the Coefficients of the Lunar Inequalities. 499 



+ £ (F z F m + 2ciGjF to ) cos (c, > c TO J* • 



'"■ 5 [ 2p2 S = -^W + 2ci ^ sin Cz ' 



- S(<*± c m ) (FJ W + 2c £ G J,,) sin fa ± *>)/. 



o 2 

 (2) F«/we 0/ 3»i, £g sin 2 (v - v,) : 

 Pi 



3-£ = 3£ + 23ftB z cos<tf, 



Pi 



/ 9 2 =l+22F z cosc^, 

 sin 2(u— »,) = sin (c { t + 2D* sin c t t) 

 (by Taylor's theorem) 



= sin cj + cos c L * . 2Dj sin c^ 



= sinc^ + 2JD;sin {c^c^t— 2 J D z sin (c^ c?)/; 



o 2 

 /. 3m / £ s sin2(t;--t; i ) = 3A;sinc 1 * + 2f £(Bz + 2F z + Dj) sin (c l -{-e l )t 

 Pi 



+ 2f*(B, + 2F,-D,) sin (c^cfr. 

 Consequently, collecting these values, equation (8) becomes 

 0=3^8^^/ 

 -tc l (4$i + 2c l G l )smcit 

 + %%k(B l +2F l + J) l ) sinfo-f c$t 

 + 2§ *(B, + 2F|-D,) sin'fo-ci)* 

 -Z(c l ±c m )(F,¥ m + 2c l G l F m ) sin {c t ±c m )t ; 



and applying this equation to the case of a given argument cj, 

 we shall have 



0=3& in the case when a?=l, 



-c x (4F x +2c iC G a; ) 



+ |&(Bj-f 2F* + Dz) when c x + ci=c x , 



+ f/:(B z 4-2F^-D i ) when c,— c t = c x , 



~5c,(F,F TO + 2ciGiF„) when cj + c m = c*. 

 2K2 



