46 



c. cos CI + |i)S + c, sin (1 -i- f , 



= -£cos. co,<il-'-^y'+~^m.,m{l-'-^y,= 



Consequently by the application of the first and third cases of 

 the preceding article, the integration of equation (d) 



gives u = „-:p^, - 5T +11 ^°<<^ 1 -— ) *-'^) + T" ^^^^ (^ v-2m.) 



4- £i cos (>>— 2 HI f + 'r) +-^ e cos (3 v— 2 ?n ^ —t) (f) 



where j'°'=-'-|^(^)(-i:^^^ and therefore is of the order 



of HI" 



\ — # J 



A = 3 )n-< „,i ;• i-d-'im)" ^'"^ therefore on account of the 

 divisor 1 — (1 — 2 m)- = 4™— 4 m" is of the order of ni 



I — m 



1 1^ 



^(2) = 3m'-J „ o.. r i_i3_3w)' and therefore is of the order 



of m' 



Hence u — — J-— cos (> -t^ + — cos [v — m v + tt) will be a new 



value of u, exact to the order of m. 



It is clear by the third term of the value of u given by equa- 

 tion (f) that when (1 — ^-^) " — "^ = o or a multiple of the circum- 

 ference, the moon is at perigee, not regarding the periodic terms 

 depending on the place of the sun, and therefore the mean motion 



of the moon : mean motion of its perigee : : 1 : -£- The pro- 



