232 STUDIES ON NEWTON'S LUNAR THEORY. [13 



Hence we have the equation 



-JQ = cos (0 - OT) \ - m- cos (6 - CT) M 1 + 2 cos 20 } 

 Ciu ^ 2* \ 



+ 2 sin(0 T3-){Gm 2 sin 2# cos (0 cr)} 



go o ic g 



= - nf + - m 2 cos 2 (^ CT) + - ?n 2 cos 2^ + ?u 2 cos 2ta- - m cos 



If we compare this with the result of p. 62 it will be seen that while 

 the periodic terms of short period are in error, the coefficients of the terms 

 which produce most effect are correct to the order to which we have carried 

 them, remembering that at the beginning we neglected the Sun's motion. 

 If we simply take 



3 15 



-,>,- = - in' + - m- cos 2ry, 

 aO 4 4 



we find the mean rate of change of ra- with respect to 6, by the methods 

 of Lecture XIII. , 



3 225 



