328 Secular Acceleration of the Mooii's Mean Motion. 



fractions.* Now since AB may be considered a straight line, 



EF' and EF'' are each equal to FM, and EF''' to EF, so that 



the numerators added together make 6EF2 +6FM^ =(Euc. 1. 47) 



6ME2, which divided by 4 gives for the mean of the numerators 



IJME^. The mean value of the denominators is SExME = 



SxEB- 

 SExEBj and the mean value of m is (1) — opT" * Hence the 



mean value of the above expressions for the ablatitious force at 



HEB=^ SxEB^ USxEB^ 

 the four points taken, is g^xEB ^ — SE^" ~' — SE* ^^ 



this expression is independent of the position of M, it must be 



true for the whole orbit, and is the mean ablatitious force required. 



r^, ' SxEB^' 



The mean addititious force (2) was — mj,^ , which subtracted 



from the mean ablatitious force leaves for the net mean force di- 



JSxEB^* 

 mmishing the moon's gravity toward the earth, — ap^T~ t (^)' 



Restoring the numerical value of S, viz. 35493G, and giving 

 to EB and SE their mean numerical values, viz. EB=237577 

 miles and SE = 95024608 miles, the value of the expression last 

 found becomes .0027735. But this value needs a correction, for 

 the reason that the sun does not lie in the plane of the moon's 

 orbit, as we have thus far assumed, which renders its disturbing 

 influence less. Applying the necessary correction from this cause, 

 the value of the expression is reduced to .0027689, or about gij, 

 so that the moon gravitates toward the earth only ^^\ as much as 

 it would do if the sun were absent from the system. 



^SxEB^* 

 In the fractional expression — spT" *^^ numerator is con- 

 stant, since the moon's orbit is regarded as a circle in this inves- 

 tigation ; but SE, the distance of the sun from the earth, varies 

 in different seasons of the year. Hence the value of the expres- 

 sion must be reciprocally proportional to SE% and consequently 



* The error when a maximum amounts to only about ^'b^'u' 



t Most writers who have treated of this subject, have preferred to represent the 



earth's attraction on the moon by , instead of representing it by unity as I 



S V FT} 

 have done, which converts this expression into ^ , or, as it is usually ex- 



, mr* 

 pressed, - — . 



