LECT. XI.] THE EQUATION OF THE CENTRE AND THE EVECTION. 47 



When there is no disturbance the four arbitrary constants are n and 

 e, which denote quantities similar to those expressed by the same symbols 

 above, and the two elliptic elements e and CT, of which e denotes the 

 eccentricity of the orbit and TS the longitude of the apse. 



We will now shew how to complete the solution by introducing into 

 log(a/r) and 6 additional terms depending on quantities similar to e, or, 

 of which the former is constant and the latter varies slowly and uniformly 

 with t ; and for the sake of simplicity we will suppose at first that e is 

 so small that its square and higher powers may be neglected though it 

 is otherwise arbitrary in magnitude. 



Let us assume then 



a ^ cos 2^ + e cos (nt TO-), 

 = nt + t + !) sin 2i/ + 2e (1 + ?> ) sin (nt CT), 



in which the elliptic terms are of the same form as in the undisturbed 

 orbit, and TO- is supposed to be slowly variable, so that 



where p is supposed to be a small quantity of the order of the disturbing 

 force. 



We will now substitute these assumed values in the differential equations. 

 We have 



-j- = 2(n n') a., sin 2\ft + (n p) e sin (nt OT), 

 = 4 (n n'Y a., cos 2\ft + (n p)"" e cos (nt CT), 

 -T- = n + 2 (n n') b., cos 2\jt + 2 (n p) (l+b a )e cos (nt CT), 



-, - 2 = 4 (n n') & 2 sin 2\jt 2(n p)- (l + b a )e sin (nt zr). 



Hence 



4(n n'Y 2 cos 2i/ + (n pY e cos (nt w) 



4- 2 (n n') (n p) a^e [cos (2i/ nt + CT) cos (2i/> + nt w)] 

 {w 2 + 4n (n n') b., cos 2i/ + 4 (n p) (l + b a )e cos (w OT) 



+ 4 (n - n') (n -p) ( 1 + 6 ) 6 2 e [cos (2i// - nt + w) + cos (2^ + nt- m]} 



