Secular Acceleration of the Moon's Mean Motion. 331 



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

 ity, and a; the diminution in the distance occasioned thereby. 

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



and 7 Tg -\-m the gravity, viz. g increased by the laws of 



{r — x) 



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, 



Therefore - : j^ri^, : '.g : ^-^, 4 m. 



Expanding, omitting the higher powers of x on account of 

 their smallness, and multiplying by r^, we have 

 1 1 gr 



Hence 



r ' r — 'dx"^ ' r — 2x^ 



g g m 



r — 'dx r—2x~ r 



Clearing of fractions and omitting the higher powers of .r, 

 gr^ —2>grx=gr'^ —Zgrx-\-mr^ —5mrx. 

 Hence grx=inr^ —5inrx. 



And gx — mr — 5mx. 



Reconverting the equation into a proportion, 

 g : mWr — Bx : x. 

 And by multiplication and composition, 

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



Now m is -^-^^'^^^^^ of g, and therefore x is ^^--^\^^^- 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 

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

 causes combined its sidereal period must decrease annually by 

 about 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 -^-^^2_____, 



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 3Iathematical and Physical Tracts. 



