100 U. S. COAST AND GEODETIC SURVEY. 



From (406) 



^^^A^-XF. cos {Hh-<^;r-,0-a^l 

 cos [a^ — a) 



Substituting the value Fp from (396) and the equivalents R^(A), 

 R{A), R{B),-Y{A) -r(vl), and-f(5) for A\ A, B, a\ a, and j8, re- 

 spectively, we have by (407) and (408) 



tan[r(^)-r(-4)] 



^ 180 sin 4-(6 — a)T „,-r>v . ,,,, , ,-r,. w.x, 



^^ 1(6- a) r ^^^^ ®''' {i(6-a)r-r(B)+r(^)} 



i?H-4) -2 ^ ^^?1^^^ i?(5) cos {i(&-a)r-r(5) +rM^) } 



(409) 



ffl(^) -S ^ ''x|^!~)f " R^B) cos {K&-a)r-f(5)+fH^)} 



i?(^) = 



cos[r(^)-r(-A)] 



(410) 



Formula (409) gives an expression for obtaining the difference to be 

 applied to the uneliminated fH-4) in order to obtain the true f(^), 

 and formula (410) gives an expression for obtaining the true amplitude 

 R{A). These formulas can not, however, be rigorously applied, 

 because the true values of R{B) and ^{B) of the disturbing compo- 

 nents are, in general, not known, but very satisfactory results may be 

 obtained by using the approximate values of R{B) and ^{B) derived 

 from the analysis or by inference. 



By a series of successive approximations, using each time in the 

 formulas, the newly eliminated values for the disturbing components, 

 any desired degree of refinement may be obtained; but the first 

 approximation is usually sufficient, and all that is justified because 

 of the greater irregularities existing from other causes. 



Form 245 (fig. 32) provides for the computations necessary in 

 applying formulas (409) and (410). 



In these formulas the factors represented by jjt- — x— , and the 



angles represented by 1(& — o)r will depend upon the length of series; 

 but for any given length of series they will be constant for all times 

 and places. Table 29 has been computed to give these quantities 

 for different lengths of series. The factor as directly obtained may be 

 either positive or negative, but for convenience the tabular values are 

 all given as positive, and when the factor as directly obtained is 

 negative the angle has been modified by ± 180° in order to compensate 

 for the change of sign in the factor and permit the tabular values to be 

 used directly in formulas (409) and (410). 



An examination of formulas (409) and (410) will show that the 

 disturbing effect of one component upon another will depend largely 



upon the magnitude of the fraction — ttI \— • Assuming that h is 



•^ " \{b — a)T ^ 



not equal to a, this fraction and the disturbing effect it represents will 



360° 

 approach zero as the length of series r approaches in value tt — y or 



