Prof. Challis on the Theory of the Moon's Motion, 521 



with this limitation is that of which the solution is usually called 

 the lunar theory, although the introduction of the limitation 

 into the reasoning is not usually pointed out. 



Let M, rrij w! be the attractions of the earth, moon, and sun 

 respectively, at the unit of distance, and at the time t reckoned 

 from a given epoch ; let x, y, z, r and a?, y\ z\ r' be the rectan- 

 gular coordinates and radius-vectors of the moon and sun, 

 referred to the earth's centre as a fixed origin, and to the plane 

 of the ecliptic and the first point of Aries. Then putting />t for 

 M -f m, and P for(^ — a?')^ + {y—y^f + {z—s^Y, we have the known 

 equations, 



J-t-f+$'+-'(.-y)V-«=o, 



As it is not my object to obtain exact numerical results, but 

 to exhibit a method of solution, I shall suppose for the sake of 

 simplicity that the sun describes a circular orbit in the plane of 

 the ecliptic at its mean distance («'), and with its mean angular 

 velocity {n^) , Thus a/ = a' cos {n't + e') , y'=a' sin (n't -f e^) , and 

 z' = 0. Hence dx' = — n'y'dt, and dy' = n'x'dt. By taking account 

 of these values of da/ and dy', and putting a' for r', the following 

 result is readily obtained : 



m^.dr'-?^d.{xa/+yy') + 2m'd.(a'^-^2(a!a/-{-yy')-hrT^. 



Hence by integration (putting ^ for the angle between the 

 radius-vectors of the sun and moon), 



^: dt^ + dt^ ^dt^ ^"""^ dt ^^""^dt ^^'- 



^-~cos^ + 2m'{a'^-2a'rcoscj> + rY^, 



It thus appears that the pi^oblem of three bodies admits of an 

 exact first integral in the case in which the relative orbit of one of 

 the bodies is a circle. I am not aware that this proposition has 

 been proved before. 



To simplify the question still further, I shall now suppose the 

 moon to move in the plane of the ecliptic, as the principle of the 

 proposed method of solution equally admits of being exhibited 



Phil, Mag, S. 4. No. 55. Suppl Vol. 8. 2 M 



