VIII.] THE PARALLACTIC INEQUALITY. 35 



If we equate to zero the coefficients of cos \)t and cos 3t/ in the first, 

 and those of sin \jj and sin 3|> in the second, we obtain the following 

 equations for a lf &,, a 3 , 6 3 ; the terms in 5i/ remain outstanding, and the 

 effect of a 5 , 6 5 in modifying the other coefficient is neglected. 



+ a -b, 

 ^/J 



+ a 3 Tea., + 1 ^ a.,] - 6 3 [~6&, - | i /s (l + &.)! = | n' 2 (3 - 96, - 4a.), 

 L ^JL * J o 



[2 ( 1 + w') - 26 J - &, |1 + 1 ' 3 - 2a 2 + 8w"6,l 



L ^ J 



[66,] - 6 3 ea s - \n'- + ? ' 2 6 2 = - n" (1 - 76, - 2a 3 ), 



a J - 2a, + 1 J J + 6, f - 26, - 1 n- + | 

 L ^ " J L * ^ 



+ 3 [6 (1 + w')] - 6 3 [9 + 3;/ 2 6J = - w' 3 s + 6, - a, . 



o y ^j Zt I 



If we require the formal values of a lt 6 I; 3 , 6 3 , we must substitute 

 for a 2 , 6 2 , /A/rt 3 the expressions we have found for them, and it will then 

 be best to develope the coefficients in ascending powers of n' . But 

 it is difficult to obtain by this process such good numerical results as we 

 can get by substituting the numerical values of a.,, 6,, /Lt/a 3 immediately 

 in the equations above. If we do so we get the equations 



4-56672a 1 -2-l72326 1 + '08093a 3 - "051376,= - n'- x 2'87937, 



8 



2-14128ai-0-995G4& 1 + -06127a 3 - '033386 3 = -f w' 2 x 0-91416, 



8 



02349^- 03012&.+ 12-51451a 3 -6'485086 3 = | w /2 x 5'00455, 



8 



02042^+ -024066J+ 6'48508a 3 - 9'000'206 3 = -- n'~ x 5'00152. 



52 



