in the Secondary Systems. 491 



same point of its surface in perpetual conjunction with the 

 primary. 



In the case of a solid satellite composed of imperfectly elastic 

 materials, it is necessary to take into consideration the slight 

 change of density attending the constant alteration of form. 

 Had this been done, the expressions (1) and (2) would be reduced 

 to four-fifths of their value ; while instead of formula (3) we 

 should find 



T/_ 5TA M\ 



127r(A-B)sin2^' ' ' ' \ [ > 



If the angle ty were equal to 90 degrees, no change would be 

 indicated in the rotation; but the angle could not have this 

 magnitude except in the case of a solid satellite all parts of which 

 were perfectly elastic, or in the case of one, consisting wholly or 

 partially of fluid, which performed its tidal oscillations without 

 friction. 



By another investigation, which brevity compels me to omit, I 

 have arrived at the same results in regard to the secular changes 

 which the rotation of a secondary body must experience until it 

 keeps pace with the orbital revolution. It will also readily appear 

 that the ultimate effect of these changes is not affected by the 

 inclination of the equator of the satellite to the plane of its orbit. 

 But it will be necessary to show that the inclination is doomed 

 to undergo a slow permanent diminution when the synchro- 

 nism of the rotation and revolution is once established. For this 

 purpose we may proceed in a manner similar to that employed 

 in investigating the mutation of the earth's axis. Let I be the 

 inclination of the equator of the satellite to the plane of its orbit, 

 which for simplicity may be regarded as circular, and let L be 

 the longitude of the satellite reckoned from the point of their 

 intersection. Regarding the body as an ellipsoid, the tendency 

 of the disturbing force to move the axis towards the plane of the 

 orbit will be 



|^(A-C)sinHsin2L (5) 



If 81 denote the change of inclination from this cause, then 



d*hl 3M/A-C\ . 2T . 2T ,„ 



On substituting nt for L, and regarding A and C as constant, 

 the integration will give only periodical quantities; so that no 

 permanent change would be indicated if the form were absolutely 

 immutable. But supposing A— C to vary, either from the pre- 

 sence of large collections of fluid on the surface of the satellite, 

 or from the necessary elasticity of its solid matter, the quantity 

 2K2 



