3] NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. Ill 



These numbers have been calculated from the datum m = 0'0748013. 

 It is important to find how they are modified by a small change in m. 



Start from the equations of Lecture VII. , 



d-l fdlY ide\ 2 |~1 3 ."I 



df + (dt) ~ (dt) + V ~ n [2 + 2 C S 2 ( ~ '<> J = ' 



#0 dl dd 



where l = log e r and the unit of length is taken as before, and suppose 

 there is a small increase in TO, introducing increases 81, 80 in I, 6, the 

 squares of all these quantities being supposed negligible. Let us take as 

 coordinates 81, 8w, in place of 8^, S$ where 



w = 0- n't, 

 and therefore 



- . n't. 



n 



[Now we have developed and = 6 n't in series of the shape 



r*r\c 



Syj (m) . 2i (n n') t, and we wish to find the changes in the coefficients, 

 the arguments being unchanged. Hence we take 



and therefore 



8m ,. , S 

 and 



- - - . 



m v ' n' 



n 

 Hence we find the equations 



ddd8a> . 



__ 2 



-rr 2 -T--J- - -j- -j 



dt- dt dt dt dt 



dd d8l 



j- r + - r .^ rr 2-j-- T - 

 dt dt dt dt dt 



