3] NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. 117 



Repeat the step with X z , Y* in place of X lt Y lt and continue until 

 the terms in X n , Y n which involve odd multiples of <f> above the first, 

 are insensible. 



The coefficients of the several terms in SJ, 8J, ... , S,cu, Sow, . . . thus 

 found will involve linearly the constants c^ and &,, and the same will be 

 the case with regard to the coefficients of cos < and sin <f> in the final 

 values obtained for the quantities X n and Y n respectively. Hence by 

 equating these latter coefficients to zero we shall have two simple equations 

 for determining a^ and & whence by substitution all the other coefficients 

 in the values of [8/J and [8o] may be found. 



Nov. 8/81. 



[On this plan the calculations were carried out by Miss Harrison, 

 taking the expressions 



.Y=X [-0-00705,27630,94721, 5 cos (n-n')t 



1225,34903,49610,6 cos 3(n-n')t 



14,36155,36056,7 cos 5(n-n')t 



14187,55306,0 cos 7(n-n')t 



132,36321,5 cos 9(n-n')t 



1,20188,9 cos 11 (n-n')t 



1074,7 cos 13 (n-n')t 



9,4 cos 15 (n-n')f], 



F = X[ 0-00223,75651,99439,7 sin (n-n')t 



+ 1224,60664,99386,0 sin 3(n-ri)t 



+ 14,35867,21138,5 sin 5(n-n')t 



+ 14186,08099,5 sin 7(n-n')t 



+ 132,35462,5 sin 9(n-n')t 



+ 1,20183,4 sin 11 (n-n')t 



+ 1074,7 sin I3(n-n')t 



+ 9,4 sin 15 (n-n')t]. 



