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



proves that, under these circumstances, it is not legitimate to 

 commence from a fixed ellipse. 



But from independent analytical principles (as illustrated in 

 the case already considered), it is certain that, under any 

 circumstances, it would be legitimate to commence from a fixed 

 ellipse, if the aggregate of the terms A 2 — 2//r + O 2 be of a higher 

 order of magnitude, estimated according to the disturbing force, 

 than the term —/j!? a . It is clear that they cannot be of a lower 

 order. 



It follows, therefore, inevitably, that the aggregate of the terms 

 A 2 — 2/ir + C?- 2 is of the same order of magnitude with respect to 

 the disturbing force as the term —fi'r 4 ; in other words, that they 

 contain the disturbing force as a factor. It is not difficult to 

 point out the origin of this analytical circumstance. The inte- 

 gration by which the constant C is introduced (see Phil. Mag. 

 for April 1854, p. 279), may be effected by assuming only that 

 the increment in a given time of the sun's true longitude is small 

 compared to that of the moon's. The remarkable integral I have 

 obtained in the Supplement to the Philosophical Magazine for 

 December 1854 (p. 521), shows that even that assumption is not 

 necessary for effecting an integration. The limitation that the 

 moon's true longitude differs little from a mean value, is made 

 subsequently to that integration. Hence the constant C possesses 

 greater generality than comports with the conditions of the pro- 

 blem, and must itself be subject to some limitation. Clearly, 

 therefore, the reasoning will be complete if, by means of the 

 constant C, the condition that h* — 2/u.r + C? s contains the di- 

 sturbing force as a factor can be fulfilled. But this may be 

 readily done as follows. 



Let /i 2 C=/i 9 , and r= £ +fv. Then 



A*_2/«- + Cr 2 -/ 4 'r 4 =c(r-g-) -/u/r 4 



Hence the required condition is satisfied if h^C — fi 2 , the unknown 

 quantity / being determined by the equation pzzzfjj. 



This point being settled, it remains to indicate the process of 

 approximation that must be adopted. Resuming the equation 



i S- = - /i2 + 2/Xr ~ Cr2 + ^' 4 ' ' * • (C) 

 it is evident, since — A 2 + 2/ir — Cr 2 contains fjj as a factor, that 

 the approximation must proceed according to the eccentricity, 

 and not according to the disturbing force. For the same reason, 



