204 Prof. Challis on the Theory of the Moon's Motion. 



These results show, since may be supposed positive, that 

 0+B will be less than 90°, and that, as A is positive, u has a 

 maximum value if 2q be less than unity, but cannot in that 

 case have a minimum value. Hence as q is by supposition a 

 small quantity, it is clear that the maximum does not apply to 

 the elliptical orbit, but to the third apse ; for if it applied to the 

 elliptical orbit, u would also have a minimum value. 



If, however, the above approximation, viz. u = b + be cos (0 + y) , 

 be derived by integration from the approximate equation 



dd i + u A 2 ' 



the second approximation can have no reference to the third ap- 

 sidal distance, and ought to be true for all values of 0. But this 

 cannot be the case, unless the above integral be considered to be 

 derived by a diverging expansion from the equation, 



The foregoing reasoning shows that when the integral of 

 equation (1) contains a term which involves as a factor, this 

 circumstance is due to the particular process of integration, by 

 which the values of the radius-vector for points contiguous to 

 the third apse are taken into account, and that although the 

 same integral approximates to an orbit of small eccentricity, it 

 is attended with the inconvenience of presenting the approxi- 

 mation under a divergent form. I have already pointed out 

 one method of obviating this inconvenience, and excluding the 

 third apse from the investigation. The same purpose is an- 

 swered by approximating to the two nearly equal roots of the 

 apsidal equation (2), which is the principle of the process I 

 have adopted for finding in a direct manner the mean motion 

 of the apses of the moon's orbit. A third method will be seen 

 in my article in the February Number (p. 132), in which the 

 approximation commences from a fixed ellipse. The reason of 

 the success of this method is, that by so commencing the ap- 

 proximation, the elliptic apsidal distances are taken account of, 

 to the exclusion of the third apsidal distance. 



All the above considerations, which for simplicity have been 

 applied to the case of a central force, are equally applicable to 

 the moon's orbit. I think I have now clearly pointed out the 

 origin, in the Lunar Theory, of terms which admit of indefinite 

 increase, and how they may be avoided. It appears that they 

 have no relation whatever to the circumstance of commencing 

 the approximation from a fixed ellipse. I confess that when I 

 entered upon these researches, I little expected such a result. 



