NEWTON S PRINCIPIA. 91 



variations of the centripetal force actually take place. The 

 action of the sun, for example, upon the moon, while 

 she is acted upon by the earth, coincides with that of 

 the earth in some parts of her orbit, and in some parts 

 opposes this action, thus alternately adding to and taking 

 away from the force of her gravitation towards the earth ; 

 and this increase and diminution is greater at the greater 

 distances of the moon from the earth. Hence the pro 

 portion of the centripetal force which keeps her in her 

 orbit, is somewhat different from the exact ratio of the 

 i inverse square of the distance. There is more taken 

 away from this centripetal force by the sun s action 

 while the bodies are placed towards each other in one 

 direction, than there is added when in the other position ; 

 and therefore there is a total diminution of the moon s 

 gravitation, or the centripetal force decreases in a somewhat 

 higher ratio than as the square of the distance increases ; 



in other words, the denominator of the expression -^ 1S 



greater than this exact power of d, which we have seen 

 keeps the orbit and its axis fixed with respect to the 

 centre, which in this case is the centre of the earth. 

 Hence this axis of the moon s orbit revolves in the 

 direction of the moon s motion, and in a certain period 

 makes a complete revolution. So that at one time, 

 half this period, the moon s greatest and lea:t distances 

 from the earth (her apogee and perigee) have changed 

 places, and at the end of the period they resume their 

 former position. The amount of this motion of her 

 apsides is about 3 in each revolution, or 39 in a year ; 

 so that the axis of her ellipse revolves in nine years ; 



and the centripetal force is not as -^ but , nearly 



