410 Mr. D. Vaughan on the Form of Satellites 



present article, some mathematical investigations of the chief 

 cases of planetary instability. 



The peculiar arrangement which is supposed to prevail in all 

 secondary systems, for keeping the same sides of satellites always 

 turned to their primary, is well calculated to preserve these 

 second-rate worlds from the injurious effects of excessive tides, 

 since they must always have their oceans elevated and their forms 

 elongated at the same localities. The exact synchronism of the 

 rotary and the orbital motion mainly contributes to secure this 

 important end ; but, to remove entirely the dynamic effects of 

 the great disturbance, it is necessary that the satellite should 

 rotate around an axis perpendicular to the plane in which it re- 

 volves, and that its path should be a true circle. In describing 

 an ellipse sufficiently small for the production of a very great 

 disturbing force, the planetary structure would have its safety 

 much endangered by the oscillatory movements of its parts; and 

 the oscillations would be attended with more fatal results if, from 

 a want of the other conditions, it presented different sides alter- 

 nately to the primary. In calculating the dimensions of the 

 smallest orbit in which it is possible for a secondary planet to 

 hold its parts together by the tie of gravity, we are necessarily 

 restricted to the cases most favourable for stability ; and I shall 

 accordingly suppose that the body has its axis perpendicular to 

 the plane of its orbit, that the latter is exactly circular, and that 

 the rotation takes place in the same time and in the same direc- 

 tion as the periodical revolution. In such circumstances the 

 relative direction of the primary from every part of the satellite 

 must ever remain Unchanged, and its powerful attraction must 

 be productive of the least derangement on the surface of the 

 latter body. 



In a very small orbit there appears to be a physical necessity 

 for a synchronism of the orbital and rotary movements of a satel- 

 lite similar in constitution to our globe; for the rotation would 

 be gradually changed by the action of enormous tides, until it 

 finally occupied the same time as the orbital revolution. Nor 

 can our conclusion be very different if we agree with Lagrange 

 in ascribing the arrangement to the deviation of these humble 

 worlds from true spheres, and to the consequent tendency of 

 their longest diameters to point towards the central body. In 

 two secondary planets of the same size, form, and density, this 

 tendency would be inversely proportional to the fourth power of 

 their distances from the primary ; but were the ellipticity of both 

 bodies such as the attraction of the latter might occasion on a 

 yielding solid mass, the tendency would be in inverse proportion 

 partly to the seventh, and partly to the higher powers of the 

 same quantities. There are accordingly sufficient grounds for 



