106 NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. [3 



= n + 0-01021,13629,5 sin 2(n-n')t 

 + 0-00004,23732,7 sin 4(n-n')t 

 + 0-00000,02375,7 sin G(n-n')t 

 + 0-00000,00015,1 sin 8(n-n')t 

 + 0-00000,00000,1 sin W(n-n')t. 



-= 1-00090,73880,5 



r 



+ 0-00718,64751,6 cos 2(n-n')t 

 + 0-00004,58428,9 cos 4(n-n')t 

 + 0-00000,03268,6 cos G(n-n')t 

 + 0-00000,00024,3 cos 8(n-n')t 

 -0-00000,00000,3 coslO(n-n')t. 



[These are the values quoted from an older MS in 1877 (Mon. Not. 

 xxxviii., p. 46; Works, Vol. i., p. 184). The original calculation has not 

 been found, but it must have had a date earlier than 1860, for the above 

 numbers are referred to in a MS of that year.] 



In order to obtain corresponding series in which 6 is the independent 

 variable, first transform these two functions or rather the functions nt 



and log - , by means of Lagrange's Theorem ; use these results as ap- 

 proximations and emend them by substitution in the equations of Lecture 

 VI., viz.:- 



d~ (aii) 

 dP 



du 



dt 



de 



