of the Moon's Motion. 89 



4 1 + e' = i +/(v') = m v + is + w/(v), "1 

 ^V{1 +/'(vO} = rorfv{l +/'(v)}, h • (A.) 

 or dv 1 =zmdv {1 +f l (y l v')} J 



Then integrating by parts, 



f: 



d v sin (* v -f z 7 v' + /3) = r cos (i v 4- i' v' + /3) 



— -i- -Tdy'sm (ii + i' J + /3), 



/■ 



</ v cos (* v 4- £' v' + /3) = -r- sin (/ v + **' i/ + /3) 



— 4- /d v' cos (t v + * 7 v' + |8). 



Put for e? v' its value given by (A.), reduce and integrate again 

 by parts ; and so on till the new terms become so small that 

 they may be neglected. 



To integrate (1 .) make p = S A . „ cos (/ v + i' v' + /3) ; then 



g= -2A. r (i + ^v) 5 cos (,' , + / i + f) 

 d 2 v' 



But r 1 (PS + 2dr'dv' = 0, or u 1 d 2 v' - c 2dn' dv' = 0, and 

 d 2 v' = j — - = dv'^fz^'). We can therefore eliminate 



both dv' and e? 2 v'; and after the necessary reductions have 

 been made, it may be convenient to put all the terms after the 

 first with those which arise from the disturbing force. Thus 

 all the integrations required can be performed, and if there be 

 more labour in the integrations, there will be less in the pre- 

 vious developments. 



Such is a brief outline of the method 1 would propose for 

 determining the moon's coordinates. It is easier and requires 

 less labour than the old method ; certainly it would require 

 very much less than M. Hansen's theory, which is far more 

 laborious than the common method. But general formulae 

 should be constructed for the determination of the coefficients 

 in the development of the disturbing force after the manner 

 of that author. I may add that it may be applied also to the 

 planets. 



Gunthwaite Hall, near Barnsley, 

 November 25th, 1843. 



