282 On two new Theorems relating to the Moon's Orbit. 

 such that h^G=fj/^. Hence it follows that 



^/2P' 

 which proves Theorem II. 



The third of the foregoing equations gives by a direct process 

 the known value of the motion of the apse to the first approxi- 

 mation. The periodic time is found to be the same as that in 

 an elliptic orbit whose mean distance is a, the force tending to 



the focus being /Lt(l — ^^j at the unit of distance, which is 



otherwise known to be true. 



The numerical value of — 4r- for the moon is 0'0529, and the 



known eccentricity of her orbit is 0'0548. The difference 0"0019 

 is not more than might be expected from the degree of approxi- 

 mation embraced by the analysis. For Jupiter's four satellites 



thevaluesof^^are000029,0-00058,0-001168,and0-002724. 

 ^2P 



Respecting the eccentricities of their orbits, observation shows 

 that those of the first and second are too small to be sensible, 

 and that those of the third and fourth are only just sensible. 

 The eccentricities of the orbits of the other satellites of the solar 

 system are known with too little precision to admit of compa- 

 rison with the theory, excepting that of Titan's orbit, which is 

 stated to be 0'0293. But in this instance the large inclination 

 of the orbit to the plane of Saturn's orbit forbids making a com- 

 parison. 



The following considerations appear to me sufficient to prove 

 that the eccentricity of the moon's orbit must be a function of 

 the disturbing force. A straight line being drawn from the 

 earth's centre in any direction in the plane of the moon's orbit, 

 the radius- vector at the instants the moon passes this line in 

 successive revolutions has different values. The fluctuations of 

 value, which, as is known, do not exist in the elliptic theory, 

 depend on the disturbing force in such a manner that the func- 

 tion by which they are expressed would vanish if the disturbing 

 force were indefinitely small. The total fluctuations, in the case 

 of a uniform apsidal motion, are the same in all directions, and 

 take place about the same mean distance ; and the difference 

 between the extreme values of the radius-vector in any given 

 direction is equal to the diffei'ence of the two apsidal distances, 

 and therefore proportional to the eccentricity. Hence as the 

 fluctuations of the radius-vector in a given direction depend on 

 the disturbing force, it follows that the eccentricity is also a 

 function of the disturbina; force. 



