3] NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. 115 



2 n 



* 



,7 w j n n 



Now choosing the unit of time so that 



n n'= I, 

 and calling 



(7 ; =/-' - 4n 2 + 3c, 



we find the following numerical values for -^ and : 



1 2n 



J TT 



c i J 



1 -6-31259,13816,12770,45 2-16169,78061,03705,0 



3 +0-12752,52152,30991,45 0-72056,59353,67901,7 



5 0-04194,35175,60279,42 0-43233,95612,20741,0 



7 0-02090,23168,78861,83 0'30881, 39723,00529,3 



9 0-01252,48012,27559,34 0'24018,86451, 22633,9 



11 0-00834,43488,15871,11 0-19651,79823,73064,1 



13 0-00595,79989,74753,66 0-16628,44466,23361,9 



15 0-00446,74451,06389,87 0-14411,31870,73580,3. 



Hence the values of a lt ^ will be considerably larger than the co- 



1 2n 



efficients p l} q 1 in consequence of the large values of j^ and for j = l; 



and since the same multipliers will reappear in the successive corrections 

 to the first values found for a 1} b lt our approximation to those quantities 

 will be slow. It will be better to avoid this inconvenience, as we may 

 by the following device. 



Assume 



81 = ! cos <f> + SJ +SJ + ...=a 1 cos <j) + [81], 

 8<a = &j sin (f> + S,<w + S.,cu + ... = b l sin 



where 8J,, 8J,, ..., 8^, 8 2 o>, ... consist of cosines and sines of higher odd 

 multiples of <f> than the first, and <f> is written for (n n') t. 



152 



