LECTURE XI. 



THE EQUATION OF THE CENTRE AND THE EVECTION. 



WE have seen that the equations of motion 



d6dl 



are satisfied very approximately by the values 



7 i r 



I = loo- - = a., cos zu>, 



a 



= nt + e + b, sin 2i/, 



where \jj = nt + e (n't + e'), 



and a.,, b,, are small quantities depending upon the ratio n'/n, and a is a 

 quantity depending upon n in such a way that 



1 Q -n' 4 



u. _ i . y it, 



while w, e are arbitrary, though subject to the assumption that the ratio 

 n'/n is small. 



This solution, then, expresses a possible case of motion ; nevertheless 

 it is no more than a particular case because it involves only two arbitrary 

 constants, whereas the complete and general solution must contain four, in 

 order that it may be able to satisfy any given initial conditions, that is, 

 in order that the initial coordinates and their initial velocities may have 

 any given values. 



