40 LECTURES ON THE LUNAR THEORY. [LECT. IX. 



. 



-T-T ---j- -3 ^ 

 at- at a 



Now suppose the following is a set of terms that appear 

 in X Pi cos (it + y), in Y q i sa\(it-\-y), 



SI a { cos (it + y), 86 b { sin (it + y) ; 



then as in Lecture VII, we find 



a,-= 



.-_ + *' 



2 n 

 n 1 



Therefore if i differs little from n, the divisor in c^ will be small, 

 and a small error or omission in the numerator of a,- will appear magnified 

 in the values of both a, and ?>,. In the case of the first term of the 

 Parallactic Inequality, 



i = n-n', 



i"--tf + -n*=-2nn' + -ri- 



Zt Zi 



and if we take 



pi =- 9 -n, 3i = *n* 



o 8 



which differ from the correct values by quantities of the fourth order, then 



_ 2 n _3 ,, 5n - 3ri 



and the formulae give 



_3 n'(5n-3n') 



2n 3 n'- 



n- 



Now if we develope these expressions in ascending powers of m, i.e. n'/n, 

 the first terms are 



15 15 



ai= Tfi m ' b t=--^ m > 



1O 



and these are the only terms which the formulae derived from our method 

 of approximation will give correctly. 



