Prof. Challis on the Theory of the Moon's Motion. 371 



part of the argument, and add such additional elucidation as it 

 may seem to require. In the Supplement to the Philosophical 

 Magazine for December 1854 (p. 521), an exact first integral of 

 the differential equations is obtained by neglecting the eccen- 

 tricity of the sun's orbit. Then, after neglecting also the incli- 

 nation of the moon's orbit to the plane of the ecliptic, two equa- 

 tions (a) and (b) are found, which, if we omit terms multiplied 

 by the first and higher powers of the ratio of the moon's radius- 

 vector to the sun's mean distance, reduce themselves to the fol- 

 lowing : — 



*Vf + C,_«^ + ^ (1 + co s2 *> . (A) 



'""(I +»') = -^W (B) 



In these equations r is the moon's radius-vector at the time t 

 and, being the moon's true longitude at the same time, A is 

 the excess of above n't + J the sun's mean longitude; fi is the 

 sum of the attractions of the earth and moon, and m' is the sun's 

 attraction at the unit of distance ; a' is the sun's mean distance, 

 so that a'W = m' ; and C, is the constant introduced by the in- 

 tegration above mentioned. 



The above limitations are made for the sake of simplicity my 

 object being to exhibit a method of solution rather than to obtain 

 numerical results. By taking account of the eccentricity of the 

 sun s orbit and terms of the fourth order containing the ratio of 

 r to a , the investigation becomes much more complicated but 

 involves no processes differing in principle from those required 

 under the proposed limitations. The terms depending on the 

 moon s latitude may be considered separately on a future occasion 



I he equations (A) and (B) not admitting of exact integration 

 are to be integrated approximately, and the approximation is to 

 proceed both according to the eccentricity of the orbit and the 

 disturbing force. Hence the first approximation to the orbit is 

 a circle described by the action of the central force £. If a be 

 the radius of the circle, it will be readily seen that 



a ~G~' **—!*&* 0=nt + e. 



These values are to be used for substitution in the small terms of 

 (A) and (B) in order to proceed to the next approximation, which 

 1 have hitherto called the first approximation, and for the sake 

 ol uniiormity shall continue to designate in the same way. In 

 this approximation it is proposed to obtain values of the radius- 

 vector and true longitude to the first order of small quantities 

 2B2 



