90 U. 8. COAST AND GEODETIC SURVEY 
Yo =A cos [24(d—1)a+ a] 
y, =A cos [24(d—1l)a+a+a] 
Yo =A cos [24(d—1)a+a-+2a] (397) 
Yo3—=A cos [24(d—1)a+a+23a] 
Representing the mean of these 24 ordinates for d day by yz, we have 
Va=54 A cos {24(d—1)a+a} [1-++cos a+cos 2a+____+ cos 23a] 
a A sin {24(d—1)a+a}[sin a+sin 2a+___.+sin 23a] 
1 >, ‘sm 12¢ 23 
=54 A ae ra | cos {24(d—1)a+a} cos 9 @ 
3 aD 
—sin {24(d—1)a+a} sin = a 
rt ily sin 12a 
2455 sim sa 
261. Formula (398), representing the average value of the constitu- 
ent 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 A. It will be noted that if we let 
A represent any of the solar constituents, 5,, S:, $3, S,, ete., the 
factor sin 12a, and consequently the entire formula, becomes zero for 
all values of d. By formula (398) clearances for each of the disturbing 
short-period constituents for each day of series may be computed and 
these clearances then applied individually to the daily means, or, if 
first multiplied by the factor 24, to the daily sums. 
262. The labor involved in making independent calculations for 
the clearance of the effect of each short-period constituent 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 yg, which must apply to the 11.5 
hour of d day, 
cos {24(d—1l)a+a+11.5a} (398) 
Pad = eeties 
and 
at=24 (d—1)a+11.5a (399) 
If the above equivalent is substituted in (398) and yq replaced by 
Ya, We have 
it sin 12a 
Ya=o4 Zak eine cos (at+a) (400) 
which represents a continuous function of ¢; and for any value of ¢ 
corresponding to the 11.5 hour of d day the corresponding value of 
y, Will be yg. This formula is the same as that for the short-period 
