the Coefficient s of the Lnar Inequalities. 495 



calculation being based on equations (11) and (12), the earth's 



orbit was supposed circular, and v t and -3 were assumed equal 



Pi 



to their mean values n.t+e. and — =*. Now both v, and — b are 



..'.**. Pi 



subject to inequalities, the chief ot which depend on the sun's 



mean anomaly. If these inequalities be taken into account, the 



values of v. and — s will be nearly 

 ' P? J 



v { = nf + e t 4- (2e i — J <? y 3 ) sin c 2 t, - 3 = — 3 [1 + {Se l — § e'f) cos c 2 t~\ , 



Pi a > 

 in which c 2 t denotes the sun's mean anomaly, and e t the excen- 



tricity of the earth's orbit. The effect of these inequalities in the 

 values of v t and —3 will be to introduce into the differential equa- 

 tions, and consequently into the values of v and p, terms having 

 the sine and cosine of the sun's mean anomaly for arguments. 

 Let it be required to find the coefficients of these terms. 

 Solution. — Assume 



p = l + F 2 cos c 2 t, v = t -\-e + Gc, sin c^t, 

 and substitute these values of p and v in equation (7), 



Ul) ft l) 



p -rp — 1 -f (F 2 + 2c 2 G 2 ) cos c 2 t, — jj~ = c 2 2 F 2 cos c 2 t, 

 g- = — 1 + 2F 2 cos c 2 1, 



2 m i ^-3 = 2 -3 (1 + F 2 C0S C 4) I 1 + B 2 C0S C 4) = 2 * + P ( B 2 + F a) COSC 2 ^, 



Pi a i 



§ m l -^cos2(v — v^} = ^kcosc l t, 

 Pi 

 where 



r» 2 _o^ 8 e, . 



Collecting the coefficients of cos c 2 t and putting their sum equal 

 to zero, we shall have 



(c 2 2 +p + 3)F 2 + 2c 2 G 2 + pB 2 =0. , . . (A) 



* It may be observed that a t is not exactly equal either to the semiaxis 

 major of the earth's orbit, or to the mean value of p,. If the semiaxis 



major be denoted by a H the value of —3 will be equal to —5 (1+fe, 2 ) 



nearly. Hence, though the semiaxis major is a constant quantity, the value 



of — , and consequently that of k, will be affected with a small secular varia- 



af 

 tion chiefly arising from the decrease in the excentricity e t . 



