49 



Hence collecting the terms (1) (2) (3) (4) (5) the co-efficient of 

 the term e cos (*— tt) in the variation of the equation (rf) obtained 

 by Ui = - cos(^— Smc+T) will be found = 



3j»» 



— -^ (5+ 5 m) A e cos (j — sr) 

 consequently the coefficient of e cos (v— «-) ni the differential equa- 

 tion arising- from the substitution of 

 u = L + i cos (v—ir) + — cos (► — m » + «•) 



is thus found = — ^ (1 + (5+ 5m) A) 



Hence by case 3 Art. (HI) the corresponding term in the in- 

 tegral or value of zt =: e cos ((1 — ^' (1+ (5-1-5™) A)i>—f) 



Therefore the mean motion of the perigee is 



f m' (1+ (5+5nO A) that of the Moon being unity. 



A = i(a) C-~ + &E) = 4 -(5+19-) nearly; 



Perinrlic time moon • r\ 0kf 40 



now m = „ . ■ , - ! ■-■ ^ nearly = 0,0748. 



Periodic time earth ■' 



Therefore A=, 1801 nearly. 



Hence the mean motion of the perigee thus found by the se- 

 cond approximation =, 00826 or 2''.58',4 each revolution, the ob- 

 served mean motion ==, 00845 = 3. 2, 5 each revolution : the dif- 

 ference is about ^'-th part of the whole, whereas the first approxi- 

 mation was only about one half of the whole. 



It is evident, that if we had also found the variations arising 

 from the other terms of u besides Ae cos (» - m ►+»), the results 

 would, (as will easily appear by considering the values of 

 these quantities in comparison of Ae cos [v — mv+Tr) and the na- 

 ture of the combinations required to form the angle [v—Tf) have 

 had only a small effect on the mean motion of the perigee. 



The above, it is conceived, is sufficient for two purposes. 



VOL. xiii. I 



