138 MR. LUBBOCK ON THE THEORY OF THE MOON. 



The second term on the right-hand side of this equation gives no constant quan- 

 tity : hence 



X= \/— sjl - m^ - ^ m^ e'^ - ^ m^ eA t + kc. 

 If the constant quantity which multiplies t be called 



{23 "I " 



1 -{• -J m^ -\- -J m^ e^ -{- m^ e^ > 



r r 2 4 4' 



Reverting to the equation 



dxA 1 rdR 



The second term of this equation gives no term multiplied by cos | ; therefore, 



2 h 



d X = n d ^ + -^ (1 + Tq) e cos I d ^ 



= nd^ + 2n(l - y) (1 +m2)ecos|d^ 

 c=l — T-m^ cn = cw c=l jwi^ 



4 4 



X = n^ 4- 2 (l + |-m2 -I- I') esin^ 



X=n# + 2(1 +^m'^)esm^ + Six:. 



T= ^ +"6 +"4^ +^V^ +-3 w2Jcos?-f &c. 



These are the expressions for X and for — , when the quantity e is retained ; but if 



the coefficient of sin |, in the expression for X, be called 2 e, after the manner adopted 

 for the planets in the first volume of the M^canique Celeste, so that 



X = n/ + 2esin|+ &c., then 



y=l+^ + ^+e (l -j^m^J cos ^+ kc. 



