Secular Acceleration of the Moon's Mean Motion. 325 



It is well known that the sun's attraction diminishes the moon's 

 gravity toward the earth, and thus increases its periodic time ; 

 and if it can be shown that the effect is greater than it would be 

 if the earth revolved in a circle at the same mean distance from 

 the sun, it is manifest that so long as the present change in the 

 shape of the earth's orbit goes on, the moon's periodic time must 

 grow less and less. 



Let S (fig. 1) represent 

 the sun, E the earth, AEB 

 a portion of the earth's orbit, 

 and BCAD the moon's orbit 

 regarded as a circle. 



As gravitation varies in- 

 versely as the square of the 

 distance, when the moon is 

 at D it is drawn away from 

 the earth by the difference 

 in the attractive forces of 

 the sun on the earth and 

 moon, and when it is at 0, 

 the earth is drawn away 

 from it by the same cause. 

 In both cases the tendency 

 is to diminish the moon's 

 gravity toward the earth. 

 But when the moon is at 

 A or B, its distance from 

 the sun is the same as that 

 of the earth, and conse- 

 quently the sun's attraction 

 on them is equal, but not 

 being in the same direction 

 it tends to draw them near- 

 er together, and thus to in- 

 crease the moon's gravity 

 toward the earth. 



The moon being thus 

 drawn toward the earth in some parts of its orbit, and from it in 

 others, it might seem rather difficult to determine whether on the 

 whole it was drawn toward or from the earth. If we reflect 



Vol. xLvii, No. 2.— July-Sept. 1844. 42 



