the Coefficients of the Lunar Inequalities. 493 



m 



and 3-+ p* sin 2 (v—v,). — Since — £ is a small fraction ( = -0056 



nearly), we may at the outset omit the inequalities of v and p in 



the terms f— §p cos 2(v—v^) and 3— £ p 2 sin 2(v—v t ), and assume 

 a t a t 



v = nt + e, p=l, these being their mean values. The equations 



(9) and (10) will thus be reduced to these forms, writing k for 



m 



— and c Y t* for 2(nf — nt + e t — e), 



dv*_d*p I 

 [V^] +Skpsm Cl t=0. .... (12) 



P^2~ ^~-^+ikp+^kpcosc l t = J . (11) 



2^Vs£ approximation. 



4. Variation computed. — Inspection of these last equations, 

 (11) and (12), shows that the values of v and p must contain 

 terms involving sin c x t and cos c t t. Let it be required to com- 

 pute approximately the coefficients of these terms. 



Solution. — Assume 



p=l-\-¥ l cosc 1 t, v = nt + e + G l sinc^, 

 F,and G x being the coefficients whose values are to be determined. 

 It will be convenient to put n (which represents the moon's 

 mean daily motion in longitude) = 1, t denoting the number of 

 days which have elapsed from the epoch. Hence c lt which de- 

 notes the daily motion of the argument 2(nt— nf + e— e ; ), will 

 be equal 2 — 2/1,, n l being the ratio of the sun's mean daily mo- 

 tion in longitude to that of the moon. The values of p and v 

 will therefore be 



p=l +F, coscj^, v=t + e-\-G l smc l t. 

 Substituting these values of p and v in equation (11), rejecting 

 the squares and product of F x and G„ we shall have 



dv dv 2 



■j t = 1 + c l G l cos c x t 3 ,\ r-- a 1 -(- 2c l G l cos c^, 



dv 2 

 '** p dt 2 =1 + ^ 1 + 2c i G i) cosc i t > 



— - J £-=c l 2 F ] cosc l t, 2-= — l+2F 1 cosc 1 ^, ±kp = ±k+ ±k? l cose,/. 



ai p 



Hence the differential equation (11) becomes 



= l + (F 1 + 2c ] G 1 ) cosc^ + c^cosc^— 1+2F, cos Cj/ -hi A 



-f- \ k¥ 1 cos c x t + l&cos c x t ; 



* Or rather ci£+2e / — 2e ; but the constant part 2e, — 2e is omitted, as 

 not affecting the computation of the inequalities. 



