oflnequalitks of Longitude in the Lunar Theory, 169 



X denotes the moon*s longitude, f and f ^ the mean anoma- 

 lies, e and Cj the eccentricities of the moon and sun. 



X = Xi4^^^sin(f-f,) + Ai5^^^sin(2T-f + y + &c. 



± = ri4 e e, cos (J-f,) +^5^^,003 (2 r-f + f/) +&c. 



f-f^ = cniJ-mnif 2T-? + g, = 2nif-w n^— en/ 



I find from the equation 



("21 1065 o 12615 Q . > Ti ■ '7 o 1 



Ai4 = {■^^'^ + ^2" ^^ + ~6^^ +^'-| L^^"^"^ ^ ^'+&c.] 

 21 1233 g , 15333 ... 



= 4■^+"32-^+'-§r^+^^• 

 15 15 53 3^49321 ^3 , ^_ 1 T, . ».. . ^' 

 ^i5=:r~T^-32^+-38l- 



TW^ + Sccj ["1+?^^+ ^ +&C. 1 



15 173 2^4.8325 3 . 



Lubbock on the Lunar Theory, p. 193. 



15333 

 It is evident that the divergence of the coefficients 



and ~7 is not due chiefly to the quantities 1 -J- tw + — w^ 



and l4-m^+ —- which arise from the expansion of divisors 



introduced by integration, nor is the expansion of these divi- 

 sors according to powers of m a step attended with labour or 

 difficulty, and on that account desirable to be avoided. The 



12615 49321 



diverging quantities - ■ and arise from the sum- 



mation of the following quantities in the value of — p-; 



12615 __675 315 _^_^.^. 1687 

 64 "^ 32 "^ 64 4 32 16 8 



49321 __ 9 39 ^ 63 13 4927 7 91 

 384 "" 16 16'"*' 4 "^32 16 384 24' 



1687 49277 



The principal terms are evidently and ~-^^^ : these 



arise from the coefficients r^^ and r,5 belonging to the same 



