of Inequalities of Longitude in the Lunar Theory, 171 



Equating coefficients, 



4221 1113 ^ . _8049__267, _9 _9265__759_3 



l28~"l28" '*~"'"64" 16 "^ 8 32 32 2 

 3015 485 _ 6133 , ?21 _ 15 _ 12289 , 3 , 9 , 87 

 128" 128 -^^1^ - 384 "^ 16 8 64 "*" 4 "^ 4 "*" 16 



1687 J _ 49277 



In order to judge of the relative magnitude of these several 

 fractions, I divide the numerators by the denominators, and 

 retaining only whole numbers, I get 



_32_8 — 2^14 = 125 — 16+1—289 — 23—1-48 

 - 23 - 3 - 2 ^,5 = 17 + 13 — 1 — 192 + + 1 + 6 

 Au = 105 ^15 = 64. 



The leading terms --289 and —192 arise from the deve- 

 lopment of R, and virould be included in the following ex- 

 pression : 



d^a^Sl . 



-j-^ -ix8- + 4^ = 0. 



cfr ^ r 



This approximation would save the calculation of some 

 quantities, but it evidently could not be safely adopted, and it 

 would still leave the calculation of i?, which is extremely 

 troublesome. Very little trouble would be saved by not de- 

 veloping the divisors introduced by integration, or the quan- 



titles in square brackets. Ihe quantities - and , 



which belong to the development of R, arise as follows, from a 

 multitude of diverging fractions, andl think it would be impos- 

 sible to give any safe rule for selecting the principal terms. 



9265 _ 1161 __ 243 525 ,^__63 ^.^.j- ?^11 _ 1^ 



T28"~ 128 256"^ 128 "^ 4 8 8 "^ 8 "*" 256 128 



122^__ 81^ 3483 1 27 5]I3_ 3^ _3 _3^ 3699 15 

 256 " 128+256 + 4 + 8 + 128 16+8 4 + 128 16 



5523 

 The term -^-- arises from the combination of the term 



1 ^o 



263 



-— ?»* e sin (2 T— f ) in the longitude (5 a) with 



