52 U. S. COAST AND GEODETIC SUEVEY. 



From (191) and (192) 



cos Ou , Tr. COS / ^^"1 , , 



sin Qu 



Q. 



= isin2P (200) 



Substituting (199) and (200) in (198), the mean value of variable 

 factors of the coefl&cient is as follows : 



I ^ sin / cos^ ^I 3 — 2~T7 cos v + cos 2P cos v — sin 2P sin j^ 



(201) 

 = [3/4 sin 2/ cos v + i sin / cos=^ ^I cos {2P + v)]^ 



For reasons similar to those given on page 49 the mean value of the 

 last term in the above is zero for an infinite series. The mean value 

 of sin 2/ cos v is given by formula (159), which when substituted 

 in the above gives the following : 



p° y *^ os fr + Q.)]^ (202) 



= 3/4 sin 2co [1 - 3/2 sin2 i] = 0.5410 



For the factor of reduction we have 



^ o ^jr 3/4 sin 2co[l -3/2 sin^ i] 0.5410 ^ 



' sin / cos^ j/ ~ sin / cos^ ^/^ ^» (203) 



The factor F for reduction of Mj, originally adopted and now in 

 general use for analysis made in accordance with the system of Sir 

 George H. Darwin, is as follows : 



sin 0} cos^ i 0} cos^ j i 0.38005 



sin / cos^ i I [5/2 + 3/2 cos 2P]^ ~ sin / cos^' ^ /[5/2 + 3/2 cos 2P]^ ^^^ 



In the above the factor rg/2_LQ/o 9PW^ ^^ *^^ approximate 



equivalent of the factor Q^ in (203). The ratio of (203) to (204) is 



5410 

 therefore approximately n Qgno'^ ^-^'^^ (205) 



This discrepancy appears to be due principally to the accidental 

 omission of the factor -yj2.5, or 1.58, from the original formula. (See 

 Scientific Papers by Sir George H. Darwin, vol. 1, p. 39.) 



The effect of this error has been that all the mean amplitudes for 

 component M^ obtained by the formula of Darwin are too small and 

 should be increased by nearly 50 per cent in order to be theoretically 

 correct. 



Since the primary purpose of reducing the amplitudes to their mean 

 values is to render the results from different series comparable with 

 each other, this purpose has not been frustrated by the introduction 



