98 U. S. COAST AND GEODETIC SUEVEY. 



30. ELIMINATION. 



Because of the limited length of a series of observations analyzed 

 the amplitudes and epochs of the components as obtained by the 

 processes described in the preceeding sections are only approximately 

 freed from the effects of each other. The separation of two compo- 

 nents from each other might be satisfactorily accomplished by having 

 the length of series ec^ual to a multiple of the synodic period of the 

 two components. To completely effect the separation of all the com- 

 ponents 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 component. The length of such a series would be too great 

 to be given practical consideration. In general, it is therefore desir- 

 able to apply certain corrections to the constants as directly obtained 

 from the analysis in order to eliminate the residual effects of the com- 

 ponents upon each other. 



Let A be the designation of a component for which the true con- 

 stants are sought and let B be the general designation for each of the 

 other components in the tide, the effects of which are to be eliminated 

 from component A. 



Let the original tide curve which has been analyzed be represented 

 by the formula 



y = A cos {at + a) +^B cos {U + j8) (389) 



in which 



?/==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 component A for the time 



covered by series of observations. 

 B = R{B) =true amplitude of component B for the time cov- 

 ered by series of observations. 

 a= —^{A) =true initial phase of component A at beginning of 

 series. 



/?= — f(5) =true initial phase of component B at beginning of 



"series, 

 a = speeds of component A. 

 & = speed of component B. 



Formula (389) may be written 



y = A cos a cos a^ + S B cos {(Z» — a.)^ + /?}cos at 

 — A sin a sin at — S 5 sin {(b — a)t + ^}sin at 

 = [A cos a + S 5 cos{(b — a)^ + i5}] cos at 

 -[A sin a + S 5 sm{{h-a)t + ^}] sin at (390) 



The mean values of the coefficients of cos at and sin at of formula 

 (390) correspond to the coefficients Cp and Sp of formulas (316) and 

 (317) which are obtained from the summations for component A. 



Let A^ and a^ = the uneliminated amplitude and initial phase, 

 respectively, of component A, as obtained directly from the analysis. 



The equation of the uneliminated component A tide may be written 



y = A'^ cos (at-j-a^) =A^ cos a^ cos at — A'^ sin a^ sin at (391) 



