Secular Acceleration of the Moon's Mean Motion. 331 



mean distance ; also let m represent the annual increase of grav- 

 ity, and x the diminution in the distance occasioned thereby. 

 Then will r— x represent the distance on the succeeding year, 



r & 

 and (^T^y* + m the g rav ity, viz. g increased by the laws of . 



gravitation in the inverse ratio of the square of the distance, and 

 also by the quantity m. Now it has been demonstrated,* that in 

 circular orbits where the radius varies from a variation in the force 

 which retains the revolving body in its orbit, the centrifugal force 

 will vary in the inverse ratio of the cube of the distance ; and 

 since the centripetal and centrifugal forces must be equal, 



1 1 



r'g 



Therefore — : , ; :g ; -. — ^_ + m 



r 3 [r — xy a (r—x) 2 



Expanding, omitting the higher powers of x on account of 

 their smallness, and multiplying by r 3 , we have 



1 . * .. #r 



r ■ r 



3x : '^ l r^2x +m ' 



Hence _£_=_£_,!?. 



r— 3x r—2x~r 



Clearing of fractions and omitting the higher powers of x, 



gr 2 — 2grx =gr 2 — 3grx -f- mr 2 — 5mrx. 



Hence grx=mr 2 —5mrx. 



And g X = mr _ 5 mXm 



Reconverting the equation into a proportion, 



g : ml :r—5x : x. 

 And by multiplication and composition, 



g-{-5m : m: \r : x. 



Now m is TT ---'_ 7 _ ¥ - of g, and therefore x is jp ^Wojtt of 

 r ' Hence the contraction of the lunar orbit, even if the moon's 

 absolute velocity remained unchanged, would reduce its periodic 

 time by the amount of the latter fraction annually. But we shall 

 Presently show that the absolute velocity is increasing annually 

 ° v the same fraction. If this is so, it follows that from both 

 causes combined its sidereal period must decrease annually by 

 a bout double the aforesaid fraction, so that the moon must pass 

 over a greater number of degrees every year than it did on the 

 year preceding by about riti$ 



8"?3'22 2 9 5* 



At the commencement of the present century the moon's mean 

 Motion was such as to carry it through 13 sidereal revolutions 



* See Stewart's Mathematical and Physical Tracts. 



