46 U. S. COAST AND GEODETIC SURVEY. 



For terms (A)^, to {A),^^ 



[sin 2 7]o [cos v]o= (147) X (153) =sm 2 co [1-3/2 i^] 

 = sin 2 oj [1-3/2 sin^ i] = 0.7214 



For terms (A)^^ to (J.) 35 



[sin2 /j^ fcos 2 ^]o= (144) X (154) =sin2 co [1 - J ^2] 

 = sin2cocos*i^ = 0.1578 



For terms (A) 59 to (A) gg 



[1-3/2 sin^' 7]o= (148) = (1-3/2 sin^ w) (1-3/2 i^) 

 = (1 - 3/2 sin^ co) (1 - 3/2 sin^ i) = 0.7532 



(159) 

 (160) 

 (161) 



The mean value of the coefficient of each term of (100) may now 

 be readily obtained by substituting for the variable factor the corre- 

 sponding mean value from formulas (155) to (161). Table 3 contains 

 a compilation of such mean values. 



12. FACTORS OF REDUCTION. 



For the analysis of a series of observations not exceeding 369 

 days, the coefficient may be considered as practically constant, with 

 / having a value corresponding to the middle of the series, but 

 in order that the results from several years of observations may be 

 comparable it is necessary to take account of the changes in / and 

 apply a factor of reduction to the amplitude' as obtained from any 

 particular series of observations. For these factors it is assumed that 

 the variations in the actual tidal components due to the changes in 

 the longitude of the moon's node will be proportional to the corre- 

 sponding variations in the coeificients of the terms of (100). 



Representing the mean value of any component amplitude by H 

 and the amplitude for any particular time by R let 



F^^, or H=FR (1C2) 



and 



f=^,oTR=fH (163) 



Then 7^ will be the factor for reducing an amplitude for a particular 

 series to its mean value and/ the , reciprocal of F, the factor for 

 adapting a mean amplitude to a particular time. 



Using the notation of the preceding section, this factor may be 

 expressed 



^^ Jo [cos u]o ^^g^^ 



u 



The constant factors of the coefficients of (100) being common to 

 both the numerator and denominator of (164) need not here be 

 considered. Making the substitutions of the mean values represented 

 by Jo [cos u]o from formulas (155) to (161) the following factors of 

 reduction for the components represented by the terms of (100) 

 may be readily obtained. 



Ff or terms (^) ito (J.) 7, including components M2,N2, 2N, U2, ^2y andjU2j 



_ cos^ ^co cos^ ^ i _ 0.9154 , . 



~ cos^iZ ~cos^iI ^^^^^ 



