LECTURE VII. 



CORRECTION OF APPROXIMATE SOLUTIONS. 



WE may simplify the equations we have been dealing with, by a 

 proper choice of units. Let the unit of distance be the radius of the 

 circular orbit which the Moon, if undisturbed, would describe about the 

 Earth in its actual periodic time ; then 



jj. = ri~. 

 Also choose the unit of time so that 



n n'=l, 



so that, if we take as the result of observation of the mean motions of 

 the Sun and Moon, 



n' : n= "07480,13, 



we get n' = '08084,9 = m 1 , 



where m, is the quantity so called in Lecture IV; and 



/A = 1-16823, 4. 

 We shall frequently adopt these simplifications in what follows. 



Now let l=log g (r/a), 



so that 



1 dr _dl 1 d 3 r _ d"l | /dl\~ p 



r dt dt ' r dt" dt 3 



_. ^ ' p 



and the equations discussed in Lecture IV become 



^" + (dt ' (dt + a 3& 



n dide ,,r 3 . , 



2 T, TT + n \ o sm ' 2<a = 



dtdt \_ 2 



where 



