Prof. Challis on the Eccentricity of the Moon's Orbit. 135 



the equation which gives the apsidal distances must be solved as 

 a biquadratic, the term fih A being of equal significance with the 

 others. Since, therefore, from the foregoing reasoning, it 



appears that r is nearly equal to ^, and the apsidal distances by 



hypothesis differ little from each other, it follows from the theory 



of equations, that to obtain their approximate values, ^ + (r — ^) 



is to be substituted for r in the last term of the equation 



-A 2 + 2/ir-C/- 2 + / LtV 4 = 0, 



and then, after expanding to the second power of r— £, the equa- 

 ls 



tion is to be solved as a quadratic. The same process of substi- 

 tution must for the same reason be gone through to prepare the 

 equation (C) for approximate integration. I need not pursue 

 the investigation further, as the results of the subsequent steps 

 are given in the Philosophical Magazine for April 1854, p. 279, 

 and in the Supplement to the Philosophical Magazine for De- 

 cember 1854, p. 526. By far the most important result is, that 

 the eccentricity of the moon's orbit has a special value depending 

 on the disturbing force. 



The following considerations appear to me proper for pi'oving 

 that, whatever be the law of the disturbing force, and whether it 

 be central or not, the motion of the apse is uniform if the eccen- 

 tricity be a function of the disturbing force. Let the undisturbed 

 orbit be an ellipse described about the focus, and let the three 

 bodies be always in a given plane, the central body having a fixed 

 position. A straight line being drawn from the centre of the 

 fixed body in any direction in the plane of motion, the radius- 

 vector of the disturbed body at the instants it passes this line 

 has different values in successive revolutions. The change of 

 value in each complete revolution depends on the disturbing 

 force, in such a manner, that the function by which it is expressed 

 would vanish if the disturbing force vanished. The total fluc- 

 tuations, in the case of a uniform apsidal motion, are the same 

 in all directions ; and the difference between the extreme values 

 of the radius-vector in any given direction depends both on the 

 partial fluctuations, and on the rate of the angular motion of the 

 apses. Hence as the partial fluctuations, as well as the motion 

 of the apses, would vanish with the disturbing force, it follows 

 that the difference between the extreme values, and consequently 

 the eccentricity of an orbit nearly circular, is a function of the 

 disturbing force. 



I adduced an argument similar to the above in the Philoso- 

 phical Magazine for April 1854 (p. 282), which I subsequently 



