Prof. Challis on (he Theory of the Moon's Motion. 529 



the values of r and given by the second approximation will be 

 employedj^ 



On going through the process exactly in the manner already 



indicated, it is found that the expression for -^ contains the term 



which on account of the small denominator m is of the third 

 order. This gives rise in the value of r, to the term 



-'2 -15^2 co&2{q-p) 



2n^eJ ~ 



4<m mn^p ' 



or 



But 



15me 



ercos2{q-p)^^^ 



/ 



sm^^ nsmp '' 



Hence the additional term in the value of r is 



15«me -_ . 

 g— cos(2^-;?), 



and we have, inclusive of all terms of the second order, 



r e^ e^ ^ 20 l^^^^ /o ^ 

 -=l— ecos|? + 7r — ^cos2jo--m^cos2fl' Q-cos(2fl'— »). 



This value, substituted in the equation 



gives 



/I ^ n • ^6^ • r» llm^ . ^ 16me . .^ . 



0=:nt-{-€ + 2e&mp + —rSin2p-\ — 5— sm2g'H -r—sm{2q—p), 



4 o 4 



We have thus arrived at the well-known expressions for the 

 radius-vector and the longitude to small quantities of the second 

 order. Thus my equation (C), which was pronounced to be 

 unfit for giving accurate information, has given in a very direct 

 manner precisely the same information as the most approved 

 methods, and has added other information, not previously ob- 

 tained, which I believe to be equally trustworthy. 



I purpose to apply the same method to the third, and possibly 

 the fourth approximation, as soon as I can get leisure for the 

 large amount of calculation which this investigation will require. 



Cambridge Observatory, 

 October 20, 1854. 



