104 U. S. COAST AlsTD GEODETIC SURVEY. 



Representing the mean of these 24 ordinates for d day by y^, we 

 have 



ya = -^A cos {2^{d—l)a + a) [1 + cos a 4- cos 2a+ • • • +cos 23a] 



24 

 24 



A sin {24(^— l)a + o;} [sin a + sin 2a+ • • • +sin 23a] 



1 , sin 12a r ,^,,, ,, , 23 



""24 sinla^L^^^ {24(rf-l)a + «} cos y a 



23 "1 



sin {24i{d— \)a + a] sin ^ a 



= -^-A^^^cos{24((^-l)a + «+11.5a} (418) 



Formula (418), representing the average value of the component A 

 ordinates contained in the daily mean for d day, is the correction or 

 clearance that must be subtracted from the mean for that day in 

 order to eliminate the effects of component A. It will be noted that 

 if we let A represent any of the solar components, Sj, S,, Sg, S^, etc., the 

 factor sin 12a, and consequently the entire formula, becomes zero for 

 all values of d. 



By formula (418) clearances for each of the disturbing short-period 

 components for each day of series may be computed and these clear- 

 ances then applied individually to the daily means, or, if first multi- 

 plied by the factor 24, to the daily sums. 



The labor involved in making independent calculations for the 

 clearance of the effect of each short-period component for each day 

 of series would be considerable, but this may be avoided to a large 

 extent by means of a tide-computing machine. 



If we let t = time reckoned in mean solar hours from the beginning 

 of the series, then for any value of ya, which must apply to the 11.5 

 hour of d day, 



t = 24:{d-l) + n.5 

 and (419) 



a^ = 24((^-l)a-M1.5a 



If the above equivalent is substituted in (418) and y^ replaced by 

 T/a, we have 



1 . sin 12a , ^ , v /.r./^\ 



2/^ = 24 ^^iEl^^«^^^^ + «) (^20) 



which represents a continuous function of t; and for any value of t 

 corresponding to the 11.5 hour of d day the corresponding value of 

 ?/a will be ^d- This formula is the same as that for the short-period 



component A, except that it includes the factor kt — = — i — in the 



coefficient. The speed a is, of course, a known constant, and the 

 values of A and a are presumed to have already been determined 

 from the harmonic analysis of the short-period components. Simi- 



