110 U. S. COAST AND GEODETIC SUEVEY. 



The numerical value of the first member of each of the above 

 normal equations is obtained from the observations by taking the 

 sum of the product of each daily mean by the cosine or sine of the 

 angle indicated. 



The solution of the equations give the values of A', A", B', B" , 

 etc. , from which the corresponding values of amplitudes A and a, B 

 and /3, etc., of formula (424) are readily obtained, since 



A = -^lA'Y+{A"Y and a-tan-^ ^=^. 



In calculating the corrected epoch, it must be kept in mind that the 

 t in this reduction is referred to the 11.5 hour of the first day of 

 series instead of the preceding midnight. 



Before solving equations (436), if the daily means have not al- 

 ready been cleared of the effects of the short-period comxponents, it 

 will be necessary to apply corrections to the first member of each of 

 these equations in order to make the clearances. 



The disturbance in a single daily mean due to the presence of a 

 short-period component is represented by equation (418). Intro- 

 ducing the subscript s to distinguish the symbols pertaining to the 

 short-period component, the disturbance in the daily mean of the 

 n^^ day of series due to the presence of the short-period component 

 ^s may be written 



[2/s]n = KT^s^^J-X^'cos {24(n-l)as + 11.5as + as} (437) 



The disturbances in the products of the daily means by 

 cos 24(n — l)a and sin 24(n — l)a 

 may therefore be written 



[y^\ cos 24(71 - l)a = ^^s ^-7-1:7-" cos {24(n-l)as + 11. 5as + as} 



^t: Sill 2 ^3 



= 24 ^s g-^ ^^ ^ h [cos {24(71-1) (as + a) + 11.-5 a.s + as} 



+ cos {24(71-1) (a.s-a')+11.5as + o;s}] (438) 

 and 



[?/s]n sin 24(71-1) a 



= 9Z ^s TTz-i"^ i [sin {24(71-1) {a^ + a) -{-ll.b a^ + a^] 



^'i bin '2'tt's 



sin {24(71-1) (as-a) +11.5 ds + 0:8}] (439) 

 Then, referring to formulas (284) and (285) 



n=365 



S [ysln COS 24(71 — l)a = 



n=l 



1 . sinl2asrsin 12x365(as + a) f-,ov,o«H^ , ^ , n r , 1 



48 ^^ iml^ L~^S^2(^H^^)-- ^^^ { 12 X 364 (a. + a) + ll.Sas -F a.) 



+ ^^^^^3|fr^^=^cos {12x364(a3-a)+11.5a. + a3}] (440) 



