49 



phase of each component are computed from its hourly heights by a 

 process based on the arithmetical integration of the coefficients of 

 Fourier's series. Components having the same component hour are 

 separated by the process. 



92. Using the form indicated in equation (29), the height, at any 

 time t, of the resultant of a group of components having the same 

 component hour is given by the equation: 



y=H^-\-A, cos (a^-rO+A cos {2at-l2)+A^ cos {?>at-^^)+ . . .(43) 



where Ho is the height of mean sea level above datum, Ai the ampli- 

 tude of the diurnal component, a its speed, and f/a the time of its 

 high water (par. 49); A2 is the amplitude of the semidiurnal compo- 

 nent and f/2a, the time of its high water; and the other terms similarly 

 represent minor components and overtides. But a few terms are 

 needed in any group of components having the same component hour, 

 and for single components the equation reduces to one variable term 

 in addition to the constant term Hq. 



The expansion of the cosines in equation (43) gives the equation: 



y=Ho-\-Ai cos at cos fi+^i sin at sin fi+-A2 cos 2at cos fo 

 -{-A2 sin 2at sin ^2+^3 cos Sat cos ^3-\-As sin Sat sin ^3 . . . (44) 



Placing: 



Ai cos fi = Ci, A sin fi = Si, A2 cos ^2=02, A2 sin ^2=52, etc. (45) 



equation 44 becomes: 



y=Ho-\-Ci cos at-\-Si sin at-\-C2 cos 2a^+S2 sin 2at-\-Cs cos Sat 

 +S3 sin Sat+ . . . (46) 



The values of the angles and coefficients in equation (43) may be 

 found readily from the coefficients Ci, Si, C2, §2, C3, S3, etc., of equation 

 (46), since, from equations (45): 



tan ti^sjci tan f 2=^2/^2 tan ^3=83/03, etc. (47) 



and: 



-4i=Ci/cos fi=Si/sin fi ^2=<?2/cos f2='S2/sin ^o, etc. (48) 



It may be noted that by expressing equation (43) in the form indi- 

 cated by equation (29), rather than equation (27), negative signs are 

 avoided in equations (46), (47), and (48). 



