90 NEWTON S PRINCIPIA. 



able orbit moves only through the same space with the 

 axis of the fixed orbit, that is if the moveable orbit coin 

 cides with the fixed, then the centripetal force is as 



d 1 - 3 = d~ 2 -y- 2 ; and conversely, if the central force is 



as Tg, the line of the apsides has no motion whatever. 



Hence the important proposition, that the inverse square 



f.of the distance, the actual law of gravitation, is the only 



/ proportion which prevents the line of the apsides from 



I moving at all. Again, if &amp;lt;p : :: 363 : 360, or the line of 



the apsides advances three degrees in each revolution, 



then the centripetal force is between the inverse square and 



inverse cube of the distance, but much nearer the for- 



2 

 mer; for -^ 3 becomes nearly equal to 2 | 7 , or about 



2- 5 - 1 T . But suppose the excess of the angle between the 

 axes in one orbit over that angle in the other orbit to 



2 

 be only 11&quot; 53 &quot;,* then - 2 3 becomes equal to 2 JTJ 77r , 



or say 2^^^, and the force as g , -. In like man 



g , 



ner, if some extraneous force is impressed upon the revolv 

 ing body, from knowing the amount of that force we 

 can find the motion of the apsides, and conversely. It 

 is found by following the method of Sir Isaac Newton, 

 that the advance in a single revolution on the suppo 

 sition of the disturbing force being to the centripetal force 

 as 1 to 357-45, is equal to 1 31 28&quot;. 



Now it is found that in the planetary motions these 



* The amount of 12&quot; is often given for the advance of the axis of the 

 earth s orbit ; but we have followed Laplace s number of 36&quot; 7 &quot;, which 

 on the sexagesimal scale is 1 1&quot; 89 &quot;, or 1 1&quot; 53 &quot;. This small difference makes 

 a difference of 1000 years in the total revolution. 



