110 On certain Questions relating to the Moon's Orbit. 

 oscillations in latitude the equation 



27ra* 

 The time of an oscillation is therefore —;=-. Also the known 



VfJb 



approximation to the moon's periodic time, as derived from the 

 first two equations, is 



2'7ra^ / m'(^ 



As this period is longer than the period of oscillation in latitude, 

 the node regresses, and the amount of regression in one lunation 

 is the arc which the moon describes in the difference of the 

 periods ; that is, putting p and P for the periodic times of 



the moon and sun, the regression is 27r x -^ nearly. This, 



however, as is well known, is a false result, and it is necessary, 

 therefore, to retrace our steps. The last term of the third equa- 

 tion ought not to have been omitted, if that equation contains a 

 small quantity of the first order as a factor, which will be found 

 to be the case. For, putting a for r in the third equation, and 

 integrating inclusively of the last term, we have 



r=:Acos(N^ + B), 

 which shows that the small factor of the equation is A, the maxi- 

 mum value of z. The period of the moon's oscillation in lati- 

 tude is nearly 



which is less than the moon's period by 



Hence in one revolution of the moon the node regresses through 

 the arc 27r x ^^> which is the known first approximation. Con- 

 sequently, 



the result it was required to obtain. This investigation presents 

 several points of analogy with that for finding the mean motion 

 of the apse. 



I am. Gentlemen, 



Your obedient Servant, 

 Cambridge Observatory, J. Challis. 



July 18, 1854. 



