seer ee Bane eee ree —0. 1708 +0. 0417 —2. 840 —0. 0018 —0. 0035 
(CS) Re at eee one oe me oe ea +0. 0441 | +0. 0105 —5. 727 -++0. 0048 -++0. 0096 
INC (0/)) Se 5 ae Sinem —0.0588 | +0. 0368 +0.0294 | —0.0176 —0. 0176 
COL) Fes ARO ee ee a | —0.0776 | —0. 2236 —0. 1938 +0. 0025 +0. 0025 
(Sas ae eee ee oe oo ee ! —0.0206 | —0.1526 —0.1221 | +0. 0002 +0. 0004 
(SE) a a oh ee Sr ee ae cae | +0. 1138 —0. 0854 —0. 0808 -++0. 0001 +0. 0002 
ORC) ee see SR fee eee ce { —0.0648 | -+0.0166 +0. 0157 —0, 1924 —0. 1934 
(COS) EE ih CE Eh Ee. | He apne fe —0. 3476 —0. 0778 —0. 0816 —0. 1826 —0. 1831 
(Sak eo ee ee. eas ye ee ost —0.3452 | +0. 0841 +0. 0875 —0. 0046 —0. 0093 
(Oi) eo Be ein aes eee eee | +0. 0405 -++0. 0338 -++0. 0331 -++0. 0090 +0. 0180 
In the above table the sign is so taken that the values are to be 
applied to the sums directly as indicated. 
279. After the clearances have been applied and the normal equa- 
tions (417) solved and the resulting amplitude and epoch obtained for 
each of the long-period constituents, the reductions will be completed 
in accordance with the processes already outlined, but it must be kept 
in mind that in this reduction the initial value of ¢ is taken to corre- 
spond to 11:30 a. m. on the first day of series. In obtaining the nu- 
merical values of such quantities as Dy, cos 24(n—1)a and 2y, sin 
24(n—1)a, in order to avoid the labor of separate multiplications for 
each day, the following abbreviations have been proposed by the 
British authorities. The values of cos 24(n—1)a and of sin 24(n—l)a 
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 dis- 
tributed into 11 corresponding groups, so that all values in one group 
will be multiplied by 0, another group by 0.1, ete. The cos 24(n—l)a 
and sin 24(n—1)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 values apply. 
280. 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. It is therefore desirable to 
*From Scientific Papers by Sir George H. Darwin, Vol. I, p. 64. 
