96 



V. S. COAST AND GEODETIC SURVEY. 



From (374) 



^ = taii 



B sin (f) 



Aj^+B cos (p 



- = tan-i 



sin (f} 



^^ L 



"dt + cos 



(376) 



In the special cases under consideration the ratio t is near unity, 



and the difference between d and tan~^ . , p : is therefore very 



A + B cos (f) -^ 



small, so that the cosine may be taken as unity. 



The resultant amplitude may therefore be expressed by 



/ D2 D 



V^' + 52 + 2AB cos <^ = ^-x/ 1 +^ + 2 ^ cos <j) 



W 



A' 



A 



(377) 



The true amplitude of the component sought being A, the resultant 

 amplitude must be divided by the factor 



' B^ B 



^1+32 + 2 Z^^^"^ 



(378) 



in order to correct for the influence of the disturbing component. 



The corrections for acceleration and amplitudes as indicated by 

 formulas (374) and (378) may to advantage be applied to the con- 

 stants for component K^ for an approximate elimination of the effects 

 of component P^ and to the constants for 83 for an approximate 

 ■elimination of the effects of components Kj and T2. By taking the 



B 



relations of the theoretical coefficients for the ratios -r and the differ- 

 ences in the equilibrium arguments as the approximate equivalents 

 of the phase differences represented by (j), tables may be prepared 

 giving the acceleration and resultant amplitudes with the arguments 

 referring to certain solar elements. 



Thus, from Table 3, the following values may be obtained. 



Effect of Pi on Kj 

 Effect of K2 on Ss. 

 Effect of TsonSs. 



0. 33086 

 0. 27213 

 0. 05881 



Aa 

 Bb 



3. 03904 

 3. 66469 

 17. 02813 



-^h+u'-lSO" 

 2h-2v". 



Substituting the above in (344) and (378) we have 

 Effect of Pi on K^ 



sin (2Ji-v') 



Acceleration = tan-^ 



3.0390 - cos (2/^-/) 

 Resultant amplitude = 0.813 Vl.6767 - cos {2h - vO 

 Effect of K, on S, 



Acceleration = tan-^ 



sin i27h-2v") 



(379) 

 (380) 



(381) 



3.6647 + 008(2^-2/') 

 Resultant amplitude = 0.738 V1.9734 + cos {2h-2v") (382) 



