332 Secular Acceleration of the Moon's Mean Motion. 



and 132° in a year. 



7 3 2/29 5 of 13 revolutions and 132°, 



Hence the annual acceleration is equcil to 

 which is about -^\-^ of 

 V] a quantity quite too small to be noticed in a single year, but 

 which increasing like other uniformly accelerated motions as the 

 square of the time, becomes quite conspicuous in the lapse of cen- 

 turies. According to our calculation it would amount to over 4° 

 in 2000 years, though from the comparison of ancient with mod- 

 ern observations this appears to be too great. 



It remains to demonstrate the truth of the principle which was 

 employed above hypothetically. 



To show generally that the absolute velocity of the moon is 

 increased when the orbit contracts, and diminished when it di- 

 lates, let us suppose it, revolving in the larger circle DAH, to 

 have arrived at the point A, when ow- 

 ing to an increase in the attractive 

 power of E it is drawn along the curve 

 AB into the circle BIG. As a line 

 drawn from E to any point F in the 

 line AB, makes the angle EFB acute, 

 '^; ' ^ the motion must be accelerated along 

 B ; so that when the moon arrives at 



it will be moving with a greater ve- 

 locity than when it left A. And this 

 increased velocity will be retained so 



long as it revolves in the smaller circle. When it has arrived at 

 0, let the attracting power of E be diminished so as to allow the 

 moon to recede along the curve CD into its former orbit. The 

 angle EGD being now obtuse, the attraction of E holds the moon 

 back and retards its motion just as much as it accelerated it along 

 the curve AB. 



The principles of central forces enable us to find the precise 

 ratio of the velocities in the two circles, which we will suppose 

 to be the orbit of the moon in two successive years. 



We have already observed, that the centrifugal force varies in- 

 versely as the cube of the distance. But it is well known that 

 it is also proportional to the square of the velocity divided by the 

 distance. Therefore if V represent the velocity in the outer cir- 

 cle, and V in the inner, we have 



1 1 V^ YV2 



EEP • ET^-'EH • ET' 



