276 NOTE ON THE ELLIPTICITY OF MARS, AND ITS [35 



to exist at the present time, would be merely fortuitous ; but this appears 

 a priori to be very improbable. 



It is well known that, if there were no external disturbing force, the 

 ellipticity of a planet would cause the nodes of a satellite's orbit to retro- 

 grade on the plane of the planet's equator, while the orbit would preserve 

 a constant inclination to that plane. Laplace has shewn that, when both 

 the action of the Sun and the ellipticity of the planet are taken into 

 account, the orbit of the satellite will move so as to preserve a nearly 

 constant inclination to a fixed plane passing through the intersection of 

 the planet's equator with the plane of the planet's orbit, and lying between 

 those planes, and that the nodes of the satellite's orbit will have a nearly 

 uniform retrograde motion on the fixed plane. The angles which this fixed 

 plane makes Avith the planes of the planet's equator and its orbit respec- 

 tively will depend on the ratio between the rates of the above-mentioned 

 retrogradations of the nodes produced by the Sun's action and by the 

 ellipticity of the planet. If the latter of these causes would produce a much 

 slower motion of the nodes than the former, as in the case of our Moon, 

 the fixed plane will nearly coincide with the planet's orbit ; but if, as in 

 the case of the inner satellites of Jupiter, the ellipticity of the planet 

 would produce a much more rapid motion of the nodes than the Sun's 

 action, then the fixed plane will nearly coincide with the planet's equator. 



The ratio of the motion of a satellite's node to that of the satellite 

 itself, when the Sun's action is the disturbing force, varies, ceteris paribus, 

 as the square of the satellite's periodic time, that is as the cube of its 

 mean distance from the planet. On the other hand, the ratio of the same 

 two motions, when the ellipticity of the planet is the disturbing cause, 

 varies inversely as the square of the mean distance. Hence, for different 

 satellites of the same planet, the motion of the nodes caused by the ellip- 

 ticity will bear to the motion caused by the Sun's action the ratio of the 

 inverse fifth powers of the mean distances. 



Now, the distance of the inner satellite of Mars from the planet's 

 centre is only about 2f radii of the planet, a greater comparative proximity 

 than is known to exist elsewhere in the Solar System, and the distance 

 of the outer satellite from the same centre is only about 7 radii of the 

 planet, while the periodic times of both are very small compared with the 

 periodic time of Mars. Hence the effect of a given small ellipticity of 

 Mars on the motion of the nodes of the satellites will be greatly magnified. 



It is true that the ellipticity of Mars is still unknown, and is pro- 

 bably too small to be ever directly measureable; but we are not without 



