84 U. S. COAST AND GEODETIC SURVEY. 



accidental errors that are not periodic, it may be readily shown by the 

 method known as the least square adjustment, using the observational 

 equations represented by (279) , that the most probable values of the 

 constant Ho and the coefficient Cp and S^ are the same as those given 

 by formulas (306), (310), and (314), respectively. 



Since in tidal work the value of So, which is the elevation of 

 mean sea level above the datum of observations, is generally deter- 

 mined directly from the original tabulation of hourly heights, formula 

 (306) is unnecessary except for checking purposes. Formulas (310) 

 and (314) are used for obtaining the most probable values of the 

 coefficients Cp and ^p from the component hourly means obtained 

 from the summations. 



When 24 hourly means are used n = 24, and u = 15°, and the formulas 

 may be written 



1 a=23 



Cp=i% S /i-acos 15ap (316) 



1^ a=o 



1 



a =23 



'S^p=To S K sin 15 a p (317) 



i-Z a=o 



in which the angles are expressed in degrees. 

 If only 12 means are used, the formulas become 



a=ll 



Gv = R S ^a cos 30 a p (318) 



O a=o 



1 



a=U 



'Sp = ^ S ^a sin 30 a p (319) 



6 a=o 



The upper part of Form 194 (fig. 29) is designed for the compu- 

 tation of the coefficients Cp and Sp in accordance with formulas (316) 

 and (317) to take account of the 24 component hourly means. 



It is now desired to express each component in the form 



y = A cos {'p e + a) (320) 



or using a more specialized notation by 



2/ = ^ cos ifd-^) (321) 



By trigonometry 



A cos {'p 6-^) =A cos r cos pd + A sin f sin p 9 (322) 



= Cp cos pd + Sp sin p d 



(323) 

 (324) 



cos I: sm f 



(325) 



By substituting the values obtained for Cp and S^ by formulas (316) 

 and (317) in formulas (324) and (325), the corresponding values of A 

 and (^ for formula (321) may be obtained. 



In (321) we now have an harmonic expression, which, with its 

 constants A and (^ determined by the methods already described, is 



