ON THE LAWS OF GRAVITATION. 413 



consequence of which, according to Dr. Bradley's original observations, 

 the pole of the equator describes in the heavens a little ellipsis, of which the 

 diameters are 16 and 20 seconds. The same cause is also concerned in 

 modifying the secular variation of the obliquity of the ecliptic : and on the 

 other hand, this variation has a considerable effect on the apparent preces- 

 'ion of the equinoxes. On account of the different quantity of the preces- 

 sion at different times, the actual length of the tropical year is subjected to 

 * slight variation : it is now 4 or 5 seconds shorter than it was in the time 

 of * fttpparchus. The utmost change, that can happen from this cause, 

 amounts to 43 seconds. 



The exact computation of the moon's motion is one of the most difficult, 

 as well as the most important problems in astronomy ; but it is easy to 

 understand, in general, how the difference in the quantity and direction of 

 the sun's actions on the moon and earth, may cause such a derangement of 

 the moon's gravitation towards the earth, that the inclination of the orbit 

 must be variable, that the nodes must have a retrograde, and the apsides a 

 direct motion ; and that the velocity of the moon must often be different 

 from that which she would have, according to the Keplerian law, in a Sim- 

 ple elliptic orbit. 



For, the sun's attraction as far as it acts equally on the earth and the 

 moon, can have no effect in disturbing their relative position, being always 

 employed in modifying their common annual revolution ; but the difference 

 of the forces, occasioned by the difference of distances, always tends to 

 diminish the effect of their mutual attraction ; since the sun acts more 

 powerfully on the nearer than on the remoter of the two bodies. The dif- 

 ference of the directions, in which the sun acts on the earth and the moon, 

 produces also a force, which tends, in some degree, to bring them nearer to- 

 gether ; but this force is, on the whole, much smaller than the former ; and 

 the result of both these disturbing forces is always directed to some 'point 

 in the line which joins the earth and the sun, on the same side of the earth 

 with the moon. It is obvious that when the nodes are also in this line, the 

 disturbing force can have no effect, either on their position, or on the in- 

 clination of the orbit, since it acts wholly in the plane of that orbit ; but 

 when they are in any other situation, the disturbing force must cause a 

 deviation from the plane, towards the side on which the sun is situated, so 

 that the inclination of the orbit increases and decreases continually and 

 equally ; but whatever may be the position of the nodes, it will appear 

 that they must recede during the greater part of the moon's revolution, and 

 advance during the smaller. (Plate XXXIV. Fig. 487.) 



When the disturbing force tends to separate the earth and moon, it de- 

 ducts from the gravitation of the moon towards the earth a portion which 

 increases with the distance, and therefore causes the remaining force to 

 decrease more rapidly than the square of the distance increases ; and the 

 reverse happens when the disturbing force tends to bring the earth and 

 moon nearer together ; but the former effect is considerably greater than 

 Jbhe latter. Now in the simple ellipsis, when the body descends from the 

 mean distance, the velocity continually prevails over the attractive force, 

 so as to turn away the direction of the orbit more and more from the 



