Prof. Challis on the Theory of the Moon's Motion. 203 

 the integral of which is, 



=l-f+M^^+4 



neglecting powers of /j! above the first. It is plain that by- 

 supposing the arbitrary constant A to be small, we satisfy the 

 condition that w shall be small for all values of 0, and conse- 

 quently that the orbit shall be nearly circular. In this mode of 

 integrating we have excluded the third apsidal distance, by giving 

 to u the general form b + iv, which admits of satisfying, by one 

 of the arbitrary constants, the case of approximate circularity. 



But when a particular form is given to u, the analysis will 

 include the case of a third apsidal distance, so far as the form 



admits of satisfying the condition that ~ be a small quantity 



for values of the radius-vector contiguous to that apsidal di- 

 stance. Thus let u=b + be cos {6 + y) nearly, the second term 

 being supposed to be small compared with the other. Substitu- 

 ting this value of u in the last term of equation (I), and retain- 

 ing only the first power of e cos (0 + 7 ), we have 



d q u a u! 



^ + M ~& + ^(l-3 (? cos(0 + 7 ))=O. 



This equation gives by integration, 



Hence by comparison with the assumed value of u, 



b= h~i7i?=h-^F nearl y' 



be=A, and y = B. 

 Consequently the integral of equation (1) in this case is 



Now since the application of this integral is limited by the con- 

 dition that u must differ little from a constant value, it follows 

 that the angle 6 must be taken within certain limits, otherwise 

 that condition is violated. Putting for the sake of brevity q for 

 3/x'A 4 . . (fa 



2~t> lfc will be found that 73=0 gives 



1 

 andJ^Acos^ + Bj^-l + Ory-^}. 



