35] 



EFFECT ON" THE MOTION OF THE SATELLITES. 



277 



means of determining, within not very wide limits, its probable amount, 

 and we shall presently see that, in all probability, in the case of both the 

 satellites the motion of the nodes produced by the ellipticity greatly 

 exceeds the motion caused by the Sun's action, so that the fixed planes 

 for both satellites are only slightly inclined to the planet's equator. 



From measures of the planet's diameter and of the greatest elongations 

 of the satellites, combined with the known time of rotation of Mars and 

 the periodic times of the satellites, it is found that the ratio of the centri- 

 fugal force to gravity at Mars' equator is about ^ 7 . Hence it follows 

 that if the planet were homogeneous its ellipticity would be about j^g-. 

 If, instead of the planet being homogeneous, its internal density varied 

 according to the same law as that of the Earth, so that the ellipticity 

 would bear the same ratio to the above-mentioned ratio of centrifugal force 

 to gravity at the equator as in the case of the Earth, then the ellipticity 

 would be about ^-g. In all probability the actual ellipticity of Mars lies 

 between these limits. 



The following Table shews the annual motions of the nodes of the 

 two satellites, caused by the Sun's action and by the planet's ellipticity 

 respectively, for the above values of that ellipticity, and also for the ellip- 

 ticity x^-g-, which has been deduced from Professor Kaiser's observations, 

 although I have no doubt that this value is too great. The Table like- 

 wise contains the corresponding inclinations of the fixed planes, so often 

 mentioned above, to the planet's equator. 



Satellite I. 



Annual motion of the node due to the 

 Sun's action, 0'06. 



Supposing ellipticity 



111 

 118 176 228 



the annual motion of the node due to 

 that ellipticity will be 



333 182 113 



Corresponding inclinations of fixed 

 plane to planet's equator : 



17" 31" 50" 



Satellite II. 



Annual motion of the node due to the 

 Sun's action, 0'24. 



Supposing ellipticity = 



1 1 1 



118 176 228 



the annual motion of the node due to 

 that ellipticity will be 



13'4 7'3 4'5 



Corresponding inclinations of fixed 

 plane to planet's equator : 



27' 50' T 19' 



