84 U. S$. COAST AND GEODETIC SURVEY 
Le Gnd Oy ye SED 
Aceeleration 77) tan TAB (367) 
Bp vee oy) 
Also because an these cases is small compared with unity, the 
following may be ‘taken as the approximate equivalent of (366): 
ae eee 
Resulting amplitude=1 +453} Re (32) +2 - cos ¢— 1 (368) . 
To allow for the effects of the longitude of the moon’s node, the 
tabular value of the acceleration should, therefore, be multiplied by 
the ratio — and the amount by which the resultant amplitude 
differs from unity by the same factor. In the particular cases under 
consideration the factor f, for constituents P, S:, and To, S unity for 
: 1 
each. Therefore, for the effect of P; on Ki, ue TAO To FID 
= (K,), and for the effect of Ky upon S,, this ratio is f(K.). For the 
effect of T, upon S, the ratio is unity. 
ELIMINATION 
244. Because of the limited length of a series of observations 
analyzed the amplitudes and epochs of the constituents as obtained 
by the processes already described are only approximately freed from 
the effects of each other. The separation of two constituents from 
each other might be satisfactorily accomplished by having the length 
of series equal to a multiple of the synodic period of the two con- 
stituents. To completely effect the separation of all the constituents 
from each other by the same process would require a series of such a 
length that it would contain an exact multiple of the period of each 
constituent. The length of such a series would be too great to be 
given practical consideration. In general, it is therefore desirable to 
apply certain corrections to the constants as directly obtained from 
the analysis in order to eliminate the residual effects of the constituent 
upon each other. 
245. Let A be the designation of a constituent for which the true 
constants are sought and let B be the general designation for each 
of the other constituents in the tide, the effects of which are to be 
eliminated from constituent A. 
Let the original tide curve which has been analyzed be represented 
by the formula 
y=A cos (at+a)+z B cos (b¢+ 8) (369) 
in which 
y=the height of the tide above mean sea level at any time ¢. 
t=time reckoned in mean solar hours from the beginning of 
the series as the origin. 
A=R(A)=true amplitude of the constituent A for the time 
covered by series of observations. 
B=R(B)=true amplitude of constituent B for the time cov- 
ered by series of observations. 
a=—¢(A)=true initial phase of constituent A at beginning of 
series. 
