LECTUBE VI. 



THE VARIATION, (continued). 



LET us consider the problem of the Variation over again, taking now 

 6 as independent variable. 



The equations of motion are given in Lecture II : 



d*w J^ T_ du 



dP* ~HV HV d0' 



n dH_T 



~d0 ~." 



P 1 n'~ 3 n 1 ' 2 



where -=/" -.cos 2(0 ff], 



u- 2 u* 2 M 3 



T 'i n'- 



1 sin 2 (0-00, 



u 2 u* 



so that the second equation may be written 



3,1* Wain 2 tf ^ 



H- de (M) 



Our aim is to express t and u in terms of # and constant quantities, 

 Now since the orbit of the Moon does not differ widely from a circle 

 we may write the difference of nt + e from 6, and the difference of au 

 from unity as series of small periodic terms depending upon 6. Inspect- 

 ing the form of the equations, it is evident that these periodic terms are 

 of argument 2 (6 0') and its multiples ; that is 



nt + e = 6 + periodic terms of argument 2 (6 6'), &c. ; 

 but rit + <!=ff; 



therefore 



0-6' = (I -m)6-f$ + periodic terms of argument 2 (1 -m) 0-2/3, &c., 



