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XIV. Differential Equations of the Moon's Motion. 

 By the Rev. Brice Bronwin*. 



TF the moon's coordinates were expressed in terms of the 

 true elliptic longitude instead of the mean, it seems very 

 probable, if not quite certain, that there would be fewer equa- 

 tions. Their development would also be easier. These consi- 

 derations have led me to seek suitable formulae for the purpose, 

 in which I think I have been successful. The object of this 

 paper is to exhibit them. 



Let ft =5 m + 7n', the sum of the masses of the moon and 

 sun, x and y the rectangular coordinates of the former on 

 the plane of her orbit, x', y' and 2/ those of the latter, 



r = (x z + y ) i r — (x u + y li + * ) » R = — 5 — — /§ — ? - JL - L 



m' 



The equations of motion are 



{r 12 - 2 (x x' + y y') + r 2 }* 



(fix ■ ux - dR „ d?y - ay dR 

 dr r* dx dt 2 r 3 dy 



From these we easily derive 

 dx 1 + dy 2 



dt 2 



_^ + ii + 2 f d R = 0) 

 r a J 



, D dR . . dR . dR < . dR , 



rfx rfy dr dv . 



V being the moon's true longitude. Multiplying the first of 

 the above by x, the second by y, and adding them to the third, 

 we have 



yyyfft' " * . * , P _ 



„ tfR , rfR rt /» ■ </R „ /»,„ 



P = x-t— + y-= |-2/rfR=r— + 2 / dR. 



dx dy J dr J 



Similarly, we find 



xdy — y dx ,'.'A „ dv , „ 



Now let g = rad. vect. y = long, in the elliptic or undis- 

 turbed orbit. These are given by the equations 



whence results 



* Communicated by the Authoi . 



