6 



Mr. G. H. Darwin. 



[Dec. 18,. 



is to increase the distance of the satellite from the planet, and to- 

 transfer m. of m. from that of planetary rotation to that of orbital 

 motion . 



If then the direction of the planet's axis of rotation does not 

 change, it follows that the normal to the lunar orbit must approach 

 the axis of resultant m. of m. By drawing a series of parallelograms 

 on the same diameter and keeping one side constant in direction, this 

 may be easily seen to be true. 



The above statement is equivalent to saying that the inclination of 

 the satellite's orbit will decrease. 



But this decrease of inclination does not always necessarily take 

 place, for the previous investigations show that another effect of tidal 

 friction may be to increase the obliquity of the planet's equator to the 

 invariable plane, or in other words to increase the inclination of the- 

 planet's axis to the axis of resultant m. of m. 



Now if a parallelogram be drawn with a constant diameter, it will 

 easily be seen that by increasing the inclination of one of the sides 

 to the diameter (and even decreasing its length), the inclination of 

 the other side to the diameter may also be increased. 



The most favourable case for such a change is when the side whose- 

 inclination is increased is nearly as long as the diameter. From this 

 it follows that the inclination of the satellite's orbit to the invariable 

 plane may increase, and also that the case when it is most likely to< 

 increase is when the m. of m. of planetary rotation is large compared 

 with that of the orbital motion. The analytical solution of the 

 problem agrees with these results, for it shows that if the viscosity of 

 the planet be small the inclination of the orbit always diminishes, but 

 if the viscosity be large, and if the satellite moves with a short periodic 

 time (as estimated in rotations of the planet), then the inclination of 

 the orbit will increase. 



These results serve to give some idea of the physical causes which,- 

 according to the memoir, gave rise to the present inclination of the 

 lunar orbit to the ecliptic. For the analytical investigation shows, 

 that the inclination of the lunar orbit to its proper plane (which 

 replaces the invariable plane when the solar attraction is introduced) 

 was initially small, that it then increased to a maximum, and finally 

 diminished, and that it is still diminishing. 



But the laws above referred to would, by themselves, afford a very 

 unsatisfactory explanation of the inclination of the lunar orbit, 

 because the sun's attraction was found to be a matter of much 

 importance. 



It was stated above, that if the viscosity of the planet be small, the 

 inclination of the orbit of the solitary satellite to the invariable plane 

 will always diminish; but when solar influence is introduced, the 

 corresponding statement is not true with regard to the inclination of 



