326 Secular Acceleration of the Moon's Mean Motion. 



however, that by increasing the sun's attractive power, it might 

 draw the moon entirely away from the earth, causing it to as- 

 sume an independent orbit, Hke Venus, it is obvious that the in- 

 fluence, whether great or small, is on the whole to draw the moon 

 away from the earth, or diminish its gravity toward it as already 

 remarked. 



We will demonstrate that such must be the influence of the 

 sun's attraction in a more rigid manner, and show how great it 

 is compared with the earth's attraction on the moon. And let us 

 first see what ratio exists between the attractive forces of the sun 

 and of the earth on the moon when it is near quadrature, as at 

 A or B. These points are selected because the moon is then at 

 its mean distance from the sun, and consequently the sun's at- 

 traction is a mean, and equal to its attraction upon the earth. 



If we take the quantity of matter in the earth as the unit by 

 which to measure other quantities of matter, that of the sun will 

 be 354936, which number we will call S. Then since gravita- 

 tion is directly as the quantity of matter and inversely as the 



1 S 

 square of the distance, y^^ : ^^i opi * '. the earth's attrac- 

 tion : the sun's attraction. Or if we take the earth's attraction 

 on the moon as the unit by which to measure other attractions, 



the proportion becomes g^j : ^^7 : : 1 : the sun's attraction on 



the moon at the point A or B, or its mean attraction for the whole 



SxEB^ 

 orbit, which is therefore equal to , qp^ (1). 



To investigate the subject generally, let the moon be at any 

 point M of its orbit, and let the sun's attraction on it at that 

 point be represented by m. Resolving this force into two others 

 in the directions ME and ES, the proportion for the former or 

 addititious force will read SM : ME;:m : the addititious force 



ME 

 ~SM^" '^^ ^^^^ ^^^ mean addititious force for the whole of 



the moon's orbit, we may substitute for m its mean value above 



SxEB^ 

 obtained (1), viz. g^^ , and for SM its mean value SE, and 



for ME its equal EB, The expression for the mean addititious 



S xEB^ 

 force will then read g-n,3 (2). 



