128 



MR. LUBBOCK ON THE THEORY OF THE MOON. 



Astronomical observations are now made with so great precision, that the numerical 

 values of the coefficients are wanted to at least the tenth of a second of space : very few, 

 however, of the coefficients of MM. Damoiseau and Plana agree so nearly, and some 

 differ much more, as may be seen in the following comparison of the numerical values 

 of the coefficients of some of the arguments in the expression for the true longitude 

 of the moon in terms of her mean longitude, being indeed those which differ the most. 



When the coefficients of the inequalities have been determined analytically, it 

 remains to determine with corresponding precision the numerical values of the arbi- 

 trary quantities m, e, and y. The quantity m is already accurately known, but the 

 quantities e and y must be obtained from the coefficients of sin | in the expression for 

 the longitude, and of sin rj in the expression for the latitude, by the reversion of series ; 

 and it seems to me that the manner in which these arbitrary quantities are to be 

 determined must be carefully and rigorously defined. 



I propose to obtain the expression for the radius vector by means of the equation, 



2 d ^^ ~" r + a + V ^ ^ T^ ^ dr - ^• 

 In order to integrate this equation, I suppose 



y = 1 -|- Tq + r^ cos 2r -\- ell g- j cos | + &c. 



If r be used to denote the terms in r which are found in the elliptic expression, so 

 that 



■^=: 1 + e(l --^)cosi-i-e2(l - ^)cos2|+-|^cos3^ + -|e4cos4? + &c.. 



H 



and 



a ff . V 1 



— = \- al — 



r r ' r 



