45 



Then d''u=d' z = c, (— i)' p+f Vdi/ , » even and v = o 



n— 2 w— 1 



d"M = d"z = c, (_i;'^n +1) 2 rf / , « odd and ► = o 

 These two equations are the particular fluxions of 



i i 



u = c, COS (1 + ^\ » + c. sin (1 +i\ v + G 



where p zz m' Kcos x and g = m - X sin -n 



A first approximation therefore gives the value of u in period- 

 ical terms, instead of in terms without the periodical signs as occur 

 in the complete solution of the differential equatiou obtained in 

 case the 2d. 



(IV.) 

 Further application to the Lunar Orbit. 



It was found by Equation (18) Art. (II.) that without the dis- 

 turbing force of the Sun 

 u = Ifl + e cos ( II — ■!r)\ = 1 + £. cos f cos a- + i. sin f sin tr, re- 



a \ J a a a 



garding only the first power of the excentricity. 



Also without the disturbing force of the Sun by the preceding 

 article the integral of equation (d) of article (II.) becomes 

 ^= c, cos ► + c , sin V + J_ 



a 



comparing these values of u 



c. =_l cos 3- Cj = £ sin TT 



a a 



Hence in case 3 of the preceding article, because 



■V- S e J iU e 3 m'' e cos ir 3 m^ e sin a- 



K = ~ —- and therefore p =z — , n = — ; 



