Prof. Challis on the Theory of the Moon's Motion. 373 



1854 (p. 527), I have shown how by including small quantities 

 of the third order, the following equation is obtained : 



df- + ^~T~-2r + L - 



n'*a*e(3coa{2q-p)-coa(2q+p)), . (C) 



q being put for nt + e — n't + e', and p for nt + e— vs. Now the 

 mean apsidal distances may be defined to be the values of r for 



dr % 

 which the part of -53-, which is independent of the sun's longi- 

 tude, vanishes, the mean distance to be the mean between the 

 two apsidal distances, and the mean eccentricity to be the ratio 

 of the difference between the mean apsidal distances to their sum. 

 Hence by equating to zero the left-hand side of the above equa- 

 tion, we have the equation of an orbit in which the mean distance 

 and eccentricity are the same as the mean distance and mean 

 eccentricity of the actual orbit, and which is proper for determi- 

 ning the motion of the apses. Putting the equation under the 

 form 



r*^+h*-2fjr-^f-+Q*=0, 



it will be seen that the integration cannot be exactly performed 

 if the general value of r be retained in the small term. The 

 equation might, however, be integrated by conducting the ap- 

 proximation solely according to the disturbing force, in the 

 manner I have exhibited in the Philosophical Magazine for last 

 February (p. 132). But that process would here be illegitimate, 

 because the equation (C) was obtained by approximating accord- 

 ing to the eccentricity as well as the disturbing force. The fol- 

 lowing is the process required by the strict rules of approxima- 

 tion. If a be the mean between the two apsidal distances, and 

 r=a + v, by hypothesis v will be a small quantity. Hence put 

 a + v for r in the foregoing equation, and in the small term ex- 

 pand to the first power of v, because the approximation includes 

 small quantities of the third order. Or, what is equivalent, 

 simply put a + r— a for r in the small term, and expand to the 

 first power of r— a. The equation will then become 



'/• 2 ^5 + / i ' 2 -2^ + C^=0, . . . . (D) 



where h' 2 = h 2 -\ — - — , and /// = /A + n' 2 « 3 . By integration this 

 equation gives 



